Attitude Dynamics and Control of Space Debris During Ion Beam Transportation 0323992994, 9780323992992

Table of contents :
Front Matter
Acknowledgment
Copyright
Introduction
Basics of space flight mechanics and control theory
Mathematical and mechanical preliminaries
Vector and matrix operations
Fundamentals of classical mechanics
Coordinate systems
Two-body problem: Motion in the orbital plane
Planetary equations for perturbed motion
Relative orbital motion
Rigid body attitude motion in orbit
Orbital and attitude perturbations
Nonspherical central body gravitational field
Moon and Sun gravity
Atmospheric influence
Solar radiation pressure
Earths magnetic field torques
Lyapunov stability theory for the search for control laws
Chaotic dynamics of nonlinear system
References
Space debris problem
Review of the space debris problem
Space debris threat
International legal aspects and space debris prevention guidelines
Space debris rotation properties and inertial parameters estimation
Postmission disposal
Active space debris removal
Just-in-time collision avoidance
Target selection for active space debris removal
Contact methods of active space debris removal
Space debris capturing
Contact space debris removal methods
Contactless space debris removal approaches
Ion beam-assisted transportation
Transportation by electrostatic interaction
Transportation by lasers
Transportation using gravity fields
Contactless detumbling using eddy currents
References
Ion beam physics
Mathematical modeling of an in-beam
Simplified ion beam models
Physics of ion beam interaction with a body surface
Calculating the forces and torque generated by the ion beam
Examples of ion force and torque calculations
Calculation assumptions and methodology
Sphere in an ion beam
Cylinder in an ion beam
Rectangular prism in ion beam
Comparison of rocket stage and cylinder
Cylindrical satellite with solar panels in ion beam
References
Dynamics of relative translation motion of spherical space debris during ion beam transportation
Mathematical model describing ion beam transportation without taking into account space debris attitude motion
General assumptions
Gauss planetary equations of space debris motion
Equations of motion in a spherical reference frame
Calculation of fuel consumption and thruster plume parameters
Estimation of the thruster exhaust velocity and propellant mass for space debris deorbiting from a circular orbit
Active spacecraft relative position stability and control
Stability of the active spacecrafts relative motion in a quasicircular orbit
Control of the active spacecrafts relative motion in a quasicircular orbit
Control of the active spacecrafts relative motion in an elliptical orbit
Space debris removal: Preliminary mission design
References
Dynamics of passive object attitude motion during ion beam transportation
Mathematical models of a passive space debris object during its contactless transportation by an active spacecraft
Mathematical model of the planar motion of space debris
Mathematical model of the spatial motion of space debris
Mathematical model of the spatial motion of a symmetrical space debris object
Stationary motions of a symmetrical space debris object
Attitude dynamics of the uncontrolled motion of a passive space debris object at a constant relative position of the ...
Phase portraits and bifurcation diagrams
Chaotic motion of the object relative to its center of mass in an elliptical orbit in the planar case of motion
Dynamics of the controlled motion of a passive object
Control approaches: Control of beam rate and direction
Control of the space debris attitude motion in a planar case
Thrust control for the case when LIzu=1(0)=0 and β=β
Thrust control for the case when LIzu=1(0)0 and β=βmin or β=βmax
Ion beam direction control
Comparison of the effectiveness of control methods
Ion beam control based on energy estimation
Control of the space debris attitude motion in a spatial case in GEO
Detumbling of axisymmetric space debris in a spatial case
Fuel costs estimation for ion beam-assisted space debris removal mission with and without attitude control
Equations of motion
Control strategies
Comparison of fuel costs for different control strategies
References
The use of contactless ion beam technology
Orbital flight in one plane
Deorbiting of space debris object into the atmosphere
Transportation of space debris into disposal orbit
References
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
R
S
T

Citation preview

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation Vladimir Aslanov Theoretical Mechanics Department, Samara State Aerospace University, Samara, Russia

Alexander Ledkov Samara State Aerospace University, Samara, Russia

Acknowledgment

The authors thank the Russian Science Foundation for offering financial support (Project No. 19-19-00085).

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www. elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-99299-2 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Rafael Guilherme Trombaco Production Project Manager: Kamesh R Cover Designer: Matthew Limbert Typeset by STRAIVE, India

Introduction

This book is devoted to one of the most important problems of modern astronautics. Space debris is a huge threat to future generations. Mutual collisions of large objects in orbit can set off a chain reaction known as Kessler syndrome, which will lead to the formation of a cloud of small debris around the Earth and can close the way for humankind into space. The scientific community offers many different ways to solve the space debris problem. This book is dedicated to one of them. Transportation by an ion beam assumes the contactless impact on a large space debris object, which is a derelict satellite or rocket upper stage, through an ion beam generated by an electric thruster of an active spacecraft. The impact of an ion beam can be used both directly for transporting a space debris object, and as one of the stages of an active space debris removal mission, aimed at detumbling of a target before docking or capturing. The idea of ion beam-assisted transportation itself is simple and elegant. It was formulated in 2011 independently by C. Bombardelli, J. Pelaez, and S. Kitamura. The idea was developed in the framework of an international FP7-SPACE competition project, “Improving Low Earth Orbit Security With Enhanced Electric Propulsion” (LEOSWEEP). The proposed scheme assumes that one of an active spacecraft’s thrusters creates an ion beam and directs it to a target space debris object. Ions hit the surface of space debris and generate a force, which is used to transport space debris. The indisputable advantage of contactless methods in comparison with methods involving direct or indirect mechanical contact of an active spacecraft and space debris is their safety. The likelihood that inaccuracies and errors during the mission will lead to the breakdown or destruction of the active spacecraft is extremely small. In addition, contactless methods can be used in multipurpose active spacecraft designed for transporting a large number of various space objects, since they do not require the development of special docking assemblies for each specific target object. Besides their obvious advantages, contactless methods also have specific disadvantages. The ion beam transport scheme assumes additional fuel consumption required to compensate for the thrust created by the ion engine blowing the space debris object. Moreover, long-term irradiation of the object with a high-speed ion flow causes surface erosion and sputtering, which in turn can lead to contamination of sensors and solar panels of the active spacecraft with a backflow of sputtered particles, and the formation of secondary space debris fragments as a result of the detachment of various elements from the irradiated surface. One of the key features of ion beam-assisted transportation of a space object is the mutual influence of its attitude motion and the magnitude and direction of the force and torque generated by the ion beam. The position and orientation of the space debris

x

Introduction

object inside the ion beam determine these force and torque. The point of application of the resultant force does not coincide with the center of mass of the object. This leads to the appearance of an ion beam torque relative to the center of mass, which tends to rotate the object in the ion flow. This rotation, in turn, will lead to a change in the magnitude and direction of the ion beam force. This book is devoted to a comprehensive study of this phenomenon within the framework of a space debris removal mission. Various mathematical models will be developed, an analysis of the regular and chaotic dynamics of a space object under the action of an ion beam will be carried out, and control laws, ways, and methods for the active space debris removal mission and space debris detumbling will be developed. The book consists of six chapters. Chapter 1 gives the reader the necessary theoretical basis, including a description of the mathematical apparatus, the main theorems and models of classical and celestial mechanics, methods of stability theory, and chaotic dynamics. Chapter 2 is devoted to an overview of the space debris problem. Particular attention is paid to various methods of active space debris removal. Chapter 3 describes the physics of ion propagation and gives the simplest models for calculating the force and torque generated by the ion beam. Chapter 4 describes mathematical models and analyzes the motion of space debris under the action of an ion beam without taking into account its attitude motion. Chapter 5 examines the dynamics of space debris and active spacecraft, taking into account the rotation of space debris relative to its center of mass. Chapter 6 presents the results of the analysis of numerical simulation of a mission to remove space debris and transfer it to a disposal orbit. The authors consider it their duty to express their gratitude and respect to the scientists who had a great influence on them, and who laid a solid theoretical and methodological foundation, which ultimately made it possible to obtain interesting and important results for a complex and multifaceted problem of space debris contactless transportation by an ion beam. We express our respect to Viktor Melnikov, who made a huge contribution to the development of modern chaotic dynamics, proposing a fundamental method for determining the boundaries of chaos; to Vladimir Beletsky, who made a great contribution to the development of the fundamental theory of space flight mechanics; and to Vasiliy Yaroshevsky, whose results and approaches on the study of the attitude motion of a spacecraft in a resisting environment made it possible to obtain important results for the problem of contactless ion beam-assisted transportation. This book is based on the results of research obtained by the authors with the financial support of the Russian Science Foundation (Project No. 19-19-00085).

Basics of space flight mechanics and control theory 1.1

1

Mathematical and mechanical preliminaries

This section briefly describes some basic mathematical operations with matrices and vectors, and the fundamentals of classical mechanics, which are used in this book to build mathematical models.

1.1.1 Vector and matrix operations Mechanics studies the motion of bodies in three-dimensional, homogeneous, isotropic space. The location of a point P can be given by a position vector r connecting some reference point O with P. Consider some Cartesian reference frame OXjYjZj, the direction of the axes of which are given by unit vectors ej1, ej2, ej3. These vectors are mutually orthogonal and form the basis of the coordinate system. Any vector r can be represented as a linear combination of the orthogonal basis vectors (Fig. 1.1) rj ¼ xej1 + yej2 + zej3 : The superscript of a vector specifies the reference frame in which its coordinates are given. This vector can be represented as a column vector rj ¼ [x, y, z]T. Let us introduce one more reference frame OXiYiZi with basis vectors ei1, ei2, ei3. The vector r has in this frame the coordinates ri ¼ [xi, yi, zi]T, and it can be represented as ri ¼ xi ei1 + yi ei2 + zi ei3 : The transition from one reference frame to another is carried out using the transition matrix ri ¼ Mij rj , where Mij is a directional cosine matrix, which transform coordinates of the vector from the reference frame OXjYjZj to frame OXiYiZi. The inverse transform can be performed using an inverse matrix, which is equal to a transpose matrix due to its orthogT onality Mji ¼ M1 ij ¼ Mij. Matrices, which transforms coordinates from jth reference frame to ith frame by the jth frame rotation around its axes OXj, OYj, OZj (Fig. 1.2), have the form.

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation. https://doi.org/10.1016/B978-0-323-99299-2.00003-3 Copyright © 2023 Elsevier Inc. All rights reserved.

2

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 1.1 Coordinate system and basis vector.

Fig. 1.2 Coordinate system rotation.

2

1

6 Mij ¼ 4 0 0 2

0 cos φx  sin φx cos φz

6 Mij ¼ 4  sin φz 0

3

0

2

cos φy

7 6 sin φx 5, Mij ¼ 4 0 sin φy cos φx

sin φz cos φz 0

0

0 1 0

 sin φy

3

7 0 5, cos φy

3

7 0 5: 1

The mathematical operations of the scalar and cross product for vectors a ¼ [ax, ay, az]T and b ¼ [bx, by, bz]T are defined as. 2

s ¼ a  b ¼ a x bx + ay by + az bz ,

3 ay bz  az by 6 7 c ¼ a  b ¼ 4 az bx  ax bz 5 : ax by  ay bx

Vector c, which is the result of the cross product of vectors a and b, is orthogonal to these vectors. If the lengths of the vectors a, b and the angle between them α are known, then the scalar product s and the modulus of the vector product c can be found as. s ¼ ab cos α,

c ¼ ab sin α:

Basics of space flight mechanics and control theory

3

Let us note some properties of basis vectors, which are used below. The scalar product of a vector with itself gives one, while the scalar product of one basis vector with another basis vector gives zero due to their orthogonality.  ei  ej ¼

1, 0,

when i ¼ j; when i 6¼ j:

Since all reference frames considered in this book are right-handed and orthogonal, the following properties for the cross product of basis vectors can be formulated: ei  ei ¼ 0,

e1  e2 ¼ e3 ,

e2  e3 ¼ e1 ,

e3  e1 ¼ e2 ,

ei  ej ¼ ej  ei :

1.1.2 Fundamentals of classical mechanics The basics of classical mechanics and the Lagrange equations are used in the following sections of this book when developing mathematical models. Without going into details, let us provide some key points and formulas. To describe the motion of a rigid body center of mass, Newton’s second law is used: m€r ¼ F,

(1.1)

where m is the mass of the body, r is the position vector of its center of mass, and F is the sum of all external forces acting on the body. Vectors r and F are given in the inertial coordinate system. To describe the motion of a body around its center of mass, the change of the total angular momentum theorem is used. If a rigid body is considered as a system of N particles, then the rate of change of the total angular momentum H of a system of particles about its center of mass is equal to the sum of the external torques L on the system: _ ¼ L, H

(1.2)

where the total angular momentum is H¼

N X

ρi  mi Vi ,

(1.3)

i¼1

where ρi is the vector connecting the center of mass with the ith particle, mi is the mass of the ith particle, and Vi is the velocity of the particle. In the case of a rigid body, when the distances between particles are constant, the momentum vector is determined by the expression H ¼ [I]ω, where [I] is the inertia matrix and ω is the column vector of the body angular velocity (Schaub and Junkins, 2014).

4

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

The equations of motion of a mechanical system can be obtained in the form of the Lagrange equations of the second kind: d ∂L ∂L  ¼ Qi , dt ∂q_ i ∂qi

(1.4)

where L ¼ T  P is the Lagrangian, T is the kinetic energy of the system, P is the potential energy of the system, qi is ith generalized coordinates, which are a set of parameters representing the state of a system, and Qi is the generalized force defined by the equation Qi ¼

n X j¼1

Fj 

∂rj , ∂qi

(1.5)

where Fj is the nonpotential force applying on the system and rjis the force application point position vector expressed in terms of generalized coordinates. The kinetic energy of a mechanical system can be found using Koenig’s theorem, which states that the kinetic energy of a system of particles is the sum of the kinetic energy of the system’s center of mass TC and the kinetic energy associated to the motion of the particles relative to the center of mass Tr. T ¼ TC + Tr ¼

N mV 2C 1X m V2 , + 2 i¼1 i ir 2

(1.6)

where VC is the absolute velocity of the system’s center of mass and Vir is the velocity of the ith point of system relative to its center of mass. In the case of a rigid body, Tr can be calculated as 1 1 T r ¼ H  ω ¼ ωT ½Iω: 2 2

(1.7)

Since the positiveness of Tr follows from Eq. (1.6), it can be concluded from Eq. (1.7) that the angle between the vectors H and ω is always acute.

1.2

Coordinate systems

Several coordinate systems are used below in developing mathematical models and analyzing the motion of space debris and active spacecraft. The planetary nonrotating inertial frame OXpYpZp is a Cartesian frame. Its origin is the center of the Earth. The axes OXp and OYp lie in the equatorial plane, and the axis OZp is directed along the Earth’s rotation axis toward the Earth’s North Pole. The axis OXp is directed to the vernal equinox. This frame is known as the Earth-centered-inertial (ECI) frame

Basics of space flight mechanics and control theory

5

Fig. 1.3 Reference frames.

(Schaub and Junkins, 2014). The Earth itself rotates relative to this frame. The unit vectors of the inertial frame are ep1, ep2, ep3. Fig. 1.3 shows the inertial coordinate frame OXpYpZp. The perifocal frame OXfYfZf is associated with the orbital plane of the considered space object moving around the Earth (Fig. 1.3). The location of the object is determined by a position vector r that connects the center of the Earth O to the object’s center of mass. The axis OXf is directed to the orbit periapsis, which is the point of the orbit closest to the attracting center O. The axis OZf is perpendicular to the plane of the orbit. It is directed along the vector h ¼ r  r_ ,

(1.8)

which is the massless angular momentum of the object relative to the point O. The axis OYf completes the right-handed coordinates system. In the case of unperturbed motion, the object’s orbit does not change with time and the reference frame OXfYfZf is inertial. The transition from the inertial to the perifocal coordinate system can be performed using three successive Euler rotations through angles Ω, i, and ω around the z-x-z axes, where Ω is the argument of the ascending node, i is the orbit inclination angle, and ω is the argument of the periapsis. The rotation matrix Mfp, which provides transformation from the frame OXpYpZp to OXfYfZf, has the form Mfp ¼ Mf 2  M21  M1p where the rotation matrices corresponding to each Euler rotation are defined as.

6

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

2

M1p

cos Ω 6 ¼ 4  sin Ω

sin Ω cos Ω

0

0

2

cos ω 6 Mf 2 ¼ 4  sin ω 0

sin ω cos ω 0

3 2 0 1 7 6 0 5, M21 ¼ 4 0 1

0

0 cos i  sin i

3 0 7 sin i 5, cos i

3

0 7 0 5: 1

Matrix multiplication gives 2

cos ω cos Ω  cos i sin ω sin Ω

6 Mfp ¼ 4  sin ω cos Ω  cos i cos ω sin Ω

cos i sin ω cos Ω þ cos ω sin Ω cos i cos ω cos Ω  sin ω sin Ω

sin i sin Ω

 sin i cos Ω

sin i sin ω

3

7 sin i cos ω 5: cos i (1.9)

Since the rotation matrices are orthogonal matrices, the reverse transition from the frame OXfYfZf to OXpYpZp can be done using the matrix transpose operation T Mpf ¼ M1 fp ¼ Mfp :

The unit vectors of the perifocal frame ef1, ef2, ef3 can be found from the expressions e ef 1 ¼ , e

h ef 3 ¼ , h

ef 2 ¼ ef 3  ef 1 ,

where e is the orbit plane eccentricity vector (Sidi, 1997) e¼

r_  h r  , μ r

which is parallel to the line of apsides and directed to the periapsis, μ is the gravitational parameter of the Earth, e ¼ | e |, h ¼ | h|, r ¼ | r | are the lengths of the corresponding vectors, and e is the orbital eccentricity. In addition to coordinate systems whose origin is located at the center of the Earth, let us introduce several orbital reference frames with origin at the center of mass of a space object B moving around the Earth. The Hill’s reference frame BXHYHZH can be obtained by transferring the origin and then rotating the axes of the frame OXfYfZf around axis OZf by the true anomaly angle f (Fig. 1.3). The corresponding rotation matrix is 2

cos f

6 MHf ¼ 4  sin f 0

sin f cos f 0

0

3

7 0 5: 1

Basics of space flight mechanics and control theory

7

The unit vectors of the Hill’s reference frame eH1, eH2, eH3 can be found from the expressions r eH1 ¼ , r

h eH3 ¼ , h

eH2 ¼ eH3  eH1 :

(1.10)

The rotation matrix MHp, which provides transformation from the frame OXpYpZp to OXHYHZH, can be found as MHp ¼ MHfMfp. It has the following components: 2

cos u cos Ω  cos i sin u sin Ω

cos i sin u cos Ω þ cos u sin Ω

6 MHp ¼ 4  sin u cos Ω  cos i cos u sin Ω

cos i cos u cos Ω  sin u sin Ω

sin i sin Ω

 sin i cos Ω

sin i sin u

3

7 sin i cos u 5, cos i

where u ¼ ω + f. The Hill’s reference frame BXHYHZH described is very convenient for solving applied problems of orbital mechanics. This frame rotates relative to the inertial frame OXpYpZp with the angular velocity   _ p3 + ie _ 11 + ω_ + f_ eH3 : ωHp ¼ Ωe This vector has the following components in the inertial frame: 2 3 2 3 0 i_ 6 7 6 7 6 7 ¼ 4 0 5 + Mp1 4 0 5 + MpH 4 0 5, _ ω_ + f_ Ω 0 2

  _ p + ie _ p + ω_ + f_ ep ωpHp ¼ Ωe p3 11 H3

0

3

where Mp1 ¼ MT1p. Using the rotation matrix MHp, let us obtain the coordinates of the angular velocity vector ωHp in the Hill’s frame: 3 3 2 _ sin i sin u + i_ cos u ωHpx Ω 7 6 _ 7 6 ¼ 4Ω sin i cos u  i_ sin u 5 ¼ 4 ωHpy 5: _ cos i + ω_ + f_ ωHpz Ω 2

p ωH Hp ¼ MHp ωHp

(1.11)

Besides the Hill’s reference frame, the local vertical, local horizontal (LVLH) frame BXlYlZl is often used (Sidi, 1997). This frame differs from the Hill’s coordinate system in the order in which the axes are numbered (Fig. 1.4): el1 ¼ eH2 ,

el2 ¼ eH3 ,

el3 ¼ eH1 :

The spherical orbital reference frame BXoYoZo is obtained from the planetary frame OXpYpZp by two turns and transfer of the origin to point B (Fig. 1.5). The first rotation is performed at an angle ϑ around the axis OZp counterclockwise. The second rotation is performed at an angle ν around the new OYo axis in a clockwise direction. The axis OXo of the obtained reference frame is directed along the position vector r. The axis

8

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 1.4 Hill’s and local vertical, local horizontal reference frames.

Fig. 1.5 Spherical orbital reference frame.

OZo lies in the plane formed by the position vector and the axis OZp. The axis OYo completes the right-handed coordinate frame. The rotation matrix, which provides transformation from the OXpYpZp frame to OXoYoZo, has the form 2

Mop ¼ Mo3  M3p

cos v cos ϑ 6 ¼ 4  sin ϑ  sin v cos ϑ

cos v sin ϑ cos ϑ  sin v sin ϑ

3 sin v 7 0 5, cos v

where 2

Mo3

cos v 6 ¼4 0  sin v

0 1

3 sin v 7 0 5,

0

cos v

2

M3p

cos ϑ 6 ¼ 4  sin ϑ 0

sin ϑ cos ϑ 0

3 0 7 0 5: 1

The unit vectors of the spherical orbital reference frame PXoYoZo are

Basics of space flight mechanics and control theory

r eo1 ¼ , r

eo2 ¼ ep3  eo1 ,

9

eo3 ¼ eo1  eo2 :

To transfer from the frame BXoYoZo to the Hill’s frame, it is necessary to perform rotation around the BXo axis by an angle φHo ¼ arccos

  h  eo3 signððeo3  hÞ  eo1 Þ: h

The corresponding rotation matrix has the form 2

1

6 MHo ¼ 4 0 0

0

0

cos φHo  sin φHo

3

7 sin φHo 5: cos φHo

The origin of the body fixed frame BXbYbZb is located at the object’s center of mass (Fig. 1.6). The axes of this frame are the principal body axes. The transition from the orbital reference frame BXOYOZO, which is the spherical orbital reference frame BXoYoZo or Hill’s frame BXHYHZH, to the body frame BXbYbZb can be performed using three Euler rotations through the angles γ, θ, and φ (y-x-y). The corresponding rotation matrices have the form. 2

cos γ

0

 sin γ

3

2

1

6 7 6 M4O ¼ 4 0 1 0 5, M54 ¼ 4 0 sin γ 0 cos γ 0 2 3 cos φ 0  sin φ 6 7 Mb5 ¼ 4 0 1 0 5: sin φ 0 cos φ

0 cos θ  sin θ

The rotation matrix to transfer from BXOYOZO to BXbYbZb is

Fig. 1.6 Orbital and body fixed reference frames.

0

3

7 sin θ 5, cos θ

10

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

2 6 MbO ¼ 4

1.3

cos γ cos φ  cos θ sin γ sin φ

sin θ sin φ

 sin γ cos φ  cos θ cos γ sin φ

sin γ sin θ

cos θ

cos γ sin θ

cos θ cos φ sin γ þ cos γ sin φ

 sin θ cos φ

cos γ cos θ cos φ  sin γ sin φ

3 7 5:

Two-body problem: Motion in the orbital plane

The two-body problem is one of the basic issues of celestial mechanics. It consists in determining the motion of two particles that interact only with each other by means of the universal gravitational law. In addition to the mutual gravitational attraction, some disturbance forces act on the particles Fd1 and Fd2. The particles’ motion is considered in some inertial reference frame. The particles’ positions are defined by the vectors r1 and r2 (Fig. 1.7). The vector r ¼ r2  r1 determines the position of m2 point relative to m1. Writing down Newton’s second law for each particle, and finding the difference between these equations, the equation of relative motion is obtained in the form €r ¼ 

μr + ad , r3

(1.12)

where μ ¼ G(m1 + m2) is the gravitational coefficient, G is the universal gravity constant, and ad is the disturbance acceleration vector ad ¼

Fd2 Fd1  : m2 m1

Although Eq. (1.12) is nonlinear in the case of unperturbed motion ad ¼ 0, it can be integrated analytically. Let us construct the fundamental integrals of an orbit, which remain constant due to the equations of motion. The derivative of the massless angular momentum vector h ¼ r  r_ , calculated using the chain rule, taking into account Eq. (1.12), has the form

Fig. 1.7 Two-body problem geometry.

Basics of space flight mechanics and control theory

  μr h_ ¼ r_  r_ + r  €r ¼ r   3 + ad ¼ r  ad : r

11

(1.13)

In the case of unperturbed motion, the right-hand side of Eq. (1.13) turns to zero, which means that h is a constant vector. Since the vector h, due to the properties of the cross product, is perpendicular to the plane formed by the vectors r and r_ , this plane remains fixed in space. Let us place a rotating reference frame O1XHYHZH with unit direction vectors (1.10) on the first particle (Fig. 1.8). The velocity vector r_ can be represented as the sum of two projections on the axes lying in the plane of the orbit _ H1 + r f_eH2 , r_ ¼ re

(1.14)

where f_ is the angular velocity of the position vector, which is r ¼ reH1. The vector r_ length, according to Eq. (1.14), is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V ¼ |_r| ¼ r_2 + r 2 f_ :

(1.15)

Direct calculation of the massless angular momentum vector gives   _ H1  eH1 + r 2 f_eH1  eH2 ¼ r 2 f_eH3 , _ H1 + r f_eH2 ¼ r re h ¼ reH1  re

(1.16)

since eH1  eH1 ¼ 0, eH1  eH2 ¼ eH3. From Eq. (1.16), it follows that h ¼ r 2 f_:

(1.17)

The total energy per unit mass is the sum of kinetic energy per unit mass (V2/2) and the potential energy per unit mass ( μ/r): ξ¼

V2 μ  : r 2

Fig. 1.8 Orbital plane.

(1.18)

12

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

To show that the energy ξ is constant in the unperturbed case (ad ¼ 0), scalar multiplication of both sides of Eq. (1.12) by r_ yields r_  €r ¼ 

   μ μ _eH2  reH1 ¼  μ r_ ¼ d μ : _ _ r  r ¼  + r f re H1 dt r r3 r3 r2

(1.19)

On the other hand,  2   _ V + W ? eV? ¼ V V_ ¼ V dV ¼ d V , r_  €r ¼ V  V_ ¼ VeV  Ve dt dt 2

(1.20)

where eV is the unit vector directed along the velocity vector, eV? is the unit vector orthogonal to eV, and W? is the projection of the acceleration onto the direction orthogonal to the vector eV. Equating expressions (1.19) and (1.20) gives    d V2 d μ : ¼ dt 2 dt r Shifting to the left-hand side of the equation yields     d V2 d μ d V2 μ dξ ¼ ¼ ¼ 0:   dt 2 dt r dt 2 r dt whence it follows that in unperturbed motion the specific energy ξ is constant, and the energy conservation equation can be written in the form ξ¼

V2 μ  ¼ const: r 2

(1.21)

μ If the value ξ0 ¼  2a is taken as the energy constant, then the vis-viva equation can be obtained from Eq. (1.18) in the form

V2 ¼

2μ μ  , r a

(1.22)

where a is the semimajor axis of the elliptical orbit. To obtain eccentricity vector integral, let us calculate the derivative of the vector product, taking into account Eq. (1.12): d μr μ _ Þ ðr_  hÞ ¼ €r  h + r_  h_ ¼ €r  h ¼  3  ðr  r_ Þ ¼  3 ðr r_  rr dt r r   d r ¼μ , dt r where the following relationship for a vector triple product was used:

(1.23)

Basics of space flight mechanics and control theory

r  ðr  r_ Þ ¼ ðr  r_ Þr  ðr  rÞ_r:

13

(1.24)

Rewriting Eq. (1.23), and dividing the right-hand and left-hand sides by μ, gives    d r_  h d r ¼ : μ dt dt r This equation can be integrated as follows: r_  h r ¼ + e, μ r where e is the integration constant, called the eccentricity vector: e¼

r_  h r  ¼ const: μ r

(1.25)

This vector lies in the orbital plane. The scalar product of eccentricity vector (1.25) with the position vector r gives er¼

 r_  h rr r : μ r

Making circular shift of the scalar triple product r  ðr_  hÞ ¼ ðr  r_ Þ  h ¼ h  h ¼ h2 , the last expression takes the form er cos f ¼

h2  r, μ

where f is the true anomaly, which is the angle between the vectors e and r, and e is the eccentricity. Solving this equation for r, we get r¼

p , 1 + e cos f

(1.26)

where p is the semilatus rectum, which is defined as p¼

h2 : μ

(1.27)

Eq. (1.26) is the trajectory equation. It describes a conic section, which is written in polar coordinates (r, f) with the origin at a focus. A conic section is the intersection curve of a plane and a right circular cone. It can be a circle, ellipse, parabola, or

14

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

hyperbola. Calculation of the time derivative of expression (1.26) taking into account Eq. (1.17) gives r_ ¼

pe sin f he sin f : f_ ¼ p ð1 + e sin f Þ2

(1.28)

For the purposes of this book, we are primarily focused on circular and elliptical orbits. Fig. 1.9 shows the geometry of an elliptical orbit. Let us write down some geometric relations that will be used below. The semimajor axis a and the semiminor axis b can be found through the semilatus rectum and the eccentricity as. a¼

p , 1  e2

p b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  e2

The distance from a focus to the minor axis is c ¼ for periapsis and apoapsis are. rp ¼

p , 1+e

ra ¼

(1.29) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  b2 ¼ ae. The radial distances

p : 1e

The above expressions determine the orientation and shape of the orbit. To determine the position of the particle in this orbit at any instance of time, it is necessary to solve the equation of motion. Consider the classical case of unperturbed motion, described by the nonlinear equation €r ¼ 

μr : r3

From the massless angular momentum conservation integral Eqs. (1.17) and (1.27), it follows that Fig. 1.9 Orbital geometry.

Basics of space flight mechanics and control theory

f_ ¼

rffiffiffiffiffi μ ð1 + e cos f Þ2 : p3

15

(1.30)

This differential equation can be reduced to ðf rffiffiffiffiffi μ df ðt  t0 Þ ¼ : 3 p ð1 + e cos f Þ2

(1.31)

f0

For the cases of circular or elliptical orbits e  [0, 1), the integral on the right-hand side of Eq. (1.31) can be calculated analytically: pffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi  1  eð1  cos f Þ μ 2 e sin f 2 ffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffi p ð Þ ¼ 1  e  t  t arctan 0 ð1 + e cos f Þ p3 1 + e sin f 1  e2 ! pffiffiffiffiffiffiffiffiffiffiffi 1  eð1  cos f 0 Þ e sin f 0 2 pffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi arctan + : ð1 + e cos f 0 Þ 1 + e sin f 0 1  e2 (1.32) Finding the dependence f(t) from the transcendental Eq. (1.32) is a difficult task. The transition from the true anomaly f to the eccentric anomaly E allows us to remove this complexity. The relationship between these angles can be written as E tan ¼ 2

rffiffiffiffiffiffiffiffiffiffiffi 1e f tan : 1+e 2

(1.33)

To clarify the geometric meaning of the angle E, let us construct an auxiliary circle, the center of which is located at the intersection of the major and minor axes of the elliptic orbit, and of which the radius is a. Then draw a perpendicular to the major axis of the ellipse through the point m2 until it intersects with the circle in point A (Fig. 1.9). The eccentric anomaly is the angle between the semimajor axis of the ellipse and the line connecting the center of the circle with point A. Using the trigonometric relation tan(f/2) ¼ sin f/(1 + cos f), expression (1.33) can be rewritten as E tan ¼ 2

rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  e 1  cos 2 f , 1 + e 1 + cos f

Let us find cosf from this equation: cos f ¼

cos E  e : 1  e cos E

It follows from Eqs. (1.33), (1.34) that

(1.34)

16

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

df ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  e2 dE, 1  e cos E

1 + e cos f ¼

1  e2 , 1  e cos E

(1.35)

Taking into account these expressions, the integral on the right-hand side of Eq. (1.31) can be calculated as ðf rffiffiffiffiffi ðE  e sin EÞ ðE0  e sin E0 Þ μ df ðt  t 0 Þ ¼ ¼  : p3 ð1 + e cos f Þ2 ð1  e2 Þ3=2 ð1  e2 Þ3=2

(1.36)

f0

This equation is known as Kepler’s equation. It can be rewritten in the form M ¼ E  e sin E,

(1.37)

where M ¼ M0 + n(t  t0) is the mean anomaly, M0 ¼ E0  e sin E0 is the mean anomaly at t ¼ t0, and n is the mean angular rate: n¼

pffiffiffiffiffiffiffiffiffiffi μa3 :

(1.38)

Using the mean or eccentric anomaly instead of the true anomaly angle is more convenient for specifying the position of the particle in orbit. Eq. (1.33) can be used to determine the true anomaly angle. In the case of using the eccentric anomaly instead of the true anomaly angle, the equation for radial distance (1.26) becomes r ¼ að1  e cos EÞ:

(1.39)

The orbital period on elliptic orbit can be obtained from the solution (1.36). Let us take t ¼ T + t0 and E ¼ E0 + 2π, then rffiffiffiffiffi ðE + 2π  e sin ðE0 + 2π ÞÞ ðE0  e sin E0 Þ μ 2π  ¼ : T¼ 0 3=2 3=2 2 2 p3 ð1  e Þ ð1  e Þ ð1  e2 Þ3=2 Using Eq. (1.29), the period is expressed as rffiffiffiffiffi a3 T ¼ 2π : μ

(1.40)

Since the two-body problem is described by a second-order vector differential equation (1.12), the solution to the two-body problem depends on six arbitrary constants. It can be three position vectors’ initial coordinates and three components of the velocity vector that define the vectors r0 ¼ r(t0) and V0 ¼ r_ ðt0 Þ. However, these constants can

Basics of space flight mechanics and control theory

17

be specified in another way. Six scalar constants called orbit elements are widely used in practice: ½a, e, i, Ω, ω, M0 :

(1.41)

These constants determine the geometry of the orbit in the two-body problem. The three Euler angles Ω, i, and ω define the orientation of the orbit plane. The shape of the orbit is determined by the semimajor axis a, the eccentricity e, and the mean anomaly M0. The values of these constants can be found from the initial conditions. The position of the particle in orbit for any time instance t can be found by calculating the value of E as a solution to the nonlinear equation (1.37), and then determining the angle of the true anomaly f from expression (1.33). To find the angle of the true anomaly, the differential equation (1.30) can also be used directly. Besides the set of constants (1.41) defining the orbit, alternative sets of constants may be used. Let us determine the orbit elements corresponding to the initial position r0 and initial velocity V0, which are given in the inertial reference frame OXpYpZp. For the semimajor axis a and eccentricity e, the following expressions can be used: a¼

μr 0 , 2μ  r 0 V 20

where r 0 ¼ |r p0 | ¼

ep ¼

Vp0  ðhp Þ rp0  , r0 μ

e ¼ |ep ,

hp ¼ rp0  Vp0 ,

ffi pffiffiffiffiffiffiffiffiffiffiffiffi rp0  rp0 and V0 ¼ | V0 |. The initial mean anomaly can be found as

M0 ¼ E0  e sin E0 , where E0 ¼ atan2



rp0  Vp0 pffiffiffiffi μa

 , 1  ra0 . To determine the angles, let us write the coordinates

of unit vectors epf1 ¼

ep ¼ C11 ep1 + C12 ep2 + C13 ep3 ¼ ½C11 , C12 , C13 T , e

epf3 ¼

hp ¼ C31 ep1 + C32 ep2 + C33 ep3 ¼ ½C31 , C32 , C33 T , h

epf2 ¼ epf3  epf1 ¼ C21 ep1 + C22 ep2 + C23 ep3 ¼ ½C21 , C22 , C23 T : Then, taking into account Eq. (1.9), the angles can be found as 

i ¼ arccos ðC33 Þ,

C33 Ω ¼ arctan , C32

 C13 ω ¼ arctan : C23

Thus, all orbit elements are defined in terms of the initial position vector and its velocity.

18

1.4

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Planetary equations for perturbed motion

Orbital elements can be used to describe the perturbed motion of a material point. If the perturbing forces acting on the system are not potential, then Gauss’ variational equations can be used to describe the evolution of the orbital parameters. Consider the case when the position vector r, the velocity vector V, and the disturbance acceleration ad are given by their coordinates in the Hill’s frame BXHYHZH (it is assumed that the point O in Section 1.2 coincides with the particle m1, and B coincides with the particle m2). rH ¼ ½r, 0, 0T ,

T _ r f_, 0 , VH ¼ r,

T aH d ¼ ½ar , aθ , ah  :

(1.42)

To calculate the components of the velocity vector, the orbit equation (1.26) and the expression for the angular momentum h ¼ r 2 f_, and h2 ¼ pμ can be used: r_ ¼

pef_ sin f μ ¼ e sin f : ð1 + e cos f Þ2 h

(1.43)

whence it follows that VH ¼

h

iT μ μp e sin f , ,0 : h hr

(1.44)

Let us obtain an equation for the semimajor axis a time derivative. Expressing the semimajor axis a from the vis-viva equation (1.22) and substituting V2 expressed from Eq. (1.21), we find that a¼

μr μ ¼ : 2ξ rV  2μ 2

Taking derivative of this equation gives a_ ¼

μ _ 2a2 _ ξ: ξ¼ μ 2ξ2

(1.45)

From classical mechanics, it follows that the power per unit mass can be found as H ξ_ ¼ aH d  V :

(1.46)

Substituting Eq. (1.46) into Eq. (1.45), taking into account Eqs. (1.42) and (1.44), gives a_ ¼

  2a2 p ar e sin f + aθ : r h

(1.47)

Basics of space flight mechanics and control theory

19

To obtain the equation of the eccentricity evolution, let us find the derivative of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi angular momentum h ¼ μað1  e2 Þ with respect to time: h_ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi μð1  e2 Þ μa _ a_  ee: 4a 1  e2

(1.48)

The equation of motion of a point around the center of the Earth under the action of a perturbing torque has the form p h_ ¼ rp  apd :

(1.49)

The vectors included in this equation can be represented as hp ¼ hepH3, rp ¼ repH1, and apd ¼ arepH1 + aθepH2 + ahepH3. Since epH1  epH1 ¼ 0, epH1  epH2 ¼ epH3, and epH1  epH3 ¼  epH2, substituting them into Eq. (1.49) gives   p _ p + h_ep ¼ rep  ar ep + aθ ep + ah ep h_ ¼ he H3 H3 H1 H1 H2 H3 ¼ raθ epH3  rah epH2 :

(1.50)

It follows from Eq. (1.50) that h_ ¼ raθ ,

(1.51)

and e_ pH3 ¼ 

rah p e : h H2

(1.52)

Let us find the derivative e_ from Eq. (1.48): 1  e2 a_  e_ ¼ 2ae

rffiffiffiffiffiffiffiffiffiffiffiffiffi 1  e2 raθ , μa e

and substituting Eqs. (1.47) and (1.51) into the resulting expression gives the Gauss planetary equation of e:          1 e_ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1  e2 ar sin f + a 1  e2 + r cos f + re aθ : μað1  e2 Þ

(1.53)

where r is defined by Eq. (1.26). As described in Section 1.2, the inclination angle i is the angle between the unit vectors ep3 and eH3. From the scalar product, it follows that cos i ¼ eH3  ep3 ¼

h  ep3 : h

20

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Differentiating the left-hand and right-hand sides of the last expression gives h_  epp3 hp  epp3 h_ h_  epp3 h_ cos i di sin i ¼ ¼ :   dt h h h h2 p



p

Let us rewrite the last equation, taking into account Eqs. (1.50), (1.51): 

raθ epH3  epp3  rah epH2  epp3  raθ cos i di ra sin i ¼ ¼  h epH2  epp3 : dt h h

(1.54)

The vector epH2 coordinates can be obtained using the rotation matrix MpH ¼ MTHp given in Section 1.2:

epH2 ¼ MpH eH H2

2 3 2 3 0  sin u cos Ω  cos i cos u sin Ω 6 7 6 7 ¼ MpH 4 1 5 ¼ 4 cos i cos u cos Ω  sin u sin Ω 5: 0 sin i cos u

(1.55)

Calculating the scalar product in Eq. (1.54) and dividing both parts of the expression by sini yields di r cos u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ah : dt μað1  e2 Þ

(1.56)

Let us obtain an equation that describes the change of the argument of the ascending node during perturbed motion. Consider a unit vector en directed along the line of nodes. Using the rotation matrix Mp1 ¼ MT1p, the coordinates of this vector in the inertial reference frame OXpYpZp can be obtained based on its coordinates e1n ¼ [1, 0, 0]T in the frame OX1Y1Zp: 2

cos Ω

6 epn ¼ Mp1 e1n ¼ 4 sin Ω 0

 sin Ω cos Ω 0

32 3 2 3 1 cos Ω 76 7 6 7 0 54 0 5 ¼ 4 sin Ω 5: 1 0 0 0

(1.57)

Since the angles Ω, i, and ω are chosen so that the line of nodes is perpendicular to the angular momentum vector h, then eH3  en ¼ 0. This equality must always hold. The derivative of this product is   d epH3  epn ¼ e_ pH3  epn + epH3  e_ pn ¼ 0: dt This equation can be rewritten, taking into account Eq. (1.52), in the form epH3  e_ pn ¼

rah p e  ep : h H2 n

(1.58)

Basics of space flight mechanics and control theory

21

The coordinates of epH3 vector can be found using the rotation matrix MpH:

epH3 ¼ MpH eH H3

2 3 2 3 0 sin i sin Ω 6 7 6 7 ¼ MpH 4 0 5 ¼ 4  sin i cos Ω 5: 1

(1.59)

cos i

Vector e_ pn can be calculated by direct differentiation of epn coordinates given by Eq. (1.57):

 _ sin Ω, Ω _ cos Ω, 0 T : e_ pn ¼ Ω

(1.60)

Dot products in expression (1.58) are calculated using coordinates (1.55), (1.57), (1.59), and (1.60). After simple trigonometric transformations, it follows from expression (1.58) that r sin u _ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω ah : μað1  e2 Þ sin i

(1.61)

Let us do a few preliminary calculations before proceeding to the obtaining the derivative of ω. The eccentricity vector can be represented as e ¼ eef1, and it has the coordinates ef ¼ [e, 0, 0]T in OXfYfZf. In the inertial frame, the same vector has coordinates 2

eð cos ω cos Ω  cos i sin ω sin ΩÞ

3

2

ep1

3

6 7 6 7 ep ¼ Mpf ef ¼ 4 eð cos i sin ω cos Ω + cos ω sin ΩÞ 5 ¼ 4 ep2 5: e sin i sin ω ep3 Differentiating the coordinate ep3 with respect to time gives e_ p3 ¼ e_ sin i sin ω + ei_ cos i sin ω + eω_ sin i cos ω: From Eqs. (1.53) and (1.56), it follows that        sin i sin ω   e_p3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1  e2 ar sin f + a 1  e2 + r cos f + re aθ + 2 μað1  e Þ er cos u + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ah cos i sin ω + eω_ sin i cos ω: μað1  e2 Þ (1.62) Direct calculation of the derivative for the eccentricity vector e given by Eq. (1.25) yields e_ ¼

€r  h + r_  h_ rðr  r_ Þ  r 2 r_ rr_  r r_ €r  h + r_  h_ + + ¼ , 2 μ μ r r3

22

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Here the derivative of the position vector modulus r_ was found from the scalar product r  r_ , calculated taking into account Eq. (1.14):   _ H1 + r f_eH2 ¼ r r: _ r  r_ ¼ reH1  re

(1.63)

Using the vector triple product (1.24) and Eqs. (1.12), (1.13), the eccentricity time derivative can be rewritten as   μr  3 þ ad  h þ r_  ðr  ad Þ r  ðr  r_ Þ r e_ ¼ þ μ r3 r  ðr  r_ Þ ad  ðr  r_ Þ r_  ðr  ad Þ r  ðr  r_ Þ þ þ ¼ þ μ μ r3 r3 a  ðr  r_ Þ þ r_  ðr  ad Þ ¼ d μ rðad  r_ Þ  r_ ðad  rÞ þ rðr_  ad Þ  ad ðr_  rÞ ¼ μ 2rðad  r_ Þ  r_ ðad  rÞ  ad ðr_  rÞ ¼ μ Substituting Eq. (1.42) yields     _ H1 + r f_eH2 ar r  r r_ðar eH1 + aθ eH2 + ah eH3 Þ _ r + r f_aθ  re 2reH1 ra e_ ¼ μ _ θ ÞeH2  r ra _ h eH3 2haθ eH1  ðhar + r ra ¼ : μ The last expression can be written in coordinate form:

T _ θ r ra _ h 2haθ har + r ra , , : e_ ¼ μ μ μ H

In the inertial reference frame, this vector has coordinates e_ p ¼ MpH e_ H . Let us write the projection of this vector onto the axis OZp: e_p3 ¼

_ θ _ h 2haθ ha + r ra r ra sin i sin u  r sin i cos u  cos i: μ μ μ

(1.64)

Equating expressions (1.62) and (1.64), and then expressing ω_ from the result, we obtain after simplification ha cos f ω_ ¼  r + μe



2 + e cos f 1 + e cos f



haθ sin f rah cos i sin u  : μe h sin i

(1.65)

Basics of space flight mechanics and control theory

23

To obtain the equation for the mean anomaly, let us find the time derivative of Eq. (1.37): M_ ¼ E_  e_ sin E  eE_ cos E:

(1.66)

The derivative of the eccentric anomaly can be found from the time derivative of Eq. (1.39): r_ ¼ ð1  e cos EÞa_  ae_ cos E + aeE_ sin E Expressing E_ from this equation using Eqs. (1.28), (1.35), and (1.38) gives E_ ¼

ð1  e cos EÞ a_ n cos E e_  + : e sin E 1  e cos E sin E e a

(1.67)

Substituting expression (1.67) into Eq. (1.66), we write an equation describing the change in the mean anomaly: pffiffiffiffiffiffiffiffiffiffiffiffiffi

2 _ 1 r r a_ a 1  e2 M_ ¼ n + ð cos E  eÞe_  3 ¼ n + e_ cos f  2 : e sin E e sin f a a

(1.68)

Taking into account Eqs. (1.47) and (1.53), the last expression can be written as M_ ¼ n +

pffiffiffiffiffiffiffiffiffiffiffiffiffi 

 2er r 1  e2 h  1  sin f : cos f  a a e sin f μ að 1  e 2 Þ r að 1  e 2 Þ θ

(1.69)

The mean anomaly at epoch M0 can be found as ðt M0 ¼ M 

nðτÞdτ:

(1.70)

0

Eqs. (1.47), (1.53), (1.56), (1.61), (1.65), and (1.69) form a closed system known as Gauss’ planetary equations. These equations describe the change in the orbit elements under the action of the perturbing acceleration ad. Assuming that the mass of the first particle greatly exceeds the mass of the second m1 ≫ m2, it can be written that. μ ¼ Gm1 ,

ad ¼

Fd2 : m2

This assumption is quite justified for studying the motion of a spacecraft around a planet. It should be noted that the system of equations (1.47), (1.53), (1.56), (1.61), (1.65), and (1.69) cannot be used for describing circular and equatorial orbits, since at e ¼ 0 or i ¼ 0, some denominators becomes zero. To overcome this singularity, a different set

24

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

of orbit elements must be used. The obtained equations are only valid for elliptical orbits. For equatorial orbits, an alternative set of constants can be used as orbit elements (Bau` et al., 2015; Cohen and Hubbard, 1962; Walker, 1986). If all perturbing forces acting on the particle are potential, then the disturbance acceleration ad is related to the disturbance potential function R by the expression ad ¼

h iT ∂R : ∂r

(1.71)

If the potential function is zero, then the orbit is Keplerian, and the orbit elements are constants. Otherwise, the evolution of the orbit elements is given by a system of differential equations known as Lagrange’s planetary equations (Schaub and Junkins, 2014): dΩ 1 ∂R ¼ , dt nab sin i ∂i

(1.72)

di 1 ∂R cos i ∂R ¼ + , dt nab sin i ∂Ω nab sin i ∂ω

(1.73)

dω cos i ∂R b ∂R ¼ + 3 , dt nab sin i ∂i na e ∂e

(1.74)

da 2 ∂R ¼ , dt na ∂M0

(1.75)

de b ∂R b2 ∂R , ¼ 3 + 4 dt na e ∂ω na e ∂M0

(1.76)

dM0 2 ∂R b2 ∂R + 4 , ¼ na ∂a dt na e ∂e

(1.77)

The Lagrange equations can be useful for estimating the perturbing effect of potential forces on the spacecraft’s orbit. For example, they can be used to study the influence of the J2 gravitational perturbation on the motion of the spacecraft (Schaub and Junkins, 2014), or to analyze the effect of a spacecraft’s electrostatic charge on its orbit (Gangestad et al., 2010).

1.5

Relative orbital motion

During transportation of a passive space object by an ion beam, the object and the active spacecraft, which generates the beam, should be located at a relatively short distance from each other. This distance can vary from a few meters to several tens of meters. Using the difference in the objects’ position vectors to calculate their relative positions is not very accurate, since this difference can be compared with

Basics of space flight mechanics and control theory

25

rounding errors in their calculation. The study of the dynamics of such a mechanical system requires the development of mathematical models that describe the relative motion of two objects on two neighboring orbits. As considered in this book, the ion beam-assisted transportation problem relates to the spacecraft formation flying problem of maintaining the relative orbit of different type objects. A detailed analysis of the spacecraft formation flying problem can be found in Alfriend et al. (2010) and Schaub and Junkins (2014). The simplest spacecraft formation flight geometry is considered here. The space debris plays the role of the chief satellite (point B on Fig. 1.10), and the active spacecraft is the deputy satellite (point A on Fig. 1.10). The deputy satellite’s onboard control system maintains its required position relative to the chief. Since the chief object is exposed to the ion beam, the parameters of its orbit change significantly over time. The deputy satellite of mass mA moves relative to the chief satellite of mass mB in a gravity field of a primary body under the action of control and perturbation forces having resultant forces FA and FB for each satellite. It is assumed that the mutual gravitational attraction of satellites is negligible. To describe the motion of the satellites, the inertial coordinate system OXpYpZp and Hill’s noninertial rotating reference frame BXHYHZH are used. The location of the chief and deputy satellites is given using the position vectors r and rA, respectively. The relative position vector of the deputy satellite is ! ρ ¼ BA ¼ rA  r:

(1.78)

Newton’s second law can be used to describe satellites’ motion. After dividing by the masses, the equations of motion of the satellites have the form €rpA ¼ μ

rpA + apA , r 3A

€rp ¼ μ

rp + apB , r3

(1.79)

where μ is the gravitational parameter of the primary body, and aA ¼ FA/mA and aB ¼ FB/mB are accelerations from disturbing and control forces. The superscript

Fig. 1.10 Spacecraft formation flight geometry.

26

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

“p” denotes the inertial reference frame OXpYpZp in which the coordinates of the vector are given. Subtracting Eq. (1.79) gives €p ¼ €rpA  €rp ¼ μ ρ

rp rp + ρ p μ + apA  apB , 3 r r 3A

(1.80)

where the magnitude of the position vector rpA ¼ rp + ρp can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r A ¼ |r + ρ | ¼ ðrp + ρp Þ  ðrp + ρp Þ ¼ r p

p

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 2rp  ρp : 1+ 2 + r r2

Assuming that the distance between the satellites is small compared to the distance to the center of the primary body ρ ≪ r, the second term can be expanded into a series: repr + ρepρ rp + ρp ¼     2 p p 3=2 2 2ρep  ep 3=2 r 3 1 + ρr2 + 2r r2 ρ r 3 1 + ρr2 + rr ρ    repr + ρepρ rp ðrp  ρp Þ ρ p p ρ 2 rp ρp ¼ 1  3 er  eρ + O ,  3 + 3 3 3 r r r r r r5 rp + ρp ¼ r 3A

(1.81)

where epr and epρ are unit vectors directed along r and ρ.Taking into account last series expansion, let us calculate the difference of the first two terms on the right-hand side of Eq. (1.80): μ

rp ðrp  ρp Þ rp rp + ρ p ρp  μ ¼ μ + 3μ : r3 r3 r5 r 3A

(1.82)

Substituting the obtained approximate expression (1.82) into Eq. (1.80) yields €p ¼ μ ρ

rp ðrp  ρp Þ ρp + 3μ + apA  apB : r3 r5

(1.83)

This vector equation (1.83) describes the relative motion of the deputy satellite in an inertial reference frame. In practice, working with Eq. (1.80) is not very convenient, since the projection of the vector ρ on the axes of the inertial reference frame OXpYpZp is not very visual, and before using Eq. (1.80), for example, for the needs of the control system of an active spacecraft, additional calculations are required to interpret the results obtained with these equations. It is much more convenient to write the equations of motion for a rotating noninertial orbital reference frame BXHYHZH. The time derivative of a vector in an inertial reference frame can be expressed in terms of the deviation with respect to a rotating coordinate system as MHp

dρp dρH H ¼ + ωH Hp  ρ : dt dt

(1.84)

Basics of space flight mechanics and control theory

27

This equation expresses the transport theorem, which is one of the basic theorems of kinematics (Schaub and Junkins, 2014). According to this theorem, the time derivative of ρ in an inertial frame is the sum of the derivative of this vector in a noninertial rotating frame and the vector product of the angular velocity ωH Hp of the rotating frame relative to the inertial frame and the vector ρ. All vectors on the right-hand side of Eq. (1.84) are given by their coordinates in the rotating frame. The second derivative of the vector ρ, taking into account Eq. (1.84), has the form MHp

  dωH d 2 ρp d 2 ρH Hp H H H H ¼ + + ω  ω  ρ  ρ + 2ωH Hp Hp Hp dt dt2 dt2 H dρ  : dt

(1.85)

Multiplying the left-hand and right-hand sides of Eq. (1.83) by the matrix MHp and using Eq. (1.85) gives

MHp

  dωH d 2 ρp d2 ρH dρH Hp ¼ 2 +  ωH  ρH + 2ωH   ρH + ωH Hp Hp Hp 2 dt dt dt dt MHp ρp MHp rp ðrp  ρp Þ + 3μ + MHp apA  MHp apB : ¼ μ r3 r5

(1.86)

Expressing the term ddtρ2 from the resulting equation, taking into account the fact that the scalar product is invariant with respect to the rotation of the coordinate system (rp  ρp ¼ rH  ρH), we get the equation of the deputy satellite relative motion, given in the orbital coordinate system 2 H

  dωH d 2 ρH dρH Hp ¼  ωH  ρH  2ωH   ρH  ωH Hp Hp Hp 2 dt dt dt  H H H r ð r  ρ Þ μ H  3 ρH + 3 + aH A  aB : r r2

(1.87)

Without yet imposing restrictions on the chief satellite motion, let us find the projections of Eq. (1.87) on the axes of Hill’s reference frame BXHYHZH. The chief satellite position vector has the coordinates rH ¼ [r, 0, 0]T in the BXHYHZH frame, and the deputy satellite relative position vector has the coordinates ρH ¼ [x, y, z]T. The accelH T T eration vectors are aH A ¼ [aAx, aAy, aAz] and aB ¼ [aBx, aBy, aBz] . Expression (1.11) H determines the components of the angular velocity vector ωHp ¼ [ωHpx, ωHpy, ωHpz]T. The terms on the right-hand side of Eq. (1.87) can be written in the form

T dωH Hp  ρH ¼ ω_ Hpy z  ω_ Hpz y, ω_ Hpz x  ω_ Hpx z, ω_ Hpx y  ω_ Hpy x , dt

28

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

 3   ωHpx ωHpy y + ωHpz z  x ω2Hpy + ω2Hpz   6  7 6 7   H H H 2 2 6 ωHp  ωHp  ρ ¼ 6 ωHpy ωHpx x + ωHpz z  y ωHpx + ωHpz 7 7, 4  5   ωHpz ωHpx x + ωHpy y  z ω2Hpx + ω2Hpy 2

2ωH Hp  ρH  3

  T   dρH  ¼ 2 ωHpy z_  ωHpz y_ , 2 ωHpz x_  ωHpx z_ , 2 ωHpx y_  ωHpy x_ , dt

rH ðrH  ρH Þ ¼ ½2x, y, zT : r2

Substituting these expressions into Eq. (1.87), we obtain a system of nonlinear equations describing the relative motion of the deputy satellite in the Hill’s reference frame.     x€ ¼ ω_ Hpz y  ω_ Hpy z  ωHpx ωHpy y + ωHpz z + x ω2Hpy + ω2Hpz   2μx 2 ωHpy z_  ωHpz y_ + 3 + aAx  aBx , r     y€ ¼ ω_ Hpx z  ω_ Hpz x  ωHpy ωHpx x + ωHpz z + y ω2Hpx + ω2Hpz   μy 2 ωHpz x_  ωHpx z_  3 + aAy  aBy , r     z ¼ ω_ Hpy x  ω_ Hpx y  ωHpz ωHpx x + ωHpy y + z ω2Hpx + ω2Hpy €   μz 2 ωHpx y_  ωHpy x_  3 + aAz  aBz : r

(1.88)

(1.89)

(1.90)

If the chief satellite moves along an unperturbed Keplerian orbit, Eqs. (1.88)–(1.90) T _ become much simpler. In this case, aB ¼ 0, ωH Hp ¼ [0, 0, ωHp] , and ωHp ¼ f . x€ ¼ ω_ Hp y + xω2Hp + 2ωHp y_ +

2μx + aAx , r3

y€ ¼ ω_ Hp x + yω2Hp  2ωHp x_  z¼ €

μz + aAz : r3

μy + aAy , r3

(1.91) (1.92) (1.93)

These equations can be rewritten in a more compact form using the approach described by Szebehely and Giacaglia (1964) by introducing new dimensionless coordinates: x ¼ x=r,

y ¼ y=r,

z ¼ z=r

(1.94)

Basics of space flight mechanics and control theory

29

and using the true anomaly angle as an independent variable instead of time. The derivatives of an arbitrary function s can be written as h s_ ¼ f_ðsr 0 + s0 r Þ ¼ ðse sin f + s0 ð1 + e cos f ÞÞ, p s€ ¼

h2 ð1 + e cos f Þ2 h2 h2 e cos f ðð1 + e cos f Þs00 + es cos f Þ ¼ 3 s00 + s, 3 pr 2 p r

(1.95)

where s ¼ s=r is the dimensionless function. Prime means the derivative with respect to the angle of the true anomaly s0 ¼

ds : df

The time derivative is related to the derivative with respect to the true anomaly angle by the following relationship: ds ds df s0 h ¼ ¼ s0 f_ ¼ 2 : dt df dt r pffiffiffiffiffi Parameters h ¼ μp, p, and e define the orbit of the chief satellite. The distance from the center of the primary to the chief satellite is determined by Eq. (1.26). Eqs. (1.91)– (1.93), allowing for the change of variables (1.94) and expression (1.95), take the form x00  2y0 

3x a r3 ¼ Ax2 , 1 + e cos f h

(1.96)

y00 + 2x0 ¼

aAy r 3 , h2

(1.97)

z00 + z ¼

aAz r 3 : h2

(1.98)

If the chief satellite follows a circular orbit, then e ¼ 0, r ¼ const, and the true anomaly rate is equal to the mean orbital rate (1.38), which is constant too: ωHp ¼ n ¼ pffiffiffiffiffiffiffiffiffiffi μr 3 ¼ const. For the case of a circular orbit, the equations of the deputy satellite relative motion (1.91)–(1.93) take the form x€  2ny_  3n2 x ¼ aAx ,

(1.99)

y€ + 2nx_ ¼ aAy ,

(1.100)

z + n2 z ¼ aAz : €

(1.101)

30

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

This system of equations is known as the Clohessy-Wiltshire equations (Clohessy and Wiltshire, 1960). The dimensionless form of these equations, taking into account the change of variables in Eq. (1.94), is x00  2y0  3x ¼

aAx , n2 r

y00 + 2x0 ¼

aAy , n2 r

z00 + z ¼

aAz : n2 r

(1.102)

The Clohessy-Wiltshire equations are valid only for the case when the deputy satellite relative coordinates x, y, and z are small compared to the chief satellite circular orbit radius r. These equations, due to their simplicity, are widely used for preliminary analysis and development of satellite formation control laws. The system of linear differential equations (1.91)–(1.93) can be integrated analytically in the absence of perturbations aAx ¼ aAy ¼ aAz ¼ 0. The analytical solutions for the bounded deputy satellite unperturbed motion are given by Schaub and Junkins (2014) in the form. x ¼ A0 cos ðnt + αÞ,

y ¼ 2A0 sin ðnt + αÞ + C0 ,

z ¼ B0 cos ðnt + βÞ,

where A0, B0, C0, α, and β are integration constants that can be determined from initial conditions.

1.6

Rigid body attitude motion in orbit

For the considered problem of space debris contactless transportation, the motion of a space debris object around its center of mass is of great importance. The Euler’s equations describing a body motion relative to its center of mass can be directly used: _ ¼L H

(1.103)

where H is the rigid body angular momentum vector about its center of mass and L is the external torque about the same point. To calculate the angular momentum of the body, let us take an infinitesimally small piece of mass dm and calculate the integral (Schaub and Junkins, 2014): ð H ¼ ρ  ρ_ dm,

(1.104)

where ρ is the vector connecting the center of mass to the piece of mass dm. For a rigid body, ρ_ ¼ ω  ρ, and Eq. (1.104) can be rewritten as ð H ¼ ρ  ðω  ρÞ dm ¼ ½Iω, where [I] is the inertia matrix

(1.105)

Basics of space flight mechanics and control theory

2 2 2 ð ρy + ρz 6 ½I ¼ 4 ρx ρy

ρ2x + ρ2z

7 ρy ρz 5 dm:

ρx ρz

ρy ρz

ρ2x + ρ2y

ρx ρy

ρx ρz

31

3 (1.106)

The inertia matrix [I] and the angular velocity vector ω must be given in the same reference frame. It is most convenient to set the inertia matrix in the body fixed reference frame BXbYbZb, since in this case the matrix elements are constants. It is assumed that the center of mass of the body is at point B. The body fixed reference frame axes in which the inertia matrix is a diagonal matrix are called the principal axes, and the diagonal elements of this matrix are called principal inertias. To translate the inertia matrix from the ith coordinate system to the jth system, the rotation matrix Mji can be used:

j

 I ¼ Mji Ii MTji : Using the transport theorem, Euler’s equation (1.103) can be written in body fixed frame: dHb + ωb  Hb ¼ Lb , dt

(1.107)

where Lb ¼ [Lx, Ly, Lz]T is the resultant external torque about the body center of mass given in the body frame, ωb ¼ [ωx, ωy, ωz]T. Substituting Eq. (1.105) into Eq. (1.107), taking into account that the inertia matrix is constant in the body frame, gives ½ I

  dωb ¼ ωb  ½Iωb + Lb : dt

(1.108)

If the axes of the body fixed reference frame BXbYbZb are aligned with the principal body axes, the inertia matrix is diagonal, and the vector equation (1.108) is reduced to   I x ω_ x ¼  I z  I y ωy ωz + Lx , I y ω_ y ¼ ðI x  I z Þωx ωz + Ly ,   I z ω_ z ¼  I y  I x ωx ωy + Lz ,

(1.109)

where Ix, Iy, and Iz are principal moments of inertia. The components of the torque Lb can depend on the orientation of the body, which can be specified using three Euler angles, components of the direction cosine matrix that determines the transition from the inertial frame to the body fixed coordinate system, or quaternions. Consider the case when the orbital motion of the body’s center of mass allows us to calculate the orientation of the Hill’s reference frame BXHYHZH through the orbit elements (1.41), and the orientation of the body fixed frame BXbYbZb

32

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

relative to the Hill’s axes is specified using three Euler angles γ, θ, and φ (y-x-y). The angular velocity ωbp, which defines the BXbYbZb frame rotation relative to the inertial frame OXpYpZp, is determined by the vector equation   _ b + φe _ b + ie _ b + ω_ + f_ eb + γ_ eb + θe _ bb2 : ωbbp ¼ Ωe p3 11 H3 H2 51 Using the rotation matrices given in Section 1.2, we obtain

2 3 2 3 2 3 2 3 3 θ_ ωx 0 0 i_ 6 7 6 7 6 7 6 7 6 7 6 7 4 ωy 5 ¼ Mbp 4 0 5 + Mb1 4 0 5 + MbO 4 γ_ 5 + Mb5 6 0 7 4 5 _ Ω ω_ + f_ ωz 0 0 2 3 0 6 7 + 4 φ_ 5: 2

(1.110)

0 If the equations of the body’s center of mass motions allow us to determine the angles that specify the position of the spherical orbital frame BXoYoZo, and the Euler angles γ, θ, and φ establish a connection between the spherical and the body fixed frames, then the angular velocity vector ωbp has the form _ b  ν_ eb + γ_ eb + θe _ b + φe _ bb2 ωbbp ¼ ϑe p3 o2 o2 51 2 3 2 3 2 3 2 3 0 0 0 θ_ 6 7 6 7 6 7 6 7 ¼ Mbp 4 0 5 + MbO 4 γ_  ν_ 5 + Mb5 4 0 5 + 4 φ_ 5: 0 ϑ_ 0 0

(1.111)

The system of three differential equations obtained by projecting Eqs. (1.110) or (1.111) onto the axes of the frame BXbYbZb establishes a relationship between the angular velocity projections and derivatives of Euler angles. When moving in the central gravitational field, different points of the body experience different gravitational attraction. This dissimilarity of the attraction forces leads to the emergence of a gravity gradient torque. The gravity gradient torque of the body relative to its center of mass can be calculated as  ð ðr + ρÞ μrP LG ¼ ρ  dFG ¼ ρ   3 dm ¼ μ ρ  dm rP r 3P ð ρ ¼ μr  3 dm, rP ð

ð

(1.112)

! where rP ¼ OP ¼ r + ρ is the position vector of an infinitesimal body element relpffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ative to the Earth’s center, r p ¼ |rP | ¼ rP  rP ¼ r 2 + 2r  ρ + ρ2 is the element’s ! position vector magnitude, and r ¼ OB is the body center of mass position vector

Basics of space flight mechanics and control theory

33

Fig. 1.11 Gravity gradient torque calculation.

(Fig. 1.11). Expanding the factor in the integrand in a series, considering ρ/r to be small, yields ρ ¼ r 3P



ρ

3=2 ¼

2rp

ρp



ρeρ

3=2 2ρe  e r 3 1 + r2 + rr ρ r 3 1 + r2 + r2    ρeρ ρðr  ρÞ ρ ρ 2 ρ ¼ 3 1  3 er  eρ + O :  33 r r r r r5 ρ2

ρ2

Ð Given this expression, and the fact that ρdm ¼ 0, since the ρ is measured relative to the body’s center of mass, the gravity gradient torque can be rewritten as LG ¼

ð  ð  μr rρ 3μ  ρ 1  3 r  ρðr  ρÞdm: dm ¼  r3 r2 r5

(1.113)

It follows from the vector triple product property a  (b  c) ¼ (a  c)b  (a  b)c that ðρ  rÞρ ¼ ρ  ðρ  rÞ + ðρ  ρÞr ¼ ρ  ðr  ρÞ + rρ2 : Substituting this expression into Eq. (1.113) and taking into account expression for the inertia matrix (1.105) give LG ¼

3μ r  ½Ir: r5

(1.114)

If the inertia matrix has a diagonal shape, then the gravity gradient torque vector can be written in the body fixed coordinate system 2

 3 Iz  Iy ry rz 3μ 6 7 LbG ¼ 3 4 ðI x  I z Þr x r z 5: r   Iy  Ix rx ry

(1.115)

34

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

where r i ¼ r i =r is dimensionless coordinates of position vector rb ¼ [rx, ry, rz]T. The body center of mass motion has a direct effect on the gravity gradient torque (1.114) through the position vector. This is especially noticeable in elliptical orbits, where the magnitude of this vector changes significantly over the orbit. In this regard, we can talk about the presence of a coupling effect between the rotational and translational motion of a rigid body in the central gravitational field. Since gravitational attraction depends on the distance to the attracting center, and the forces acting on different points of the body differ, the effect of the body rotational motion on its translational motion in orbit can be observed. Let us assess this effect. The gravitational force acting on the rigid body is ð FG ¼ μ

r+ρ dm: r 3P

Performing a series expansion similar to Eq. (1.81), but keeping a larger number of terms and using the inertia matrix definition (1.106), the expression for the gravitational force in the form (Schaub and Junkins, 2014) can be obtained  h  T  i μm 3 1 ½I r, (1.116) 1+ tr ð ½ I  Þ  5 e ½ I  + ½ I e r r r 2 mr 2 Ð where er ¼ r/r is the column unit vector and tr([I]) ¼ 2 ρ2dm. The second term inside brackets is due to the effect of the gravitational gradient. For space vehicles, this term is much less than unity. This allows us to ignore the influence of rotational motion on the body translational motion in the first approximation. FG ¼ 

1.7

Orbital and attitude perturbations

The right-hand sides of the equations of motion obtained above contained the accelerations of the perturbing forces ad. This may include gravitational perturbations due to the nonsphericity of the Earth, the attraction of other bodies, solar pressure, aerodynamic resistance, and geomagnetic field. A detailed analysis of these perturbations can be found in many books (Beletskii, 1966; Kluever, 2018). Let us briefly consider these perturbations in a volume sufficient for the considered problem of contactless transportation of space debris.

1.7.1 Nonspherical central body gravitational field The mathematical models and formulas developed above were obtained for an ideal central gravitational field. In reality, the shape of the Earth is not a sphere. The difference between the gravitational field and the ideal central field can be taken into account through the perturbing forces and torque. To model the gravitational field and the shape of the Earth, expansions in terms of spherical harmonics are used. The gravity field can be defined by a scalar geopotential function U(r, λ, ϕ), where

Basics of space flight mechanics and control theory

35

Fig. 1.12 Gravity potential of the Earth.

r is the radius, λ is the longitude, and ϕ is the geocentric latitude (Fig. 1.12). The function parameters set the point at which the gravitational field is determined. The gravitational acceleration aG can be found as the gradient from the geopotential function aG ¼ rU, where r is the vector differential operator, which for the inertial coordinate system OXpYpZp is r¼

∂ ∂ ∂ e + e + e , ∂xp p1 ∂yp p2 ∂zp p3

and r ¼ xpep1 + ypep2 + zpep3. The geopotential function U(r, λ, ϕ) can be represented as the sum U ðr, λ, ϕÞ ¼

μ + Rðr, λ, ϕÞ, r

(1.117)

where μ/r is the potential of an ideal central gravitational field and R(r, λ, ϕ) is the disturbing potential function, which is caused by the nonsphericity and heterogeneity of the Earth.  n RE Pn ð sin ϕÞ r n¼2 ∞ X n   X RE n ðkÞ + Pn ð sin ϕÞðCnk cos kλ + Snk sin kλÞ, r n¼2 k¼1

Rðr, λ, ϕÞ ¼ 

μ r

∞ X

Jn

(1.118)

where RE is the semimajor axis of a common terrestrial ellipsoid, Pn is the Legendre polynomial of n order, P(k) n is the associated function of Legendre (Atkinson and Han, 2012), Jn is the coefficient of zonal harmonic, and Cnk and Snk are dimensionless coefficients of tesseral harmonics (when n 6¼ k) or sectoral harmonics (when n ¼ k). These coefficients characterize the difference of the shape of the Earth from the sphere.

36

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

There are models of the Earth gravitational field, which are tables with groups of numbers that allow us to calculate the coefficients Jn, Cnk, and Snk (Lerch et al., 1994; Tapley et al., 1996). The second zonal harmonic is the major disturbing term in the decomposition of the geopotential. The J2 term defines oblateness of the Earth at the poles and exceeds all remaining terms Jk on three orders. Restricting ourselves to considering only this term, and taking into account that P2(u) ¼ (3u2  1)/2 and sin ϕ ¼ z/r, Eq. (1.117) can be rewritten in the form   J 2 R2E 3z2 μ U ðr, λ, zÞ ¼ 1 1 , r 2 r2

(1.119)

where μ ¼ 3.986  1014 m3/s2, J2 ¼ 0.0010826267, and RE ¼ 6378140 m. Taking into account Eq. (1.119), Newton’s equations of motion of a particle around the Earth in the inertial reference frame OXpYpZp take the form "   μxp 3 RE 2 x€p ¼  3 1  J 2 2 r r "   μyp 3 RE 2 y€p ¼  3 1  J 2 2 r r "   μzp 3 RE 2 zp ¼  3 1  J 2 € 2 r r where r ¼

5z2p 1 r2

!# ,

!# 5z2p 1 , r2 !# 5z2p 3 , r2

(1.120)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2p + y2p + z2p .

According to Eq. (1.120), the perturbing acceleration due to the J2 harmonic in projections onto the axes of the inertial coordinate system has the form !3 2   3 RE 2 5zp 1 7 6 J2 2 r r2 7 6 6 !7 7 6 2 7 6 3 RE 2 5zp 7: J apd ¼ 6  1 2 7 6 22 r r 7 6 7 6 ! 7 6 2   2 5zp 5 4 3 RE J2  3 2 r r2 2

(1.121)

In order to use this vector in Gauss’ planetary equations, it is necessary to transpose it p into the Hill’s coordinate system aH d ¼ MHpad, where zp ¼ r sin i sin u is calculated from T [xp, yp, zp] ¼ MpH[r, 0, 0]. For use with Lagrange’s planetary equations (1.72)–(1.77), the disturbing potential function must be rewritten as

Basics of space flight mechanics and control theory

R¼

2a3 ð1

37

  μ J R2 ð1 + e cos f Þ3 3 sin 2 u sin 2 i  1 : 3 2 E 2 e Þ

In the case of a nonspherical central body gravitational field, the gravity gradient torque also changes. For the geopotential function defined by expression (1.119), the torque takes the form !! 3 2 R2E 7zp ry rz 7 6 I z  I y 3  5J 2 r 2 r 2  1 7 6 7 6 !! 7 6 2 2 7 6 7z RE μ p 7 LbG ¼ 3 6 ð I  I Þ 3  5J  1 r r x z 2 2 x z 7: 2 6 r 6 r r 7 7 6 !! 7 6 2 2  RE 7zp 5 4 I y  I x 3  5J 2 2  1 r r x y r r2 2





1.7.2 Moon and Sun gravity When moving around the Earth, the satellite experiences the perturbing effect of the gravitational fields of the Moon and the Sun. Let us consider a mechanical system consisting of the Earth, the Moon, and a satellite, which are mass points E, M, and B with masses mE, mM, and mB respectively (Fig. 1.13). The position of these points is given by the position vectors rE, rM, and r in the inertial coordinate system. The relative positions of the points are given by the vectors ρij ¼ rj  ri. Newton’s second law for points B and E has the form mB€rB ¼ 

GmE mB GmM mB ρEB  ρMB , ρ3EB ρ3MB

Fig. 1.13 Three body system.

mE€rE ¼

GmE mB GmE mM ρEB  ρME : ρ3EB ρ3ME

38

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

To obtain an equation describing the motion of the satellite relative to the Earth, let us express the second derivatives from the equations written above and find their difference. €EB ¼  ρ

 GðmE + mB Þ ρMB ρME ρ  Gm  : M EB ρ3MB ρ3ME ρ3EB

(1.122)

Comparing Eqs. (1.122) with (1.12), and given that ρEB plays the role of r in Section 1.3, the equation for the vector of perturbing acceleration ad due to the Moon’s attraction can be written as  ad ¼ μM

ρMB ρME  , ρ3MB ρ3ME

(1.123)

where μM ¼ GmM is the gravitational parameter of the Moon. The perturbing acceleration due to the attraction of the Sun can be written similarly by formally replacing the point M with the point S. ad ¼ μS

 ρSB ρSE  , ρ3SB ρ3SE

(1.124)

where μS ¼ GmS is the gravitational parameter of the Sun, and ρSB and ρSE are the relative position vectors defining the position of the satellite and the Earth relative to the Sun. The perturbing accelerations defined by expressions (1.123) and (1.124) are known as the effective attractions of the Moon and the Sun on the satellite (Kaplan, 2020), which are the differences between the lunar or solar gravity acting on the satellite and lunar or solar gravity acting on the Earth. For a satellite on a circular orbit with a radius of 300 km, the perturbing acceleration of the Moon is 1.18  106m/s2, and the acceleration of the Sun is 5.29  107m/s2. As the height of the orbit increases, the magnitude of the accelerations increases.

1.7.3 Atmospheric influence In low Earth orbit, the impact of the atmosphere is tangible. Aerodynamic forces and torques act on a body moving in a resisting medium. These forces and torques depend on the shape of the body, its relative velocity, and the parameters of the medium. All aerodynamic forces generated on the surface of the body can be reduced to a resultant force Fa, which is applied to the center of pressure P. Aerodynamic forces and torques vectors can be represented as projections on the axes of the body fixed reference frame BXbYbZb: Fba ¼ ½FaA , FaY , FaN T ,

T Lba ¼ Lax , Lay , Laz ,

(1.125)

Basics of space flight mechanics and control theory

39

where FaA is the axial force, FaY is the side force, FaN is the normal force, Lax is the rolling torque, Lay is the pitching torque, and Laz is the yawing torque. Often, to set the projections of the aerodynamic force, an air-path coordinate system BXVYVZV is used. The axis BXV is directed along the vector of the body velocity relative to the atmosphere Vrel ¼ V + ωE  r,

(1.126)

where V is the absolute velocity of the body, ωE is the Earth rotation angular velocity, and r is the body’s center of mass position vector. The axis BZV lies in the local vertical plane and directed to a planet, and the axis BYV completes the right-hand system. Aerodynamic force can be represented as FVa ¼ ½FaD , FaC , FaL T ,

(1.127)

where FaD is the drag, FaC is the lateral force, and FaL is the lift force. To determine the aerodynamic forces and torques, experimental methods for blowing models in a wind tunnel, finite element calculations of computer models (Chen et al., 2015), and approximate analytical methods can be used. The components of the resultant aerodynamic force Fa and the torque La take the following form: FaA ¼ CA qS,

FaY ¼ CY qS,

FaN ¼ CN qS;

(1.128)

FaD ¼ CD qS,

FaC ¼ CC qS,

FaL ¼ CL qS;

(1.129)

Lax ¼ CLx qSl,

Lay ¼ CLy qSl,

Laz ¼ CLz qSl;

(1.130)

where Ci is the dimensionless coefficient of the ith component, S is the reference area, l is the length of the body, and q is the dynamic pressure q¼

ρa V 2rel , 2

(1.131)

where ρa is the atmosphere density. In estimated calculations, the density of the atmosphere can be approximated by the exponential dependence ρa ¼ ρa0 exp ðβðh  h0 ÞÞ,

(1.132)

where ρa0 is the density of the atmosphere at the reference altitude h0 and β is the coefficient. At present, quite detailed models are available that allow us to calculate atmospheric parameters, including density, up to a height of 1000 km (Picone et al., 2002). If the body has an axisymmetric shape, its vector Fa lies in the plane of the total angle of attack, which is the angle between the velocity vector and the axis of symmetry. The dimensionless coefficients of the resultant aerodynamic force and the aerodynamic moment depend on the orientation of the body relative to the velocity vector,

40

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Mach number M, Reynolds number Re, and dimensionless angular velocity of the body. In the case of the axisymmetric body, the aerodynamic moment depends mainly on the total angle of attack and the dimensionless angular velocity. If the mathematical model does not take into account the body attitude motion, when calculating the aerodynamic force, the body is considered as a sphere and it is assumed that the entire effect of the atmosphere is expressed by aerodynamic drag. In this case, the atmospheric drag acceleration can be defined as ad ¼

ρa V rel SCD Vrel , 2mB

(1.133)

where mB is the mass of the body. Since the drag force is in the opposite direction of the velocity, it dissipates the orbital energy. This eventually leads to the body re-entry. Aerodynamic forces and torques are nonconservative perturbation.

1.7.4 Solar radiation pressure Solar radiation also affects the motion of satellites. The magnitude of the perturbing force created by solar radiation depends on the area and material of the irradiated surface, the distance to the Sun, and the intensity of solar energy. Unlike the aerodynamic drag force, which always reduces the orbital energy, the solar radiation pressure force can also increase this energy when the direction of orbital motion coincides with the direction of propagation of the Sun’s rays. Solar radiation pressure is determined by the following expression (Kluever, 2018): PSRP ¼

Is ¼ 4:54  106 N=m2 , c

(1.134)

where Is ¼ 1.361 W/m2 is the solar intensity for the mean Earth-Sun distance and c ¼ 3  108 m/s is the speed of light. Using expression (1.134), perturbation acceleration due to solar radiation pressure can be written as ad ¼

PSRP SCR , mB

(1.135)

where S is the area of the satellite surface exposed to the Sun and CR is the reflectivity of the satellite surface. The value of CR ¼ 1 corresponds to a “black body,” which completely absorbs radiation, the value of CR ¼ 0 corresponds to a translucent body, through which light passes unhindered, and the value of CR ¼ 2 corresponds to a “pure mirror,” which reflects all radiation. The size of the area S depends on the movement of the body relative to the center of mass. In addition, it is difficult to determine accurately the reflectivity coefficient of the satellite. The solar radiation pressure torque can be calculated as LSRP ¼ roc  ad mb es ,

(1.136)

Basics of space flight mechanics and control theory

41

where es is the unit vector directed from the Sun to the Earth and roc is the position of the optical center of pressure.

1.7.5 Earth’s magnetic field torques A satellite in orbit is disturbed by the Earth’s magnetic field. The satellite equipment may contain permanent magnets and current-carrying conductors, the satellite body is magnetized, and eddy currents are induced in metal surfaces. All these factors lead to the appearance of magnetic torques. An external magnetic flux density B exerts a torque LM on a satellite. This torque is determined by the vector product (Fig. 1.14) LM ¼ IM  B:

(1.137)

where IM is the magnetic moment of the spacecraft’s intrinsic magnetic field. Similar to the gravitational field, the near-Earth geomagnetic field is usually modeled with a potential function UM expressed as a series of spherical harmonics (Langel et al., 1996) ∞ X n   X  RE n+1 m gn cos mλ + hm sin mλ Pm n n ð cos ϕÞ r n¼1 m¼0 ∞ X n   X  RE n m qn cos mλ + sm sin mλ Pm + RE n n ð cos ϕÞ, r n¼1 m¼0

U M ðr, λ, ϕÞ ¼ RE

(1.138)

m m m where RE is the mean radius of the Earth, and gm n , hn , qn , and sn are Gauss coefficients. The Earth’s magnetic field can be calculated as

B ¼ rUM :

(1.139)

Fig. 1.14 Earth’s magnetic field.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

In the simplest case of magnetic dipole, the magnetic field vector is determined by the expression B¼

1 ð3ðk0  er Þer  k0 Þ: r3

(1.140)

where k0 ¼ μ0m(4π)1 ¼ 8  1015Tm3, where μ0 is the vacuum permeability, k0 is a vector in the direction of the geomagnetic dipole axis, and er is a unit vector along the radius-vector of the orbit. There are various methods for determining the magnetic dipole moment IM of a satellite. Some of them calculate moment based on magnetic field measurement data (Moskowitz and Lynch, 1964). Static torque calculation methods imply dipole moment calculation based on a measurement of the static torque reaction to an applied magnetic field. The method described by Tossman (1967) is based on the analysis of resonant oscillations of the suspended by torsion wires spacecraft. The most significant contribution to the magnetic torque is made by current-carrying devices, permanent magnets in instruments, and magnetization of the satellite hull in the geomagnetic field (Beletskii, 1966).

1.8

Lyapunov stability theory for the search for control laws

Lyapunov stability theory is widely used by scientists in the development of nonlinear control laws for space systems. Lyapunov divides all methods for solving stability problems into two groups. The methods of the first group involve finding a general or particular solution to a differential equations system. The second group of methods do not require finding solutions to the equations, but are based on the construction and analysis of some functions having specific properties. Lyapunov and his followers formulated a number of theorems that make it possible to draw a conclusion about the stability of motion based on an analysis of the properties of these functions, which are usually called Lyapunov functions. Below are 12 theorems for an autonomous and nonautonomous system with periodic coefficients. For a deeper introduction to this theory, it is recommended that the reader consults Hahn et al. (1963) and Malkin (1959). Consider the motion of an arbitrary dynamical system defined by the differential equation y_ ¼ Fðy, tÞ,

(1.141)

where y ¼ [y1, y2, …, yn]T is a column vector of n components, F(y, t) ¼ [F1(y, t), F2(y, t), …, Fn(y, t)]T. This vector equation can be represented as a system of n scalar equations. Let ys ¼ f(t) be some particular solution of Eq. (1.141). Introducing a new variable

Basics of space flight mechanics and control theory

x ¼ y  ys ,

43

(1.142)

a new system of equations, called the differential equation of perturbed motion, is obtained:     x_ ¼ F x + ys , t  F ys , t ¼ f ðx, tÞ,

(1.143)

The origin of the system x ¼ 0 is an equilibrium point, which means that f(0, t) ¼ 0, where 0 is a column vector consisting of zeros. A scalar function V(x) is called positive (negative) semidefinite if V(0) ¼ 0, and the inequality V(x)  0 (V(x)  0) is satisfied in a spherical neighborhood of the origin | x |  h, where h is a fairly small positive number. A scalar function V(x) is called positive (negative) definite if V(0) ¼ 0, and the inequality V(x) > 0 (V(x) < 0) is satisfied in a spherical neighborhood of the origin | x |  h (Hahn et al., 1963). Examples of positive definite and positive semidefinite functions for the case n ¼ 1 are shown schematically in Fig. 1.15. The total derivative of V for Eq. (1.143) is V_ ¼

n X ∂V i¼1

∂xi

f i ð xÞ +

∂V : ∂t

(1.144)

The equilibrium of Eq. (1.143) is stable if a number δ > 0 exists for each ε > 0 such that the inequality | x(t0) | < δ implies | x(t) | < ε for t  t0. The equilibrium point is asymptotically stable if it is stable and a small δ can be found for which all perturbed motions obey the condition lim xðtÞ ¼ 0. The equilibrium is unstable if a number ε > 0 with the t!∞

following property exists. There exist a sequence of numbers τ1, τ2, … ! ∞ and a sequence of initial points x01, x02, … ! 0 such that | p(t0 + τi, x0i, t0) |  ε, where p(t, x0, t0) is the solution of Eq. (1.143) for the initial point x(t0) ¼ x0. Let us formulate some Lyapunov theorems for the case of an autonomous system given by the equation x_ ¼ f ðxÞ:

(1.145)

Fig. 1.15 Examples of positive definite (PDF) and positive semidefinite (PSDF) functions.

44

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Theorem 1. The equilibrium is stable if it is possible to find such a positive definite function V(x) that its total time derivative for the equation of perturbed motion is negative semidefinite or is equal to zero. Theorem 2. The equilibrium is asymptotically stable if it is possible to find such a positive definite function V(x) that its total time derivative for the equation of perturbed motion is negative definite. The largest region of initial points x0 ¼ x(t0) for which the conditions of Theorem 2 are satisfied is called the region of asymptotic stability. Theorem 3. The equilibrium is unstable if it is possible to find such function V(x) that its total time derivative for the equation of perturbed motion is a positive definite function, while the function V(x) is not negative semidefinite. Theorem 4. The equilibrium is unstable following conditions are met: there is a function V(x) such that its total time derivative for the equation of perturbed motion in domain 6A x 6A  h has the form dV ðxÞ ¼ CV ðxÞ + W ðxÞ, dt where C is a positive constant and the function W(x) is either equal to zero or a positive semidefinite function. In the latter case, the function V(x) is not a negative semidefinite function. Quite often, the stability problem can be solved by analyzing the first approximation equations, which is obtained by expanding the right-hand side of the system of equations (1.143) into the Maclaurin series and discarding the nonlinear part x_ ¼ AðtÞx,

(1.146)

where A(t) is an n  n matrix. The elements of this matrix can be defined as  ∂f i ðx, tÞ aij ¼ : ∂xj x¼0 In the case of an autonomous system (1.145), the matrix A is constant. To study stability, it is necessary to analyze eigenvalues λi of the matrix A, which are the roots of the characteristic polynomial detðA  λEÞ ¼ 0, where E is the identity matrix. The following theorems can be formulated. Theorem 5. The equilibrium of Eq. (1.145) is asymptotically stable if all eigenvalues of the first approximation equations matrix A have negative real parts.

Basics of space flight mechanics and control theory

45

Theorem 6. The equilibrium of Eq. (1.145) is unstable if at least one of the eigenvalues of the first approximation equations matrix A has positive real parts. Theorem 7. If among the eigenvalues of the first approximation equations matrix A there are no numbers with positive real parts, but there are numbers with zero real parts, then the nonlinear terms in the perturbed equation can be chosen so that unstable or stable motion is obtained at will. The cases when it is not enough to consider the first approximation equation (1.146) to determine the stability of the equilibrium of the nonlinear equation (1.145) are called critical. Critical cases are common in the mechanics of controlled systems. The equations of motion of a spacecraft in an elliptical orbit can be reduced to systems of differential equations with periodic coefficients. This class of nonautonomous differential equations has been fairly well studied in stability theory. To formulate stability theorems, it is necessary to supplement the definitions. Consider a scalar function V(x, t) defined in the region t  t0 > 0, |x| < h,

(1.147)

where t0 and h are constants. This function has continuous partial derivatives in the region (1.147), and it turns to 0 at the point x ¼ 0. A function V(x, t) is said to admit an infinitesimal upper bound if for every λ > 0 there exists a μ > 0 such that | V(x, t) |  λ for all x values satisfying inequalities t  t0 and | x | < μ. A scalar function V(x, t) is called positive (negative) definite if in region (1.147) for sufficiently large t0 and sufficiently small h, the inequality V(x, t)  W(x) (V(x, t)  W(x)) is satisfied, where W(x) is a positive (negative) definite function. The following theorems can be formulated for the case of a nonautonomous system with periodic coefficients. Theorem 8. The unperturbed motion is stable if it is possible to find a positive definite function V(x, t) for Eq. (1.143), for which the time derivative for the equation of perturbed motion (1.144) is negative semidefinite or is equal to zero. Theorem 9. The equilibrium is asymptotically stable if the conditions of Theorem 8 are satisfied, the derivative V_ is a negative definite function, and the function V(x, t) admits an infinitesimal upper bound. Theorem 10. The equilibrium is unstable if a function V(x, t) that admits an infinitely small upper bound exists. Its time derivative V_ for the equation of perturbed motion (1.144) is positive definite. The function V(x, t) for arbitrarily small values of x and _ arbitrarily large values of t can take values of the same sign as the derivative V. In the case of a nonautonomous system of differential equations with periodic coefficients, an apparatus for studying stability can be constructed using the equations of the first approximation. In contrast to the case of an autonomous system, where the matrix A was constant and the conclusion about stability was made on the basis of its eigenvalues, the case of a system with periodic coefficients is more laborious, since

46

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

in order to compile the characteristic equation, it is necessary to find the fundamental system X(t) of the linear differential equation (1.146). X(t) is an n  n matrix, each column of which represents a linearly independent solution of the linear differential equation. The characteristic equation has the form detðS  λEÞ ¼ 0,

(1.148)

where S is defined by the matrix equation X(t + T) ¼ X(t)S and T is the period. It is assumed that the nonlinear part B(x, t) of the expansion of the right-hand side of the perturbed equation (1.143) in the Maclaurin series x_ ¼ AðtÞx + Bðx, tÞ

(1.149)

obeys the following conditions. There is a region (1.147) in which the inequality |Bðx, tÞ|  C

n X

|xi |

(1.150)

i¼1

holds, where C is some constant. In the region (1.147), the vector B(x, t) components are continuous and satisfy the requirements for the uniqueness of the solution of the differential equation (1.149). Theorem 11. If all roots of the characteristic equation of the first approximation system (1.146) have absolute value less than unity, then the unperturbed motion for Eq. (1.149) is asymptotically stable for any function B(x, t), when the constant C in inequalities (1.150) is sufficiently small. Theorem 12. If the characteristic equation of the first approximation system (1.146) has at least one root with a modulus greater than unity, then the unperturbed motion for Eq. (1.149) is unstable for any function B(x, t), when the constant C in inequalities (1.150) is sufficiently small. The critical cases are the cases when the characteristic equation has no roots with modulus greater than unity, and has roots with modules equal to one. The above theorems can be used in the development of control laws. If it is necessary to bring the system to an equilibrium position, then the control vector u, which is included on the right-hand side of the differential equations of perturbed motion, should be chosen in such a way that the conditions of the asymptotic stability theorem are satisfied. If it is required, on the contrary, to bring the system out of equilibrium, then the control vector u must be chosen based on the conditions of the instability theorem.

1.9

Chaotic dynamics of nonlinear system

The behavior of the mechanical systems considered in this book is described by nonlinear differential equations. The possibility of the existence of chaotic modes of motion is one of the key features of nonlinear systems. This means that in the phase

Basics of space flight mechanics and control theory

47

space of the dynamical system, there are regions filled with nonperiodic trajectories that never intersect. Inside these regions, phase trajectories are characterized by a strong sensitivity to initial conditions. The loss of information about the initial conditions is one of the main features of chaotic trajectories. From a practical point of view, if a phase trajectory falls into the region of chaotic motions, it means that it is impossible to predict the behavior of the system over long time intervals. Here we are not talking about the presence of external random perturbations. All forces acting on a mechanical system are predetermined. Numerical integration of a system of differential equations from a given starting point with given parameters of the system and the integration algorithm will always give the same result. However, the system will be very sensitive to initial conditions and parameters. In addition, numerical integration errors will play a dramatic role. In practice, the motion parameters of a mechanical system and its mass-geometric characteristics are known approximately, which means that the results of numerical integration cannot be used over a long time interval in the presence of chaos in the system. In this regard, the identification of chaotic motion regions and the search for ways to suppress chaos for a particular mechanical system are relevant tasks from a practical point of view. Chaotic dynamics is currently an actively developing scientific field. The following references are recommended to introduce this theory: Magnitskii and Sidorov (2006), Rasband (2015), and Wiggins (2003). To date, several effective methods to detect chaos in a dynamic system have been developed. Two of them are considered below: the construction of the Poincare sections and the calculation of the Lyapunov exponents spectrum. Both of these methods are easy to program and involve the numerical integration of a system of differential equations. The construction of the Poincare section is a relatively effective way to identify chaotic trajectories in the study of dynamical systems of small dimensions. To construct a section in the phase N-dimensional space, a secant surface S with dimension N  1 is selected. The choice of this surface is largely arbitrary. An essential requirement is the repeated intersection of this surface by the studied trajectories. Then some point X ¼ [x1, x2, …, xN]T is selected on this surface. This point is used as the initial point when integrating the considered system of differential equations. The phase trajectory is calculated from this point to the intersection with the secant surface S. Let us denote this intersection point as X0 . Thus, we obtain a mapping of the point X to the point X0 . Now, taking X0 as the initial point, the next point X00 can be obtained in a similar way, and so on (Fig. 1.16). If the system of differential equations has the analytical solution, then the dependence X0 ¼ F(X) can be written analytically, where F is the succession function. If one of the phase coordinates of the considered dynamic system changes periodically, then it is possible to consider the intersection of phase trajectories not with one secant surface, but with a family of periodically located surfaces, with the subsequent display of all intersection points on one surface. The construction of the Poincare section allows us to focus on studying a simpler mapping instead of analyzing the original dynamical system. With this approach, information about the behavior of the system between successive intersections of the secant surface S by the phase trajectory is lost, but the possibility of analyzing the nature of the trajectories remains. In the simplest case of a system of third-order differential equations, the secant surface is a two-dimensional plane. In this case, the equilibrium positions (point A in

48

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 1.16 Poincare section.

Fig. 1.17) and periodic trajectories of the system are represented on the section as fixed points. Periodic trajectories can also be represented as several points or lined up in curves positions (section B in Fig. 1.17). Chaotic trajectories are displayed on the Poincare section as a cloud of points (section C in Fig. 1.17). It should be noted that a nonautonomous system of second-order differential equations can be reduced to a system of third-order differential equations by introducing an auxiliary phase coordinate proportional to time. One of the most reliable ways to detect chaos is to determine the rate at which close trajectories move away from each other. This rate can be estimated using the Lyapunov exponents. Let us consider an N-dimensional dynamical system described by differential equations: x_ ¼ FðxÞ:

(1.151)

The phase trajectory x(t) is a solution to Eq. (1.151). The phase trajectory yðtÞ ¼ xð t Þ + e xðtÞ is a close trajectory to x(t), where e xðtÞ is mutual deviation of two trajectories. The equation for e xðtÞ , taking into account Eq. (1.151), after linearization can be written as

Fig. 1.17 Different types of trajectories in the Poincare section.

Basics of space flight mechanics and control theory

e x_ ¼ Ae x,

49

(1.152)

where A is the matrix composed of partial derivatives of the components of the vector function F(x): 2 ∂F

1

6 ∂x1 6 6 ∂F2 6 AðtÞ ¼ 6 ∂x1 6 6 ⋮ 4 ∂FN ∂x1

∂F1 ∂x2 ∂F2 ∂x2 ⋮ ∂FN ∂x2

⋯ ⋯ ⋱ ⋯

∂F1 3 ∂xN 7 7 ∂F2 7 7 ∂xN 7: 7 ⋮ 7 5 ∂FN ∂xN

The Lyapunov exponent λi exists for any solution of Eq. (1.152), and it is defined as the upper limit: λi ¼ lim

1

T!∞ T

log ðkxei ðT ÞkÞ:

(1.153)

The set N of Lyapunov exponents λ1 > λ2 > … > λN forms the spectrum of Lyapunov exponents. The geometric meaning of the Lyapunov exponents is that two solutions whose initial values are located in a certain neighborhood of ε radius in the time T will diverge into an N-dimensional ellipsoid along the N principal semiaxes, and at time t the distances are determined by the values εi ¼ εeλit (Fig. 1.18). The spectrum of exponents is useful in that it allows us to determine the type of the phase trajectory from the signs of the Lyapunov exponents. The presence of at least one positive Lyapunov exponent in the spectrum means that the considered phase trajectory is unstable. If all exponents are negative, then the phase trajectory is asymptotically stable. Any periodic trajectory necessarily has at least one zero Lyapunov exponent. If the considered phase trajectory belongs to an attractor consisting of a

Fig. 1.18 Phase volume deformation.

50

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

set of phase trajectories (a torus or a strange attractor), then the spectrum of Lyapunov exponents for this trajectory also characterizes all other attractor trajectories. An attractor is a compact subset of the phase space, all trajectories from some neighborhood of which tend to it as time tends to infinity. The sum of all Lyapunov exponents of the attractor is negative. An attractor other than a stability point has at least one zero exponent. The following combinations of signs of the Lyapunov exponents can be observed in three-dimensional space: (,  , ) is an attractive equilibrium position, (0,  , ) is an attractive limit cycle, (0, 0, ) is a two-dimensional torus, and (+, 0, ) is a strange attractor (chaos). Thus, positive, zero, and negative Lyapunov exponents must be present in the spectrum of a chaotic trajectory. For most dynamical systems, the calculation of the Lyapunov exponent is possible only numerically. The algorithm based on the Gram-Schmidt orthogonalization is widely used (Wolf et al., 1985). Consider the procedure for determining the spectrum of Lyapunov exponents. Take a point r0 ¼ [r01, r02, …, r0N]T on the considered phase trajectory. Let us also choose N disturbed points xj0 ¼ [xj01, xj02, …, xj0N]T, so that the perturbation vectors Δxj0 ¼ xj0  r0 have the length of ε and they are mutually orthogonal. Using the numerical integration of Eq. (1.151), we obtain new positions r1 and xj1 after a short time interval T. The new perturbation vectors take the form Δxj1 ¼ xj1  r1. Reorthogonalization of these vectors by the Gram-Schmidt method gives Δx0j1  Δx00j1 ¼   0  Δxj1  Δx011 ¼ Δx11 ,

  Δx021 ¼ Δx21  Δx21 , Δx0011 Δx0011 ,     Δx031 ¼ Δx31  Δx31 , Δx0021 Δx0021  Δx31 , Δx0011 Δx0011 , … N 1  X   Δx0N1 ¼ ΔxN1  ΔxN1 , Δx00k1 Δx00k1 : k¼1

After this orthogonalization, the resulting perturbation vectors Δxj100 will be mutually orthogonal and have unit length. Multiplying these vectors by ε gives Δxj1000 ¼ ε Δxj100 . Consider a new set of perturbed points Δxj10 ¼ xj1 + Δxj1000 . After that, the process is repeated, but instead of points r0 and xj0, points r1 and xj10 are taken. Repeating the described operation M times, the Lyapunov exponents are calculated as. λj ¼

 M  1 X   log Δx0jk  , MT k¼1

j ¼ 1, 2, …, N:

A certain difficulty is the choice of the time interval T. Choosing too large a value of the interval causes a shift in the values of all Lyapunov exponents to the maximum exponent.

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51

References Alfriend, K.T., Vadali, S.R., Gurfil, P., How, J.P., Breger, L.S., 2010. Spacecraft Formation Flying. Butterworth-Heinemann, Oxford, https://doi.org/10.1016/B978-0-7506-85337.00209-8. Atkinson, K., Han, W., 2012. Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Lecture Notes in Mathematics, https://doi.org/10.1007/978-3-64225983-8_1. Bau`, G., Bombardelli, C., Pela´ez, J., Lorenzini, E., 2015. Non-singular orbital elements for special perturbations in the two-body problem. Mon. Not. R. Astron. Soc. 454, 2890– 2908. https://doi.org/10.1093/mnras/stv2106. Beletskii, V.V., 1966. Motion of an Artificial Satellite About its Center of Mass, NASA TT F-429. Chen, B., Zhan, H., Zhou, W., 2015. Aerodynamic design of a re-entry capsule for high-speed manned re-entry. Acta Astronaut. 106, 160–169. https://doi.org/10.1016/j.actaastro. 2014.10.036. Clohessy, W.H., Wiltshire, R.S., 1960. Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27, 653–658. https://doi.org/10.2514/8.8704. Cohen, C.J., Hubbard, E.C., 1962. A nonsingular set of orbit elements. Astron. J. 67, 10. https:// doi.org/10.1086/108597. Gangestad, J.W., Pollock, G.E., Longuski, J.M., 2010. Lagrange’s planetary equations for the motion of electrostatically charged spacecraft. Celest. Mech. Dyn. Astron. 108, 125–145. https://doi.org/10.1007/s10569-010-9297-z. Hahn, W., Hosenthien, H.H., Lehnigk, H., 1963. Theory and Application of Liapunov’s Direct Method. Prentice-Hall, Englewood Cliffs, NJ. Kaplan, M.H., 2020. Modern Spacecraft Dynamics and Control. Courier Dover Publications. Kluever, C.A., 2018. Space Flight Dynamics, second ed. John Wiley and Sons Ltd, Hoboken, NJ. Langel, R.A., Sabaka, T.J., Baldwin, R.T., Conrad, J.A., 1996. The near-earth magnetic field from magnetospheric and quiet-day ionospheric sources and how it is modeled. Phys. Earth Planet. Inter. 98, 235–267. https://doi.org/10.1016/s0031-9201(96)03190-1. Lerch, F.J., Nerem, R.S., Putney, B.H., Felsentreger, T.L., Sanchez, B.V., Marshall, J.A., Klosko, S.M., Patel, G.B., Williamson, R.G., Chinn, D.S., Chan, J.C., Rachlin, K.E., Chandler, N.L., McCarthy, J.J., Luthcke, S.B., Pavlis, N.K., Pavlis, D.E., Robbins, J.W., Kapoor, S., Pavlis, E.C., 1994. A geopotential model from satellite tracking, altimeter, and surface gravity data: GEM-T3. J. Geophys. Res. Solid Earth 99, 2815–2839. https://doi.org/10.1029/93JB02759. Magnitskii, N.A., Sidorov, S.V., 2006. New Methods for Chaotic Dynamics. World Scientific Series on Nonlinear Science Series A, World Scientific, Singapore, https://doi.org/ 10.1142/6117. Malkin, I.G., 1959. Theory of Stability of Motion. US Atomic Energy Commission, Office of Technical Information. Moskowitz, R., Lynch, R., 1964. Magnetostatic measurement of spacecraft magnetic dipole moment. IEEE Trans. Aerosp. 2, 412–419. https://doi.org/10.1109/TA.1964. 4319617. Picone, J.M., Hedin, A.E., Drob, D.P., Aikin, A.C., 2002. NRLMSISE-00 empirical model of the atmosphere: statistical comparisons and scientific issues. J. Geophys. Res. Space Physics 107. https://doi.org/10.1029/2002JA009430. SIA 15-1-SIA 15-16. Rasband, S.N., 2015. Chaotic Dynamics of Nonlinear Systems. Dover Publications.

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Schaub, H., Junkins, J.L., 2014. Analytical Mechanics of Space Systems, third ed. American Institute of Aeronautics and Astronautics, Inc., Washington, DC, https://doi.org/ 10.2514/4.102400. Sidi, M.J., 1997. Spacecraft Dynamics and Control. Cambridge University Press, https://doi. org/10.1017/CBO9780511815652. Szebehely, V., Giacaglia, G.E.O., 1964. On the elliptic restricted problem of three bodies. Astron. J. 69, 230. https://doi.org/10.1086/109261. Tapley, B.D., Watkins, M.M., Ries, J.C., Davis, G.W., Eanes, R.J., Poole, S.R., Rim, H.J., Schutz, B.E., Shum, C.K., Nerem, R.S., Lerch, F.J., Marshall, J.A., Klosko, S.M., Pavlis, N.K., Williamson, R.G., 1996. The joint gravity model 3. J. Geophys. Res. Solid Earth 101, 28029–28049. https://doi.org/10.1029/96JB01645. Tossman, B., 1967. Application of resonance technique for measuring satellite magnetic dipole moment. Nev. Univ. Sp. Magn. Explor. Technol., 379–395. 69–33978. Walker, M.J.H.H., 1986. A set of modified equinoctial orbit elements. Celest. Mech. 38, 391– 392. https://doi.org/10.1007/BF01238929. Wiggins, S., 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, Springer-Verlag, New York, https://doi.org/10.1007/b97481. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a time series. Physica D 16, 285–317. https://doi.org/10.1016/0167-2789(85) 90011-9.

Space debris problem 2.1

2

Review of the space debris problem

Space debris is one of the most pressing challenges of modern astronautics. In an unfavorable scenario, mutual collisions of satellites and upper rocket stages will lead to the formation of a cloud of small space debris fragments that move at high velocities and can make it impossible to launch new spacecraft into space. This section provides a general overview of the various aspects of this problem.

2.1.1 Space debris threat It is no coincidence that the space debris problem is attracting close attention of scientists all over the world. Described in 1978, the Kessler syndrome paints a rather pessimistic scenario for the orbital environment evolution, when the collision of spacecraft causes a chain reaction, leading to the formation of a cloud of debris around the Earth (Kessler and Cour-Palais, 1978). Fragments of destroyed satellites and upper rocket stages have tremendous velocity. In a collision with a satellite or a spacecraft being launched into orbit, this small space debris can significantly damage and even incapacitate them, turning this expensive equipment into new space debris. In 2009, the collision of Cosmos 2251 and Iridium 33 led to the formation of clouds of debris, some of which will remain in orbit until 2090 (Pardini and Anselmo, 2011). Such collisions can set off a chain reaction that may result in humanity losing the possibility of practical use of space for many decades. Since the beginning of the space age, more than 630 incidents, resulting in generation of new space debris fragments, have occurred in orbit, including breakups, explosions, and collisions (ESA, 2021). An adequate assessment of the threats posed by space debris and the development of effective measures to combat them require tools that allow obtaining of information about the objects located in near-Earth space and the parameters of their motion. Such information can be provided by space debris environment models. There are several models (Krisko et al., 2015; Manis et al., 2021; Sdunnus et al., 2004): ORDEM 3.0 by NASA, MASTER-8 by ESA, and SPDA by RSA. Space debris is usually divided into three categories: small (10 cm). According to the data provided on the ESA website (ESA, 2021), at the time of writing, there are 36,500 large space debris objects, about 1 million medium-sized objects and 330 million small objects in Earth’s orbit. Another source of statistics about the current state of near-Earth space is the regularly issued Orbital Debris Quarterly News of the NASA Orbital Debris Program Office (NASA, 2021). The response to the space debris threat has included the holding of regular scientific conferences, the creation of international associations, and changes in national standards (Stokes et al., 2020). Several areas can be identified regarding methods to combat orbital pollution: improvement of spacecraft, rockets and their flight programs, Attitude Dynamics and Control of Space Debris During Ion Beam Transportation. https://doi.org/10.1016/B978-0-323-99299-2.00004-5 Copyright © 2023 Elsevier Inc. All rights reserved.

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managing space traffic and prevention of on-orbit collisions, postmission disposal, active space debris removal, and “just-in-time” collision avoidance. Methods for dealing with space debris depend on its size. To protect spacecraft from small space debris, they are covered with special shields with a multilayer structure. Laboratory tests of various space debris shielding configurations for manned modules protection were carried out by Destefanis et al. (2006). There are design solutions aimed at protecting inflatable structures from micrometeorites and small particles of space debris (Buslov et al., 2019). Unfortunately, shields are useless against medium to large space debris. The only defense is to deviate trajectories to avoid collisions. This can be achieved through space traffic control measures, and by using “just-in-time” collision avoidance. Postmission disposal and active space debris removal measures are aimed at reducing the probability of such collisions. Of great concern are the plans of commercial companies SpaceX and OneWeb to deploy their large satellite constellations in low Earth orbit (LEO). The studies carried out by Le May et al. (2018) predicted a high probability of collision of satellites of the constellation with space debris at least once in 5 years of operation. Moreover, neither deorbit of all stages that launch new spacecraft, nor total deorbit of rocket bodies and payloads within 25 years, which is the current postmission disposal requirement for spacecraft to be launched, diminish this potential hazard. The authors of the work (Pardini and Anselmo, 2020) came to similar conclusions. According to them, the creation of a constellation of 6000 additional satellites on LEO will lead to an increase in the total collision rate among cataloged objects by nearly 20%–30%.

2.1.2 International legal aspects and space debris prevention guidelines One of the obstacles to solving the problem of space debris is the lack of legal regulation mechanisms in this area. The term “space debris” itself is not defined in the current international treaty documents on outer space. Instead, the term “space object” is used to refer to spacecraft, orbital stages, and their fragments (Stelmakh et al., 2019). This causes legal difficulties both with the investigation of incidents related to space debris and with the development of active space debris removal missions. Physical manipulation of an object in space legally requires permission from the owner country of the object. The issue is compounded by the plans of private companies to create their own satellite constellations in near space. The problem of insufficient elaboration of the international legal framework cannot be solved without close international cooperation. Despite the presence of obvious difficulties, it would be wrong to say that no action is being taken in the legal area. Currently, several voluntary standards and guidelines have been developed: –

Inter-Agency Space Debris Coordination Committee (IADC) Space Debris Prevention Guidelines (IADC, 2021);

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– – – –

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Space Debris Mitigation Guidelines of the UN Committee on the Peaceful Uses of Outer Space (UNOOSA, 2010); European Code of Conduct on the Prevention of Space Debris (ESA, 2004); Recommendation of the International Telecommunication Union ITU ITU-R S.1003 “Protection of the geostationary satellite orbit as an environment”; and ISO 24113:2019 “Space systems: space debris mitigation requirements” (ISO, 2019).

The analysis of this standard and other space debris-related ISO documents is given by Stokes et al. (2020). These standards are reflected in the national legislation and documents of the space agencies of various countries. The Inter-Agency Space Debris Coordination Committee is an intergovernmental forum that was created to coordinate international efforts to deal with the space debris problem. All standards and guidelines mentioned above are based on the IADC Space Debris Prevention Guidelines (IADC, 2021). This document offers debris mitigation guidelines with a focus on cost-effectiveness. It is assumed that the proposed measures will be taken at the design stage of spacecraft and launch systems and at their flight programs development. The measures are aimed at reducing the generation of space debris during operation in orbit and after the completion of the missions. The IADC guidelines are based on the following fundamental principles: – – –

preventing explosive and collisional on-orbit breakups; removing spacecraft and orbital stages after their missions from the useful orbit regions; and restriction on ejection of objects during normal operation.

The document identifies two regions in near-Earth space that are of great practical importance and therefore should be protected. The first one is called the low Earth orbit (LEO) protected region, and is a spherical region around the Earth RLEO ¼ 2000 km above its surface. The second region is called the geosynchronous orbit (GEO) protected region, and has a more complex shape. This region of space is bounded by four surfaces: two spheres and two cones. The spheres have a radius of RGEO  200 km, where RGEO ¼ 42,164 km is the radius of the geostationary Earth orbit. The cones are located symmetrically relative to the equatorial plane of the Earth. Their vertices coincide with the center of the Earth, and the apex angles are equal to 15 degrees. The regions described are shown schematically in Fig. 2.1. Different strategies for removing space objects after the end of their life from these regions are prescribed.

Fig. 2.1 IADC protected regions.

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The IADC guidelines recommend working out debris mitigation measures at the stage of development and planning of the future space mission. The guidelines state that the measures should be described in a document called “Space Debris Mitigation Plan,” which should contain six items: 1. a management plan that describes space debris mitigation activities; 2. a plan for assessing and reducing space debris-related risks; 3. a description of measures aimed at minimizing the potential hazards related to malfunctions, which can lead to formation of new space debris; 4. a disposal plan, which describes the removal of the spacecraft and rocket stages from working orbit after the completion of their mission; 5. justification of choice when there are several possibilities; and 6. guidelines compliance matrix.

When designing spacecraft and orbital stages, engineers should anticipate possible emergencies and minimize the likelihood of accidental explosive breakups. During operation, it is necessary to monitor the state of the spacecraft or orbital stage. In the event of a malfunction that cannot be corrected, passivation and disposal measures shall be conducted. Passivation involves the elimination of all stored energy on a spacecraft or orbital stages. It suggests all fluids that can lead to breakups by overpressurization or chemical reaction. Solar power lines must be turned off. The pressure in the high-pressure vessels should be reduced to a safe level. Self-destruct systems must be designed with protection against unintentional tripping caused by various reasons. Flywheels and momentum wheels must be de-energized. Disposal phase includes the actions to reduce the hazards that the spacecraft or orbital stage poses to other space objects. This phase can end with re-entry or the movement of the spacecraft or orbital stage into special regions of near-Earth space. Postmission disposal suggests different strategies for different regions. Spacecraft and orbital stages that move inside the geosynchronous protected region, after completing their mission, must be placed outside this region into orbits that guarantee the nonreturn of these objects to the region for at least 100 years. The eccentricity of the new orbit should be less than 0.003. Guidelines give the following formula for calculating the increase in orbit perigee: Δr π ¼ r b +

1000CR A m

(2.1)

where rb ¼ 235 km is the sum of the attitude of the GEO protected region and maximum descent caused by perturbations, CR is the solar radiation pressure coefficient, A is the aspect area, and m is the dry mass. If the orbits of the spacecraft or the stage pass through the low Earth orbit protected region, then after the completion of their mission phase, they must be transferred to orbit, where their expected residual orbital lifetime is 25 years or shorter. The probability of success of the disposal maneuver should be no less than 90%. For large satellite constellations, a shorter lifetime and a higher probability of success may be needed. Direct re-entry is encouraged whenever possible, but this maneuver should

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not pose an undue risk to people or property. In addition, possible ground environmental pollution must be taken into account. Spacecraft and stages outside the protected regions after the end of their mission phase must be moved into orbits with the low Earth orbit lifetime requirements, or into orbits that do not intersect with highly utilized orbit regions. The deliberate destruction of spacecraft or stages is possible only in low orbits, where the orbital fragments generated after destruction will be short-lived. When developing a spacecraft and its flight program, the probability of collision with known space objects during the entire period of its existence in orbit must be taken into account. Wherever possible, avoidance maneuvers and the selection of safe launch windows should be provided. In addition, the design of the spacecraft should reduce the risk of incapacitation of the spacecraft after collision with small space debris. The IADC guidelines define a number of requirements that must be met for designed spacecraft and rockets, as well as in the development of flight programs in order to reduce the generation of space debris during operation in orbit and after the completion of the mission. The implementation of measures to reduce orbital pollution is a necessary step toward preserving the space environment for future generations.

2.1.3 Space debris rotation properties and inertial parameters estimation Currently, the characteristics of many objects in orbit, such as size, shape, massinertial, and rotation parameters, remain unknown or imprecise. This significantly limits the ability to predict the near-space environment evolution, since this data is required for an adequate assessment of the influence of nonconservative forces. For example, the drag force and solar radiation pressure depends on the shape and orientation of an object. In addition, when designing an active space debris removal mission, information about the target object angular motion and its inertial parameters is of great importance. The rapid rotation of space debris can become a serious obstacle to the use of some orbital transport technologies. For instance, it is almost impossible to grab a rapidly rotating object with a robotic arm. The lack of information on the inertial parameters of the space debris object does not allow developing of effective control laws for an active spacecraft that performs the space debris removal. In this regard, the determination of the characteristics and the motion parameters of space debris objects is an important practical task. The technique to determining the motion parameters of a space object depends on its type. If a space object can somehow inform the active spacecraft about its position and orientation, then it is called cooperative. The information can be transmitted by an active radio transmitter, or passive artificial markers on the surface of the object can be used, making it easier to recognize the object’s position. An uncooperative or noncooperative spacecraft is unable to transmit this information. Space debris falls into this category. For some noncooperative objects, information about their size and shape is

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available with varying accuracy; for some, this information is completely unknown and it should be obtained directly during the active space debris removal mission. There are several approaches to determining the characteristics and the motion parameters of space objects. The first concerns the interpretation of the ground-based observations results, the second consists in the processing of information obtained by cameras and sensors of an active spacecraft located in close proximity to the target object, and the third involves measuring and analyzing the result of a controlled mechanical impact on an object from an active spacecraft. To implement the second and third approaches, the active spacecraft must be equipped with electro-optical sensors. Opromolla et al. (2017) provided an overview of existing sensors. All sensors can be divided into active ones, which include light detection and ranging systems, and passive monocular and stereo cameras (Fig. 2.2). Passive sensors can only register radiation, while active ones are themselves a radiation source. Ground-based observations make it possible to obtain some information about the parameters of a space object motion without the need to deliver measuring equipment in close proximity to it. Optical ground stations, based on the results of observations of a space object, can build its light curve, which is a series of the space debris object’s measurements over time. This data can be used to determine an object’s size, shape, orientation, and material composition. In addition, it allows the rotation axis direction and the apparent rotation period of the space debris to be found. The light curve of a rotating object has a periodic character when observations are made often enough, and the observation time exceeds the period of the object’s rotation. In the article by Sˇilha et al. (2018), 135 rotating objects, with angular velocities ranging from 0.42 to 400.9 deg/s, were identified based on light curve data from the Astronomical Institute of the University of Bern using the phase-diagram reconstruction method. In total, the study examined 397 objects located in the low Earth orbit, global navigation systems orbits, high eccentricity orbit, and geosynchronous Earth orbit. In a study by Yanagisawa and Kurosaki (2012), a rotational axis direction, rotation period, precession, and a compositional parameter for the Cosmos 2082 orbital stage, which moves

Fig. 2.2 Electro-optical sensors.

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in LEO, were determined only from the light curve data obtained by the JAXA’s 35 cm optical telescope. Observations show that the angular velocity of space debris objects can vary significantly over time. For example, the GOES 8 satellite in GEO was in uniform rotation at the end of 2013, in 2014 the period increased significantly, and according to 2018 data, the period again decreased by an order of magnitude (Benson et al., 2020). Approaches based on machine learning have great potential for orbital objects characterization. Yao and Chang-yin (2021) used the machine learning method to classify space debris objects based on observed light curves. Only four types of space debris were considered in the paper: diffuse cylinder, specular cylinder, box-wing satellite, and cube satellite. A six-layer deep neural network was developed by Liu and Schreiber (2021) for automatic classification of observed objects and analysis of their rotational motion evolution. The raw observation data in this study was taken from the archive of Russian Mini-Mega TORTORA system database (http://astroguard.ru). The main difficulty in training neural networks is the relatively small amount of real observational light curve data for training. Allworth et al. (2021) proposed using a computer simulation to obtain the data set necessary for training. Using the Blender 3D modeling software, photo-realistic images were generated, and the light curves were calculated for them. After the network was trained, the authors used it to initialize a different neural network. This network has been fine-tuned using real observational data. The developed neural network can be used to improve the accuracy of determining the characteristics of objects based on the results of their observations. Information about the position and the translational and angular velocities of a space debris object can be obtained using electro-optical sensors installed on the active spacecraft. A laser-vision system mounted on an active spacecraft can perform noisy pose measurements of the near located space debris object. Based on these measurements and the comprehensive dynamics model, accurate estimates of the object’s position, orientation, and translational and angular velocities can be made using an adaptive Kalman filter (Aghili and Parsa, 2009). Passive stereovision systems that use two cameras to obtain 3D information about the position of points of an object have great potential. A technique for assessing the state and the tensor of inertia of an object based on stereovision measurements using several Kalman filters was proposed by Segal et al. (2014). A stereovision-based algorithm using an adaptive unscented Kalman filter, which is more economical in terms of computational costs, was developed by Wang et al. (2019) to determine the relative states and moment-ofinertia ratios of a noncooperative unknown object. Deep learning algorithms make it possible to develop effective systems for estimating the parameters of a space object from its photographs. For example, Phisannupawong et al. (2020) proposed using computer vision technologies based on deep convolutional neural network to determine the position and angular orientation of a space object. The described methods are based on processing the result of observations. Although some of they provide information on the normalized inertial parameters of an object, methods for determining real moments of inertia are based on the processing of information obtained as a result of the mechanical action of an active spacecraft on the target object. They allow determination of both the kinematic and

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the inertial parameters of the object including mass, mass center location, and tensor of inertia. The literature mainly discusses two different approaches to the definition of inertial parameters. The first approach involves the use of the conservation of momentum equation and the measurement of velocities (Ma et al., 2008; Nguyen-Huynh and Sharf, 2013). The second approach involves the use of dynamic Newton-Euler or Lagrange equations and requires the measurement of accelerations or forces (Zhang et al., 2015). Regardless of which method is used, the result is a system of linear identification equations that includes the sought inertial parameters as unknown variables. The coefficients of these equations depend on the kinematic motion parameters, which can be measured by the sensors, and the inertial properties of the spacecraft and the robotic arm, which are known in advance. Chu et al. (2017) developed a scheme for determining the inertial parameters of space debris after it is captured by a manipulator, taking into account errors in measuring the kinematic parameters and inaccuracies of data on the spacecraft’s inertial parameters. Direct use of such distorted data leads to large errors in determining the inertial parameters of space debris. The authors proposed equipping the manipulator with sensors and measuring the force effect on the space debris object’s surface and the force or torque generated by the end-effector. The data on the momentum obtained from the conservation of momentum equation is replaced by data on the linear or angular impulse, which can be obtained by integrating the resulting force or torque of the object. This in turn can be calculated based on the information about the force reacting on the object’s surface and the force or torque of the end-effector. The identification equations obtained on the basis of this modification provide better accuracy in determining the inertial parameters, since they contain less inaccurate data. Besides modifying the identification equations, Chu et al. used the hybrid immune algorithm combining recursive least squares and an affine projection sign algorithm in their scheme to solve the problem of the colored noise and intense spike pulse noise. This allows improvement of a stable identification property. An interesting concept of a system for determining the parameters of a space object was proposed by Meng et al. (2019). It is based on the use of data from visual and force-moment sensors of an active spacecraft in the presence of measurement noise. The flexible rod is used for gentle mechanical action on an object (Fig. 2.3). An impact of the order of 10 N is sufficient to determine all the inertial parameters of a 1000 kg object. Meng et al. (2020) proposed a scheme for identifying inertial parameters of an object during its detumbling by eddy current generated by magnetic rotor mounted on the end of an active spacecraft’s robotic arm. This scheme does not imply direct mechanical contact. The impact on the space debris object is carried out through an electromagnetic field. This method works if the surface of the object is made of conductive nonmagnetic metal. To observe the object, the use of a floating camera is proposed, which is a depth camera. It moves independently of the spacecraft and the object under study. Thus, the determination of motion parameters and mass-inertial characteristics of noncooperative space objects is a complex and multifaceted issue. Various methods and approaches are used to solve it, including methods of mathematical statistics, control theory, mathematical modeling, and machine learning. The normalized inertial

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Fig. 2.3 Using a flexible rod for determining space debris parameters.

parameters can be calculated from the results of orbital observations. Determination of real inertial characteristics of an object is a more complicated and unsafe task, since it requires an analysis of the result of the controlled force action of an active spacecraft on the target object.

2.1.4 Postmission disposal One of the obvious and necessary measures to combat future space debris is to include the stages of removing spacecraft and their rocket stages from operational orbits after the completion of their main flight programs. This is the essence of postmission disposal. In the case of LEO objects that are deorbited, in the case of GEO, they are transferred to the disposal orbit, which was described in more detail in Section 2.1.2. An obvious technical solution is to use the remnants of the fuel and the propulsion system of the spacecraft itself or the rocket stage to carry out this transport operation. The disposal can be carried out by installing an additional system based, for example, on the use of solar wind, the resistance force of the atmosphere, or the electromagnetic field of the Earth. An overview of possible solutions can be found in Eichler et al. (1997). These systems do not differ structurally from those described in Section 2.2 for the purpose of active debris removal, but they are installed not on the active spacecraft, which is removing the target object, but directly on this target object itself. Kerr et al. (2017) proposed developing a universal postmission disposal module that can be installed on various satellites. In addition to its main task, this module can be used as a backup propulsion system in the event of a failure of the satellite’s main engines. The study analyzes possible concepts for the implementation of this module and compares their advantages and disadvantages. The authors concluded that drag augmentation, solar sailing, electrodynamic tether, low thrust propulsion, and high thrust propulsion look the most promising. Regardless of the chosen concept,

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

the satellite must be detumbled before using the module. Each system had its advantages and challenges, and the choice should be based on the specific mission of the satellite.

2.1.5 Active space debris removal According to existing estimates, humanity should remove from orbit some of the millions of kilograms of intact derelict mass. Among them, large space debris poses a particular danger, a collision with which can generate a large cloud of debris. According to some estimates, in order to stabilize the situation in orbit, it is necessary to remove at least five large space debris objects annually (Bonnal et al., 2013). The scientific literature discusses many different ways of large space debris removal. Most of them are based on the use of a propulsion system of an external active spacecraft. An overview of existing research and directions on this topic is provided by Hakima and Emami (2018), Mark and Kamath (2019), and Shan et al. (2016). An active spacecraft can transport a passive object in one of three ways (Fig. 2.4). In the first way the active spacecraft can rigidly capture, grip or dock with the passive object, after which they will move as a single rigid body. The active spacecraft’s engines can be used for the required maneuver. This method allows the use of both impulse flights, implying short-term use of high-thrust engines, and flights with constant small thrust. It is also possible to use not jet engines, but electrodynamic tethers or solar sails. This method is the simplest from the point of view of controlling the system during transportation. Its main disadvantage is the need to capture the passive object, which is a complex and dangerous operation. This method is well-suited for transporting space objects in low and medium Earth orbits. Satellites located in geostationary orbits are equipped with large-area solar panels, which make it harder to approach these satellites. Another complication is the possible rotation of a passive object, which makes it difficult or impossible to dock with it. The second way involves towing the passive object on a flexible tether. The capture operation is still necessary, but its danger is much less compared to the first method, since the probability of collision of the spacecraft and the object during the docking is small. Tethered towing does not allow for pulsed maneuvers, since it can lead to break the tether and collision of the spacecraft and the object as a result of the tethered object rebound. Successful towing requires more complex laws for controlling the engines of the active spacecraft, which take into account possible oscillations of the tether and the

Fig. 2.4 Ways of a passive object transportation: (A) as a whole rigid body; (B) tether towing; and (C) contactless transportation.

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attached object. In addition, the jet stream of the active spacecraft’s engines can burn out the tether during towing, so the engines should be slightly deviated from the tether, which reduces useful thrust. The third way does not require mechanical capture of the passive object and involves contactless action on the passive object. The active spacecraft can act on the passive object through electromagnetic fields, the gravity field, using lasers, or blowing an object with a stream of particles. All of these methods will be described in more detail below. The lack of mechanical contact between the active spacecraft and space debris increases the safety of the whole active space debris removal mission and makes it possible to transport rapidly rotating objects. The main difficulty with contactless transportation is to take into account the influence of the passive object’s orientation on the magnitude of the force impact that is transmitted to the object by the contactless method. Despite the fact that the need to creation active space debris removal system has been under discussion for a long time, the lack of significant progress in this area is associated with four issues (McKnight et al., 2021): the cost of debris removal is uncertain, but probably expensive; manipulating space debris requires resolving a number of legal issues related to property rights; lack of consensus in the procedure and order of selecting targets for debris remediation activities; and the potential danger of the space debris object destruction during the mission of its transportation will lead to additional pollution of the orbit.

2.1.6 Just-in-time collision avoidance Unlike active space debris removal missions, which involve deorbiting or transferring into disposal orbit a space debris object, just-in-time collision avoidance operations are aimed at slightly changing the orbit of a space debris object in order to prevent its colliding with another piece of space debris. Apart from changing the parameters of the orbit itself, the purpose of the operation can be changing the orbit period. A small change in the period will lead to the fact that although the orbits of the two objects will intersect, the objects themselves will pass the dangerous area of intersection of the orbits at different times and will not meet. The required period change can be achieved by very little force. While active space debris removal is a strategic solution, gradually reducing the risk of collisions in orbit of all objects, just-in-time collision avoidance is a temporary tactical solution to overcome the crisis. It is very important direction of space environment management, especially in view of the fact that among more than 36,500 orbital objects, fewer than 1500 of them are able to controllably change their orbit. Despite the importance of this issue, currently in the scientific literature there are only a few ways of adjusting the orbit of unguided space objects are discussed. An overview of these methods can be found in Bonnal et al. (2020). One of the main obstacles to the implementation of just-in-time collision avoidance operations right now is the lack of accuracy of object tracking systems. According to data from Peterson et al. (2018), the measurement error of existing systems is about 1 km. To create a just-in-time collision avoidance system, it is necessary

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to improve the accuracy of ephemerides by at least two orders of magnitude. This can be achieved by orbiting orbital laser ranging stations (Bonnal et al., 2020). Relatively light objects weighing less than 1 ton make up the majority of the space debris population. Therefore, such objects can be considered as priority targets for just-in-time collision avoidance operations based on weak force generation. Acting for a long time on an object, this force imparts a velocity increment ΔV to it, which leads to a change in the object’s orbit. As a result, the miss distance increases and space debris objects can safely fly past each other. For example, according to the estimate given by Bonnal et al. (2020) based on the Clohessy-Wiltshire equations of relative motion, continuously imparting a small velocity increment ΔV ¼ 3.5 mm/s along the object’s velocity vector during the day will increase the miss distance by 1 km. There are several ways in which such a small ΔV can be achieved. Lasers. One way to create such force is to use ground-based or orbital-based lasers. Pure photon propulsion or ablation phenomena can be used to generate a force on a space object. Laser ablation is the process of removing a substance from the surface of a body under the action of high-power laser radiation. As a result of the rapid absorption of energy by the substance, it heats up and explosively evaporates the substance. The body particles appear in the surrounding gas. For some materials this gas can ionize, resulting in the formation of a high-temperature, high-pressure plasma flow. Ejection of the body particles and plasma creates thrust. A detailed description of the physical phenomena that make up the essence of laser ablation and some computational models are provided by Phipps et al. (2010, 2017). Pure photon propulsion implies a photon momentum transfer due to radiation pressure. In this case, the magnitude of the force effect turns out to be much weaker than in the case of laser ablation (McInnes, 2004). The Laser Ablative Debris Removal by Orbital Impulse Transfer (L’ADROIT) project involves the use of the ablation effect to change the orbit of space debris (Phipps, 2014). Fig. 2.5 schematically shows the influence of a laser on space debris. The system is based on the use of a 100 ps ultraviolet pulses laser. Since the path of the laser beam does not travel through the atmosphere, the system allows the use of more compact optics for focusing the beam compared to ground stations. These optics allow small objects to be affected at a distance of 250 km and large targets at a distance of up to 600 km. It is supposed that the laser station moves in a polar elliptical orbit, whose eccentricity is e ¼ 0.028. The altitudes of the apogee and perigee of the orbit are 960 km and 560 km, respectively. The orbit inclination is i ¼ 90°, and the argument

Fig. 2.5 Use of the ablation effect to change the space debris orbit.

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of the periapsis ω ¼  180°. The laser is aimed in a close to horizontal direction, which eliminates the dazzle of sensing satellites that are aimed at the Earth’s surface. Small particles of debris can be removed from orbit by short-term laser exposure. For example, according to the calculations given by Phipps and Bonnal (2016), a 10 s laser action is enough to remove a piece of space debris with a mass of 50 g from a circular orbit with an altitude of 760 km. The laser ablation will provide a speed increment ΔV ¼ 160 m/s and requires 3.2 kW of electrical power. In the case of large space debris, the laser space station can be used for the avoidance of a collision. To change the orbit of 1 ton space debris that moves in the same circular orbit, to an altitude of 10 km, the velocity increment ΔV ¼ 0.52 m/s is required. It can be provided by the laser action with duration of 830 s from a distance of 1600 km. Several such orbit adjustments can be made in 1 day. There are projects to protect the International Space Station from medium-sized space debris using a laser. Today, to prevent potentially dangerous collisions of the station with space debris of 1–10 cm, it is necessary to adjust its orbit, which incurs significant fuel costs. A paper by Shen et al. (2014) investigated the possibility of protecting a hypothetical space station in a circular orbit of a 400 km altitude from space debris using a space-borne laser in 420 km altitude orbit. A more detailed project was presented by Ebisuzaki et al. (2015), who proposed equipping the International Space Station with a superwide field-of-view telescope (EUSO) for detection of high velocity medium-sized space debris near the station and a fiber-based laser system (CAN), which can perform tracking, characterization, and remediation of detected space debris using the ablation effect. LightForce project involves the use of a ground-based continuous wave laser, and a telescope with adaptive optics (Fig. 2.6) to change the orbit of space debris due to pure photon momentum transfer (Stupl et al., 2012). Ground lasers can only be used for LEO. Adaptive optics is needed to counteract the effects of atmospheric turbulence, which results in a significant increase in beam width. The space debris orbit’s specific energy is changed as a result of its illumination by a ground-based laser. This, in turn, is expressed in a change in the orbit’s semimajor axis and period. A method for evaluating the effectiveness of the LightForce concept for collision avoidance based on an estimate of the probability of collision for all cataloged objects was proposed by Stupl

Fig. 2.6 Ground-based laser station.

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et al. (2013). Simulation results show how an increase in the number of ground stations leads to a decrease in the expected number of collisions. Unlike LightForce, the ORION project proposes the use of the laser ablation effect. It is supposed that the ground station will use a pulsed laser with adaptive optics (Phipps et al., 1996). Nano-tugs. The idea behind nano-tugs is to use the propulsion system of cubesat to correct the orbit of a massive space debris objects that pose a great danger for environment. According to the concept proposed by Mcknight and Santoni (2019), nanosatellites of the 3U-6U format are equipped with a grappling mechanism, an electric propulsion system, embedded accelerometers, and global positioning system (GPS) transceivers. After several nano-tugs “stick” to different parts of the space debris surface, they network together to determine their relative positions. Having information about the relative position, nano-tugs can more accurately determine the parameters of space debris motion and use their engines in a coordinated manner to perform detumbling maneuvers and change the space debris orbit (Fig. 2.7). Such a system allows derelict objects to be returned to life and included in the space traffic management system. The nanosatellites’ design and the just-in-time collision mission itself were discussed by McKnight et al. (2020). According to Bonnal et al. (2020) 1.5 g of fuel is enough to deflect the orbit of the SL-8 upper stage by 200 m. The CNES concept “Mosquito” was also mentioned by Bonal and coauthors as a similar scheme. Mosquito cubesat can inspect space debris from a short distance using a macrocamera, and can physically affect the space debris using deployable legs. The propulsion system of the cubesat can be used to prevent collisions. It should also be noted that short electrodynamic tethers can be used instead of thrusters to change the orbit of a large space debris object after a cubesat is attached and fixed on the object’s surface. Articles by Hoyt et al. (2003) and Zhu (2016) considered the possibility of using short electrodynamic tethers on the nanosatellite platform. Since space debris orbit adjustments need to be made infrequently, the tether deployment system must be capable of repeatedly deploying and retracting the tether to reduce the probability of colliding with other space debris. Cloud of particles. The idea is somehow to place a cloud of particles in the path of space debris. When space debris passes through this cloud, it experiences a resistance force (Fig. 2.8). The cloud must be delivered to a suborbital trajectory. Calculations by Bonnal et al. (2020) showed that to deflect a space debris object with a mass of 1400 kg/1 km from an orbit of 1200 km, an effective mass of 3 g is needed, and the

Fig. 2.7 Nano-tugs application.

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Fig. 2.8 Cloud of particles usage scheme.

impact must occur 12 h before the predicted collision. The main challenge of this method lies in the placement of the cloud to ensure that enough particles hit the surface of space debris. This requires precise phasing of the sounding rocket and space debris. To build an effective system capable of creating a cloud of particles at various points in low Earth orbit, it will be necessary to create a number of ground bases for launching sounding rockets. An alternative approach involves using an air-launched system, which can be built on the basis of aircraft Dassault Aviation Falcon 7X or Gulfstream IV. An interesting question is the choice of particle parameters and ejector design. A paper by Jarry et al. (2019) proposed using a solid rocket motor as a particle generator. Calculations show that solid particles should be released at a low rate to avoid dispersion. Bonnal et al. (2020) suggested using a modified bladder tank from Ariane 5, which is filled with 50 μm copper particles. The total mass of the filled ejector system is 65 kg. Ion beam interceptor for collision avoidance. This book proposes a new concept for using ion thrusters for just-in-time collision avoidance missions (Fig. 2.9). This is a kind of symbiosis of the concepts of the nano-tugs described above and the “ion beam shepherd,” to whom this book is dedicated. As in the case of the nano-tugs, when a potential collision hazard is detected, it is proposed to bring the small interceptor spacecraft to the target object, but not to dock to it. The small spacecraft acts on the object contactlessly. To do this, it directs the jet plume of its thruster to the object. Crashing into the surface of the object, the particles of the plume exert a force on it, changing its orbit. After the space debris orbit is sufficiently altered, the spacecraft returns to the base station for refueling and service. At the initial stage of project implementation, spacecraft interceptors can be based on the International Space Station, protecting it from possible collisions. However, for effective protection from space debris, it makes sense to place several base stations to provide quick access

Fig. 2.9 Ion beam interceptor scheme.

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of spacecraft to the widest possible range of space debris orbits. The selection of the optimal number of base stations and their orbits, taking into account the parameters of the orbits of the existing space debris, is a difficult task, which the authors plan to tackle in the near future.

2.1.7 Target selection for active space debris removal A huge number of potential targets for active space debris removal missions are currently in orbit. Reasonable choice of the target object is one of the defining stages of the development of the mission, since the mass-inertial characteristics, the parameters of the orbit, and the rotation of the space debris object have a direct impact on the method of capture and the cost of subsequent transportation. The potential danger posed by the target to other orbital objects is also of great importance. In recent years, a number of works have appeared in the scientific literature on calculating indices that allow ranking of objects of space debris and enabling a suitable choice of target to be made. A large international team of scientists from various organizations conducted a joint study with the aim of identifying 50 “statistically-most-concerning” space debris objects in low Earth orbit (McKnight et al., 2021). In the study, the authors focused on four key factors: the mass of the space debris object; how often objects of this type are found in orbit; orbital lifetime; and proximity to operational satellites. Space debris remediation from this list of 50 objects will stabilize space sustainability in the long term and make low Earth orbit safer. Rather than developing yet another new index for the “correct” ranking of space debris, McKnight et al. used the diversity prediction theorem for the already existing 11 different indices. This theorem states that collective performance of different decision schema exceeds the collective performance of a single scheme (Hong and Page, 2004). Very brief descriptions of the used indexes and algorithms are given below. The last names of the authors of the corresponding index are indicated in bold. 1. McKnight. The rank of the space debris object is calculated as the product of probability and consequence of collision. The probability is proportional to the annual collision rate of the cluster and to the approximate area of the object. The consequence factor is proportional to the mass of the object, the lifetime of an intact object at the center of the cluster, and the spatial density of operational satellites within 150 km of each cluster (McKnight et al., 2018). 2. Witner. The rank is the product of consequence and probability, which is based on the miss distance of each conjunction. The consequence is the sum of the combined mass of the objects, approximate persistence based on the orbital lifetime of debris, and the number of satellites in the vicinity of the object (McKnight et al., 2021). 3. Anselmo/Pardini. The normalized ranking index is proportional to the orbital debris flux capable of breaking an object into fragments, the object’s lifetime, its mass, the time required for half of the fragments larger than 10 cm to decay after the catastrophic fragmentation of the object, and the ratio between the debris flux, which is able to cause catastrophic fragmentation, on the object’s orbit and the flux on the equatorial orbit. For the purpose of

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4.

5.

6.

7.

8.

9.

10.

69

normalization, the above values are divided by the corresponding values of the reference object, which is some average intact object in LEO (Pardini and Anselmo, 2018). Letizia/Lemmens. The Environmental Consequences of Orbital Breakups risk indicator is a complex index that takes into account not only the current threat to the object, but also the integral threat to the orbital environment throughout the entire period of the object’s existence from its placement to removal from orbit or its fragmentation during postmission disposal. The indicator is based on the space debris object’s index, which is the product of the collision probability of the object with the existing space debris population, and the severity term that represents the estimation of the object’s potential fragmentation consequences (Letizia et al., 2019). Rossi. The criticality of the spacecraft index is proportional to the mass of the object, spatial density for the orbital shell, lifetime of the object at the altitude corresponding to the shell, and function of the orbital inclination, which reflects the increased risk of collision with increasing orbital inclination. The index is normalized by the mass, spatial density, and lifetime period (Rossi et al., 2015). Baranov/Grishko. Instead of constructing a numerical index reflecting the potential danger of space debris, the authors of the study (Baranov et al., 2017) analyzed the upper stages existing in low Earth orbit and tried to rank them in terms of optimizing the active space debris removal mission fuel costs, when one active spacecraft removes several space debris objects at once. An attempt to work out the basic concept of such a mission was made by Baranov et al. (2021). The authors identified five groups of closely located large space debris objects. Within each group, the authors proposed ranking the debris by mass, since the greater the mass, the greater the potential danger that space debris poses to other objects. Lewis. The index is based on the results of long-term probabilistic modeling of the nearEarth environment evolution using the Debris Analysis and Monitoring Architecture to the Geosynchronous Environment model (Lewis, 2020). To calculate the index, 220 calculations were performed using the Monte Carlo method at 1000-year intervals. Then, for each space debris object, various average metrics were calculated from the simulation results. Ten different lists of objects ranked according to these metrics were compiled. The index of the object was calculated as the sum of points for a position in each list divided by the number of lists (McKnight et al., 2021). Kawamoto. The Near-Earth Orbital Debris Environment Evolutionary Model, developed by Kyushu University and JAXA (Kawamoto et al., 2020), was used to assess the impact of active space debris removal on the orbital environment. Based on the simulation results, 50 objects that generate the largest amount of space debris fragments in the initial time interval were selected. Nicolls. This rating is based on the processing of statistical data obtained by LeoLabs as a result of observations of objects in low Earth orbit. The LeoLabs collision impact score is proportional to the probability of collision and the combined average radar cross section of the two objects, which is an indication of space debris fragments amount in a collision event. After the number of space debris fragments is determined, a quantitative assessment of their impact on the environment is made, according to which the ranking is confirmed. Dolado Perez/Ruch. The software developed by CNES makes it possible to calculate a complex index taking into account the peculiarities of the mission, the structure of the object, and the orbital environment. It also depends on the probability of the object fragmentation as a result of collision and explosion, and effects of these fragmentations (McKnight et al., 2021).

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11. Jing/Dan/Wang. The criteria of active space debris removal target selection are calculated as the product of object mass, collision probability, and normalized spatial density of cataloged objects.

Different scoring techniques to rank debris objects were tried by McKnight et al. (2021), but these techniques yielded the same 25 first objects. As a result, the following simple ranking methodology was chosen. For each of the 11 indexes and algorithms, its own ranked list of space debris objects was compiled. The first object on this list received 50 points, the second object received 49, and so on down to 1. Then the sum of the points for the objects from all the lists was calculated. This amount was multiplied by the number of lists to which the object belongs. The following formula describes this technique: sj ¼

11 X

! pij kj

(2.2)

i¼1

where sj is the scope of the jth space debris object, pij is the points of the jth object for the ith ranged list, and kj is the number of lists to which the jth object belongs. Table 2.1 contains a list of most dangerous space debris objects grouped by their type. It is based on data given in Table 4 by McKnight et al. (2021). Among these objects, 39 are rocket bodies, and 11 are nonfunctional satellites. The orbits of these objects are concentrated in three zones, the most concentrated of which is located at an average altitude of about 830 km and has an inclination of about 72 degrees. McKnight et al. noted that although they have a ranked list, it would be wrong simply to remove space debris by moving it from the top to the bottom of the list. Removing any object from the list will change the orbital environment, and the risks will change. This will

Table 2.1 Active space debris removal targets candidates. SATNAME

Mass, kg

Number of objects

Country

SL-16 R/B SL-8 R/B COSMOS SL-12 R/B(2) ADEOS ADEOS 2 ARIANE 5 R/B COSMOS 1275 CZ-2D R/B ENVISAT H-2 R/B H-2A R/B METEOR 3M SL-3 R/B

9000 1435 3250 2440 3560 3680 2575 800 4000 7800 2700 3000 2500 1100

21 11 6 2 1 1 1 1 1 1 1 1 1 1

CIS CIS CIS CIS Japan Japan France CIS China ESA Japan Japan CIS CIS

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lead to the transformation of the list itself. In addition, effective active space debris removal systems are multipurpose, and they can remove several closely spaced objects in one mission.

2.2

Contact methods of active space debris removal

The scientific literature discusses many different methods of large space debris removal. Most of them are based on the use of a propulsion system of an external active spacecraft. There are projects involving the docking or capture of a space debris object by a robotic manipulator, harpoon, or net, and its subsequent towing on a rigid bundle or on an elastic tether. The development of contactless space debris transportation is promising. It involves the use of electrostatic, gravitational, electromagnetic fields, lasers, as well as the flow of high-speed particles created by the active spacecraft to exert a force on a space debris object. Planned active space debris removal missions usually include several phases: launching a spacecraft, rendezvous with a space debris object, evaluating and refining its motion parameters and mass-geometric characteristics, detumbling, docking or capture, active transportation phase, re-entry or space debris separation, and rendezvous with the next target. Depending on the specific implementation of the active spacecraft and mission design, certain phases may be absent. For example, if contactless transportation methods are supposed to be used, there is no need for the docking phase.

2.2.1 Space debris capturing The methods discussed in the scientific literature that ensure the capture of a space debris object by an active spacecraft are discussed in this section. Robotic arms or tentacles. Using a manipulator to grab space debris is perhaps the first thing that comes to mind when we talk about capturing space debris. Space manipulators are a fairly well-established technology. An overview of existing technical solutions and approaches can be found in the work by Flores-Abad et al. (2014). The currently used manipulators are designed primarily for working with cooperative targets, which can report information about themselves. Space debris belongs to the class of noncooperative objects, the parameters of which are known very approximately, and the space debris itself can tumble with a rather high rate. These circumstances significantly complicate the use of robotic manipulators. The need to capture space debris has meant that in recent years, scientific research has begun to be devoted to various aspects of capturing noncooperative targets (Dai et al., 2020; Stolfi et al., 2018; Zhan et al., 2020). The robotic manipulator itself is a mechanical system consisting of a chain of links that are connected to each other by motorized joints. At the end of this chain is an end-effector, which can be equipped with various sensors (Fig. 2.10). The onboard capture system of a spacecraft can consist of one or several manipulators. For example, Choi et al. (2019) proposed using four robotic arms to grip a target.

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Fig. 2.10 Robotic manipulator.

The designs of the effector can be very diverse. It could be a simple probe that must be inserted into the nozzle of a nonfunctioning upper stage and secured there (Mayorova et al., 2021; Yoshida et al., 2004), or it could be a two-finger gripper mechanism (Lanni and Ceccarelli, 2009). The design of the effector can be more complex. For example, Yang et al. (2021) proposed a flexible construction similar to a sea anemone, which consists of several arms. The kinetic energy of a randomly rotating target is absorbed by a mechanism that entangles and locks it. Li et al. (2018b) proposed a multilink mechanism, which embraces the target object for capturing. This is based on spatial Bennett linkages connected by scissor elements. A compact multimodule capture mechanism was described by Sun et al. (2019). It is based on an origami principle and consists of interconnected polygonal plates. When capturing the target, the mechanism deploys and covers it. A soft robotic arm similar to an octopus tentacle was proposed by Grissom et al. (2006), and uses air muscle extensors. In addition to mechanical gripping, alternative approaches are discussed in the scientific literature. The design of an electrostatic adhesive gripper was proposed by Schaler et al. (2017). Gecko-inspired dry adhesion can be used to create a robotic gripper to capture large noncooperative objects in space ( Jiang et al., 2017). The technical solutions described do not cover all possible designs, but are intended to demonstrate their diversity. Net capturing. This method assumes that an active spacecraft flies up to space debris and throws a net over it (Fig. 2.11). The net tangles around the target object, after which the spacecraft tows it on a tether attached to the net. The advantage of this method is the ability to capture large rotating objects. It is insensitive to the shape of the object. This method is relatively safe for an active spacecraft, since the net can be thrown over the object from a great distance. The net is very compact when folded. It is fired using a spring mechanism and deployed during flight. For stable deployment, weights are attached to various points in the net. Rotation can be used for additional stabilization. The weights can be equipped with special mechanisms that tighten one of the net threads and thus close the net, preventing the loss of the caught object (Bischof, 2003). However, capturing high angular momentum heavy space debris object with a light net is a very difficult and dangerous task. The main difficulty in modeling the dynamics of a network is ensuring adequate representation of the interaction of the network threads with the surface of space

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Fig. 2.11 Space debris capturing by a net.

debris. Deployment and capture dynamics of rectangle net was considered by Botta et al. (2017). The lumped-parameter approach was used to simulate different modes of contact of the net’s threads with the surface of space debris and other threads of the net. The numerical simulation of capturing the cylindrical object by the net was carried out in the study. A paper by Peters et al. (2018) was devoted to the study of dynamics and control of the large space debris attitude motion after its capture by the tethered net. It investigated the influence of the tether material on the safety and resource intensity of space debris tethered deorbiting. To date, several experiments have been carried out to test net capturing technology. In particular, a parabolic flight experiment was carried out, which showed good agreement between the simulation results of net deployment with the experimental results (Shan et al., 2017). An orbital experiment was performed as part of the RemoveDEBRIS mission in 2018 (Aglietti et al., 2020b). A passive cubesat was caught by a net that was shaped like a hexagonal star with masses at the vertices. These masses were equipped with internal mini winches, which allowed the net to be closed after the target’s capture. Since the cubesat was captured at a distance of 11.5 m, and not at the calculated 6 m, the data from the video cameras prohibits the assertion that the net was closed successfully. Tethered space robot. The robotic manipulator can be installed on an autonomous module, which is connected by a tether to the active spacecraft (Fig. 2.12). The tether serves as a very long, but flexible, arm of the space robot. The creation of a rigid arm of comparable length will be significantly more expensive to manufacture, deliver, and operate. Using a tether makes it possible to increase the maneuverability and reliability of the system, since an unsuccessful attempt at capture will not lead to a breakdown of the active spacecraft. At the same time, the presence of an extended elastic element significantly complicates the behavior of the system. Many works have been devoted to the study of space tether systems dynamics (Aslanov and Ledkov, 2012; Beletsky and Levin, 1993; Cartmell and McKenzie, 2008). The functioning of the tethered robot assumes the deployment and possible retrieval of the tether. An overview of

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Fig. 2.12 Tethered robot for space debris capture.

existing control laws and approaches has been provided by Yu et al. (2018), where all control laws are divided into four groups: tether length control, velocity control, tension control, and thruster control. The motion of tethered space robot at the station keeping phase after the tether deployment but before a target object was considered by Zhao et al. (2018). An adaptive sliding mode control algorithm was proposed to stabilize in-plane and out-of-plane tether oscillations. Numerical modeling has confirmed the effectiveness of using the algorithm in the presence of unknown boundary disturbances. A control scheme that ensure the detumbling of space debris after its capturing was proposed by Huang et al. (2015). Currently, the most well-known project for using a tethered space robot is ROGER. This project proposes using a 40 kg tether-gripper, which is a self-contained module equipped with two stereo cameras, and a laser range finder and cold gas propulsion system. The capturing is carried out directly by the three-finger effector mounted on a telescope arm (Bischof, 2003). Harpoon. The use of a harpoon is another promising way to capture space debris. After the spacecraft approaches a space debris object, a harpoon is shot at it. The harpoon pierces the target’s shell and gets stuck in it. A tether is tied to the harpoon, which makes it possible to tow space debris (Fig. 2.13). When choosing the point at which the shot will be made, it is necessary to take into account the rotation parameters of the target object and the peculiarities of its casing. In a study by Sizov and Aslanov (2021), it was shown that a successful choice of the target point allows the problem of space debris detumbling to be solved. The technology has passed a number of

Fig. 2.13 Space harpoon.

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successful laboratory tests of harpoons (Dudziak et al., 2015); in addition, in 2019, an orbital experiment as part of the RemoveDEBRIS project was carried out. A square aluminum honeycomb panel with a side length of 10 cm was successfully harpooned from a distance of 1.5 m. The measured firing speed was 19 m/s, and the harpoon projectile mass was 0.115 g (Aglietti et al., 2020b). The orbital experiment was preceded by a series of 27 ground tests to verify various aspects of the harpoon design (Aglietti et al., 2020a). The advantage of the harpoon capture method is its weak sensitivity to the shape of the target object and safety for an active spacecraft, since there is no need to approach space debris at a close distance. However, harpooning tanks and solar panels can lead to undesirable consequences. The possibility of formation of new space debris fragments as a result of an unsuccessful shot is the main disadvantage of harpoon capturing.

2.2.2 Contact space debris removal methods After capturing and stabilizing space debris, the active transportation phase begins. In the case of docking and hard grip, when the active spacecraft and space debris can be considered as a single rigid body, the use of impulsive orbital maneuvers using powerful thrusters of the active spacecraft is possible. The main advantages of this technology are its reliability and many years of experience in application in astronautics. If the space debris has deployed flexible elements such as solar panels or antennas, or if the capture results in a tethered connection and not to a rigid bundle, then impulse maneuvers cannot be used due to the danger of the flexible elements breakage and tether rupture. In such cases, the use of low-thrust engines is justified. The technology for using low-thrust engines is also well developed; however, some peculiarities arise with tethered towing. Possible alternative concepts for transporting space debris are discussed below. Tethered towing. The presence of a tether connection significantly complicates the dynamics of a mechanical system consisting of an active spacecraft and a space debris object. The dynamics of the space tether system when towing space debris (Fig. 2.14) has been studied by Aslanov and Yudintsev (2013a,b). The point is not only that, in the spatial case depending on the assumptions made, an additional

Fig. 2.14 Tethered towing.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

3–7 degrees of freedom are added to the mathematical model. The tether elasticity and the eccentricity of the orbit that increases during the descent from the Earth orbit create the prerequisites for the emergence of resonant and chaotic phenomena (Ledkov and Aslanov, 2019). The development of control laws for the active spacecraft propulsion system, taking into account possible tether oscillations, is necessary for the successful solution of the tether-assisted transportation problem. Preliminary design of a control system based on simple sliding-mode controllers was proposed by Linskens and Mooij (2016). For relative orbit control, classic Clohessy-Wiltshire equations are used for motion predictions. Euler equations are used in attitude controller of active spacecraft tug, which is considered as a rigid body. Wang et al. (2017) proposed controlling the spatial orientation of space debris during towing by changing the position of the tether attachment point. This change is made by a manipulator at the end of the tethered gripper. Backstepping-based hierarchical sliding mode control was proposed by the authors. A reel mechanism is used to stabilize the tether tension by changing its length. In a study by Li et al. (2019), an optimal control scheme consisting of two phases for space debris removal by tethered space tug was developed. The open-loop control trajectory, which minimizes fuel consumption for tethered towing, is determined in the first phase using a direct collocation method. The second phase includes tracking the obtained optimal trajectory using closed-loop optimal control. The behavior of the space tether system is further complicated by the presence of elastic elements attached to space debris and reservoirs with fuel residues (Aslanov and Yudintsev, 2015a, 2015b). Electrodynamic tethers. Space tether technology allows space debris removal to be performed without the use of any engines due to the interaction of the conducting tether with the Earth’s electromagnetic field. A description of various aspects of physical processes underlying the use of electrodynamic tethers, and various methods of application for these, can be found in Sanmartin et al. (2010). The concept of using an electrodynamic tether in deorbiting or thruster operation modes is based on the idea of using the Lorentz force, which acts on a conducting tether from the Earth’s magnetic field when the electric current flows in it (Fig. 2.15). This method can be used for LEO, since the magnetic field intensity decreases rapidly with increasing orbital altitude. To create a current in a single core tether, it is necessary to ensure its contact with the ionosphere. This can be done with plasma contactors, which are hollow cathodes; they are attached to the ends of the tether and collect or emit electrons into the environment. A bare conductor can be used as a passive electron collection system (Sanmartin et al., 1993). Low work-function coating of the tether can be used as an alternative to an active cathodic device, which eliminates the need for an expellant to the hollow cathode (Williams et al., 2012). The coated tether emits electrons in passive mode due to thermionic and photoelectric effects (Sanchez-Arriaga and Chen, 2018). A study by Sa´nchez-Arriaga and Sanmartı´n (2020) was devoted to modeling a low work-function electrodynamic tether in thrust mode. Electrodynamic tether is a promising technology that is in the experimental stage of confirming its key points. To date, several orbital experiments have been carried out: TSS-1R, PMG, and KITE. The equipment for the KITE experiment was brought into orbit, but the experiment itself

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Fig. 2.15 Electrodynamic tether.

failed as the tether was not deployed due to mechanical problems (Ohkawa et al., 2020). Many scientists have studied the dynamics of electrodynamic tether systems. An investigation of the feasibility of applying a reusable electrodynamic tether system to remove space debris from low Earth orbit based on the simulation results obtained using a simplified model was carried out by Ishige et al. (2004). In particular, it has been shown that space debris can be removed from orbit in a reasonable amount of time and requires a realistic power supply. Mathematical model and simulation results of satellite deorbiting using an electrodynamic tether system, taking into account different perturbations, were given by Zhu and Zhong (2011). A multiphysics finite element model of flexible bare electrodynamic tether was developed by Li et al. (2017), who proposed a robust energy control strategy, which provides deorbiting of the space tether system without loss of stability at low altitudes. A comparison of space debris removal technology using an electrodynamic tether with systems based on the use of chemical engines and drag augmentation devices was carried out by Sarego et al. (2021). Among the advantages, the authors highlighted the low mass of the removal system, the ease of control, the ability to perform collision avoidance maneuvers, and a wide range of orbits where this technology can be applied. Momentum exchange. Space debris momentum exchange methods imply the redistribution of angular momentum between an active spacecraft and space debris after capture or docking. One of the technical solutions capable of implementing this redistribution is space tether systems (Fig. 2.16). Taking into account the features of

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 2.16 Momentum exchange tethers.

the space debris removal mission, two tether-based space debris removal schemes can be implemented: space debris deorbiting by a space tether system of variable length, or by a rotating space tether system. After capture, the active spacecraft and space debris represent a rigidly connected mechanical system, or they are connected at a relatively short distance from each other. In any case, space debris and the active spacecraft must be separated over a long distance in order to perform momentum exchange effectively. The tether deployment is one of the central issues in the tether-assisted deorbiting missions. Much attention has been paid to this topic in the scientific literature. Overviews of the currently existing control laws and technologies were given in articles by Kumar (2006) and Yu et al. (2018). For payload deorbiting missions, tether length control laws can be divided into two groups (Zimmermann et al., 2005): “static,” which supposes the payload release from the tether equilibrium state, and “dynamic,” which assumes the use of the tether back oscillations near the equilibrium position for additional payload velocity reduction. The static scheme involves slow deployment of the tether, causing the tether to oscillate in the vicinity of the spacecraft’s local vertical. As a result of the angular momentum redistribution, the velocity of the payload at the lower end of the tether is significantly less than the first cosmic velocity; therefore, after separation from the tether, the payload passes to the descent trajectory. The dynamic laws are based on the idea of using the Coriolis force to deflect the tether in the orbital flight direction and then using the return oscillatory motion of the tether to reduce the payload velocity (Williams, 2008). Tether swing control law allows increasing of the amplitude of the tether oscillations by changing its length (Aslanov, 2016). The main advantage of dynamic control in comparison with static is the possibility of carrying out the payload deorbiting using considerably shorter tethers. To date, three successful experiments of a tether-assisted payload deorbiting have been conducted: SEDS-1 in 1993, SEDS-2 in 1994 (Smith, 1995), and YES2 in 2007 (Kruijff and van der Heide, 2009). To implement the scheme for space debris removing by a rotating space tether system, the system spins up after tether deployment. The direction of rotation must coincide with the direction of the system orbital motion. In this case, the relative velocity of the tethered space debris at the time when it is closest to the Earth is opposite to the velocity of the orbital motion. Thus, the absolute velocity of space debris can be significantly reduced, and if at this moment the tether is cut off, the space debris will be

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transferred to the descent trajectory. A comparison of payload deorbiting using static deployment, dynamic deployment, and descent from the rotation mode was carried out by Aslanov and Ledkov (2017a). The space tether system can be turned into rotation by thrusters (Guang et al., 2019), electric motors (Ziegler and Cartmell, 2001), or electrodynamic tether (Luo et al., 2021), or by controlling the tether length (Aslanov, 2016). The Sling-Sat Space Sweeper project is similar to a rotating space tether system, where the tether is replaced with rigid rods (Fig. 2.17), as a result of which the dimensions of the system itself are small (Missel and Mortari, 2011). The sweeper is supposed to consistently catch and slow down various space debris objects. For a flight from one object of space debris to another object, the energy obtained as a result of decelerating the first of them can be used (Missel and Mortari, 2013). Drag augmentation devices. The scientific literature discusses several ways to increase the resistance of the environment acting on space debris. All these methods imply space debris cross-section area growth, which leads to an increase in the atmosphere drag. Obviously, these methods are applicable for low Earth orbit only. After docking with space debris, an active spacecraft can inflate an inflatable balloon (Fig. 2.17) folded in it (Koryanov et al., 2019; Nock et al., 2010). The main disadvantage of this technology is its vulnerability to small space debris. Another method is based on the use of sticky hardening foam. An active spacecraft flies up to space debris and directs a stream of sticky polymeric foam at it. The spacecraft flies around the space debris object, gradually covering it with foam, thus forming a large ball. The main obstacle to implementing this method is the creation of foam that can reliably expand, adhere, and solidify in vacuum and microgravity conditions (Pergola et al., 2011). If, instead of foam, hot fiber that solidifies upon cooling is used, a ball of filaments can form around the space debris object (Wright, 2013). Solar sails. The use of a solar sail (Fig. 2.18), like drag augmentation, is a passive method of space debris removal, in the sense that it uses the external environment. A solar sail is a device that uses solar radiation pressure to generate force. It can be

Fig. 2.17 Inflatable balloon for drag augmentation.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 2.18 Solar sail concept.

deployed after docking of the active spacecraft carrying it with the space debris object. The solar sail is made of lightweight reflective material. It should have a large area, since the solar radiation pressure is very low. Overviews of existing design solutions, mathematical models and control laws were provided by Fu et al. (2016) and Gong and Macdonald (2019). The main disadvantage of a solar sail is its low efficiency. Since the generated force is small, a space debris removal mission using this method will take a long time. To use the solar sail effectively, it is necessary to control its orientation during the mission, since the solar radiation pressure can slow down an object in one part of its orbit and accelerate it in another part of the orbit (Borja and Tun, 2006). To date, orbital experiments have been carried out. In 2010, the Japan Aerospace Exploration Agency launched the IKAROS spacecraft, which was equipped with a 196 m2 solar sail. The sail was deployed successfully and the spacecraft sailed to Venus using solar radiation pressure (Tsuda et al., 2011). The LightSail-2 mission sponsored by the Planetary Society was carried out in 2015. Cubesat was equipped with a 32 m2 sail. The experiment demonstrated the ability to control the orientation of the solar sail during orbital flight (Spencer et al., 2021).

2.3

Contactless space debris removal approaches

Capturing or docking with a noncooperative object is a complex technical task. Failure at this stage is highly probable and can lead to the formation of new debris. An alternative is the use of contactless transportation methods.

2.3.1 Ion beam-assisted transportation Space debris transportation by ion beam involves the installation of an additional electrodynamic engine (impulse transfer thruster) on board the active spacecraft. This engine creates a stream of ions that pushes on the space debris. Ions hit the surface

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Fig. 2.19 The ion beam shepherd concept.

of the space debris and create a force (Fig. 2.19). The resulting force generated by the ion beam on the entire surface is called the ion beam force. This force is used to transport the space debris object. Modern ion thrusters allow a force of the order of a dozen mN to be exerted on a space debris object. In order to keep the active spacecraft near the space debris, the spacecraft includes a second ion engine (impulse compensation thruster), which is directed opposite to the first one and neutralizes its force effect. The idea of this space debris removal method was expressed by Bombardelli and Pelaez (2011) and Kitamura (2010), independently of each other. The goal of Kitamura’s project was reorbiting space debris in GEO. The project by Bombardelli and Pelaez was aimed at removing large space debris from LEO. It was named the “ion beam shepherd.” In 2013, an international team of scientists won the FP7-SPACE competition with the project “Improving Low Earth Orbit Security With Enhanced Electric Propulsion” (LEOSWEEP) (Ruiz et al., 2014). Thanks to this, the concept of the ion beam shepherd has been developed, various studies have been carried out, and a large number of scientific papers have been published. The project website is https://leosweep. upm.es. The advantage of the ion beam transportation method is that such a system can be created based on existing technologies. It can be used to transport objects of various shapes and layouts. In addition, it can transport rotating objects. One of the main disadvantages of this concept is the need to compensate for the thrust created by the impulse transfer thruster. There are projects of an economical double-sided ion engine aimed at solving this problem (Dobkevicius et al., 2020; Takahashi et al., 2018). Another significant issue is backsputtering substances from the space debris object to the active spacecraft. This may cause contamination to the spacecraft’s systems and damage its sensors and solar panels. The complexity of the ion beam transportation method lies in the fact that the magnitude and direction of the resulting ion beam force depends, among other things, on the position and orientation of the space debris object inside the ion beam. The point of application of the resultant force does not coincide with the center of mass of the object. This leads to the appearance of an ion beam torque relative to the center of mass, which tends to rotate the object in the ion flow. This rotation, in turn, will lead to changes in the magnitude and direction of the ion beam force. Ion beam force and torque calculation. Determining the forces and torque transmitted by the ion flow to a blown rigid body can be carried out using the finite element

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

method implemented in various software products. A simplified algorithm for calculating force and torque was described in detail by Alpatov et al. (2016, 2019). The idea is to break the body into triangles and calculate the force impact on each of them, and then sum up the results. The self-similar model of ion engine plume expansion and the fully diffused reflection model of ions’ interaction with the object’s surface are used to calculate the effect of ion flow on a triangle. A description and comparison of three self-similar models of plasma plume expansion was given by Merino et al. (2011). An article by Korsun et al. (2004) described the use of a self-similar plume model to simulate the effect of an exhaust plasma plume on the hull and solar panels of a spacecraft. The simulation results were compared with the results of the EPICURE space experiment (Borisov et al., 1991). A comparison of the simulation results of propagation in a far-field plume of two different Hall effect thrusters obtained using simplified selfsimilar fluid models with experimental data obtained with a single cylindrical Langmuir probe was given by Dannenmayer et al. (2012). The experimental data are in good agreement with the results of the analytical models. A more accurate asymptotic expansion method can be used as an alternative to the self-similar model (Cichocki et al., 2014). A detailed description of the physical processes and phenomena arising from the spacecraft-plasma-space debris interaction, such as the ion backscattering flow, the sputtered atom backflow from the space debris, and the electric charging effects, was given by Cichocki et al. (2018). The simulation hybrid code EP2PLUS, which was described by Cichocki et al. (2017a), was used for numerical simulation and analysis. An alternative approach for the approximate calculation of ion force and torque for a body, based on the use of its known aerodynamic characteristics, was proposed by Ryazanov and Ledkov (2019). Electric propulsion subsystem. The structure of a propulsion system of the ion beam shepherd concept is based on radio-frequency ion thrusters (Cichocki et al., 2017b). Numerical optimization study has shown that a spacecraft’s impulse transfer thruster must operate at a higher voltage than the compensation thruster to ensure total minimum dedicated mass or power. Decreasing the impulse transfer thruster divergence angle leads to higher momentum transfer efficiency. An increase in the thruster operational voltage contributes to a decrease in this angle. Balashov et al. (2017) studied the analytical and experimental determination of the characteristics of a radio frequency ion source for xenon ion beam generation. Their research results confirmed the possibility of creating a wedge-shaped ion beam with a beam expansion half-angle of the order of 2 degrees. The technology of modern ion thrusters was described in detail by Holste et al. (2020). Space debris dynamics and control. The process of contactless space debris removal by an ion beam without taking into account the motion of space debris relative to the center of mass has been investigated in detail. Simplified ClohessyWiltshire equations that determine the relative position of a spherical space debris object in a quasicircular orbit under the action of ion force were obtained by Bombardelli et al. (2012). Using these equations, the relative stability was investigated and a control law for a three-axis thruster-based feedback control system was proposed. It has been shown that the open-loop control is indispensable even for such a simplified system. The controller for the impulse compensation thruster, which

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maintains a constant distance between the active spacecraft and the space debris object, was developed by Alpatov et al. (2018) for the case of plane motion. Timeinvariant and periodic linear-quadratic regulators were used by Khoroshylov (2020) to control an active spacecraft’s compensation thruster in the case of elliptical orbit. The control law of yaw angle of the active spacecraft was proposed by Khoroshylov (2019) for out-of-plane space debris oscillations damping. Urrutxua et al. (2019) presented a preliminary design methodology based on the matching chart approach, which allows selection of active spacecraft design parameters taking into account different design and operational constraints. Within the framework of the methodology, the ion beam shepherd mission design stems from two key parameters: the distance between the spacecraft and the object, and the ion engine thrust. Optimization of the mission of sequential deorbiting of several space debris objects and optimization of the parameters of a spacecraft based on the Express-1000NV platform were considered by Obukhov et al. (2021). These studies have shown the feasibility and great practical potential of the ion beam removal method. Meanwhile, the space debris angular motion relative to its center of mass can have a significant impact on the efficiency of contactless ion beam transportation. Deep studies of space debris dynamics during its ion transportation, taking into account the influence of the angular orientation on the magnitude of the ion impact, have been carried out in a planar statement. A mathematical model describing the motion of a space debris object in the orbital plane under the action of an ion beam generated by an active spacecraft was developed using the Lagrange formalism by Aslanov and Ledkov (2017b). This model was used to analyze the influence of the angular orientation of cylindrical space debris on the descent time. According to the results of numerical simulations, the difference in the descent time from 500 km circular orbit between the most favorable and unfavorable cases for the Cosmos 3M stage without attitude control reaches 30%. The effect of the atmosphere in LEO on the space debris behavior during its removal by ion beam was studied by Ledkov and Aslanov (2018). In this work, the influence of angular oscillations of the space debris on the value of the average ion force was also investigated. In a paper by Aslanov et al. (2020), using a simplified model, the dynamics of cylindrical space debris under the action of an ion beam in a Keplerian orbit was investigated. Bifurcation diagrams were constructed, showing that in the process of decreasing the orbital altitude, bifurcations are possible, leading to a change in the structure of the space debris phase space. The possibility of the existence of chaos was also shown. A study by Aslanov and Ledkov (2020) was devoted to the problem of controlling the angular oscillations of a space debris object. Two approaches to the attitude motion control were proposed. The first approach is based on changing the thrust of the engine that creates an ion beam. For this method, a control law was proposed. Using Lyapunov’s theorems on stability and asymptotic stability in the first approximation, it was shown that the control law ensures the asymptotic stability of the equilibrium position of space debris. The second approach is based on changing the direction of the ion beam. This can be achieved by placing the ion beam-generating engine on a movable platform that is driven by an electric motor. The Bellman method was used to build the control law for this approach. The results of numerical simulation show that

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

the second approach is a more efficient way of stabilizing attitude motion in terms of minimizing the time spent. In a study by Aslanov and Ledkov (2021b), four control strategies for an active spacecraft were proposed and compared. The first strategy involves transporting the object without considering its attitude motion. The second strategy involves stabilizing the object in a stable angular equilibrium position. The third strategy assumes transportation the space debris in the oscillation mode with the maximum average force. The fourth strategy involves stabilizing the object in an angular position corresponding to the maximum ion force. The control laws for the active spacecraft thrusters and for the direction of the ion beam axis, implementing these strategies, were proposed. The results of numerical simulation of rocket stage transportation showed that the second strategy is the most preferable from the point of view of minimizing fuel consumption. The fourth strategy is ineffective. The planar motion case is ideal and cannot be realized in practice; however, the revealed patterns of motion and found control laws can be useful in the analysis of the general case of 3D motion. In Chapter 2 of the book by Alpatov et al. (2019), the equations of the attitude of motion of space debris in the three-dimensional case were obtained using the Euler’s equations and Rodrigues-Hamilton parameters. A simplified mathematical model describing the three-dimensional motion of a symmetrical space debris in GEO under the influence of ion beam only was constructed by Aslanov et al. (2021). The control law for the active spacecraft engine thrust, which creates the ion beam, providing the stabilization of the spatial motion of space debris was proposed. This control translates the space debris not into a stationary relative state, but into a regular precession mode. A control law for space debris detumbling in GEO, taking into account gravity gradient torque, was proposed by Aslanov and Ledkov (2021a). Nakajima et al. (2018) described a control algorithm for axisymmetric rocket body detumbling based on interpolation of precalculated ion force and torque obtained using high-fidelity computational fluid dynamics approach. A control algorithm for space debris detumbling, when an active spacecraft flies around the rotating space debris object, was proposed by Nakajima et al. (2020). Li et al. (2018a) described a mathematical model describing the spatial motion of a system consisting of two rigid bodies with their contactless interaction. When developing the model, the authors did not take into account the subtleties of calculating the force and moments of contactless interaction, but used the corresponding vectors in a general form. A stable backstepping control that provides elimination of the relative velocity and angular velocity was designed. The backstepping control is based on the use of a recursive Lyapunov-based scheme, which automatically ensures the stability of the controlled motion. This section was intended to acquaint readers with the current state of research concerning transport by means of an ion beam in general terms. A more detailed disclosure of key aspects of this technology will be given in the subsequent chapters of this book.

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2.3.2 Transportation by electrostatic interaction Contactless transportation by means of electrostatic interaction involves the use of Coulomb force between an electrically charged spacecraft and space debris. The required charge of these bodies can be carried out using an electron or ion gun that is installed on the active spacecraft. Since electric charges, depending on their signs, can be both attracted and repelled, two schemes of contactless electrostatic transportation are possible (Fig. 2.20). The first scheme assumes that an active spacecraft and a space debris object have different charges in signs, and therefore they are mutually attracted. In the literature, this method is called the “pull scheme” (Aslanov and Yudintsev, 2018) or “Coulomb tether” (Natarajan and Schaub, 2006). The “push scheme” assumes that the active spacecraft and space debris have charges of the same sign and therefore repel each other. Since near-Earth space is not a vacuum, but is filled with charged particles, the surrounding plasma has a great influence on the charges of an active spacecraft and space debris, as well as on the possibility of using this method of transportation. When comparing transportation schemes, for a pull scheme, it is technically easier to implement the maintenance of charge on bodies, since it is possible to direct the flow of charged particles from one body to another. The push scheme has the following advantages over the pull scheme. The push scheme is naturally stable in terms of distance between the spacecraft and debris. In the case of a change of the active spacecraft’s thrust or the charge of the spacecraft or debris, the new stable relative distance is established in a natural way. In the pull scheme, the change in the charges of the bodies or variation of the spacecraft’s thrust can lead to increasing distance between the spacecraft and the space debris object, decreasing Coulomb force, and so the control force should be applied to maintain the condition of stationary motion. Hence, the pull scheme imposes heavy demands on the spacecraft’s control system in comparison

Fig. 2.20 Schemes of contactless electrostatic transportation.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

with the push scheme. In the push scheme, the spacecraft’s thrust can be applied along the line between the tug and debris, but in the pull scheme, the thrust should be inclined to that line to avoid unwanted pressure of the spacecraft’s engine flow on the debris. Therefore, the effectiveness of the pull scheme is less than that of the push scheme. The potential difference between bodies in the push scheme is less than in the pull scheme, and so the first one is safer in the case of contact between the bodies (Aslanov, 2020). Plasma interactions. The parameters of cosmic plasma are very different in different areas of outer space. At low altitudes, the plasma density is high and it is in a state close to equilibrium. With increasing altitude, the plasma density drops sharply. Plasma in GEO is much more rarefied and colder than in LEO. Detailed information on the composition and characteristics of plasma at different altitudes, as well as on the features of spacecraft interaction with plasma environment, can be found in Hastings and Garrett (2004). If we place a charged body in plasma, then it will attract particles of the opposite sign to itself. The distance to which the electrostatic action of this body extends is called the Debye length of a charge. At a greater distance, the electrostatic force field of the body is shielded by the charge of the plasma surrounding the body (Gombosi, 1998). The Debye length is different at different heights, and its length determines the possibility of using the electrostatic field of an active spacecraft for safe contactless transportation. According to the data given in Table 2.3 of a NASA report (King and Parker, 2002), the Debye length in low Earth orbit varies from 2 to 40 cm, which obviously makes safe contactless electrostatic transportation unrealizable due to the high risk of collision of an active spacecraft and a charged space debris. In GEO, this length varies from 142 to 1456 m, which is quite enough for the implementation of transport by means of Coulomb force. When a charged body is placed in plasma, it attracts the surrounding free charged particles, which leads to the appearance of a current around the body and to a gradual equalization of the charge of the body and the surrounding plasma (Fig. 2.21). In addition, the charge of the body is affected by the emission of electrons due to the action of solar radiation. To maintain a constant charge of the body, it is required to carry out a constant emission of charged particles from the body. For example, this can be done by an electron or ion gun, and requires few Watts of electrical power and a very small amount of propellant. To charge a passive space debris object, a stream of particles can be emitted to it (Schaub and Moorer, 2012). Since the charge of a passive body tends to

Fig. 2.21 Interaction of a charged spacecraft with plasma.

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87

equilibrium with the surrounding plasma, in high orbits the body can accumulate a large absolute charge, which can be used for contactless transportation. The effects of charge storage by a spacecraft and the ability to control the charge using special devices have been demonstrated experimentally in the frameworks of missions SCATHA, ISEE, Geotail, and Cluster (Riedler et al., 1997; Torkar et al., 2016). Modeling of a body electrostatic charging.An important approach in the study of the behavior of bodies under the influence of Coulomb force is the multisphere method. It allows taking into account of the influence of a body shape on the charge distribution. The complete body electrostatic charging model is represented as a collection of spherical conductors dispersed through the body to provide induced charging effects (Fig. 2.22). This approximate analytical method was developed by Stevenson and Schaub (2013a). To determine the radius and location of charged spheres that simulate the charge of a particular body, they proposed using the finite element model obtained in Ansoft Maxwell 3D software. The parameter selection algorithm was described in detail by Stevenson and Schaub (2013a). An alternative approach involves placing the spheres uniformly inside the object and optimizing their radius only (Stevenson and Schaub, 2013b). Once the location and parameters of the spheres are determined, and this model can be used to analyze the attitude motion of the charged body in the electrostatic field of another body. From a mathematical point of view, this method is a simplified version of the boundary element method (CA, 1978). The multisphere method involves significantly lower computational costs than the finite element method. In addition, the method allows an analytical analysis of the equations of motion to find approximate analytical solutions (Aslanov, 2017). To illustrate the core idea of the method, let us consider a simplified case when a spacecraft is modeled by one sphere, and a space debris object is a set of m  1 spheres. According to the multisphere method, the ith body voltage can be written as. φi ¼ k c

qi + Ri

m X j¼1, j6¼i

kc

qj r i,j

Fig. 2.22 Charged bodied are represented as a collection of spherical conductors.

(2.3)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

where i ¼ 1 corresponds to the spacecraft, i ¼ 2 is the space debris object, kc ¼ 8.99  Nm2/C2 is the Coulomb’s constant, qi is the charge of the ith sphere, Ri is the radius of the ith sphere, and rij is the distance between the centers of the ith and jth spheres. Expression (2.3) can be represented in matrix form: 3 2 φ1 1=R1 6 7 6 6 φ2 7 6 1=r 2,1 6 7 6 6 ⋮ 7 ¼ kc 6 ⋮ 6 7 6 6 7 6 4 φ2 5 4 1=r m1,1 φ2 1=r m,1 2

1=r 1,2 1=R2

⋯ ⋱

1=r 1,m1 ⋯

⋱ ⋯

⋱ ⋯

⋯ 1=Rm1

⋯

⋯

1=r m,m1

32 3 q1 1=r 1,m 76 7 ⋮ 76 q2 7 76 7 76 ⋮ 7 ⋮ 76 7 76 7 1=r m1,m 54 qm1 5 1=Rm

(2.4)

qm

With known potentials φ1 and φ2 provided by the active spacecraft’s charge control system, the charges of the spheres can be found as a solution to the matrix Eq. (2.4). The distances between the centers of the spheres belonging to the same body are constants. The method assumes that Coulomb electrostatic force acts between the spheres. After the charges are determined, the total force and torque acting on space debris from the active spacecraft can be found as. F ¼ k c q1

m X

kc

j¼2

L O ¼ k c q1

m X j¼2

qj rj,1 r 3j,1

kc

 qj  r1,j  rO  rj,1 3 r j,1

(2.5)

(2.6)

where ro is the vector connecting the center of the first sphere, modulating an active spacecraft, and the origin O, relative to which the torque is calculated. The described technique for calculating forces and torque can be easily extended to the case when the first body, like the second, is represented by a set of spheres. The possibility of using the multisphere method to simulate the charge of flexible bodies was considered by Maxwell et al. (2020). Comparison of the analytical model with the experimental results showed that the method describes well time-varying shapes of pure conductors and can be used to simulate the charge of deployed solar arrays of a spacecraft. When trying to simulate thin mixed dielectric-conductor structures, the method showed large discrepancies with the experimental results. When a thin aluminized Mylar strip moves in an electrostatic field, the effects of dielectric polarization and self-emission charges that are not taken into account by the method played an important role. An example of using the multisphere method for studying the dynamics of space debris with attached solar panels can be found in the work by Aslanov (2018). It shows that elastic oscillations of the solar panels during contactless electrostatic transportation can lead to chaotic behavior of the space debris object. In addition to the described method, when preparing and analyzing space missions, software packages are used. Descriptions of some of these have been given by King and Parker (2002) and Novikov et al. (2016). The main disadvantages of these

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packages are high computational costs and long calculation time, which preclude their use in onboard control systems of spacecraft. Space debris dynamics and control. The space debris removal concept based on the use of Coulomb force for the contactless transportation of space debris in GEO was first proposed in the GLiDeR project, and involved the use of a pulling scheme. Calculations show that the milli-Newton force level provided by 10 kV of absolute charge is sufficient to raise the orbit of a 1000 kg space debris object by 300 km within several months. Simulations took into account solar radiation pressure. It is shown that due to its cyclical nature, the influence of solar radiation on the transportation process can be neglected (Schaub and Moorer, 2012). An autonomous relative-motion-control algorithm for an active spacecraft thruster was developed by Hogan and Schaub (2013), placing moving passive spacecraft into a desired relative position using electrostatic attractive force. Dynamics and stability of two charged spacecraft formation flying taking into account exponential decrease in Coulomb force in plasma with increasing the ratio of the relative distance over the Debye length were studied by Yamamoto and Yamakawa (2008). In these works, the motion of the transported object relative to its center of mass is not considered. The dynamics of a charged spacecraft attitude motion in the case of onedimensional rotation were studied by Schaub and Stevenson (2013). It was shown that the equilibrium position of a cylindrical object, whose charge was modeled using the multisphere method, depends on whether the system is in a pull or a push configuration. A control law for an active spacecraft potential, aimed at arresting the rotation of a passive object, was developed using Lyapunov theory. It was assumed that the active spacecraft and space debris are at a short distance of 3–4 radii of the spacecraft (Schaub and Stevenson, 2013). This control law was generalized to the three-dimensional case of an axisymmetric cylindrical object by Bennett and Schaub (2015). The key assumption of this work is that angular motion occurs under the action of an electrostatic moment only. In a study by Bennett and Schaub (2018), the assumptions that the charged objects are at a small distance from each other, the influence of gravitational forces and gravity gradient are negligible, and the constant relative position of the space debris object and the active spacecraft is provided by its control system are collectively called the “deep space scenario.” The paper demonstrated the possibility of using the charge control law developed for the deep space scenario to solve the problem of space debris detumbling in GEO using a closed-loop thrusting controller to provide constant separation distance. The study of the dynamics of space debris during its contactless pushing by an active spacecraft has been the subject of works by Professor Aslanov. A feedback control law that stabilizes the relative motion of an active spacecraft and space debris using Coulomb force was proposed by Aslanov and Yudintsev (2018). In a study by Aslanov and Schaub (2019), a family of control laws for the active spacecraft charge, ensuring detumbling of space debris during electrostatic transportation, was proposed and investigated. Aslanov (2019) proposed a feedback control laws for active spacecraft thrust and its charge for the contactless transportation and detumbling of an axisymmetric space debris object in 3D case. The possibility of chaotizing the attitude motion of a space debris object with solar panels was

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

demonstrated by Aslanov (2018). For the push configuration, equations of spatial attitude motion in the canonical form were provided by Aslanov (2017). Exact analytical solutions were obtained using Jacobi elliptic functions, in the case where the distance between the space debris and the tug remains unchanged. If the distance and charge voltage change slowly over time, adiabatic invariants are found in terms of the complete elliptic integrals. To use the developed control laws, it is necessary to know the charges of an active spacecraft and space debris. Determination of the spacecraft charge can be carried out by onboard sensors. Wilson et al. (2020) considered two methods for the remote determination of the space debris object charge, based on observing X-rays and spectra of emitted electrons. Series of vacuum chamber experiments were performed. The study showed that the simultaneous use of both methods improves the measurement accuracy.

2.3.3 Transportation by lasers Contactless transportation by means of lasers can be based on the physical phenomenon of radiation pressure or on the phenomenon of ablation. Let us estimate the forces generated by laser radiation on the surface of space debris. Photon momentum transfer. The idea of using light pressure for propulsion in space has been independently expressed by Tsander (1924), Tsiolkovsky (1926), and Oberth (1923). Eugen S€anger proposed using a nuclear-pumped gas laser to create radiation pressure for his photon rocket concept (S€angeru, 1959). Basic equations for nonchemical propulsion were provided by Moeckel (1975). In 2010, the Japan Aerospace Exploration Agency launched the IKAROS spacecraft to Venus, which is the first interplanetary vehicle equipped with a solar sail (Tsuda et al., 2011). Photons have no own mass, but it is a well-known fact that they can carry momentum. Reaching the surface of the body, the photons knock out the electrons from the substance, which leads to the appearance of the momentum. Photons carry momentum in the direction of their motion. The resultant force of a laser beam due to the photons momentum transfer can be found as follows: F¼

Cr c

Z I ðx, yÞdA

(2.7)

where Cr is the radiation pressure coefficient, the surface integral is taken over the illuminated area, c is the speed of light in vacuum, and I is the intensity of the radiation. The radiation pressure coefficient can take values from 0 to 2. For a translucent object, Cr ¼ 0, for a black body absorbing all photons, Cr ¼ 1, and for a flat mirror reflecting all photons, Cr ¼ 2. The intensity distribution at the surface point with coordinates x, y depends on the parameters of the laser and the medium through which the laser beam passes. In the simplest case of axisymmetric beam with Gaussian distribution, the radiation intensity can be found as follows (Siegman, 1986):

Space debris problem

I ðL, r Þ ¼ I 0 e2r

91

2

=wðLÞ2

(2.8)

where r is the distance from the laser beam axis to the considered point on the surface, I0 is the maximum intensity of the laser beam on the surface, and w is the width of the beam, which is the radius where the intensity is I0e2. The width w is a function of a distance L from the laser to the surface. It depends on the laser parameters, its optics, and the characteristics of the medium. The lower limit of the width for an ideal laser and vacuum is determined by the expression. w min ðLÞ ¼

λL D

(2.9)

where λ is the laser’s wavelength and D is the diameter of the focusing optic (Siegman, 1986). In the case of a ground-based laser, the beam propagates through the atmosphere. This has a great influence on the parameters of the photon beam and the force generated. The beam width changes significantly, and the energy decreases due to absorption and scattering by the atmosphere. The extended Gaussian model can be used to account for the effect of the atmosphere. This model assumes that the beam width can be found as the product of the minimum width wmin(L) and a beam propagation factor (Mason et al., 2011): 2Ssum ðLÞr 2 2P I ðL, r Þ ¼ Ssum ðLÞτtr ðLÞ exp  πw min ðLÞ2 w min ðLÞ2

! (2.10)

where Ssum is the Strehl factor, τ is the atmospheric transmittance, and P is the output power of the laser. The Strehl factor is the ratio of the real radiation intensity to the ideal intensity: Ssum ¼

I real : I ideal

(2.11)

It takes into account all the factors influencing the decrease in the real intensity due to the increase in the width of the laser beam. The atmospheric transmittance can be determined using the Beer-Lambert law: Z τtr ¼ e



L αðzÞdz

0

(2.12)

where α(z) is the local absorption coefficient, which is a cumulative factor. The factors Ssum and τtr depend on the atmospheric path, which changes as space debris moves across the sky. Therefore, when simulating the motion of space debris under the action of a laser beam, these coefficients must be recalculated at each integration step. More

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

detailed information on this subject can be found in studies by Stupl and Neuneck (2010, Appendix A) and Mason et al. (2011). Laser ablation. The idea of using laser ablation for propulsion was first proposed by Kantrowitz (1972). Laser ablation propulsion involves irradiating the body with an intense laser beam that melts the surface, the particles of which produce a jet of vapor or plasma. Claude Phipps made a huge contribution to the development of this direction, having published a large number of works with theoretical and experimental results. The ablation effect can be used to create laser thrusters. Flight tests of the Lightcraft engine, which was the first functioning laser-powered engine, were carried out in 1999–2000 (Myrabo, 2003). According to the estimates given by Phipps et al. (2010), ablation allows a force to be generated on the body surface that is four to six orders of magnitude greater than the force of pure photon action. The ablation phenomenon can be used to create space engines, and examples can be found in works by Phipps and Luke (2002) and Zhang et al. (2016). Pulsed lasers are more efficient than continuous wave ones when using the ablation effect (Phipps, 2014). The force generated by a pulsed laser can be calculated as follows: F¼η

Cm AΦ τ

(2.13)

where η is the impulse transfer efficiency factor, Cm is the momentum coupling coefficient, which is an indicator of the quality of the laser ablation, A is the area of the laser spot, Φ is the on-target laser fluence, and τ is the pulse duration. A pulse duration is related to the intensity of the irradiation I as Φ ¼ Iτ. The momentum coupling coefficient for a pulsed laser is defined as the ratio of impulse density to the laser fluence: Cm ¼

μV e Φ

(2.14)

where μ is the target areal mass density and Ve is the material exhaust velocity. Numerous experiments show that the coupling coefficient depends on the intensity of irradiation. It has a maximum located near the threshold between vaporization and a plasma generation regime (Phipps et al., 2017). The decrease in Cm upon increasing the intensity is due to the fact that most of the energy is spent not on vaporization, but on the creation and acceleration of the plasma. Phipps et al. (2010) offer a summary of the laser ablation propulsion theory and subtleties of analytical calculating momentum coupling in various ablation regimes. The fundamental difference between the force generated by laser ablation and light pressure is that in the first case, the force is directed perpendicular to the surface of the object, and in the second case, the force is directed along the laser beam mainly, although there are off-beam components due to diffuse and specular reflection. Liedahl et al. (2013) studied the influence of the body shape on the magnitude and direction of the force generated by a laser beam. They proposed a technique of linear momentum calculation for the case when generated force is not parallel to the laser

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beam. The surface material plays an important role in the ablation process. In addition, laboratory tests (Loktionov et al., 2021) have shown that irradiation of glass-covered solar cells by laser pulses can lead to the ejection of a large number of small glass fragments. Thus, when deorbiting large space debris by a laser, the beam must be focused on the metal parts and avoid hitting the solar panels. This creates additional difficulties for creating laser-based active large space debris removal systems. It makes sense to develop laser active large space debris removal systems only on the basis of the use of the ablation effect, since pure photon pressure generates very weak force. It should be noted that laser-based space debris removal systems are effective primarily for removing medium-sized space debris of 1–10 cm. Lasers can operate at distances of hundreds of kilometers, which makes it possible to use one spacecraft to remove various targets. In this case, the spacecraft does not spend fuel for the flight from one target to another. Space debris removal projects. Papers by Phipps (2014) and Phipps and Bonnal (2016) considered large space debris removal from GEO and LEO by an ultraviolet L’ADROIT laser system mounted on an active spacecraft. This system includes two telescopes. One of them is used for target detection. It consists of two conical mirrors and a wide field of view passive sensor, which uses sunlight for target acquisition. The second telescope has a narrow field of view and it is used for laser operations. This telescope tracks the target, receives response signals from it, focuses on these, and produces repetitive shots. The telescope includes a zoom lens and two mirrors having the off-axis Cassegrain design. To analyze the returned signal, a 9M pixel detector is used. Phipps and Bonnal, (2016) proposed two schemes for space debris removal for the GEO. The first one assumes the removal of a derelict satellite into a disposal orbit using an active spacecraft, which follows the target space debris object and then returns to GEO. The second scheme involves the use of two active spacecraft that move in one elliptical orbit but are at an angular distance of 180 degrees from each other. The researchers calculated the fuel costs of the propulsion system and the required power for laser and electrical engines of the spacecraft. In the case of using the first scheme with one spacecraft, the transfer of a 3 ton object from GEO to a disposal orbit, which is 300 km higher, will require 14 h of laser illumination and 1.3 kg of fuel for the electrodynamic engines. The required ΔV is 11 m/s, and the average power required for the entire maneuver including the return phase to GEO is 25 kW. If the cleaning is carried out from a weakly elliptical orbit with an inclination of 15 degrees, a similar maneuver will require 75 kg of fuel and 25 days of operation of the system at ΔV ¼ 780 m/s. According to the calculations of Phipps and Bonnal, to clean 10 spacecraft, 2 of which are in orbit with an inclination of 15 degrees, an active spacecraft equipped with a L’ADROIT laser and solar panels providing 36 kW power and carrying 320 kg of fuel is required. In the case of using the second scheme with two spacecraft in an elliptical orbit, the fuel is required only for their reorientation. It will take 75 days to remove a 3 ton object from GEO using this method. Moreover, since the elliptical orbit itself rotates with a period of about 1 month, several objects can be transported in parallel in this way. The L’ADROIT laser system can also be used in low Earth orbit (Phipps, 2014) for

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medium-sized space debris removal. The system is designed for single shot re-entry, as tracking a small object after impact is very difficult. In the case of interaction with large space debris, prolonged exposure is required, since the force generated by ablation is small. For example, to reduce the height of the orbit of space debris with 1 ton mass by 40 km, 625 interactions of a 16 kW average power laser are required. The mass of the spacecraft with the L’ADROIT system is estimated at 10,000 kg. The power supply for a spacecraft with an onboard pulsed laser was discussed by Avdeev et al. (2019). For diode-pumped solid-state laser with 25 kW optical power and 1 GW peak power, 10 kW power solar cell panels with area 25 m2 were combined with an energy storage system. Another alternative is a thermionic nuclear power plant, which produces 40 kW of electrical power and has a mass of 4.4 tons. The use of a space-based nanosecond pulse laser to remove medium-sized space debris in low Earth orbit was discussed by Fang and Pan (2019). ANSYS/LS-DYNA finite element software was used to simulate the laser-induced plasma shock wave propagation in the space debris material. Equations describing the change in the radius of perigee and apogee, local orbit inclination, and true anomaly angle of space debris orbit under the action of orbital pulse laser irradiation were provided. The influence of the parameters of the space debris orbit on the impulse coupling coefficient was investigated. The International Coherent Amplification Network (ICAN) project, aimed at tracking and removal of space debris, was proposed by Soulard et al. (2014). It is based on fiber diode-pumped laser architecture. Its laser impulse repetition rate and average power are sufficient for medium-sized space debris removal. An array of thousands of phase-combined fibers provides excellent beam control. Solar panels are used to power the laser and other spacecraft systems. The optical system of the spacecraft consists of two mirrors, which provide the required expansion of the laser beam for its transmission over a long distance and focusing on the target space debris object (Fig. 2.23). This optical system also operates in tracking mode, collecting reflected laser light. The design features of the laser system make it possible to assess the surface of space debris and adjust the impact, taking into account the spatial orientation of the space debris surface and its type. The spacecraft can operate in scanning, tracking, or shooting mode. In the scanning mode, the search for space debris in the surrounding space is carried out at a distance of more than 100 km. In the tracking mode, when the target object is determined, the object is diagnosed in order to determine its characteristics and velocity vector parameters. In the shooting mode, focusing on the object and its irradiation with fluence, which is selected taking into account the characteristics of the object, is performed. The scanning and tracking modes involve mechanical movement of the mirrors and the use of a mode lens, which is retracted in the shooting

Fig. 2.23 ICAN spacecraft.

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mode. The mirrors and fiber array sizes determine the focal range where the energy is optimally distributed on the object, and the boundaries of the tracking and shooting area. According to the authors of the project, the mass of a spacecraft equipped with a 1 kW ICAN system with 10,000 fibers, operating with 3.33 kW input power at 30% efficiency, is about 2.4 tons. An article by Schmitz et al. (2015) examined how effective the placement of a laser equipped spacecraft in orbit will be in terms of the magnitude of removed mediumsized space debris objects. To estimate their quantity, data on all existing objects ranging in size from 1 to 10 cm was taken from the ESA MASTER-2009 model (Flegel et al., 2009), and then objects that for some reasons cannot be removed from orbit using a laser were gradually cut off. Fig. 2.24 shows the process of clipping “unsuitable” objects schematically. The main input parameters that determine the number of deorbited space debris objects are the parameters of the laser-equipped spacecraft’s orbit, mission duration, laser operating range Rmax, laser system tracking velocity and pointing accuracy, momentum coupling coefficient Cm, and efficiency factor, which is taken very roughly and takes into account the rotation and orientation of the object during the ablation process. Based on the analysis of the density of medium-sized space debris at different mean orbital altitudes, a polar orbit with low eccentricity and a mean altitude of 871 km is selected to host a space-borne laser. In the first stage of unsuitable objects clipping, all objects beyond the limits of low Earth orbit are excluded from consideration. During the second stage, space debris objects that do not cross the layer of thickness Rmax around the spacecraft altitude are removed. In the third stage, objects that during the mission do not approach the spacecraft at the distance Rmax are excluded from consideration. During the fourth stage, objects whose perigee radius are less than 200 km are excluded, since they themselves will leave orbit due to the atmosphere influence. In the fifth stage, objects that during the mission are never in a position that allows lowering their orbit are removed. In other words, at the moments when these objects are near the spacecraft, they are so located relative to it that the laser action raises rather than lowers their orbit. In the sixth stage, objects with a velocity exceeding the maximum tracking velocity of the laser system are excluded from consideration. Finally, in the seventh stage, objects, the laser action on which during the mission turns out to be insufficient

Fig. 2.24 Stages of the mission-specific performance model.

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to remove them from orbit, are excluded. The remaining objects can be deorbited using the considered space-borne laser. During the computer simulation, it was assumed that the spacecraft was equipped with an Nd:YAG laser with 1064 nm wavelength, 10 ns pulse width, a pulse energy of 372 J, and a beam quality of 12. The laser output power is 372 kW. The laser operating range is 20 km, and a maximum tracking velocity is 15 deg/s. The spacecraft’s electrical power system should provide peak powers of at least 100 kW for few seconds. The requirements for the electrical power supply system very much depend on the design and parameters of the laser used. Calculations show that over 10 years of operation, the orbital laser will reduce the density of space debris at the most critical region of LEO by 23%. The system will fire about 140,000 shots, the average time between which is about 30–40 min. Space lasers can also be used for contactless space debris detumbling. An article by Vetrisano et al. (2015) proposed a control scheme for large space debris object detumbling, in which the control torque vector was applied in the direction opposite to the instantaneous angular velocity vector to stop the object’s rotation. Since the force generated by ablation is directed perpendicular to the local normal of the surface where the spotlight is located, the set of directions of the generated control torque vector available at a given moment is limited by the peculiarities of the shape of the space debris object. The authors proposed to select a point on the surface in such a way that the misalignment of the generated torque with respect to the angular velocity will be the minimum. Determination of the angular velocity of rotation of an object is carried out by tracking several points on the object’s surface. The results of numerical simulations have shown that the shape of the space debris object is of great importance for detumbling process. For a space object with mass parameters close to the Envisat satellite and rotating with an angular velocity of magnitude 3.5 deg/s, the use of control in the case of a cylindrical shape reduces the angular velocity to 1 rad/s in 14 days. In the case of a parallelepiped shape, the angular velocity decreases to zero in less than 2 days. To implement contactless action, an 860 W laser is used. It generates a force of about 0.017 N. The satellite is located at a distance of 50 m from the target object. Kumar and Sedwick (2015) considered space debris as a three-dimensional surface mesh and proposed an algorithm for choosing a face on this mesh, where the laser action should be directed to solve the detumbling problem. The face, the irradiation of which provides the greatest change in angular momentum, is selected at each step of integration of the equations of motion.

2.3.4 Transportation using gravity fields An original method of space debris transportation in GEO by heavy spacecraft (chaser or orbital collector), which is based on the nature of the libration motion in the threebody problem, was proposed by Aslanov (2019a). In GEO conditions, where the Earth’s gravitational force is balanced by the centrifugal force of inertia, a small gravitational attraction force between a relatively light space debris object and a heavy chaser may be sufficient for contactless transportation. From the point of view of

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classical mechanics, the problem is similar to the Earth-Moon three-body problem, where the chaser plays the role of a Moon. The basic idea is to get the chaser close enough to space debris to form the Earthchaser-debris system. Space debris should be inside the Hill sphere of the chaser, where the chaser attraction dominates (Fig. 2.25). In this configuration of the three-body system, the space debris object will never leave the area bounded by the Hill sphere. Further, various scenarios are possible. Either space debris can be caught by the chaser and becomes part of it, increasing its mass and Hill sphere radius, or the space debris can stay inside the sphere and move around the chaser. The numerical simulation has shown that the gravity of the Moon has little effect on the relative motion of the chaser-debris system, since this gravity causes additional acceleration for both the space debris object and the chaser, and these accelerations are close to each other. Chaser’s mass estimation. The Hill radius is given by a well-known formula, which can be found, for example, in Schaub and Junkins (2003):  rh ¼ r

m2 3ðm1 + m2 Þ

1=3 (2.15)

where r is the distance between the Earth and chaser, m1 is the mass of the Earth, and m2 is the mass of the chaser. From Eq. (2.15) it follows that, to capture small debris, it is enough to have the chaser’s mass in the order of 10 tons. In this case, the Hill sphere radius is 3.5 m. To capture large space debris, the chaser’s mass should be 100 tons or more. In this case, the Hill sphere radius is 7.5 m or more. The idea of building such a

Fig. 2.25 Chaser’s Hill sphere.

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heavy chaser does not seem more fantastic than building a space elevator that will have a length of about 100,000 km and a mass of more than 10,000 tons. Moreover, in the future, the role of the chaser can be performed by a relatively small asteroid delivered to GEO. Space debris removal scenario. This scenario has been discussed by Aslanov (2019b). It involves three types of consecutive maneuvers: capturing of debris into an area bounded by the Hill sphere of the chaser, towing, and discharging debris in the graveyard orbit (Fig. 2.26). A mathematical model to simulate the planar motion of the chaser, which is called a heavy orbital collector in this study, relative to the Earth has been developed to substantiate and to study these maneuvers. The shape of the collector section in the plane of motion of the Earth-collector system is assumed as an inelastic disk. The proposed mathematical model allows, in the first place, in addition to the centrifugal forces of inertia, for Coriolis forces and gravitational forces to be taken into account, along with a thrust force of the collector, and, secondly, for the impact of small debris objects with the collector to be considered according to the laws of inelastic impact theory. It is shown that all of these maneuvers are completely realizable. Values of the collector’s thrust force, at which the debris can be captured and towed or discharged in the graveyard orbit, are determined. All basic ideas of the proposed approach are confirmed by the direct numerical integration of the motion equations. The requirements for the collector configuration and its attitude motion are formulated, which provide for the capture, towing, and discharging of space debris.

2.3.5 Contactless detumbling using eddy currents A description of contactless transportation systems would be incomplete without mentioning systems based on electromagnetic interaction. The shells of most satellites and rocket stages are made of nonmagnetic materials such as aluminum or titanium alloy. This makes it impossible to use magnetic force to grip them or to perform contactless transportation. However, since the shells can conduct an electric current, it is possible to use a different electromagnetic effect to solve these issues. It is known that closed loops of electrical currents, called eddy currents or Foucault’s currents, arise inside a conductive surface in a changing magnetic field. These currents create their own magnetic field that resists a change in the initial field. As a result, a force, which is caused Fig. 2.26 Three types of the maneuvers of the debris removal mission: the capture, towing, and discharging of the debris in the graveyard orbit.

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Fig. 2.27 Eddy current.

by interaction between the eddy current and magnetic field, is generated on the surface (Fig. 2.27). This force is used, for example, in electric brakes for trains. The braking force can be used for contactless impact on a space debris object. The idea of using a magnetic field to despin a satellite was proposed by Kadaba and Naishadham (1995). The authors considered two schemes and estimated the parameters of the corresponding despin systems. The first one involves placing a conducting ring around a space object. The second scheme is based on the use of a permanent magnet or U-shaped electromagnet, which is brought to the surface of a satellite by a robotic arm. A detumbling strategy for uncontrollable satellite was proposed by Sugai et al. (2013). It is based on using two coils of the eddy current brake located at the ends of robotic manipulators. An experimental laboratory study to determine the parameters of the eddy current braking system was carried out. The obtained parameters were used to perform numerical simulation of the detumbling operation. The proposed solution assumes the location of the electromagnetic coil at a small distance from the satellite surface (5 mm), since a magnetic field decays very quickly. This is a very significant drawback, since it actually negates the advantages of contactless methods. An interesting design of end-effector based on a rotating array of differently oriented permanent magnets was proposed by Huang et al. (2018). The effector (Halbach rotor) generates a rotating magnetic field that creates eddy currents on the surface of space debris. A force of 100 mN can be created at a distance of about 10 cm from the surface. This effector is robust for nonsymmetrical space debris object detumbling. A study by Gomez and Walker (2017) considered an effective eddy brake system, based on a high-temperature superconducting wires coil. It is capable of effectively impacting a space debris object at a distance of 10 m. The article developed a 3D mathematical model that uses magnetic tensor theory to explain interactions between active spacecraft and space debris. A control strategy for space debris detumbling not sensitive to errors and delays in sensors and actuators operations was proposed. Yu et al. (2021) proposed using two small satellites equipped with high-temperature superconducting coils to generate a control detumbling torque during formation flight.

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Ion beam physics 3.1

3

Mathematical modeling of an in-beam

The study of a space object’s dynamics during its transportation by an ion beam is impossible without considering some issues related to the propagation of the plasma plume, which is generated by an active spacecraft’s electric thruster, and its interaction with the surface of the object. This topic is very extensive and covers various mechanical, electrical, and chemical phenomena. The plasma emitted by a thruster makes up the ion beam and consists of ions with velocities of up to several tens of km/s and electrons with an energy of a few eV. High-energy ejected plasma plume physics has been studied by many scientists. Mathematical modeling and computer simulations of the propagation process have been investigated in various works (Cichocki et al., 2014; Merino et al., 2011; Perales-Diaz et al., 2021; VanGilder et al., 1999). To date, many laboratory studies have been carried out (Giono et al., 2017; Nakles et al., 2007; Zhang et al., 2021), and several space experiments have been performed (Gabdullin et al., 2008; Hilgers et al., 2006). Papers (Cichocki et al., 2017; Korsun et al., 2004; Shang et al., 2019) are devoted to the study of various aspects of the interaction of the plasma plume with nearby objects. The method for the approximate calculation of forces and moments generated by a plasma jet on a solid body surface is described by Alpatov et al. (2016). This book does not seek to give a deep and detailed description of all these plasma physics study directions. Its purpose is to introduce the reader to some basic concepts, models, and calculation equations related to the interaction of plasma with the surface of a space object and used at various stages of modeling to study the dynamics of an active spacecraft and a passive object during its contactless transportation by an ion beam. The plasma plume emitted by an electric thruster can be conditionally divided into a near and a far region. In the near region, the propagation of the plasma is significantly affected by the electromagnetic field of the thruster and the radiation of the cathode. In this region, the plasma inhomogeneity is potentially greater than in the far region, where the influence of the electric thruster and its fields on the plasma propagation is negligible. In the far region, plasma propagation is determined only by the parameters of the plasma itself. Usually, the dimensions of the near region are several diameters of the engine nozzle—that is, it does not exceed a meter. Near and far regions, as well as some plume parameters, are shown in Fig. 3.1. Simulation of plasma propagation in the near region is a very difficult task due to the coexistence in this area of a number of different physical phenomena that are difficult to take into account and simulate. In particular, residual electric and magnetic fields of the thruster, collisions with a large number of neutrals, three-dimensional inhomogeneity caused by the imperfection of the shape of the thruster nozzle, and collisionless shocks near the thruster centerline have a great influence in this area (Beal et al., 2005). In most of the existing studies of the near region of a plasma plume, an Attitude Dynamics and Control of Space Debris During Ion Beam Transportation. https://doi.org/10.1016/B978-0-323-99299-2.00006-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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Fig. 3.1 Plasma propagation scheme with regions.

empirical approach is used. For example, a robe array diagnostic platform consisting of 67 Faraday probes was recently used to investigate the full three-dimensional morphology of the plume’s current density from the near region to the far region of an electric propulsion stationary thruster, its divergence angle, and thrust vector angle (Zhang et al., 2021). The study of the near region is necessary to assess the thrust characteristics of the engines, as well as to analyze the interaction of the jet of an electric thruster with the sensitive equipment of the spacecraft. In contrast to the near region, the propagation of plasma in the far region can be described by relatively simple mathematical models. Research in this area ranges from an analytical description of the basic properties of plasma and flow to complete numerical modeling of more complex aspects of the plume (Boyd and Dressler, 2002; Merino et al., 2015). Experimental data on the plasma plume parameters in the far region is practically absent due to the limited size of vacuum chambers in laboratories and the complexity of experimental research in space. For the considered problem of contactless transportation of a passive object by an ion beam, it is of interest to model the plasma plume in the far region, although studies in the near region are also important, since the features of the flow propagation in the near field determine the flow parameters necessary for modeling at the boundary of the far region. Complex physical phenomena occurring in the near region determine the divergence angle of the ion beam α0 (Fig. 3.1); this is an important characteristic of the beam and it is used in simplified models of particles propagation. It is assumed that the engine nozzle has a circular cross section. Plasma spreads inside a certain cone. The divergence angle is understood as the half-angle of the cone, within which 95% of the flow particles are enclosed. It is known that the divergence angle of the plasma flow increases with increasing distance from the engine nozzle to the measurement plane (Takegahara et al., 1993). That is, the propagation of the plasma flow is not strictly conical. In addition, according to the experimental data obtained by Zhang et al. (2021), the flow generated by a circular nozzle cannot be axisymmetric and its cross section can have the shape of an ellipse. Contactless transportation involves the placement of an active spacecraft at a distance of 5–15 m from the transported object to decrease the probability of their

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collision and reduce the impact of the particle flux sputtered by the ion beam from the surface of the object. In this case, the object is in a far region of the ion plume. There are several models for far-field plasma propagation, which can be used to calculate the force impact of an ion beam on an object. These models can be divided into three groups: simple point source propagation models, self-similar models, and full models. They differ in the complexity and accuracy of the obtained results. The simplest plasma propagation models are the point source propagation models. These models can only be used for rough estimates, since the results obtained with their help are in rather poor agreement with experimental data (Korsun et al., 1999). The accuracy of these models decreases with increasing θ angle and moving away from the flow axis. Nevertheless, these models can be useful due to their simplicity and clarity. Various formulas can be used to describe the plasma density distribution n in a plasma flow as a function of polar coordinates r, θ defining a point in space relative to a point source. Exact solutions of plasma density and velocity distributions for the cases of free-molecular and nearly free-molecular diffusion flow in vacuum were obtained by Narasimha (1962): n¼


 VN v M2 γ 3 2 sin cos θ exp  θ , 2 πr 2

u ¼ u0 cos 2 θ,

(3.1)

where V N ¼ ddNt is the particles’ flow rate, N is the number of particles, v and γ are dimensionless parameters that are determined experimentally, u is the flow velocity, u0 is the particles’ velocity at the origin, and M is the flow Mach number. An even simpler equation for flow density was proposed by Roberts (1964): n¼

VN v cos k θ, πr 2 u0

(3.2)

where k is the hypersonic parameter. Another approximation can be found in Carney (1988): n¼

n0 exp ððλð1  cos θÞm Þ , r 2 cos 2 θ

(3.3)

where n0 is the particle density at the beginning of the far regions, and λ, m are the empirical parameters. Eqs. (3.1)–(3.3) are given by Korsun et al. (1999), where a comparison of these formulas with experimental data and a self-similar plume propagation model can also be found. The inaccuracy of the considered models for large θ angle is mainly due to the residual pressure inside the plasma. Self-similar models of plasma propagation are more complex. Within the framework of these models, plasma profiles are found analytically based on a set of initial hypotheses. An analytical self-similarity function h, which determines the flow parameters depending on the coordinates of the considered point, can be written. These models make it possible to simulate correctly the expansion of the plasma plume due to thermal effects and to take into account the influence of the thruster

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parameters (Korsun and Tverdokhlebova, 1997; Merino et al., 2015). To obtain a solution to the equations of fluid flow, additional simplifying assumptions are made. These assumptions introduce small errors in the case of flows with large Mach numbers. The results of self-similar models are in good agreement with experimental data, especially for points located near the ion flow axis (Bombardelli et al., 2011; Korsun et al., 2005). Self-similar models are discussed in the next section of this chapter. The group of full numerical models can include a wide range of models, ranging from the integration of simplified equations of fluid flow to hybrid modeling (Gallimore, 2001; Mikellides et al., 2002). Compared to simple and self-similar models, full models require much more computational resources and time for calculations; therefore, at the current stage of technical development, they cannot be used in onboard control systems of spacecraft. They are currently used to study the interaction of plasma with spacecraft surfaces. The report by Bombardelli et al. (2011) notes that the use of self-similar models is preferable to calculate the force effect of an ion beam on the surface of a space object. On the one hand, with greater accuracy, these models have the same computational complexity as simple point source conical plasma propagation models. On the other hand, complete models provide a disproportionately small contribution to the accuracy relative to the required computational effort increase.

3.2

Simplified ion beam models

This section describes several self-similar plasma propagation models in vacuum, taking into account the influence of the main parameters of the thruster and the residual plasma pressure. It is assumed that plasma consists of single-charged cold ions and hot electrons. The ion pressure pi is many times less than the electron pressure pe. It is supposed that that there are no collisions between plasma particles. Magnetic effects are negligible. Plasma is considered as quasineutral—that is, the concentration of ions and electrons in any small volume coincides ni ¼ ne ¼ n. The flow continuity equations   ∂ρj + r  ρj uj ¼ 0, ∂t

ðj ¼ i, eÞ

(3.4)

for the steady-state case can be rewritten in the form   r  n uj ¼ 0,

(3.5)

where ρj ¼ mjn is the jth particle density, mi is me is the electron mass, uj  the ion mass, 

is the jth particle velocity vector, and r ¼

∂ ∂ ∂ ∂x , ∂y , ∂z

is the nabla operator. The ion

momentum equation can be written as nmi ðui  rÞui ¼ enrΦ,

(3.6)

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117

where e is the electron charge and Φ is the ambipolar electric potential. The electron momentum equation in the case of neglecting electron inertia (me ! 0) can be written as r  Pe  enrΦ ¼ 0,

(3.7)

where Pe is the electron pressure tensor, which can be approximated as a diagonal tensor r  Pe ¼ r pe, and pe is the scalar electron pressure. When obtaining this momentum equation, it was assumed that the ion thermal pressure is negligible (pi  0), and the following relations are satisfied (Merino et al., 2015): me u2e , T i ≪T e ≪mi u2i : Taking into account the assumptions made, Eq. (3.7) takes the form rpe  enrΦ ¼ 0:

(3.8)

Let us introduce the cylindrical coordinates (z, r, θ) with z along the ion beam axis (Fig. 3.2). The Cartesian coordinates shown in the figure are related with the cylindrical coordinates by expressions x ¼ r cos θ, y ¼ r sin θ:

(3.9)

In cylindrical coordinates, the velocity vector uj ¼ [uxj, uyj, uzj] has the following components: uxj ¼ urj sin θ  uθj cos θ, uyj ¼ urj cos θ + uθj sin θ,

(3.10)

Fig. 3.2 Ion beam and the cylindrical coordinates (z, r, θ).

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_ A nonrotating _ uzi ¼ z,_ and uθj ¼ r θ. where urj and uθj are shown on Fig. 3.2, urj ¼ r, plasma flow is considered below: uθj ¼ 0:

(3.11)

Taking into account Eqs. (3.9)–(3.11), Eq. (3.5) can be rewritten as       ∂ nurj r ∂ nuθj ∂ nuzj r + + ¼ 0: ∂θ ∂r ∂z

(3.12)

Using the formula for calculating the derivative of the product and dividing both sides of the result by n, Eq. (3.12) can be rewritten as uzi

∂ ln n ∂uzi ∂ ln n 1 ∂ðruri Þ ¼ 0: + + + uri ∂z ∂r r ∂r ∂z

(3.13)

The vector Eq. (3.6) implies two scalar expressions: uzi

∂uzi ∂u e ∂Φ , + uri zi ¼  mi ∂z ∂z ∂r

(3.14)

uzi

∂uri ∂u e ∂Φ , + uri ri ¼  mi ∂r ∂z ∂r

(3.15)

Merino et al. (2015) noted that for the hypersonic plume of ion engines, radial plasma profiles remain similar along the axial direction near the plume axis. This creates the prerequisites for the use of the self-similar assumption for an approximate description of the beam propagation. Self-similar models are based on the assumption that all ion streamlines expand in a similar way. The plume expansion can be described by a certain dimensionless self-similarity function h(ζ): r ðζ, ηÞ ¼ ηR0 hðζ Þ,

(3.16)

where ζ ¼ z/R0 is a dimensionless axial coordinate, R0 is the characteristic length, which is the initial radius of the ion beam containing 95% of the ion current at the border of the far region (z ¼ 0), and η ¼ r(0)/R0 is the initial dimensionless streamline position at z ¼ 0. The plume propagation can be studied in the space of dimensionless coordinates (ζ, η). Time differentiation of Eq. (3.16) yields uri ¼ ηR0

dhðz=R0 Þ z_ ¼ ηh0 uzi : dz

(3.17)

Additional hypotheses are required to close the above equations. Three self-similar models using different hypotheses are discussed below. The notation and methodology for obtaining models adopted in the report (Bombardelli et al., 2011) are used.

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A self-similar model was described by Parks and Katz (1979). The basic assumption of this model is that the electron velocity spectrum is isotropic and Maxwellian. In this case, the electron pressure can be described by the equation pe ¼ nT e0 ,

(3.18)

where Te0 is the electron temperature at the origin. Substitution of Eq. (3.18) into Eq. (3.8), taking into account that 1n rpe ¼ T e0 r ln n, gives rΦ ¼

T e0 ln n: e

(3.19)

It is also supposed that the velocity component uzi is constant at small angles of the streamline deviation from z axis: uzi ¼ u0 ¼ const:

(3.20)

This equation replaces Eq. (3.14). Such an approximation is justified for a supersonic pffiffiffiffiffiffiffiffiffiffiffiffiffiffi beam near the axis, when M ¼ ui mi =T e0 ≫1. The radial velocity component (3.17) can be rewritten as uri ¼ ηh0 u0 :

(3.21)

These assumptions make it possible to find analytical expressions for the particle density profile using Eqs. (3.15) and (3.19). This profile is Gaussian, the spread of which increases with h increasing: n¼


2 n0 η exp , σ2 h2

(3.22)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where n0 is the particle density at the origin and σ ¼ 1= ln ð1  kÞ is the dimensionless variance of the density distribution (k < 1). The variance σ 2  1/3 corresponds to the case when about 95% of the total flow (k ¼ 0.95) is contained in r  R0 at η ¼ 1. From Eqs. (3.13) and (3.15), it follows that d ðh00 hÞ ¼ 0: dζ

(3.23)

This equation can be integrated: h0 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ20 + C2 ln h:

(3.24)

where C2 ¼ 4σ 2M2 0 is the constant, M0 ¼ u0/cs is the Mach number at the origin, pffiffiffiffiffiffiffiffiffiffiffiffiffi cs ¼ T e m1 is the local ion sound velocity, and δ0 ¼ h0 (0) ¼ tan(α0) is the tangent i of the ion beam divergence angle. Eq. (3.24) can also be integrated:

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Zh ζ¼ 1

dω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : C2 ln ω + δ20

(3.25)

It should be noted that for this model, the plasma plume propagation is determined by two parameters: the Mach number M0 and the divergence angle α0 at the origin. A self-similar model was described by Ashkenazy and Fruchtman (2001). For this model, it is assumed that the velocity profile in the initial plane ζ ¼ 0 is determined by the expression u0 uzi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 + r 2 d2

(3.26)

and this velocity is conserved along the ion streamline; d is the distance from the initial plane to the top of the plume cone (Fig. 3.2). Similar to the previous model, Eq. (3.26) is used instead of Eq. (3.14). The expression for the particles’ density in the beam has the form n¼

 2

n0

h 1 + η2 R20 d 2

Rd22σ2 +12

,

(3.27)

0

Eq. (3.23) and solutions in the form of Eqs. (3.24) and (3.25) can also be written for this model. The difference lies in the values of the constants: C2 ¼ (4 + 2σ 2δ20)σ 2M2 0 . The variance σ value can be found as a solution to a nonlinear equation that determines the fraction k of flux Gi inside a cone η  1. kðηÞ ¼

Gi ðηÞ < 1, G i ð∞ Þ

∗  ∗ Rη 2 where Gi η ¼ 2πu0 n0 R0 ηnuzi dη, Gi(∞) ¼ πσ 2R20n0u0, and

0

σ2 ¼ 

  d 2 ln 1 + R20 d2 : ln ð1  kÞ

Within the framework of the assumptions made, as before, it is convenient to take k ¼ 0.95. Compared to the previous model, this model more accurately describes the flow at large angles of deviation from the axis. Korsun and Tverdokhlebova described a self-similar model (1997). In contrast to the previous two models, it was assumed that the thermodynamic behavior of electrons is described by the equation Te ¼

nγ1 T 0e , nγ1 0

(3.28)

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121

where γ is the adiabatic exponent. The case γ ¼ 1 corresponds to the isothermal flow. In the case γ 6¼ 0, the potential Φ gradient can be found from the Boltzmann equation rΦ ¼

γrT e : e ð γ  1Þ

(3.29)

As in previous models, self-similarity implies the fulfillment of Eq. (3.17). Solutions are sought in the following form: uzi ðζ, ηÞ ¼ uc ðζ Þut ðηÞ

(3.30)

T e ðζ, ηÞ ¼ T c ðζ ÞT t ðηÞ

(3.31)

gz ðζ, ηÞ ¼ nuzi ¼

n0 u0 gt ðηÞ h2 ð ζ Þ

(3.32)

where the subscript “c” means the longitudinal direction along the ion beam axis, the subscript “t” means the transverse direction, and ut(0) ¼ Tt(0) ¼ gt(0) ¼ h(0) ¼ 1. Substitution of Eq. (3.30) into Eqs. (3.14) and (3.15) yields
 h0 mi uc u2t u0c + α T 0c T t  T c T 0t η ¼0 h mi ηhh0 uc u2t u0c + mi u2c u2t ηhh00 + αT c T 0t ¼ 0

(3.33) (3.34)

where α ¼ γ/(γ  1). In Eq. (3.33), the variables are not separated. To resolve this situation, Korsun and Tverdokhlebova (1997) proposed to use the equation that is obtained by differentiating Eq. (3.33) with respect to η instead of using Eq. (3.33). 2mi uc ut u0t u0c


 0 0 0 0 00 h 0h ¼ 0: + α TcTt  TcTt η  TcTt h h

(3.35)

Eq. (3.35) can be integrated, but the resulting motion will not be a solution to the momentum Eq. (3.33). The solution to Eqs. (3.34) and (3.35) has the form 0

mi huc

ðuc h0 Þ ¼ C1 ¼ 2γpq, Tc

(3.36)

α

T 0t ¼ C1 ¼ 2γpq, ηu2t

(3.37)

α

T 0c h ¼ C2 ¼ 2γp, T c h0

(3.38)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Tt C ¼ 1 + 1 η2 ¼ 1 + qη2 , C2 u2t

(3.39)

where C1, C2, p, and q are integration constants. From Eq. (3.28), it follows that ¼ uγ1 Tt: gγ1 t t

(3.40)

From Eqs. (3.37), (3.39), and (3.40), the following expressions are obtained:  pðγ12 Þ+1 ut ¼ 1 + qη2 ,

(3.41)

 pðγ1Þ , T t ¼ 1 + qη2

(3.42)

 ð1+pγÞ : ut gt ¼ 1 + qη2

(3.43)

Expressions (3.41)–(3.43) define the restrictions imposed on the initial radial profile. The axial profiles are defined as uc ¼ u0 h2p2 ,

(3.44)

T c ¼ T e0 h2pðγ1Þ :

(3.45)

The particle density can be described as   p n ¼ n0 h2 1 + qη2 :

(3.46)

The equation for self-similarity function can be obtained from Eqs. (3.36), (3.44), and (3.45) in the form  0 2γqp h2γp1 h2pp h0 ¼ 2 , M0

(3.47)

and integrated: 

h

 2pp 0 2 h

¼

δ20

  2γq h2pðγ1Þ  1  : ðγ  1ÞM20

(3.48)

Let us rewrite the obtained solutions, taking into account the choice of the integration constants p ¼ 1, q ¼ σ22 γ which, according to Bombardelli et al. (2011), provides the minimum error. This choice means that at any point in the flow uc ¼ u0. h0 ¼ δ20  2

  4 h2ðγ1Þ  1 , ðγ  1Þσ 2 M20

(3.49)

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123


γ 2η2 2 uz ¼ u 1 + 2 , σ γ

(3.50)


1γ 2η2 T ¼ T 0e h2ðγ1Þ 1 + 2 , σ γ

(3.51)

n¼

2



n0

h 1+

2η2 σ2 γ

:

(3.52)

For the isothermal case (γ ! 1), Eq. (3.49) takes the form h0 ¼ δ20 + 2

8 ln h : σ 2 M20

(3.53)

As for the previous two models, the variance σ value can be found as 2 : σ 2 ¼  2 ln ð1kÞ  γ γ e 1

(3.54)

For this model, the plasma plume propagation is determined by three parameters: the Mach number M0, the divergence angle α0 at ζ ¼ 0, η ¼ 1 and the specific heat ratio γ. Comparing the self-similar models described above, it should be noted that none of them satisfies the ion momentum equation. Each of the models solves this problem in its own way. The first model assumes that uz ¼ const at any point in the plume. The second model first selects a conical profile for uz and then captures its values along streamlines. The third model replaces the equation of momentum with the equation of its derivative. The results of a numerical comparison of these models and the methodology for calculating errors are given in the second section of the report (Bombardelli et al., 2011). Although self-similar models are approximate, their relative error is of the order of M2 0 , which makes these models suitable for describing supersonic plasma flows and, in particular, plumes of ion and Hall effect thrusters. These models demonstrate close results. Errors and discrepancies become noticeable when the considered point is at a large angular distance from the beam axis. The first two models show more accurate results in the region of interest near the ion beam axis.

3.3

Physics of ion beam interaction with a body surface

The process of interaction of high-velocity heavy particles with the surface of a body can be quite complex and be accompanied by various physical phenomena. In particular, the following can be observed (Fig. 3.3): particles backscattering, the penetration of particles into the surface and their subsequent evaporation after the establishment of thermal equilibrium, excitation of electronic transitions that can cause the release

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.3 Interaction of ions with a surface.

electrons or changes in chemical bonds, ejection of atoms from the surface, or even radioactive destruction. The predominant phenomenon depends on the kinetic energy of the beam particles bombarding the surface. The incident ion must have enough energy to overcome the surface binding forces and to eject the atoms from the surface. It is easy to calculate that for ion beam-assisted transportation projects, when the ion engine creates a plume of xenon ions with a mass of 2.18  1025 kg, having velocities ranging from 40,000 m/s to 80,000 m/s, the kinetic energy of the ions is from 1 to 4.5 keV. This energy level corresponds to a physical phenomenon known in the literature as sputtering. Sputtering refers to the process in which a particle bombards the target surface with energy sufficient to eject one or more atoms from the target surface. Sputtering can be physical or chemical. Thus, irradiating a target with a high-velocity ion beam leads to erosion of its surface. Sputtering can have a physical or chemical nature. Physical sputtering involves the ejection of atoms and molecules, as well as the emission of particles introduced during the bombardment. Chemical sputtering involves irradiating a target with chemically active ions, radicals, and atoms, leading to the formation of volatile compounds. In some cases, the processes of physical and chemical sputtering can occur simultaneously. From a practical point of view, the most important characteristic of erosion is the sputter yield Y, which is the ratio of the average number of atoms removed from the surface to the number of incident ions. This value is determined statistically and depends on many factors, including the energy of incident particles, the mass and angle of incidence of particles on the target surface, the atomic number of the irradiated substance, the heat of vaporization of the surface, the crystallographic structure, the number of atoms per unit area of the surface, its purity and roughness, and the surface binding energy. The dominant phenomenon in contactless transportation of space debris by an ion beam is physical sputtering (Cichocki et al., 2018; Nadiradze et al., 2018). This is a complex process. Particles of different mass and charge can be knocked out from the surface. They will have different directions of motion and energy levels. The surface

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Fig. 3.4 Models of sputtering.

can leave not only molecules and atoms, but also whole molecular complexes. Sputtering of particles leads to significant changes in the morphology and structure of the bombarded surface. The existing mathematical models are based mainly on the theory of Sigmund (1981), which distinguishes three main modes of sputtering (Fig. 3.4): the single knock-on regime (typical for ions with an energy level from 50 eV to 1 keV); the linear cascade mode (for ions with an energy level of several keV); and the spike regime (for ions with an energy level of more than 100 keV). In the single knock-on regime, the incident ion hits the surface atom. This atom recoils and strikes one or more neighboring atoms, which causes their displacement and impact of these atoms with their neighbors. Thus, hitting the target surface, the ion causes a chain of displacements and collisions of neighboring atoms inside the surface material. If this cascade reaches an atom located on the boundary of the surface, the atom can break off and leave the surface. In this regime, the cascade is very short and involves only a few atoms. In the linear cascade regime, the incident ion causes a longer and more branched collision cascade, but the number of atoms affected by the cascade is still relatively small. The probability that the recoil atoms will collide with the atoms already affected by the cascade is very small. All bonds between atoms inside the cascade are broken. In the spike regime, the incident ion generates a very dense and branched cascade that affects most of the atoms in some volume around the collision point. At high energies, the ion can penetrate deep into the target surface; as a result, only a few atoms on the surface are affected by the cascade and can leave the surface. The incident ion is implanted into the target volume, but the intensity of the sputtering process drops significantly. Since sputtering is closely related to the process of momentum and energy transfer from the incident ions to the target surface, several general regularities can be noted (Powell and Rossnagel, 1998). Different target materials have different binding energy, and the incident ions must have sufficient energy to overcome this barrier to emit atoms. Materials with lower binding energies are characterized by higher sputter yields. The efficiency of energy transfer between two particles depends on the ratio of the product of their masses to the sum of their masses. The highest sputter yields are

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

observed when the target and the incident ions are of the same species (self-sputter yield). An increase in the mass difference will lead to a decrease in sputter yield. At present, the results of experimental studies concerning the measurement of the yield when targets made of various materials are irradiated by ion beams with different parameters can be found in the scientific literature. There are various semiempirical formulas that allow us to calculate the value of sputter yield Y as a function of the kinetic energy of ions E in eV, for example, the Yamamura formula (Yamamura and Tawara, 1996): " rffiffiffiffiffiffi#s   0:042Qα∗ M S n ð EÞ Eth Y¼ 1 , US E 1 + Γke ε0:3

(3.55)

where Eth is the sputtering threshold energy in eV, US is the atomic heat of sublimation (Table 3.1), Q is the dimensionless empirical parameter, Sn(E) is the nuclear stopping cross section, M ¼ M2 =M1 is the mass ratio, M1 is the incident ion mass in a.m.u., M2 is the mass of the target atom, and α∗ is the dimensionless empirical parameter defined as   α M ¼

(

∗

0:249M

0:56

0:0875M

1:5

+ 0:0035M ,

0:15

when M  1,

(3.56)

when M < 1;

+ 0:165M,

Γ is the factor Γ¼

W ðZ 2 Þ , 1 + ðM1 =7Þ3

(3.57)

W(Z2) is the dimensionless empirical parameter given in Table 3.1, Z2 is the atomic number of the target atom, and ke is the Lindhard electronic stopping coefficient (Lindhard and Scharff, 1961) ke ¼ 0:079

ðM1 + M2 Þ3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi  M1 M1 M2

2=3 1=2

Z1 Z2 2=3

Z1

2=3

3=4 ,

(3.58)

+ Z2

Table 3.1 Empirical parameters for sputter yield. Target material

Z2

M2

US, eV

Q

W

s

Aluminum (Al) Titanium (Ti) Ferrum (Fe)

13 22 26

26.982 204.38 55.845

3.39 4.85 4.28

1.00 0.54 0.75

2.17 2.57 1.20

2.5 2.5 2.5

From Yamamura, Y., Tawara, H., 1996. Energy dependence of ion-induced sputtering yields from monatomic solids at normal incidence. At. Data Nucl. Data Tables 62, 149–253. https://doi.org/10.1006/adnd.1996.0005.

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127

Z2 is the atomic number of the incident ion and ε is the reduced energy ε¼

0:03255M2 E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2=3 2=3 Z 1 Z 2 ðM 1 + M 2 Þ Z 1 + Z 2

(3.59)

The sputtering threshold energy can be calculated as

Eth ¼

8 6:7U s > < γ ,

> : M2 + 5:7M1 , γM2

when M1  M2 ; (3.60) when M1 < M2 ;

where γ is the energy transfer factor in the elastic collision γ¼

4M1 M2 : ðM 1 + M 2 Þ2

(3.61)

The nuclear stopping cross section is defined by the formula 8:478Z1 Z 2 M1 Sn ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sTF n ðεÞ, 2=3 2=3 M1 + M2 Z1 + Z2

(3.62)

where sTF n (ε) is the reduced nuclear stopping power based on the Thomas-Fermi potential: sTF n ð εÞ ¼

pffiffiffi 3:441 ε ln ðε + 2:718Þ pffiffiffi pffiffiffi : 1 + 6:355 ε + εð6:882 ε  1:708Þ

(3.63)

Within the framework of the active space debris removal mission, it is assumed that an ion beam blows the body of a derelict satellite or an upper stage, which consists mainly of alloys of aluminum, ferrum, and titanium. Table 3.1 contains the values of the empirical coefficients required to calculate the sputter yield obtained in the study (Yamamura and Tawara, 1996). A study by Sugiyama et al. (2016) compared Eq. (3.55) with similar analytical formulas and experimental data. Expression (3.55) is one of several existing formulas that allow an approximate calculation of the sputtering yield. An overview and comparison of existing empirical formulas can be found in Chapter 12 of the thesis by Urrutxua (2015). Of great practical interest is work by Yim (2017), in which various experimental data is collected on the bombardment of various surfaces by xenon ions with energies up to 1 keV, and a Bayesian parameter fitting approach is used with a Markov chain Monte Carlo method to obtain parameters of Eckstein energy-dependent sputter yield (Behrisch and Eckstein, 2007) and Wei angular-dependent sputter yield (Wei et al., 2008) formulas. Eq. (3.55) is obtained for the case when the direction of the incident ion motion is perpendicular to the target surface. In the case when it is deflected by an angle θ, the value Y will change (Gorrilla et al., 2008):

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Y ðθÞ ¼

h  i Y ð 0Þ 1  1 , exp f cos θ opt cos θ cos f θ

(3.64)

qffiffiffiffiffiffiffiffiffi   Eth ð0Þ where f ¼ f s 1 + 2:5 1ζ , ζ ¼ 1  ζ E , values Y(0) and Eth(0) are sputter yield and threshold energy calculated for θ ¼ 0 by Eqs. (3.55) and (3.60), respectively, fs is an empirical parameter, and θopt is the angle corresponding to the maximum erosion. For aluminum irradiated with xenon, this angle can be calculated based on the empirical formula.

θopt ¼ 90°  286ðψ Þ0:45 ,


ψ¼

a Rc0

3=2

0

11=2

Z1 Z2 1C B @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2=3 2=3 E Z1 + Z2

,

(3.65)

where a is the screening radius and Rc0 is the average lattice constant. In the study by Gorrilla et al. (2008), the following values of the empirical constants for aluminum are given, calculated on the basis of processing the results of the experiments performed: fs ¼ 1.8, a ¼ 0.1052, Rc0 ¼ 2.56. Particles leave the surface with different energies, states of excitation, and charge at all angles. The distribution of the emission angles of particles can be described by the following law (Mahan, 2000): yðαÞ ¼

Y ðEÞ cos d α , π

(3.66)

where y(α) is the angular distribution of the ejected flux, d is a parameter, and α is the angle of emission measured from the normal to the surface. This formula works well for ions with an energy level of several keV. For an energy of hundreds of eV, the formula obtained by Zhang and Zhang (2004) is in better agreement with the experimental data:   α∗ M Sn ðEÞ cos α yðE, θ, α, ϕÞ ¼ 0:042 πU s ! rffiffiffiffiffiffi  1 Eth 3π sin θ sin α cos ϕ , χ ðαÞ cos θ +  1 4 E 2

(3.67)

where ϕ is the azimuthal angle of ejected atoms, which is shown on Fig. 3.5.  
 cos 2 α 3 sin 2 α + 1 3 sin 2 α  1 1 + sin α : + ln χ ðαÞ ¼ 1  sin α sin 2 α 2 sin 3 α Dependence y(α, ϕ) is called differential sputter yield and describes the number of atoms per incident ion, which were ejected in a particular direction. Expression (3.67) allows us to calculate the total sputter yield as

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129

Fig. 3.5 Angles for differential sputter yield determination.

Z2π Y ðE, θÞ ¼ 0

0 B @

Zπ=2

1 C yðE, θ, α, ϕÞdαAdϕ:

(3.68)

0

The energy possessed by the target particles after being ejected as a result of irradiation with a high-velocity ion beam can significantly exceed the energy obtained during evaporation. The sputtered atoms’ kinetic energy distribution has a pronounced maximum, followed by a decline close to the 1/E2 dependence. This estimate is in good agreement with experimental data for the case when the incident ions have energies of the order of several keV. For ions with energies less than 1 keV, the tail distribution differs from 1/E2; this is primarily due to the anisotropy effects, which are caused by the conservation of momentum of the impinging ions (Lautenschl€ager and Bundesmann, 2017). The location of the peak is determined mainly by the material of the target and the incident ions and, within some limits, depends weakly on the energy of the bombarding ions (Powell and Rossnagel, 1998). Interesting results of experimental studies of the influence of the ion incidence angle on the sputtered atoms’ energy distribution for the ion beam sputtering of a titanium target with Ar and Xe ions in a reactive oxygen atmosphere can be found in Lautenschl€ager and Bundesmann (2017). An in-depth study of the interaction of a flux of xenon ions with the surface of space debris was carried out by Cichocki et al. (2018). For the research, the authors wrote the EP2PLUS software package, in which ions and neutrals are modeled as macroparticles, and electrons as a fluid. This allows the simulation of collisions of heavy particles, macroparticles’ interaction with conductive or dielectric surfaces, electric currents in the plasma, quasineutral and nonneutral regions, and correct ion flux conditions at quasineutral material boundaries. The structure of the program and the used mathematical models are described in detail by Cichocki et al. (2016, 2017). The results of modeling the irradiation of an aluminum surface with xenon ions showed that, in addition to the sputtering effect, a rapid reflection of a particle from the surface (backscattering) can be observed. The probability of this event pb depends on the angle of incidence of the ion θ, and it is not sensitive to changes in the kinetic energy of the incident ion in the considered energy range. A reflection close to specular occurs. Fig. 3.6 shows the dependence of the backscattering probability on the angle of ions incidence. It can be seen that when the direction of incidence of ions

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.6 Backscattering probability when bombarding an aluminum surface with xenon ions (Cichocki et al., 2018).

Fig. 3.7 Mean sputtered atoms’ energy when bombarding an aluminum surface with xenon ions (Cichocki et al., 2018).

is close to the normal, backscattering is not observed and sputtering predominates. As the angle increases, the fraction of reflected ions increases. Fig. 3.7 demonstrates the dependence of the average energy of sputtered atoms Es on the angle of ions incidence for various values of ion energies. Dependences of sputter yield on the incidence angle for various ion kinetic energy are shown in Fig. 3.8. The curves have a pronounced maximum near an angle of 75 degrees, which is in good agreement with the results of other studies (Gorrilla et al., 2008; Wei et al., 2008). Figs. 3.6–3.8 are built on the basis of the data given in Section 2.3 of the work by Cichocki et al. (2018). The existing experimental and theoretical studies of the sputtering process concern mainly the sputtering of metal samples or polymers. A review of studies related to xenon ions irradiation with energy up to 1 keV of materials used in the manufacture of spacecraft can be found in Boyd and Falk (2001). In a study by Yamamura and Tawara (1996), a large array of experimental data was collected and processed, and the dependences of sputtering yield on kinetic energy were obtained, including for bombardment of various materials with xenon. Meanwhile, many spacecraft and upper rocket stages are covered with special blanket thermal insulation (BTI) materials, which have a complex multilayer structure. Computer modeling of such complex structures is a very difficult task. A methodology for conducting laboratory simulation, which makes it possible to extrapolate the obtained experimental results to

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Fig. 3.8 Sputter yield when bombarding an aluminum surface with xenon ions (Cichocki et al., 2018).

the conditions of orbital motion and long-term (about 100 days) assistance of the ion flow, was proposed by Shuvalov et al. (2017). For the experiment, the cylindrical vacuum chamber with a diameter of 1.2 m and a length of 3.5 m was used. It produced a static vacuum of about 105 N/m2 pressure. Two disc-shaped targets were used. The BTI target consisted of 10 layers of aluminized polymer film 5  106m thick, which were integrated into fiberglass fabric. This structure was put in an envelope bag made of alumina-borosilicate fiber of thickness about 2.5  104m. The second target was made of 12Kh18N10T stainless steel and was used to verify and compare the results with known data. The authors obtained a formula that allows calculation of the time it takes under conditions of orbital flight to give the same result in terms of erosive sputtering, as in a laboratory test. The weight loss of 1.43  103g, which was obtained as a result of 2 days of target irradiation in the laboratory, can be achieved in 97 days in orbital conditions using the LEOSWEEP project equipment (Ruiz et al., 2014). Fig. 3.9 shows the approximation curves of the sputtering yield data obtained in Fig. 3.9 Comparison of sputter yield of stainless steel and blanket thermal insulation (BTI). Based on data from Shuvalov, V.A., Gorev, N.B., Tokmak, N. A., Kochubei, G.S., 2017. Physical simulation of the longterm dynamic action of a plasma beam on a space debris object. Acta Astronaut. 132, 97–102. https://doi.org/10. 1016/j.actaastro.2016.11.039.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

the paper by Shuvalov et al. (2017) for BTI and steel targets at normal xenon ions incidence. The emitted atoms can be deposited on the surface of both the space debris itself and on a nearby active spacecraft. This could lead to film formation on solar panels and spacecraft sensors, making them unusable. Average energy is an important characteristic of the emitted flux of atoms affecting the deposition process of the material. The deposition process is also influenced by heat of sublimation, which is the binding energy of the atom. To calculate the thickness of the film deposited on the surface of an active spacecraft, which is required to assess the effect of sputtered particles’ flux on the performance of solar panels and sensors, it is necessary to have the parameters of the flux falling on the surface of the active spacecraft. On the basis of the methodology described above, the following procedure for calculating the distribution of the flux emitted when a space debris object is irradiated with an ion beam can be implemented. The surface of the space debris object is broken into triangles. Using Eq. (3.27) for the ion beam density, the number of particles that bombarded the jth triangle during a short time interval is calculated. Based on this value, using Eq. (3.64), the number of particles emitted from the surface of the triangle is determined. The direction of each of the emitted particles is determined randomly in accordance with distribution (3.66). The sphere around space debris is divided into segments and the number of particles emitted by the jth triangle and their energy and hitting each segment is recorded. Then the contributions of all triangles are summed up and the total number of particles and the average energy for each segment are calculated. The total number of particles in a segment is divided by the total number of emitted particles. Thus, the distribution of particles emitted by space debris over the sphere is calculated. As an example, let us consider the blowing of a passive cylindrical object with a radius of 1.5 m and a length of 6 m with an ion beam. The active spacecraft creating the ion beam is located at a distance of 15 m from the object. The ion beam parameters are given in Table 3.2. Fig. 3.10 shows the simulation results for different angles of cylinder axis deflection θ from the ion beam axis that passes through the active spacecraft and the center of the cylinder. The yellow areas on the sphere around the object correspond to the maximum concentration of emitted particles, and the blue ones to the minimum concentration. The figures also contain the cylinder, of which the particleemitting faces are shown in light green. Analysis of the figures shows that the flux of particles sputtered from the surface of the object occupies a rather large area, and some of these particles will settle on solar panels and sensors of an active spacecraft, interfering with their normal operation. Analysis of Fig. 3.10 makes it possible to determine the angular position of the passive object at which the flux of sputtered particles in the space area where the active spacecraft is located will be minimal. For the considered example, this is the position θ ¼ 50° (Fig. 3.11). Maintaining this angular position in the process of contactless transportation can be justified from the point of view of reducing the harmful effect of the sputtered particles’ flow on the active spacecraft and can be one of the goals of the active spacecraft control law. A study by Nadiradze et al. (2018) proposed the following formula for a rough overestimation of the level of contamination of an active spacecraft:

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Table 3.2 Ion beam parameter Parameter

Value

Plasma density n0 Mass of particle (xenon) m0 Radius of the beam at the beginning of the far region R0 Axial component of the ion velocity u0 Divergence angle of the beam α0 Particle kinetic energy E0

2.6  1016 m3 2.18  1025 0.1 m 38,000 m/s 15 degrees 982.5 eV

Fig. 3.10 Distribution of the sputtered atoms around the cylinder.

mc ¼

J i Am1 Y ðEi Þτ , πed 2

(3.69)

where mc is the mass of deposited particles, d is the distance from the active spacecraft to the transported object, A is the average area of the sputtering spot (it is assumed that the area of the spot is less than the area of the midsection of the passive object), e is the electron charge, and Ji is the ion beam density. This formula overestimates the deposited mass. It should be noted that the mass of deposited particles is inversely proportional to the distance between the active spacecraft and the space debris object d. According to estimates made by Nadiradze et al. (2016), the degradation of solar panels and optical elements properties as a result of the deposition of atoms sputtered from the surface of space debris is a serious problem that arises during the

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.11 Sputtered atoms distribution for the cylinder angular position corresponding to the least contamination of an active spacecraft.

implementation of a contactless space debris removal mission by an ion beam. When a space debris object is irradiated by a xenon ion beam with an energy of 4 keV at a current of 0.5 A, the rate of film deposition on the surface of an active spacecraft located at a distance of 40 m will be 1.2  106 g/cm2/s. This can lead to a loss of transparency of the protective surfaces of solar cells in 20 days. To reduce the negative impact of pollution on solar panels, the study proposed to change the flight program and not to irradiate space debris along the trajectory segments when it is located between the active spacecraft and the Sun (Fig. 3.12). Obviously, this will increase the duration of the entire mission. Since the transport operation of contactless space debris removal takes a significant amount of time, the matter sputtering from the surface of space debris can lead to significant erosion. The study by Nadiradze et al. (2018) was devoted to the analysis of the consequences of the erosive and polluting effects of an ion beam on a space debris object. Calculations showed that an ion beam with a kinetic energy of particles of 4 keV in 10 days will lead to surface erosion with a total depth from 13.31 μm Fig. 3.12 Active and passive sections of the trajectory.

Ion beam physics

135

(graphite) to 199.04 μm (argentum). Thus, during ion beam-assisted transportation, there is a probability of the formation of secondary space debris. If space debris is covered with thin-film screen-vacuum thermal insulation, then the film can be partially or completely sputtered. Parts of it may come off. Carbon fiber-reinforced polymer antennas can form new debris if their filaments are partially sprayed. Cable sheaths, solar interconnectors, and other delicate structural elements can also be destroyed. Thus, a thorough study of the surface of a transported space debris object and an assessment of the acceptability of the risk of secondary space debris creation during ion beam-assisted removal should be undertaken at the early stages of the mission preparation.

3.4

Calculating the forces and torque generated by the ion beam

Ion beam produces high velocity rarefied flow. Two simplified models of the interaction of ions with the surface are considered in the literature. The first one supposes ions spectral reflection when the resulting force is normal to the surface. The second one assumes fully diffused reflection when the force acting on the surface is directed along the velocity of the incoming particles. The experimental results reported by Mikellides et al. (2002) indicate that the diffuse reflection model is in better agreement with the experimental data. The sputtering of the target material and the escaping ions from the space debris surface are neglected when calculating the forces and torques generated by the ion beam on the surface. It is assumed that when ions collide with the surface of the object, they completely transfer their momentum to the surface—that is, the diffuse model of the ions interaction with the surface is used. In this case, the force dF acting on the elementary area dS of the object’s surface is dF ¼ nmi ui ðui  eN ÞdS,

(3.70)

where ui is the velocity of ions near the considered elementary area dS and eN is the unit vector of the outer normal to the elementary area. The ion force and torque acting on the whole body can be found as Z FI ¼

Z dF,

S

LI ¼

ρ  dF,

(3.71)

S

where ρ is the vector connecting the center of mass of the object with the geometric center of the elementary area dS, LI is the ion torque relative to the center of mass of the object, and S is the area of the object’s surface that is blown by the ion beam. Eq. (3.71) can be calculated analytically only for bodies of the simplest shapes. For example, Section 3.2 of the book by Alpatov et al. (2019) describes in detail the calculation of the ion force acting on a sphere. Satellites and rocket stages both

136

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

have a more complex shape. In addition, to calculate the ion force and torque, the use of an approximate technique is required. The object can be divided into N small triangles. For each of them, the ion force Fj can be calculated. Then the summation is performed and the resultant ion force and torque acting on the whole object are calculated as FI ¼

N X j¼1

δj Fj ,

LI ¼

N X

δj ρj  Fj ,

(3.72)

j¼1

where δj ¼ 0 if jth triangle is in the shadow, and δj ¼ 1 if ions hit the jth triangle. When calculating the force Fj, a number of assumptions are introduced. It is supposed that the areas of the triangles Sj into which the surface of the object is divided are so small that all the ions hitting into different parts of the triangle have the same velocity and move in parallel. Forces are applied in the barycenter of the triangles, which is denoted as Pj (Fig. 3.13). The force acting on the jth triangle, taking into account Eqs. (3.70) and (3.71), can be written as   Fj ¼ nj mj Sj V 2j eVj eVj  eNj ,

(3.73)

where nj is the particle density of ions at Pj point, Vj is the ion velocity at Pj, eVj is the unit vector directed along the flow rate at the point Pj, and eNj is the unit vector of the outer normal of the jth triangle (Fig. 3.13). Eq. (3.22) can be used to describe the particle density. Let us take σ 2 ¼ 1/3 and rewrite the equation in the form ! n0 r2 nj ¼ exp 3 2 : hð ζ Þ 2 R0 hðζ Þ2

Fig. 3.13 Surface area and ion force acting on it.

(3.74)

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137

The self-similarity function h(ζ) can be found from the differential Eq. (3.24). In the case M0 ≫ 1 and α0 > 10°, the parameter δ20 ¼ tan2α0 is almost two orders of magnitude greater than C2. Taking into account that for a distance of about 10 m from the engine, the second term in Eq. (3.24) can be neglected, and that h(0) ¼ 1, an approximate solution can be written as hðζ Þ ¼ 1 + ζ tan α0

(3.75)

Taking into account the last expression, Eq. (3.74) takes the form nj ¼

r2

n0

!

 2 exp 3 2 2  1 2 : ζ 2 ζ 1 + tan α0 ζ R0 ζ + tan α0

(3.76)

For the considered case of space debris contactless transportation by an ion beam, the value ζ 1 ¼ R0/z is small and Eq. (3.76) can be simplified: n r2 nj ¼ 2 0 2 exp 3 2 2 ζ tan α0 ζ R0 tan 2 α0

! (3.77)

Substituting Eq. (3.77) into Eq. (3.73), and then into Eq. (3.72), the ion force and torque generated by the ion beam can be calculated. In practice, to calculate ion force and torque using Eq. (3.72), it is necessary to introduce coordinate systems and project these equations onto the corresponding axes. The magnitude of the force generated on an elementary surface by an ion beam, in addition to the parameters of the ion beam itself, depends on the relative position of the beam source and the orientation of the surface in the flow. Consider an orbital reference frame BXoYoZo, the origin of which is at the center of mass of the space debris object B. The orientation of the axes of this coordinate system in space is not important at the stage of determining the ion force and torque. The orientation of the object’s body frame BXbYbZb, of which the axes are principal object axes, relative to the orbital frame BXoYoZo is specified using three Euler angles γ, θ, and φ (rotation sequences is y-x-y). The position of the center of mass of the active spacecraft (point A) relative to space debris is determined by the vec ! tor BA , which has coordinates [xA, yA, zA] in the orbital frame. In order to determine the direction of the ion beam axis, a reference frame AXsYsZs is introduced. The ! axis AZs is directed along the vector BA , and the axis AXs lies in the plane formed by the axes AXo and AZs. The transition from the reference frame AXoYoZo to AXsYsZs can be carried out by two successive rotations at the angles α1 ¼ arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 (yA/zA) clockwise and α2 ¼ arctan xA = yA + zA counterclockwise around the x and y axes, respectively (Fig. 3.14). The corresponding rotation matrix has the form

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.14 Auxiliary reference frames.

2 6 Mso ¼ 4

cos α2

 sin α1 sin α2

0 sin α2

cos α1 sin α1 cos α2

 cos α1 sin α2

3

7  sin α1 5: cos α1 cos α2

(3.78)

It is expedient to set the direction of the ion beam axis in the active spacecraft reference frame AXsYsZs. Let us introduce a coordinate system AXaYaZa associated with the ion beam axis. The axis AZa is directed along the ion beam axis. The axis AXa lies in the plane formed by the axes AXs and AZa. The transition from the reference frame AXsYsZs to AXaYaZa can be carried out by two successive rotations at the angle π + β1 counterclockwise and the angle β2 clockwise around x and y axes, respectively. The corresponding rotation matrix has the form 2

cos β2 6 Mas ¼ 4 sin β1 sin β2 cos β1 sin β2

0  cos β1 sin β1

3 sin β2 7  sin β1 cos β2 5:

(3.79)

 cos β1 cos β2

In the case when the problem is considered in a planar statement, then it can be taken that γ ¼ π/2, φ ¼  π/2, α1 ¼ 0, and β1 ¼ 0 (Fig. 3.15).

Fig. 3.15 Planar case of the system motion.

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139

Consider a jth triangle, the coordinates of the vertices of which are given in the orbital coordinate system BXoYoZo: h iT PM oji ¼ xji, yji , zji ,

i ¼ 1, 2, 3;

where the superscript indicates the reference frame in which the coordinates of the point or vector are given. The coordinates of the barycenter of the triangle are defined as

Poj

xj1 + xj2 + xj3 yj1 + yj2 + yj3 zj1 + zj2 + zj3 ¼ , , 3 3 3

T :

(3.80)

The vertices of the triangle are numbered so that the normal vector is directed to the side, looking from where the numbers of the vertices increase when going counterclockwise. The normal vector has coordinates 2

  3    yj2  yj1 zj3  zj1  zj2  zj1 yj3  yj1 6      7 6 7 Noj ¼ 6  xj2  xj1 zj3  zj1 + zj2  zj1 xj3  xj1 7: 4       5 xj2  xj1 yj3  yj1  yj2  yj1 xj3  xj1 The modulus of this vector can be found as N j ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2jx + N 2jy + N 2jz, where Njx, Njy, and

Njz are coordinates of the normal vector in BXoYoZo. The unit normal vector eNj ¼ Nj/Nj has the form 2   3    yj2  yj1 zj3  zj1  zj2  zj1 yj3  yj1 7 6 7 6 Nj 7 6     7 6  6 z + z x  x  x  z  z  x j2 j1 j3 j1 j2 j1 j3 j1 7 7: eoNj ¼ 6 7 6 N j 7 6     7 6   6 x x yj3  yj1  yj2  yj1 xj3  xj1 7 j1 5 4 j2 Nj

(3.81)

To determine the direction of the ion beam particles velocity vector at point Pj, we use Eqs. (3.17) and (3.20), taking into account Eq. (3.75): uri ¼ η tan α0 uzi ¼

r ð 0Þ R 0 u ¼ tan χuzi , R0 d zi

uzi ¼ u0 ¼ const,

(3.82)

where χ is the angle of the streamline deflection from the ion beam axis. Eq. (3.82) implies that the beam particles velocity vector Vj at point Pj is directed along the line

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

connecting the source of the ion beam to this point. In this case, the unit vector can be ! AP found as eVj ¼ APj j. Projecting this equation on the axis of the orbital coordinate system gives 3 Pjx  xA 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 7 6 Pjx  xA + Pjy  yA + Pjz  zA 7 7 6 7 6 7 6 Pjy  yA 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 2  2  2 7 eVj ¼ 6  7, 7 6 P  x + P  y + P  z jx A jy jz A A 7 6 7 6 Pjz  zA 7 6 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 5 Pjx  xA + Pjy  yA + Pjz  zA 2

(3.83)

where Pjx, Pjy, and Pjz are coordinates of the barycenter in the orbital coordinate system given by Eq. (3.80). The magnitude of the velocity vector in the barycenter, taking into account Eq. (3.82), can be written in the form Vj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2ri + u2zi ¼ u0 1 + tan 2 χ ¼ 0 , cos χ

(3.84)

where cosχ ¼ (eVjxuaxisx + eVjyuaxisy + eVjzuaxisz), eVjx, eVjy, and eVjz are the coordinates of the vector (3.83), and uaxisx, uaxisy, and uaxisz are the components of the unit vector directed along the ion beam axis in the orbital reference frame. 2

3

2 3 0 6 7 6 7 ¼ M u 4 axis y 5 oa 4 0 5 1 uaxis z uaxis x

where Moa ¼ (MasMso)T. Taking into account Eqs. (3.78) and (3.79), from the last equation it follows that uaxis x ¼ cos α2 cos β1 sin β2  sin α2 cos β1 cos β2 , uaxis y ¼ cos α1 sin β1  sin α1 cos β1 ð cos α2 cos β2 + sin α2 sin β2 Þ, uaxis x ¼ sin α1 sin β1  cos α1 cos β1 ð cos α2 cos β2 + sin α2 sin β2 Þ: To find out whether the jth triangle is in the shadow area, it is necessary to iterate over all the other triangles. For each of them, the coordinates of the point of intersection K of the line passing through point A and the barycenter Pj with the plane built on the points of the kth triangle should be found. These coordinates are the root of a system of linear equations, the first two of which are the equation of a straight line, and the last is the equation of the plane to which the kth triangle belongs:

Ion beam physics

141

y  yA xK  xA ¼ K , Pjx  xA Pjy  yA xK  xA z  zA ¼ K , Pjx  xA Pjz  zA ððzK  zk1 Þðyk3  yk1 Þ  ðzk3  zk1 ÞðyK  yk1 ÞÞðxk2  xk1 Þ + ððzk1  zK Þðyk2  yk1 Þ + ðzk2  zk1 ÞðyK  yk1 ÞÞðxk3  xk1 Þ ððyk1  yk2 Þðzk3  zk1 Þ + ðyk3  yk1 Þðzk2  zk1 ÞÞðxK  xk1 Þ ¼ 0: If a solution exists and the distance from point K to A is less than the distance from A to Pj, then the kth triangle can create a shadow for the jth triangle. In this case, it is necessary to check whether the point K lies inside the kth triangle. If this condition is satisfied, then it is assumed that the entire jth triangle is in the shadow area due to the fact that the sizes of the triangles are small. The condition that a point is located inside the triangle can be written as ðρ21  ρK1 Þ  ðρK1  ρ31 Þ > 0, ðρ12  ρK2 Þ  ðρK2  ρ32 Þ > 0, ðρ13  ρK3 Þ  ðρK3  ρ23 Þ > 0,

(3.85)

where ρij ¼ [xki  xkj, zki  zkj, zki  zkj]T and ρKj ¼ [xK  xkj, zK  zkj, zK  zkj]T. If at least one of the inequalities (3.85) is not satisfied, then the point K lies outside the kth triangle, and this triangle does not create a shadow for the jth triangle. Substitution of Eqs. (3.77), (3.81), (3.83), and (3.84) into Eq. (3.73) allows us to subtract the projections of the ion force Foj acting on the jth triangle on the axis of the orbital reference frame BXoYoZo. Then, using Eq. (3.72), the resulting ion force and torque can be calculated. This result will be valid for a given specific position of the active spacecraft, the direction of the flow axis, and the orientation of the space debris object.

3.5

Examples of ion force and torque calculations

3.5.1 Calculation assumptions and methodology This section contains the results of ion forces and torques calculations for objects of the simplest shape, and for complex-shaped satellites and rocket stages (Fig. 3.16). To carry out the calculations, considered objects were modeled in the open source 3D software Blender and then exported to STL format. The authors of this book have written a program in MATLAB that reads information about triangles and their vertex coordinates from the STL model and calculates the ion force and torque using the approach described in Section 3.4. The influence of ion beam parameters and objects’ geometry on the ion force and torque is also analyzed in this section.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.16 Three-dimensional models of objects considered in this section.

In the general case, the magnitude and direction of the ion force and torque depend on many factors including an object’s geometry, ion beam parameters given in Table 3.2, the relative position of the active spacecraft, the direction of the ion beam axis, and the orientation of the object in the ion beam. Some of these parameters remain unchanged throughout the entire contactless transportation maneuver, while some of them may change. The variable parameters include the parameters defining the relative position of the active spacecraft α1, α2, and d (Fig. 3.14), the direction of the ion flow, given by the angles β1 and β2, the initial velocity of the ions u0, which can be changed within certain limits by the ion engine voltage for control purposes, and three angles that specify the spatial orientation of the object. The calculation of the ion force and torque at each step of object’s motion differential equations integration, taking into account all this variety of input parameters, is a rather laborious process that requires large computational and time costs. For optimization purposes, it is advisable to precalculate the force and torque for a large set of input parameters and then use this database to carry out fast approximate calculation of the values for the required set of parameters. This approach was used in studies by Aslanov and Ledkov (2021) and Nakajima et al. (2018). Analysis of the dependence of the ion force and torque on the entire manifold of the parameters is a very cumbersome task, so in this section we restrict ourselves to the consideration of the planar case only (Fig. 3.15), when the orientation can be specified using one angle θ. In calculations, it was assumed that the control system of the active spacecraft, which is the source of the ion beam, keeps it in a constant position relative to the center of mass of the object. In the orbital reference frame BXoYoZo, of which the origin is the center of mass of the object (Fig. 3.14), the active spacecraft has coordinates [0, d, 0], where d is the distance between the beam source and the center of mass of the object. If the active spacecraft has other coordinates [xA, yA, zA] in BXoYoZo, being at the same distance d from B point, then the ion force and torque projections for this coordinates can be recalculated from the spacecraft position [0, d, 0] by means of three Euler rotations. For further use in the equations of motion, it is most convenient to represent the ion force in the form of projections on the axes of the orbital coordinate system BXoYoZo, and the ion torque in the form of projections on the axes of the object’s body

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143

Fig. 3.17 Ion force and torque recalculation.

frame BXbYbZb. Consider a planar case to illustrate visually the recalculation of ion force and torque (Fig. 3.17). It is assumed that the force and the torque are functions of four generalized coordinates: xA, zA, β ¼ α  γ ¼  β2, and θ. The process of calculating the force and the torque is quite laborious in terms of the cost of computing resources. These costs can be reduced by making a preliminary calculation for the case and turning the system around the axis BZo (Fig. 3.17). In this case, Lbx ¼ Lby ¼ 0, Foy ¼ 0, and Lby ðθ, α, xA , yA Þ ¼ Lby ðθ + γ, α, 0, dÞ, Fox ðθ, α, xA , yA Þ ¼ Fox ðθ + γ, α, 0, dÞ, Foz ðθ, α, xA , yA Þ ¼ Foz ðθ + γ, α, 0, dÞ, where d ¼

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2A + y2A and γ ¼ arctan xyA . A

The ion beam axis direction, which is defined in the planar case by the angle β1, has a great influence on the ion force and torque. Let us choose some point inside the object, which we will call the “geometric center.” For asymmetric objects, this can be a certain point easily determined by optical sensors. Three limiting cases are considered below: l

l

l

Case 1: the object’s center of mass coincides with the geometric center (point B1), and the ion beam axis passes through the geometric center. Case 2: the center of mass is on the surface of the object (ideal boundary case, point B2), and the ion beam axis is directed to the center of mass. Case 3: the center of mass is on the surface of the object (point B3), and the ion beam axis is directed to the geometric center. It is also assumed that in this case, the control system of the active spacecraft is oriented not to the center of mass, but to the geometric center, and it maintains a constant distance from the spacecraft to the geometric center of the object.

Since the patterns of the ion flow around the object in cases 1 and 3 coincide, in these cases the same ion forces are generated on the surface of the object, but the ion torques relative to the center of mass are different. If the ion torque relative to the geometric

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

center LI is known (case 1), then the torque relative to the center of mass in case 3 can be found by the equation LC ¼

N N  !  X ! X δj CO + dj  Fj ¼ CO  δ j Fj + L I j¼1

j¼1

! ¼ CO  FI + LI ,

(3.86)

where C is the center of mass of the object and O is the geometric center. In the planar case, the ion force and torque are 2π periodic functions of θ angle, which defines the orientation of the object relative to the orbital coordinate system. It makes sense to use the Fourier series to approximate these dependences: Lbi ¼ LImax

ðLiÞ a0

k  X

+

ðLiÞ aj

cos jθ +

ðLiÞ bj

!  sin jθ

,

(3.87)

j¼1

Foi

¼

FImax

ðFiÞ a0

+

k  X

ðFiÞ aj

cos jθ +

ðFiÞ bj

!  sin jθ

,

(3.88)

j¼1

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2     2 2 Lbx ðθÞ + Lby ðθÞ + Lbz ðθÞ where i ¼ x, y, z, LImax ¼ max is the ion torθ½0, 2π  ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2     2 2 Fox ðθÞ + Foy ðθÞ + Foz ðθÞ que extreme value, FImax ¼ max is the ion θ½0, 2π 

force extreme value, and culated by the equations ðLiÞ a0

ðLiÞ

a0

ðLiÞ b0

1 πLImax

1 ¼ max πLI

b()j are dimensionless Fourier coefficients, which are cal-

Z2π

1 ¼ 2πLImax

¼

a()j ,

Lbi ðθÞdθ,

ðFiÞ a0

0

Z2π Lbi ðθÞ cos jθdθ,

1 ¼ 2πFImax

ðFiÞ

a0

0

Z2π Lbi ðθÞ sin jθdθ, 0

ðFiÞ b0

¼

Z2π Foi ðθÞdθ, 0

1 πFImax

1 ¼ πFImax

Z2π Foi ðθÞ cos jθdθ, 0

Z2π Foi ðθÞ sin jθdθ: 0

Let us calculate the ion force and torque for the objects shown in Fig. 3.16 in the planar case, when the orientation of an object is defined by a single angle θ. Unless otherwise specified, it is assumed that the ion beam generated by the active spacecraft has the parameters given in Table 3.2. The above three cases of the center of mass location and the ion beam axis direction are considered for each object.

Ion beam physics

145

Fig. 3.18 A spherical object in an ion beam.

3.5.2 Sphere in an ion beam From a computational point of view, a sphere is the simplest case. Due to symmetry, even in the spatial case the problem of the ion force and torque calculation can be reduced to the planar case when considered plane passing through point A, which is the ion beam source, the geometric center of the sphere, and its center of mass. Let us consider three cases described in Section 3.5.1. At the angle θ ¼ 0, the geometric position of the spheres in all these cases is the same (Fig. 3.18A). When the angle θ changes (Fig. 3.18B), the positions of the spheres in the first and third cases coincide, but differ from the second case, in which the geometric center of the sphere is displaced from the ion beam axis. Let us perform the calculations for a sphere with a radius Rs ¼ 1m. Figs. 3.19 and 3.20 show the dependences of the ion force projections on the axis of the orbital reference frame BXoYoZo. The projection Foz ¼ 0, which is explained by the symmetry of the object and the beam. In the first and third cases, the change in the angle θ does not affect the change in the object position in the ion flow; respectively, the ion force will be the same for all values of θ angle. In the second case, the angle θ affects the position of the sphere inside the flow (Fig. 3.18B). Both the distance from the geometric center of the sphere to the source A and the direction of the particles hitting into different parts of the sphere change when the θ angle changes. The ion force component Foy Fig. 3.19 Dependence of the ion force projection Fox on the deflection angle θ.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.20 Dependence of the ion force projection Foy on the deflection angle θ.

has a maximum in absolute value at the angle θ ¼ π, when the sphere is capsized and its surface is closer to the ion beam source—that is, the concentration of particles hitting the surface of the sphere will be greater compared to the position corresponding to θ ¼ 0. Fig. 3.21 shows the dependence of the ion torque on the angle θ. Due to the symmetry of the ion flow, Lbx ¼ Lby ¼ 0. In case 1, the center of mass is located at the geometric center, and again, due to the symmetry, Lbz ¼ 0 is observed at any angle θ. For the planar case, axes BZb and BZo are parallel and codirectional (Fig. 3.15), thus Lbz ¼ Loz . In cases 2 and 3, a significantly different picture is observed. In case 2, the dependence Lbz (θ) is close to sinusoidal, and in case 3, it is sinusoidal by virtue of Eq. (3.86) and the equality of the torque with respect to the geometric center LI and force projection Foz to zero. Lbz3 ¼ Rs Fox cos θ + Rs Foy sin θ + LI ¼ Rs Foy sin θ:

(3.89)

An analogy with the aerodynamic restoring moment acting on a sphere with a displaced center of mass is observed here. Distortions of the graph in the second case

Fig. 3.21 Dependence of the ion torque projection Lbz on the angle θ.

Ion beam physics

147

Fig. 3.22 The boundary position of the sphere when it is completely inside the ion beam.

are caused by a change in the particle density and direction of a part of the ion flow falling on the various parts of the sphere surface with θ change. It should be noted that in all the cases considered above, the entire sphere was inside the ion beam. As the distance between the source of the ion beam and the center of mass of the sphere d or angle α0 decreases (Fig. 3.18A), a situation may arise when not the entire sphere, but only part of it, is in the beam. In the case when the ion beam is directed to the geometric center of the sphere, the minimum distance at which the whole sphere is located inside the beam is d∗ ¼ Rssin1α0 (Fig. 3.22). Let us investigate the influence of the distance between the beam source and the center of mass of the object and the angle of divergence of the beam on the value of the ion beam force. In case 1, when the center of mass is in the geometric center of the sphere, the dependence of the ion force projection Foy on d and α0 is shown in Fig. 3.23. Other force projections and ion torque are zero. The dashed line in the figure shows the boundary value, where d ¼ d∗ and the entire sphere is still inside the beam. With an increase in d or α0, part of the particles of the ion beam will pass by the sphere without hitting into its surface. This is accompanied by a decrease in the modulus of the ion force. Let us investigate the influence of the angle of the ion beam axis deflection β on the value of the ion force and torque. This angle is measured from the line connecting and the source of the ion beam with the geometric center of the sphere (Fig. 3.17). Returning to the notation of variables adopted in Section 3.4, this angle can be defined as β ¼  β2 for β1 ¼ 0 (Fig. 3.14). Graphs of the ion force and its projections are shown on Fig. 3.24. For large values of the beta angle, the ion force is zero, since the entire

Fig. 3.23 Dependence of the force projection Foy on the distance d and angle α0.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 3.24 Dependence of ion force and its projections on the angle θ.

beam passes by the sphere. The maximum force is observed at β ¼ 0. It should be noted that at this value, projection Fox is equal to zero; that is, there is no force component tending to displace the sphere from the beam axis. Fig. 3.25 shows the surface describing the dependence of the ion torque on the angles θ and β in the case when the center of mass of the sphere is at point B3 (Fig. 3.18B). The global maximum and minimum of the ion torque are observed at β ¼  0.99° and β ¼ 0.99°, respectively. Fig. 3.25 shows that by changing an ion beam axis direction depending on the angle θ, it is possible to change the value and direction of the ion torque in a fairly wide range. This circumstance can be used to control the object’s angular motion.

3.5.3 Cylinder in an ion beam In general cases, the shape of a cylinder is determined by two parameters: radius Rc and height Hc. The cylinder has an axis of symmetry and a plane of symmetry perpendicular to the axis. The geometric center of the cylinder is the point lying at the intersection of the axis and the plane of symmetry (Fig. 3.26).

Fig. 3.25 Dependence of ion torque on the angles θ and β.

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Fig. 3.26 Cylinder in an ion beam.

Consider a planar case when axis of symmetry is in the BXoYo plane. As for the sphere, consider three cases of the cylinder’s center of mass location and ion beam axis direction described in Section 3.5.1. The value of the ion torque relative to the center of mass in case 3 is determined by Eq. (3.86), which for the considered object takes form Lby3 ¼

Hc o H F cos θ  c Fox sin θ + LI , 2 y 2

(3.90)

where LI is the ion beam torque relative to the geometric center of the cylinder. It is assumed that the ion beam has the parameters shown in Table 3.2. The radius of the cylinder is Rc ¼ 1m and its height is Hc ¼ 3m. Fig. 3.27 shows projection Fox as a function of angle θ. In case 2, this projection reaches large values compared to cases 1 and 3, which is caused by the asymmetric arrangement of the cylinder in the ion beam in the third case. Fig. 3.28 shows the dependences of the ion beam projection Foy on the angle θ calculated for the above three cases. In case 2, the graph is not symmetrical with respect to θ ¼ π. The modulus of Foy projection is greater in interval θ  [0, π) than in interval θ  [π, 2π), since in the first interval the surface of the object is closer to the

Fig. 3.27 Dependence of the ion force projection Fox on the deflection angle θ.

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Fig. 3.28 Dependence of the ion force projection Foy on the deflection angle θ.

ion beam source, and the concentration of ions hitting the surface is greater. Fig. 3.29 shows the dependence of the ion torque on the angle θ. The displacement of the center of mass from the geometric center leads to an increase in the torque. If the oscillations of the cylinder occur only under the action of the ion torque, then the zeros of the dependence of the torque on the angle θ and the sign of the derivatives at these points determine the position and type of equilibrium points. The dependences obtained for cases 2 and 3 are close in amplitude; however, in case 2, position θ ¼ 3π/2 is stable, and in case 3, it is unstable, but there are two stable equilibrium positions in its vicinity. It should be noted that, in contrast to the sphere (Fig. 3.21), there are breakpoints on the graphs of the ion torque dependences of the cylinder (Fig. 3.29). The presence of these points is due to the simultaneous entry of large surfaces of cylinder into the shadow region and exit from the shadow. Fig. 3.30 schematically shows the transition of surfaces into a shadow for case 1. The green color indicates the areas blown by the ion beam. At θ ¼ 5°, particles of the ion beam hit only part of the lateral surface of the cylinder facing the beam source and do not fall on the upper and lower base planes,

Fig. 3.29 Dependence of the ion torque projection Lby on the angle θ.

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Fig. 3.30 Cylinder blowing at different orientations θ.

which are in shadow (Fig. 3.30). As the angle θ increases to 6 degrees, the upper plane leaves the shadow and begins to contribute to the ion torque. At this point, a break is observed on the graph of Lbz (θ) on Fig. 3.29 (case 1). The transition of the side surface into the shadow at θ ¼ 87° occurs more smoothly and there is no sharp break on the Lbz (θ) graph. Let us study the influence of the cylinder radius in the case when the entire cylinder is inside the ion beam, and its center of mass is in the geometric center (case 1). Figs. 3.31–3.33 show the dependences of the ion force and torque projections on the angle θ for various radii Rc. With an increase in the radius of the cylinder at a constant height, an increase in the projection module Foy is observed, which is primarily caused by an increase in the surface area of the object (Fig. 3.32). With a relatively small radius, the area of the base surfaces of the cylinder is small compared to the area of the side surface. If the cylinder motion occurs only under the action of the ion Fig. 3.31 Dependence of the ion force projection Fox on the angle θ for Hc ¼ 5m.

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Fig. 3.32 Dependence of the ion force projection Foy on the angle θ for Hc ¼ 5m.

Fig. 3.33 Dependence of the ion torque projection Lbz on the angle θ for Hc ¼ 5m when the center of mass is at the geometric center (case 1).

torque, the positions θ ¼ 0 and θ ¼ π are stable, when the axis of the cylinder is perpendicular to the axis of the ion beam, and the position θ ¼ π/2 is unstable. As the base area increases, the equilibrium positions θ ¼ 0 and θ ¼ π become unstable and two stable positions appear around them. If the center of mass is located on the lower base surface at point B3, and the ion beam axis is directed to the geometric center, then the graphs of the ion torque dependences change significantly (Fig. 3.34). The position θ ¼ π/2 becomes unstable (when the axis of the cylinder lies on the axis of the beam, and the center of mass is behind in the shadow). In the case when the area of the base surface is large compared to the area of the side surface, then the position θ ¼ 3π/2 is stable. As the radius decreases, this position becomes unstable and two stable points appear around it. As in the case of a sphere, the distance from the center to the ion beam source d and ion beam divergence angle α0 influence the ion force and torque. With an increase in d and α0, a decrease in the force projection module Foy is observed (Figs. 3.35 and 3.36), which is due to two factors: an increase in the proportion of beam particles that do not collide with the surface of the object, and a change in the direction of the beam particles motion as the distance from the barycenter of the triangle to the beam axis

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Fig. 3.34 Dependence of the ion torque projection Lbz on the angle θ for Hc ¼ 5m when the center of mass is at B3 point (case 3).

Fig. 3.35 Dependence of the ion force projection Foy on the angle θ for various distances d for α0 ¼ 15°.

increases. In cases 1 and 3, the curves on the intervals [0, π) and [π, 2π) coincide due to symmetry. In case 2, when the ion beam axis is directed to the center of mass, which is not the geometric center, the symmetry is broken, and the projection modulus on interval [π, 2π) is smaller than on interval [0, π). This is caused by the fact that the distance from the ion beam source to the surface of the object is greater at interval [π, 2π) than at interval [0, π). The same asymmetry in case 2 is also observed for projection Fox (Figs. 3.37 and 3.38) and for the ion torque Lbz (Figs. 3.39 and 3.40). For ion beam force projection Fox in cases 1 and 3, as the distance d and angle α0 increase, first an increase in the maximum modulus is observed, which is due to a change in the direction of the streamlines with distance from the flow axis; however, there is then a decrease caused by an increase in the proportion of beam particles passing by the surface. The mismatch of the Fox projection directions for α0 ¼ 4° and α0 ¼ 7° for the cylinder near angular positions θ ¼ π/2 and θ ¼ 3π/2 is due to the fact that at

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Fig. 3.36 Dependence of the ion force projection Foy on the angle θ for various angles α0 for d ¼ 15m.

Fig. 3.37 Dependence of the ion force projection Fox on the angle θ for various distances d for α0 ¼ 15°.

small α0 angles, the entire beam hits the side surface of the cylinder and does not affect its base surfaces (Fig. 3.38, case 1, 3). As the angle α0 increases, the beam hits the base surface and it begins to play a significant role. In case 2, when the beam axis is directed to the center of mass located on the lower base surface of the cylinder, the projection Foz direction is determined by the inclination θ angle due to the asymmetry of the flow around the object. Dependences of the ion torque on the angle θ for various d and α0 are shown in Figs. 3.39 and 3.40. When calculating ion torque, in contrast to forces, in addition to factors of the streamlines’ directions and the distance to the ion beam source, the distance from the point where the beam particle hits the surface to the center of mass also has an impact. In case 1, when the center of mass is in the geometric center,

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Fig. 3.38 Dependence of the ion force projection Fox on the angle θ for various angles α0 for d ¼ 15m.

Fig. 3.39 Dependence of the ion force projection Lbz on the angle θ for various distances d for α0 ¼ 15°.

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Fig. 3.40 Dependence of the ion force projection Lbz on the angle θ for various angles α0 for d ¼ 15m.

with an increase in d and α0, first an increase in the maximum value of the ion torque is observed, due to an increase in the arm of the force relative to the center of mass and a change in the directions of the streamlines, and then a decrease due to an increase in the fraction of particles passing by the surface. In case 3 (Figs. 3.39 and 3.40), the decrease in torque as d and α0 increase is determined mainly by the decrease in Foy projection shown in Figs. 3.35 and 3.36. In case 2, the increase in the modulus of the torque is due to the increase in the arms of forces relative to the center of mass as d and α0 increase. Let us consider the influence of the direction of the ion beam axis on the value of the ion torque. The direction of the ion beam axis can be set using the angle β (Fig. 3.17). Fig. 3.41 shows the surface Lbz (θ, β) obtained for case 1. The angle β has a significant effect on the magnitude and direction of the ion torque. Fig. 3.42 shows the dependency graphs that are built for different cases of the center of mass location and correspond to β angles at which the maximum and minimum values of the

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Fig. 3.41 Dependence of ion torque on the angles θ and β.

Fig. 3.42 Dependence of ion torque on the angles θ and β.

ion torque are observed. These curves define the range within which the ion torque can vary. In the general spatial case, the orientation of the cylinder relative to the orbital coordinate system can be given by three Euler angles. If the center of mass of the cylinder lies on the axis of symmetry, then its position can be specified using only two angles, since the rotation about the axis of symmetry does not play a role in the calculation of the ion force and torque.

3.5.4 Rectangular prism in ion beam The shape of a rectangular prism is determined by the three lengths of its sides Hp1, Hp2, and Hp3. The prism has three planes of symmetry. The geometric center is located at the intersection of these planes. Consider a planar case when the plane of symmetry of the prism is in the BXoYo (plane Fig. 3.43). In the case Hp1 ¼ Hp2, in addition to the

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Fig. 3.43 Rectangular prism in an ion beam.

location of the symmetry plane shown in Fig. 3.43, another arrangement of the symmetry planes of the prism is possible, when the diagonals of the base surfaces belong to the symmetry plane. As in the previous sections, consider three cases described in Section 3.5.1. The value of the ion torque relative to the center of mass in is determined by Eq. (3.86), which can be rewritten for the rectangular prism in the form Lbz3 ¼

H p3 o Hp3 o Fy cos θ  F sin θ + LI , 2 2 x

(3.91)

where LI is the ion beam torque relative to the geometric center of the prism. Consider the blowing of a rectangular prism with side lengths Hp1 ¼ Hp2 ¼ 2m and Hp3 ¼ 3m by a stream of ions with the parameters given in Table 3.2. Figs. 3.44–3.46 show the dependences of the ion force projections Fox , Foy and ion torque projection Lbz on the angle θ, calculated for the cases when the plane of symmetry of the prism is parallel to the side face (black lines) and when the plane of symmetry passes through the diagonal of the base surface of the prism (blue lines). In the diagonal case (blue lines), the area flowed around the surface turns out to be larger than in the case when the symmetry plane is parallel to the side (black lines). This circumstance mainly determines the difference in the modulus of ion force projections. It should be noted Fig. 3.44 Dependence of the ion force projection Fox on the angle θ.

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Fig. 3.45 Dependence of the ion force projection Foy on the angle θ.

Fig. 3.46 Dependence of the ion torque projection Lbz on the angle θ.

that the dependences obtained for a rectangular prism are close to the graphs obtained for a cylinder (Figs. 3.27–3.29), but the breakpoints, due to the transition of the faces into the shadow, are more pronounced in the prism case. Calculations show that the conclusions made about the influence of the distance d and the angles α0 and β on the nature of the dependences of the ion forces and torque for the case of a cylinder are also completely valid for a rectangular prism.

3.5.5 Comparison of rocket stage and cylinder In the previous sections, objects of the simplest forms have been considered. Real satellites and rocket stages have a more complex shape. Various sensors, engine nozzles, antennas, solar panels, and other equipment can be located on their surface. These irregularities affect the force and torque generated by an ion beam. In order to assess how significant this effect is, let us calculate and compare the ion force and torque obtained for the upper stage and for the cylinder corresponding to it in size. The computational grid for the rocket stage is shown in Fig. 3.47. The grid was created on the

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Fig. 3.47 Calculation grids of the rocket stage and cylinder.

basis of analysis of photographs of the Cosmos-3M upper stage, which are freely available on the Internet. As in the previous sections, three boundary cases that differ in the location of the center of mass and the direction of the ion beam axis are considered. The geometric center of the stage is a point located at the intersection of the axis and the plane of symmetry of its cylindrical tank. The radius of the cylinder is Rc ¼ 1.2 m and the height of the cylinder is Hc ¼ 4.95m. The total height of the upper stage is 6.245 m and the total width is 3.236 m. Figs. 3.48–3.50 show the dependences of the ion force and torque projections on the angle θ. The black color shows the curves corresponding to the rocket stage, and the blue color corresponds to the cylinder. It can be seen from Fig. 3.49 that the engine nozzles, additional external elements, and irregularities make a significant contribution to the ion force projection Foy compared to a cylinder corresponding in size to the stage fuel tank. The dependence curve obtained for the cylinder is close in shape to the curve for the rocket stage. Differences in absolute values are due to the fact that the surface area of the entire rocket stage significantly exceeds the area of its fuel tank. Differences in the projections Fox are less significant (Fig. 3.48). For graphs of the dependence of the ion torque Lbz on the angle θ, the points of intersection of the curves with the abscissa axis are of great importance, since these points correspond to the equilibrium positions of the angular motion of the object in the absence of external torques. In particular, in case 1, which corresponds to the case when the center of mass of the object is in the geometric

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Fig. 3.48 Dependence of the ion force projection Fox on the angle θ.

Fig. 3.49 Dependence of the ion force projection Foy on the angle θ.

Fig. 3.50 Dependence of the ion torque projection Lbz on the angle θ.

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center, the points of intersection with the abscissa for the black and blue curves are quite far apart. In addition, the blue curve, which is plotted for the cylinder, has more intersection points than the black curve. This means that the phase portrait of the cylinder has a more complex structure. The cylinder approximation of the upper stage shape is poorly suited for studying the stage dynamics under the influence of an ion beam. In cases 2 and 3, when the center of mass is on the lower base surface of the cylinder or stage fuel tank, these points of the abscissa axis intersection coincide and the curves are close to each other. In this case, the approximation of the shape of the rocket stage by a cylinder is quite acceptable.

3.5.6 Cylindrical satellite with solar panels in ion beam In modern astronautics, solar panels are widely used to provide satellites with electrical power. These panels can be located either directly on the surface of the satellites or on external deployable frames. In the latter case, the area of the outer panels can be several times greater than the surface area of the satellite itself. The ion force generated on the surface of solar panels can have a decisive influence on the resulting ion force and torque. In addition, the panels cast shadows, which also has a large effect on the resulting force and torque. To evaluate the influence of the solar panels, let us calculate the ion force and torque for a cylindrical satellite in the case of the presence of rectangular solar panels and their absence. The satellite is shown in Fig. 3.51. The radius of the cylinder is Rc ¼ 1.35 m and the height of the cylinder is Hc ¼ 3 m. The solar panels’ side lengths are Hp1 ¼ 5 m and Hp2 ¼ 3 m. It is assumed that the geometric center of the satellite is the point lying at the intersection of the axis and the plane of symmetry of the cylinder. The calculation results are given in Figs. 3.52–3.54. Calculations show that the presence of solar panels leads to a significant increase in the ion force modulus, which is due to an increase in the total surface area (Figs. 3.52 and 3.53). The black line shows the curves obtained for the satellite with deployed solar panels, and the blue line represents a cylindrical satellite without panels. The presence of panels significantly changes the form of the dependence of the ion torque on the angle θ (Fig. 3.54). The presence of panels leads to a significant increase in ion force, and hence to a decrease in the time required to complete the mission of satellite contactless transportation. However, it should be remembered that solar panels, like the satellite body, are subject to erosion, and prolonged exposure to an ion beam can lead to the formation of secondary space debris.

Fig. 3.51 The satellite with solar panels.

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Fig. 3.52 Dependence of the ion force projection Fox on the angle θ.

Fig. 3.53 Dependence of the ion force projection Foy on the angle θ.

Fig. 3.54 Dependence of the ion torque projection Lbz on the angle θ.

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Parks, D., Katz, I., 1979. A preliminary model of ion beam neutralization. In: 14th International Electric Propulsion Conference, American Institute of Aeronautics and Astronautics, Reston, Virginia, pp. 1–10, https://doi.org/10.2514/6.1979-2049. Perales-Diaz, J., Cichocki, F., Merino, M., Ahedo, E., 2021. Formation and neutralization of electric charge and current of an ion thruster plume. Plasma Sources Sci. Technol. 30, 105023. https://doi.org/10.1088/1361-6595/ac2a19. Powell, R.A., Rossnagel, S.M., 1998. PVD for Microelectronics: Sputter Disposition to Semiconductor Manufacturing. Academic Press, San Diego, CA. Roberts, L., 1964. Comments on exhaust flow field and surface impingement. AIAA J. 2, 971–973. Ruiz, M., Urdampilleta, I., Bombardelli, C., Ahedo, E., Merino, M., Cichocki, F., 2014. The FP7 LEOSWEEP project: improving low earth orbit security with enhanced electric propulsion. In: Space Propulsion Conference 2014, pp. 35–42. Shang, S., Xiang, S., Jiang, L., Wang, W., He, B., Weng, H., 2019. Sputtering distribution of LIPS200 ion thruster plume. Acta Astronaut. 160, 7–14. https://doi.org/10.1016/j. actaastro.2019.04.008. Shuvalov, V.A., Gorev, N.B., Tokmak, N.A., Kochubei, G.S., 2017. Physical simulation of the long-term dynamic action of a plasma beam on a space debris object. Acta Astronaut. 132, 97–102. https://doi.org/10.1016/j.actaastro.2016.11.039. Sigmund, P., 1981. Sputtering by ion bombardment theoretical concepts. In: Sputtering by Particle Bombardment I. Springer, pp. 9–71, https://doi.org/10.1007/3540105212_7. Sugiyama, K., Schmid, K., Jacob, W., 2016. Sputtering of iron, chromium and tungsten by energetic deuterium ion bombardment. Nucl. Mater. Energy 8, 1–7. https://doi.org/10.1016/j. nme.2016.05.016. Takegahara, H., Kasai, Y., Gotoh, Y., Miyazaki, K., Hayakawa, Y., Kitamura, S., Nagano, H., Nakamura, K., 1993. Beam characteristics evaluation of ETS-VI Xenon Ion Thruster. In: Proceedings of the 23rd International Electric Propulsion Conference, Seattle, WA, USA, 13–16 September 1993, pp. 2166–2174. Urrutxua, H., 2015. High Fidelity Models for Near-Earth Object Dynamics. Universidad Politecnica de Madrid, Madrid. VanGilder, D.B., Font, G.I., Boyd, I.D., 1999. Hybrid Monte Carlo-particle-in-cell simulation of an ion thruster plume. J. Propuls. Power 15, 530–538. https://doi.org/10.2514/2.5475. Wei, Q., Li, K.-D., Lian, J., Wang, L., 2008. Angular dependence of sputtering yield of amorphous and polycrystalline materials. J. Phys. D Appl. Phys. 41. https://doi.org/10.1088/ 0022-3727/41/17/172002, 172002. Yamamura, Y., Tawara, H., 1996. Energy dependence of ion-induced sputtering yields from monatomic solids at normal incidence. At. Data Nucl. Data Tables 62, 149–253. https:// doi.org/10.1006/adnd.1996.0005. Yim, J.T., 2017. A survey of xenon ion sputter yield data and fits relevant to electric propulsion spacecraft integration. In: 35th International Electric Propulsion Conference, IEPC-2017060. Zhang, Z.L., Zhang, L., 2004. Anisotropic angular distribution of sputtered atoms. Radiat. Eff. Defects Solids 159, 301–307. https://doi.org/10.1080/10420150410001724495. Zhang, Z., Zhang, Z., Xu, S., Ling, W.Y.L., Ren, J., Tang, H., 2021. Three-dimensional measurement of a stationary plasma plume with a faraday probe array. Aerosp. Sci. Technol. 110, 1–11. https://doi.org/10.1016/j.ast.2020.106480.

Dynamics of relative translation motion of spherical space debris during ion beam transportation 4.1

4

Mathematical model describing ion beam transportation without taking into account space debris attitude motion

4.1.1 General assumptions In this chapter, the orbital motion of a mechanical system consisting of a space debris object and an active spacecraft is considered (Fig. 4.1). The space debris object is assumed to be spherical, and its center of mass is at the geometric center of the sphere. The active spacecraft is equipped with at least two thrusters. An impulse transfer thruster generates an ion beam affecting the space debris object. The impulse compensation thruster has the main purpose of negating the effect of thrust of the impulse transfer thruster, but it can also be used for control purposes. The ion beam created by the impulse transfer thruster generates ion force on the surface of the space debris object. The issue of calculating the ion force for a sphere is analyzed in detail in Section 3.5.2. It is also supposed that the active spacecraft is equipped with an orientation system. These assumptions allow us to consider the active spacecraft and space debris as point masses and to exclude the equations of attitude motion of these bodies from consideration, since the equations of a dynamically symmetric sphere motion relative to its center of mass degenerate, and the attitude motion of the active spacecraft is determined by its propulsion and orientation systems.

4.1.2 Gauss planetary equations of space debris motion Let us compose vector equations describing the motion of the centers of mass of a space debris object and an active spacecraft. The translational dynamics equation of the space debris object in the inertial frame OXpYpZp can be described by Newton’s law: mB€rp ¼ 

μmB rp + FpI + FpBP , r3

(4.1)

where mB is the mass of the space debris, r is the space debris center of mass position vector, μ is the Earth gravitational constant, FI is the ion beam force, and FBP is the resultant of perturbation forces acting on the space debris object. The superscript in the notation of vectors denotes the coordinate system in which the vector components Attitude Dynamics and Control of Space Debris During Ion Beam Transportation. https://doi.org/10.1016/B978-0-323-99299-2.00007-0 Copyright © 2023 Elsevier Inc. All rights reserved.

168

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.1 Mechanical system consisting of a space debris object and an active spacecraft.

are given. Newton’s equation in the inertial reference frame for the active spacecraft has the form mA€rpA ¼ 

μmA rpA + FpTi + FpTc + FpTu + FpAP , r 3A

(4.2)

where mA is the mass of the active spacecraft, rA is the active spacecraft position vector, FTi is the thrust force of the spacecraft’s impulse transfer thruster, FTc is the thrust force of the impulse compensation thruster, FTu is the thrust force of control engines directed perpendicular to the ion beam axis, and FAP is the resultant of perturbation forces acting on the spacecraft. The force FTc may not coincide in absolute value with the impulse transfer thruster force FTi, since in addition to performing the function of compensating this thrust, it also, together with FTu, implements the control law that provides the required relative position of the active spacecraft. As mentioned in Section 1.4, using Eq. (4.2) to describe the relative motion of the active spacecraft is not a very good solution. The location of the center of mass of the
! active spacecraft relative to space debris can be given by the vector ρ ¼ BA ¼ rA  r ¼ ½x, y, zT . The differential equation for the vector ρ in the inertial reference frame OXpYpZp can be written using Eqs. (4.1) and (4.2) in the form p

€p ¼ ρ

μrp μrA  3 + apA  apB : r3 rA

(4.3)

For this vector, the differential equation in the form (3.87) can be written for the noninertial Hill’s coordinate system BXHYHZH:   dωH d 2 ρH dρH Hp  ρH  ωH ¼  ωH  ρH  2ωH  Hp Hp Hp 2 dt dt dt   H H H r ðr  ρ Þ μ H  3 ρH + 3 + aH A  aB , r r2

(4.4)

Relative translation motion of spherical space debris

169

H where the acceleration vectors aH A and aB , according to Eqs. (4.1) and (4.2), are determined by the expressions.

aH A ¼

 H  1  H 1  H FTi + FH + FH + FH FI + F H , aB ¼ : Tc Tu AP BP mA mB

(4.5)

The angular velocity vector ωH Hp defines the rotation of the Hill’s frame BXHYHZH relative to the inertial frame OXpYpZp. If Newton’s equation (4.1) is used to describe space debris motion, then the angular velocity vector can be found as ωpHp ¼

rp  r_ p : r2

(4.6)

Calculating the derivative with respect to time gives ω_ pHp ¼

rp  €rp r_  2 2 ωpHp : r r2

Substituting here the second derivative of the position vector from Eq. (4.1) yields ω_ pHp ¼

rp  apB r_  2 ωpHp : r r2

(4.7)

Taking into account that according to the transport theorem H H _ pHp ¼ ω _H _H MHp ω Hp + ωHp  ωHp ¼ ω Hp ,

Eq. (4.7) in a noninertial coordinate system takes the form ω_ H Hp ¼

r H  aH r_ B  2 ωH : r Hp r2

(4.8)

T The cross-product in Eq. (4.8) gives rH  aH B ¼ [0,  raBz, raBy] . To use the angular velocity (4.6) in Eq. (4.4), it is necessary to recalculate the p coordinates as ωH Hp ¼ MHpωHp. The components of the transition matrix MHp are determined by the current coordinates of the rp and r_ p vectors. The corresponding formulas can be found at the end of Section 1.3. In practice, it is much more convenient to use the Gauss planetary equations (1.47), (1.53), (1.56), (1.61), (1.65), and (1.69) instead of Eq. (4.1).

a_ ¼

  2a2 p aBx e sin f + aBy , r h

         1 e_ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1  e2 aBx sin f + a 1  e2 + r cos f + re aBy , 2 μað1  e Þ

170

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

di r cos u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aBz , dt μað1  e2 Þ

(4.9)

r sin u Ω_ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aBz , μað1  e2 Þ sin i ω_ ¼ 

haBx cos f + μe

M_ ¼ n +

  2 + e cos f haBy sin f raBz cos i sin u ,  1 + e cos f h sin i μe

pffiffiffiffiffiffiffiffiffiffiffiffiffi      2er r 1  e2 h  1  sin f , cos f  a a e sin f μ að1  e2 Þ Bx að1  e2 Þ By

Ðt where a, e, i, Ω, ω, M0 ¼ M  nðτÞdτ are the space debris object’s orbit elements, 0

which are described in Section 1.3, and acceleration vector components T aH B ¼ [aBx, aBy, aBz] are determined by Eq. (4.5). When using Eq. (4.9), the compoT nents of the angular velocity vector ωH Hp ¼ [ωHpx, ωHpy, ωHpz] can be found using expression (1.11): _ sin i sin u + i_ cos u, ωHpy ¼ Ω _ sin i cos u  i_ sin u, ωHpx ¼ Ω _ cos i + ω_ + f_: ωHpz ¼ Ω

(4.10)

Projection of the vector equation (4.4) onto the axes of the Hill’s reference frame allows us to write the equations of the relative motion of the active spacecraft in the form of Eqs. (1.88)–(1.90). To calculate the components of the derivative of the angular velocity, Eq. (4.8) can be used, where r_ is defined by Eq. (1.39) using Eq. (1.37). The only assumption made at this stage is the smallness of the distance ρ in comparison with r.

4.1.3 Equations of motion in a spherical reference frame The equations of motion written in Section 4.1.1 were obtained by directly using Newton’s second law and the transport theorem. The purpose of this section is to obtain the equations of motion of the considered mechanical system based on the Lagrange formalism. To compose the Lagrange equations in the form (1.4), it is necessary to determine the Lagrange function, which is defined as the difference between the kinetic T and potential P energies of the system. The considered mechanical system consists of two particles with masses mA and mB. The state of the system can be described by six parameters that form a vector of generalized q ¼ [ϑ, ν, r, x, y, z]. The angles ϑ, ν, and the distance r determine the position of point B, and the coordinates x, y, and z define the relative position of point A in the spherical orbital reference frame BXoYoZo (Fig. 4.2). To calculate the kinetic energy, it is necessary to determine the absolute velocities of points A and B. This can be done by computing the time derivative of the position

Relative translation motion of spherical space debris

171

Fig. 4.2 Generalized coordinates.

vectors r and rA (Fig. 4.2) given in the inertial coordinate system OXpYpZp in terms of the generalized coordinates rp ¼ ½r cos ν cos ϑ, r cos ν sin ϑ, r sin νT ,

(4.11)

rpA ¼ rp + ρp ¼ rp + Mpo ρo ,

(4.12)

Substituting Eq. (4.11) into Eq. (4.12), taking into account the rotation matrix given in Section 1.2, after simple algebraic transformations, gives 2

3 ðr + xÞ cos ν cos ϑ  z sin ν cos ϑ  y sin ϑ 6 7 rpA ¼ 4 ðr + xÞ cos ν sin ϑ  z sin ν sin ϑ + y cos ϑ 5:

(4.13)

ðr + xÞ sin ν + z cos ν Calculation of the time derivative of the vector (4.11) gives 2

3 r_ cos ν cos ϑ  r ϑ_ cos ν sin ϑ  r ν_ sin ν cos ϑ 6 7 r_ p ¼ 4 r_ cos ν sin ϑ  r ϑ_ cos ν cos ϑ  r ν_ sin ν sin ϑ 5:

(4.14)

r_ sin ν + r ν_ cos ν The magnitude of this vector is determined by the expression qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V B ¼ |_r | ¼ r_2 + r 2 ν_ 2 + r 2 ϑ_ cos 2 ν: p

(4.15)

Similarly, from Eq. (4.13) the derivative can be obtained in the form 2

   3 V A1 cos ν  V A2 sin ν  yϑ_ cos ϑ  ðr + xÞϑ_ cos ν  zϑ_ sin ν + y_ sin ϑ    6 7 r_ pA ¼ 4 V A1 cos ν  V A2 sin ν  yϑ_ sin ϑ + ðr + xÞϑ_ cos ν  zϑ_ sin ν + y_ cos ϑ 5, V A2 cos ν + V A1 sin ν (4.16)

172

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

where V A1 ¼ r_ + x_  z_ν and V A2 ¼ ðr + xÞ_ν + z._ The velocity magnitude of point A is determined by the expression   2 _ + zy V A ¼ |_rpA | ¼ ðr + xÞ2  z2 ϑ_ cos 2 ν + 2ϑ_ ðν_ yðr + xÞ  yz _ Þ sin ν       _ + x_ Þ  zν_ Þy cos ν ðr + xÞ2 + z2 ν_ 2 2ϑ_ ðr + xÞ zϑ_ sin ν  y_ + ððry 1=2   2 + 2ν_ ðz_ðr + xÞ  zðr_ + x_ ÞÞ + y2 + z2 ϑ_ + ðr_ + x_Þ2 + y_2 + z_2 : (4.17) The kinetic energy of a mechanical system is determined by the expression T¼

mA V 2A mB V 2B + , 2 2

(4.18)

where VA and VB are determined by expressions (4.17) and (4.15). The potential energy of particles in the central gravitational field is defined as P¼

μmA μmB  , rA r

(4.19)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r A ¼ ðr + xÞ2 + y2 + z2 . Substituting Eqs. (4.18) and (4.19) into Eq. (1.4) allows us to write the equations of motion which can be brought into the form r€ ¼ 

  Q Q 2 μ x + ν_ 2 + ϑ_ cos 2 ν + r , 2 mB r

(4.20)

€ν ¼ 

Q + Qx z  Qz ðr + xÞ 2ν_ r_ _ 2  ϑ cos ν sin ν + ν , r mB r 2

(4.21)

_ _ € ¼  2ϑr_ + 2ϑ_ν sin ν ϑ r cos ν Qϑ + Qx y cos ν + Qy ðz sin ν  ðr + xÞ cos νÞ  Qz y sin ν , (4.22) mB r 2 cos 2 ν   _ + yϑ_ 2r_ νz 2 2 2 _ ν + 2ϑ_ y_ cos ν  zϑ_ sin 2ν + xϑ_ cos 2 ν + 2ϑ_ νysin x€¼ ν_ x + 2_ν z  r   yQϑ zQν μ μðr + xÞ Qr 1 r 2 + y 2 + z2 + 2 + + +   Qx mA r mB r 2 cos 2 ν mB mB r 2 mB r 2 rA3 +



yððr + xÞcos ν  zsin νÞQy ðzðr + xÞcos ν + y2 sin νÞQz  , mB r 2 cos 2 ν mB r 2 cos 2 ν

ð4:23Þ

Relative translation motion of spherical space debris

y€¼

173

2ϑ_ r_ ðx cos ν  z sinνÞ 2 _ 2ϑ_ νz μy Qy + 2ϑ_ z_ sin ν + ϑ_ y  2ϑ_ x_ cos ν  3 + + r cos ν rA m A

ððr + xÞcos ν  zsin νÞðQϑ + yQx cos νÞ ððr + xÞsin ν cos ν  zsin 2 νÞyQz  mB r 2 cos 2 ν mB r 2 cos 2 ν    ðr + xÞ2  z2 cos 2 ν  zðr + xÞsin 2ν Qy , ð4:24Þ + mB r 2 cos 2 ν    2  2r_ yϑ_ sin ν + ν_ x 2 z€¼ z ϑ_ + ν_ 2  2_ν x_  2ϑ_ y_ sin ν + 2ϑ_ ν_ ycos ν  zϑ_ cos 2 ν + r 

ðr + xÞQν ðzðr + xÞcos ν + y2 sin νÞQx Qz _ 2ϑ_ νy μz y sin νQϑ  3   + 2 2 cos ν rA mB r cos ν mA mB r 2 mB r 2 cos ν    ðr + xÞ2  y2 cos 2 ν + y2 Qz ððr + xÞ cos ν  z sin νÞyQy sin ν + + : ð4:25Þ mB r 2 cos 2 ν mB r 2 cos 2 ν



To calculate the generalized forces, Eq. (1.5) is used:   ∂rp   ∂rp H + F Qj ¼ FpTi + FpTc + FpTu + FpAP  A + FH I BP  ∂q ∂qj j p ∂rA ∂rp p p ¼ mA aA  + mB aB  , ∂qj ∂qj

(4.26)

where qj is the component of the generalized coordinates vector q. Using expressions (4.11) and (4.13), the generalized forces (4.26) take the form Qr ¼ mA aAx + mB aBx , Qν ¼ mA aAx z + mA ðr + xÞaAz + mB raBz , Qϑ ¼ mA aAx y cos ν + mA ððr + xÞ cos ν  z sin νÞaAy + mA yaAz sin ν + mB raBy cos ν,

(4.27)

Qx ¼ mA aAx , Qy ¼ mA aAy , Qz ¼ mA aAz , where aoA ¼ [aAx, aAy, aAz]T and aoB ¼ [aBx, aBy, aBz]T are accelerations given by components in the orbital spherical coordinate system BXoYoZo. Substituting the generalized forces (4.27) into the equations of motion (4.20)–(4.25) gives r€ ¼ 

  μ 2 _ 2 cos 2 ν + aBx , _ + ν + ϑ r2

(4.28)

€ν ¼ 

2ν_ r_ _ 2 a  ϑ cos ν sin ν + Bz , r r

(4.29)

174

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

_ _ € ¼  2ϑr_ + 2ϑ_ν sin ν + aBy , ϑ r cos ν r cos ν   2r_ ν_ z + yϑ_ 2 2 2 _ νy sin ν _  x€ ¼ ν_ x + 2νz  zϑ_ sin 2ν + xϑ_ cos 2 ν + 2ϑ_ r yaBy + zaBz μ μ ðr + x Þ + 2ϑ_ y_ cos ν + 2  , + aAx  aBx + 3 r r rA y€ ¼

2ϑ_ r_ðx cos ν  z sin νÞ 2 _ 2ϑ_ νz + 2ϑ_ ðz_ sin ν  x_ cos νÞ + ϑ_ y + r cos ν   μy z sin ν x  3 + aAy + aBy  1 , r cos ν r rA

 2  2 z ¼ z ϑ_ + ν_ 2  2_νx_  2ϑ_ y_ sin ν + 2ϑ_ ν_ y cos ν  zϑ_ cos 2 ν €   2r_ yϑ_ sin ν + ν_ x aBy y sin ν ðr + xÞaBz 2ϑ_ ν_ y μz +  + aAz   :  cos ν r 3A r r r cos ν

(4.30)

(4.31)

(4.32)

(4.33)

The obtained system of equations describes the perturbed motion of the mechanical system consisting of the space debris object and the active spacecraft in the presence of interaction between them through the ion beam.

4.2

Calculation of fuel consumption and thruster plume parameters

One of the most important indicators of the effectiveness of an active space debris removal mission is the mass of fuel that needs to be spent to carry out the transport operation. It is known that the thrust force FT of a spacecraft engine can be expressed as follows (Kluever, 2018): _ eff , FT ¼ mV

(4.34)

where m is the fuel mass, m_ is the mass flow rate, and Veff is the thruster’s effective exhaust velocity, which can be defined by the expression V eff ¼ V e +

ðpe  pa ÞSe , m_

(4.35)

where Ve is the fuel gas exhaust velocity, pe is the exhaust pressure at the nozzle exit, pa is the atmosphere pressure, and Se is the nozzle area. In practice, a quantity called specific impulse is often used to describe the parameters of rocket engines. The specific impulse Isp is calculated as I sp ¼

FT , _ 0 mg

(4.36)

Relative translation motion of spherical space debris

175

where g0 ¼ 9.80665 m/s2 is the Earth gravitational acceleration at sea level. Substituting Eq. (4.34) into Eq. (4.36) and expressing the effective exhaust velocity gives V eff ¼ g0 I sp :

(4.37)

An ion beam-assisted active space debris removal mission assumes the use of electric propulsion low-thrust thrusters. An overview of modern electric engines can be found in papers by Holste et al. (2020) and Lev et al. (2019). The thruster _ 2eff =2 to efficiency η is the ratio of the output power of the jet exhaust Pout ¼ mV the input electric power Pin: η¼

_ 2eff Pout mV ¼ : Pin 2Pin

(4.38)

Expressing Veff from Eq. (4.34) and substituting the result into Eq. (4.38) yields η¼

FT V eff : 2Pin

(4.39)

Expressing FT from Eq. (4.39), taking into account Eq. (4.37), gives FT ¼

2ηPin : g0 I sp

(4.40)

From Eq. (4.40), it follows that the greater the thrust force created by the electric propulsion low-thrust thrusters is, the greater the input power. An increase in specific impulse leads to a decrease in force. The input electric power determines the mass of the spacecraft’s power system mps. For a rough estimate, the following approximate expression can be used: mps ¼ 2αsm Pin ¼

αsm FT V eff , η

(4.41)

where αsm is the inverse of the specific power or specific mass of the power generation system (Bombardelli and Pelaez, 2011). The paper by Urrutxua et al. (2019) proposed using more complex approximations for force and mass flow rate: FT ¼ a + bPin ,

(4.42)

m_ ðPin Þ ¼ a0 + b0 Pin + c0 P2in ,

(4.43)

where the parameters a, b, a0 , b0 , and c0 were determined empirically by the characteristics of a particular ion thruster. For example, NSTAR xenon 30 cm ring-cusp ion thruster parameters can be found in the paper by Marcucci and Polk (2000). This ion

176

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

engine was used on the Deep Space 1 spacecraft as part of NASA’s New Millennium Program. The spacecraft was launched on October 24, 1998. Its primary mission was in the study of the Braille asteroid. The Deep Space 1 spacecraft set a record for the longest operation of an engine in space. The total operating time of the engine was 16,265 h, while 73.4 kg of fuel was consumed out of 83 kg. In parallel with the Deep Space 1 mission, a ground experiment with an identical ion engine was conducted from 1998 to 2003 at the Jet Propulsion Laboratory. The time of active operation of the ion thruster was 30,352 h. During this time, 235.1 kg of xenon propellant was spent (Sengupta et al., 2004). Table VII in the work by Marcucci and Polk (2000) contains data on the dependences of the engine’s thrust force and specific impulse on input power. The data for the mass flow rate can be obtained using Eq. (4.36). The parameters a, b, a0 , b0 , and c0 of Eqs. (4.42) and (4.43) calculated by the least-squares method on the basis of these data are given in Table 4.1. Figs. 4.3–4.5 show the corresponding data and their comparison with the approximation curves, which were obtained as a result of the approximation of the dependences FT and m_ by polynomials (4.42) and (4.43). The solid black line represents approximations (case 1 in Table 4.1) that do not take into account the first two data points, which deviate sharply in Fig. 4.4. The dotted line (case 2 in Table 4.1) is the result of using the entire data set.

Table 4.1 The parameters of Eqs. (4.42) and (4.43). Parameters

a, N

b, N/W

a0 , kg/s

b0 , kg/(sW)

c0 , kg/(sW2)

Case 1 (solid line on Figs. 4.3–4.6) Case 2 (dashed line on Figs. 4.3–4.6)

2.981  103

3.831  105

7.919  108

1.171  109

1.106  1014

1.431  103

3.750  105

5.881  107

5.561  1010

1.609  1013

Fig. 4.3 Dependence of the thrust force FT on the input power Pin for NSTAR thruster.

Relative translation motion of spherical space debris

177

Fig. 4.4 Dependence of the mass flow rate m_ on the input power Pin for NSTAR thruster.

Fig. 4.5 Dependence of the specific impulse Isp on the input power Pin for NSTAR thruster.

Eqs. (4.42), (4.43), and (4.36) allow us to write the voltage dependence of the specific impulse in the form I sp ðPin Þ ¼

g0



a0

a + bPin , + b0 Pin + c0 P2in

(4.44)

Fig. 4.5 demonstrates the data given by Marcucci and Polk (2000) and approximation (4.44) with the coefficients from Table 4.1. Expressing the power from Eq. (4.42) and substituting it into Eq. (4.44) gives the dependence of the specific impulse on the thrust force I sp ðFT Þ ¼



b 2 FT

g0 b2 a0 + bb0 ðFT  aÞ + c0 ðFT  aÞ2

:

(4.45)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

The thruster efficiency η, taking into account Eqs. (4.42) and (4.44), can be written as η¼

ða + bPin Þ2  : 2Pin a0 + b0 Pin + c0 P2in

(4.46)

The thruster efficiency can be rewritten as a function of the thrust force FT by the substitution Pin ¼ (FT  a)b1. Using Eqs. (4.42) and (4.43), the dependence of fuel flow rate on thrust can be obtained as m_ ¼ a0 +

b0 ðFT  aÞ c0 ðFT  aÞ2 + : b b2

(4.47)

To determine the mass of propellant required for a maneuver, the rocket equation can be used (Kluever, 2018). Assuming that the spacecraft motion occurs under the action of the thrust force only, Newton’s equation can be written in the form msc V_ sc ¼ FT ,

(4.48)

where msc ¼ m0  m is the spacecraft mass, m0 ¼ const is the initial mass, m is the mass of fuel consumed, and Vsc is the spacecraft velocity. Taking into account Eqs. (4.34), (4.48) can be rewritten in the form _ eff mV m_ sc V eff V_ sc ¼ ¼ , msc msc

(4.49)

From Eq. (4.49), taking into account Eq. (4.37), it follows that dV sc ¼ 

g0 I sp dmsc : msc

(4.50)

Integration of Eq. (4.50) gives the rocket equation   m0 ΔV ¼ g0 I sp ln , mf

(4.51)

where mf is the final mass of the spacecraft after the propulsive burn and ΔV is the velocity increment that can be achieved by the thruster. Eq. (4.51) shows the maximum velocity increment that can be obtained by a thruster with the specific impulse Isp with a fuel consumption of mp ¼ m0  mf. From the rocket equation (4.51), it follows that the fuel mass can be obtained as    ΔV mp ¼ m0 1  exp , g0 I sp

(4.52)

Relative translation motion of spherical space debris

179

According to this equation, an increase in specific impulse leads to a decrease in the required mass. For impulse maneuvers, the required velocity increment ΔV is calculated quite simply as an instantaneous change in the velocity vector of the initial and the transfer orbit. For low-thrust thrusters, calculating ΔV is a more difficult task. The procedure for calculating the low-thrust velocity increment can be found in Chapter 9 of the book by Kluever (2018). In particular, low-thrust ΔV for a circle-to-circle transfer is greater than corresponding impulsive ΔV. Instead of using Eq. (4.52) and finding the velocity increment, Eq. (4.47) can be numerically integrated to calculate the fuel mass with a known force-time dependence. As mentioned in Section 4.1.2, the active spacecraft is equipped with several engines that generate forces FTi, FTc, and FTu (Fig. 4.6). To calculate the required total propellant mass mprop, the dependence of the thrusters’ flow rate on the thrust force is required m_ prop ¼ m_ Ti ðFTi Þ + m_ Tc ðFTc Þ + m_ Tu ðFTu Þ,

(4.53)

where mTi is the fuel mass of the impulse transfer thruster, mTc is the fuel mass of the impulse compensation thruster, and mTu is the fuel mass of control engines directed perpendicular to the ion beam axis. If Eq. (4.53) is a known function obtained experimentally or analytically, then the total mass can be obtained as ðtk mprop ¼ m_ Ti tk +

ðtk m_ Tc ðFTc Þdt +

0

m_ Tu ðFTu Þdt,

(4.54)

0

where m_ Ti ¼ const and tk is the mission duration time. Calculating the thruster parameters a, b, a0 , and b0 on the basis of experimental data and substituting Eq. (4.47) into Eq. (4.54), it is possible to determine the amount of fuel required for the mission. To calculate the ion force and torque acting on a space debris object using methodology described in Section 3.4, it is necessary to have some parameters of the ion beam, in particular, the plasma density at the beginning of the far region n0, the axial component of the ion velocity u0, the divergence angle of the ion beam α0, the ion particle mass mi, and the radius of the beam at the beginning of the far region R0. The flow behavior

Fig. 4.6 Forces generated by the active spacecraft engines.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.7 Plasma density calculation.

around the engine nozzle is quite complex and depends on many factors. It is assumed that the ion beam axial velocity is equal to the effective exhaust velocity of the thruster u0 ¼ V eff ,

(4.55)

and the thruster nozzle has the shape of a circle of radius R0. To calculate plasma density n0, consider an infinitesimal volume dV ¼ πR20dx near the nozzle, where dx ¼ u0dt is the distance over which the flow particle moves during the time dt (Fig. 4.7). During this time, N ¼ n0dV particles fly out of the nozzle. The mass of these particles is defined as dm ¼ mi N ¼ mi n0 dV ¼ mi n0 πR20 dx ¼ mi n0 πR20 u0 dt:

(4.56)

Division of the right-hand and left-hand sides of expression (4.56) by dt gives m_ ¼ mi n0 πR20 u0 :

(4.57)

Expressing the particle density from Eq. (4.57) yields n0 ¼

m_ : mi πR20 u0

(4.58)

Taking into account Eqs. (4.34), (4.55), and (4.47), expression (4.58) can be rewritten as ða0 b + b0 ðFT  aÞÞ m_ 2 ¼ , mi πR20 FT mi πR20 b2 2

n0 ¼

(4.59)

where FT ¼ FTi is the force generated by the impulse transfer thruster. Fig. 4.8 shows a comparison of the particle density (4.58) calculated using Eqs. (4.55) and (4.37) and the data from Table VII in the paper by Marcucci and Polk (2000), with the approximation curves (4.59) with the coefficients given in Table 4.1.

Relative translation motion of spherical space debris

181

Fig. 4.8 Dependence of the particle density n0 on the force FT for NSTAR thruster.

Eq. (4.59) is approximate, but it is quite suitable for rough estimates. When planning a real mission, the particle density and other engine parameters can be determined experimentally.

4.3

Estimation of the thruster exhaust velocity and propellant mass for space debris deorbiting from a circular orbit

Two of the first tasks, which were solved when developing the scheme of a contactless space debris removal mission using an ion beam, were the estimation of fuel costs and the determination of the optimal thruster exhaust velocity (Bombardelli and Pelaez, 2011). The total mass of the active spacecraft, which was called Ion Beam Shepherd, was considered as an optimality criterion. The mission of space debris removal was considered in the simplest setting, while a number of assumptions were made. It was assumed that at the initial moment of time, the active spacecraft is near the space debris object, and they move along the same circular orbit. The force generated by the impulse transfer thruster is transferred to the space debris object as FI ¼  ηBFTi, where ηB is the momentum transfer efficiency. The impulse transfer thruster and the impulse compensation thruster have the same exhaust velocities Veffi ¼ Veffc ¼ Veff and mass flow rates _ It was assumed that the thrust force FTi is constant, whence it follows also m_ i ¼ m_ c ¼ m. that m_ ¼ const and Veff ¼ const. The motion of the system occurs in the central Newtonian gravitational field, while no perturbing forces are taken into account, FAP ¼ FBP ¼ 0. The planar motion of the system is considered. It is supposed that the control system of the spacecraft keeps it in a relative position ρH ¼ [0, y, 0] (Fig. 4.9). Assuming the hypothesis that the active spacecraft follows the space debris object all the time, and the orbit is circular, it follows from Eq. (4.4) that y€ ¼

FTi  FTc F + I ¼ 0: mA mB

(4.60)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.9 Considered mechanical system.

Taking into account that FI ¼ ηBFTi, the expression from Eq. (4.60) of force FTc gives   m FTc ¼ FTi 1 + ηB A : mB

(4.61)

If the space debris object has a spherical shape, and the axis of the ion beam is directed to the geometric center of the sphere, and the ion beam is Gaussian, then the momentum transfer efficiency coefficient can be determined using the approximate analytical formula given by Ruiz et al. (2014):  ηB ¼ 1  exp 

 3χ 2 , 1  χ 2 tan 2 α0

(4.62)

where α0 is the ion beam divergence angle, χ ¼ Rsp(ρ tan α0)1, and Rsp is the sphere radius. When deriving this formula, it was assumed that the sphere goes beyond the boundaries of the ion beam cone. This condition is met when ρ  R/ sin α0. The active spacecraft mass is defined by the sum mA ¼ mprop + mps + mstr ,

(4.63)

where mprop is the total propellant mass spent by control system engines, impulse transfer and compensation thrusters, mps is the power system mass, which depends on the power consumption of the thrusters, and mstr is the structural mass. If it is assumed that mA ≪ mB, then FTc  FTi, and the total propellant mass can be found by Eq. (4.54): mprop ¼ 2m_ Ti tk ,

(4.64)

where tk is the mission duration time. Expressing Veff from Eq. (4.34) and substituting the result into (4.64) gives mprop ¼

2FTi tk , V eff

(4.65)

Relative translation motion of spherical space debris

183

Taking into account Eqs. (4.41) and (4.65), the total mass of the spacecraft is mA ¼

αsm FTi V eff 2FTi tk + + mstr : V eff η

(4.66)

As can be seen from Eq. (4.66), the mass of the spacecraft depends on the exhaust velocity Veff. To find the value of the velocity that provides the minimum spacecraft mass, let us find the derivative of the mass with respect to the exhaust velocity and equate the result to zero: 

2FTi tk α F + sm Ti ¼ 0: 2 η V eff

(4.67)

The expression from this equation of the velocity Veff gives the value corresponding to the optimal exhaust velocity V opt

rffiffiffiffiffiffiffiffi 2ηtk , ¼ αsm

(4.68)

which is known as the Irving-Stuhlinger characteristic velocity (Stuhlinger, 1964). Expression (4.68) allows us to determine the specific impulse Isp opt ¼ Vopt/g0 that the spacecraft’s thruster should have. Substituting Eq. (4.68) into Eq. (4.65) gives the required mass of fuel: mopt prop

rffiffiffiffiffiffiffiffiffiffiffiffi 2αsp tk ¼ FTi : η

(4.69)

The optimal mass of the active spacecraft is as follows (Bombardelli and Pelaez, 2011): mopt A

rffiffiffiffiffiffiffiffiffiffiffiffi 2αsp tk ¼ 2FTi + mstr : η

(4.70)

Expressions (4.68) and (4.70) are approximate, since they do not take into account the change in the mass of the active spacecraft caused by the fuel burnout during contactless transportation and the change in the thrust of the impulse compensation thruster. However, these formulas are very useful for initial estimation of spacecraft mass and thrusters’ specific impulse before detailed mission design. Expressions (4.68) and (4.70) show that in order to calculate these estimates, it is required to know the mission duration time tk. For the simplest case of a space debris object in a circular orbit, this time was found analytically in the study by Bombardelli and Pelaez (2011). It is assumed that the ion force FI generated by the ion beam is constant in magnitude and directed along the tangent to the orbit. In the Hill’s reference frame BXHYHZH, the T ion force vector has coordinates FH I ¼ [0,  FI, 0] (Fig. 4.9). During the entire space debris deorbiting mission, its orbit evolves in a quasicircular manner. These

184

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

assumptions are quite justified, since many space debris objects move in close to circular orbits, and the ion force is small in absolute value (about 0.2 N), which leads to a small change in the eccentricity for heavy space debris objects. It follows from the first equation in the perturbed Gaussian equations (4.9) that the semimajor axis changes under the action of the ion force as a_ ¼ 

2a2 FI : h mB

(4.71)

Taking into account that for a circular orbit the massless angular momentum is deterpffiffiffiffiffi mined by the expression h ¼ aμ, expression (4.71) can be rewritten as da 2a3=2 FI ¼  pffiffiffi : dt μ mB

(4.72)

The variables in this differential equation can be separated: ðtk

pffiffiffi ðrk mB μ da dt ¼  , 2FI a3=2

(4.73)

r0

0

Calculation of integrals in Eq. (4.73) gives  pffiffiffi  mB μ 1 1 pffiffiffiffi  pffiffiffiffi , tk ¼ FI rk r0

(4.74)

where r0 is the radius of the space debris object orbit before the mission and rk is the final orbit radius. Substituting the obtained mission duration time (4.74) into expressions (4.69) and (4.70), taking into account that FI ¼ ηBFTi, allows us to determine the required propellant mass and the entire spacecraft’s mass: mopt prop

mopt A

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi  2FTi mB αsm μ 1 1 pffiffiffiffi  pffiffiffiffi : ¼ ηηB rk r0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi  2FTi mB αsm μ 1 1 pffiffiffiffi  pffiffiffiffi + mstr : ¼2 ηηB rk r0

(4.75)

(4.76)

Using expressions (4.68) and (4.74), the expression for the thruster exhaust velocity can be written in the form

V opt

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi  2ηmB μ 1 1 pffiffiffiffi  pffiffiffiffi : ¼ αsm ηB FTi rk r0

(4.77)

Relative translation motion of spherical space debris

185

In order to estimate the magnitude of the required fuel mass and the specific impulse of the active spacecraft’s thruster for the mission of an active space debris deorbiting from a circular orbit, let us carry out a series of calculations using the obtained above equations. As the parameters of the active spacecraft and its thruster, the numerical values from the article by Aslanov and Ledkov (2021) are used. It is assumed that the thruster efficiency is η ¼ 0.7, the impulse transfer thruster force is FTi ¼ 0.2 N, the power generation system specific mass is αsm ¼ 75 kg/kW, the momentum transfer efficiency is ηB ¼ 0.1615 (which corresponds to the ion force FI ¼ 0.0323 N), the initial orbit radius is r0 ¼ 7, 871, 000 m (which corresponds to an altitude of 1500 km), and the final orbit radius is rk ¼ 6, 471, 000 m (which corresponds to an altitude of 100 km). For comparison, we also consider the ideal case when ηB ¼ 1 and the transfer of momentum from the ion beam particles to space debris occurs without loss. The calculation results for space debris objects of different masses are shown in Figs. 4.10–4.13. Calculations show that at a constant engine thrust of 0.2 N and momentum transfer efficiency ηB ¼ 0.1615 descent of space debris of 5 tons mass requires 986 kg of fuel. The entire maneuver takes 1312 days. The optimal specific impulse is 4690 s. The total power consumed by two engines is 13.142 kW. The high fuel consumption and the Fig. 4.10 Duration time of the space debris removal mission.

Fig. 4.11 Mass of spent propellant for the space debris removal mission.

186

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.12 Optimal specific impulse for the space debris removal mission.

Fig. 4.13 Input electric power for the space debris removal mission.

duration of the mission are caused by the fact that most of the ion beam is spent inefficiently. Part of the beam’s particles may pass by the space debris object. The curvature of the surface also negatively affects the magnitude of the ion force generated by the beam. In the ideal case, when ηB ¼ 1, the duration of the mission is just 212 days. In this case, the fuel mass consumption is 396 kg. The optimal specific impulse is 1885 s, and the power consumption is 5.281 kW. Due to the assumptions made, expression (4.75) gives underestimated values for the fuel mass. In order to understand how accurate these estimates are, let us numerically integrate the Gauss planetary equations (4.9), which for the considered case take the form a_ ¼

2a2 aBy ð1 + e cos f Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , μað1  e2 Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1  e2 Þðe + e cos 2 f + 2 cos f ÞaBy e_ ¼ , pffiffiffi μð1 + e cos f Þ ω_ ¼

aBy

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1  e2 Þ sin f ð2 + e cos f Þ , pffiffiffi μeð1 + e cos f Þ

(4.78)

Relative translation motion of spherical space debris

f_ ¼

pffiffiffi μð1 + e cos f Þ2 a3=2 ð1  e2 Þ3=2

187

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1  e2 Þð2 + e cos f ÞaBy sin f  , pffiffiffi e μð1 + e cos f Þ

where aBy ¼  ηBFTi/mB. The inclination angle i and the argument of the ascending node Ω are constant. To calculate the fuel consumption, the system of equations (4.78) must be supplemented by Eq. (4.53), which, taking into account Eq. (4.61), can be written in the form m_ prop ¼

FTi V eff

  m 2 + ηB A : mB

(4.79)

The mass of the active spacecraft changes with time as mA ¼ mstr + mpsi + mpsc + mprop0  mprop ðtÞ,

(4.80)

where mprop0 is the initial fuel supply, and mpsi and mpsc are the mass of impulse transfer and compensation thrusters power system, which are determined using expression (4.41). Since the mass of an active spacecraft mA decreases with time as a result of fuel burnup, the magnitude of the compensation force FTc determined by expression (4.61) is also decreased. Therefore, the value of the maximum force FTc should be used for the mass mpsc calculation, which corresponds to the initial moment of time when all the fuel is intact and the mass of the spacecraft is maximum. αsm FTi V eff , η   αsm FTi V eff mstr + mpsi + mpsc + mprop0 1 + ηB ¼ , η mB

mpsi ¼ mpsc

(4.81)

The expression mpsc from the last expression gives mpsc

   αsm FTi V eff mB + ηB mstr + mpsi + mprop0 ¼ : ηmB + αFTi V eff ηB

(4.82)

The results of numerical integration of the system of differential equations (4.78) and (4.79) showed that the mission duration time tk coincides with the analytical solution (4.74). Two calculations were performed for a space debris removal mission with a mass of mB ¼ 5000 kg from an altitude of 1500 to 100 km for ηB ¼ 0.1615 and ηB ¼ 1. Since Eq. (4.78) has a singular point at zero eccentricity, the value e0 ¼ 106 was chosen as the initial value. According to Fig. 4.12, in the first case, Veff ¼ g0Isp ¼ 45995 m/s, and in the second case, Veff ¼ 18484 m/s. It is assumed that mstr ¼ 150 kg. In the first case, mpsi ¼ 985.6 kg, mpsc ¼ 1022.6 kg, and mprop0 ¼ 1050 kg, and in the second case, mpsi ¼ 396.1 kg, mpsc ¼ 440.1 kg, and mprop0 ¼ 450 kg. The results of numerical integration for these cases give mprop ¼ 1036.9 kg and mprop ¼ 444.1 kg, respectively. Graphs of the space debris orbit semimajor axis change in time are shown in Fig. 4.14. The dependence of the eccentricity change on time is shown in Fig. 4.15. It can be seen that the orbit of space debris remains quasicircular throughout the entire mission.

188

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.14 Dependence of the semimajor axis on time.

Fig. 4.15 Dependence of the space debris orbit eccentricity on time.

4.4

Active spacecraft relative position stability and control

In the previous section, it was shown that the use of an ion beam makes it possible to carry out the mission of space debris deorbiting. The mission requires control of the relative position of an active spacecraft, since this position largely determines the magnitude of the momentum transfer efficiency ηB and the direction of the ion force FI generated by the ion beam, which is created by the spacecraft’s impulse transfer thruster. This control is implemented by the propulsion system of the active spacecraft. The issue of developing the control laws without taking into account a space debris object’s attitude motion in the framework of the contactless ion beam-assisted space debris removal mission has been fairly well investigated. The main results in this area are given in studies by Alpatov et al. (2018, 2019), Bombardelli et al. (2011, 2012), Khoroshylov (2019, 2020), and Obukhov et al. (2022).

Relative translation motion of spherical space debris

189

4.4.1 Stability of the active spacecraft’s relative motion in a quasicircular orbit Before proceeding to development of control laws for an active spacecraft’s thruster, let us study the stability of its motion relative a spherical space debris object. As in Section 4.3, it is assumed that the object moves in a quasicircular orbit under the action of ion force, which is created by the impulse transfer thruster of the active spacecraft. The thrust force magnitude FTi is constant during the entire maneuver. The system is not affected by external disturbances. The equations of the active spacecraft motion can be written in the Hill’s reference frame, of which the origin is at the center of mass of the space debris object (Fig. 4.16). In the general case, the motion of the spacecraft is described by Eqs. (3.88)–(3.90). In the case of a quasicircular orbit T _ Hp ≪ω2Hp , and the angular velocity is equal to the time-varying ωH Hp ¼ [0, 0, ωHp] , ω pffiffiffiffiffiffiffiffiffiffi orbit mean motion ωHp ¼ n ¼ μr 3 . The equations system can be reduced to the form of the perturbed Clohessy-Wiltshire equations: x€  2ny_  3n2 x ¼ aAx  aBx , y€ + 2nx_ ¼ aAy  aBy ,

(4.83)

€z + n z ¼ aAz  aBz : 2

Substituting zeros into Eq. (4.83) instead of velocities and accelerations, we obtain T expressions for determining the equilibrium position ρ H ∗ ¼ [x∗, y∗, z∗] in the form x∗ ¼

aBx  aAx , 3n2

aAy  aBy ¼ 0,

z∗ ¼

aAz  aBz : n2

(4.84)

For a space debris object deorbiting mission, the most efficient way to direct the generated ion force in terms of minimizing fuel consumption is to direct it opposite to the object’s velocity vector (Bombardelli et al., 2012). In this case, the position of the Fig. 4.16 Direction of the ion beam when the active spacecraft is displaced from the equilibrium position.

190

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

T active spacecraft is determined by the coordinates ρ H ∗ ¼ [0, y∗, 0] , and it follows from Eq. (4.84) that.

aBx ¼ aAx ¼ aAz ¼ aBz ¼ 0, aAy ¼ aBy , and FTux ¼ 0, FTuz ¼ 0,

FTi  FTc F ¼ I, mA mB

(4.85)

whence formula (4.61) follows. It is assumed that when the active spacecraft moves out from the position ρ∗, the ion beam axis direction does not change and remains parallel to the axis BYH. In this case, the forces magnitudes FTi, FTc, and FTu do not change, and correspond to the equilibrium position ρ∗. The ion force can be calculated by the analytical expression T FH I ¼ ½FTi f r cos β, FTi f x , FTi f r sin β  ,

(4.86)

where fr and fx are normalized forces, which were obtained analytically in the article by Bombardelli et al. (2012) in the form !  4 9χ 2 ðχ 2 tan 4 α0  tan 2 α0 + 2Þ 2 α +O α f x ¼ ηB 1  , (4.87) 2 tan 2 α0 ð1  χ 2 tan 2 α0 Þ3  f r ¼ ð1  η B Þ

  3 3χ 2 α + O α , 1  χ 2 tan 2 α0

(4.88)

where ηB is defined by Eq. (4.62) and χ ¼ Rsp(ρ tan α0)1. When calculating these forces, it is assumed that all particles of the ion beam hit the surface of the spherical space debris object. The angles α and β can be determined in terms of the active spacecraft’s coordinates in Hill’s reference frame BXHYHZH: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + z2 y , cos α ¼ , sin α ¼ ρ ρ x z cos β ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , sin β ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (4.89) 2 2 2 x +z x + z2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ρ ¼ x2 + y2 + z2 . Substituting Eqs. (4.85) and (4.86) into Eq. (4.83) gives x€  2ny_  3n2 x ¼  y€ + 2nx_ ¼

FTi f r cos β , mB

ηB0 FTi FTi f x  , mB mB

z + n2 z ¼  €

FTi f r sin β : mB

(4.90)

Relative translation motion of spherical space debris

 where ηB0 ¼ 1  exp  tan 2 α

3R2sp

191



is the momentum transfer efficiency calcuðy2∗ R2sp Þ lated for the spacecraft position ρ∗. To analyze the stability of the equilibrium position, the first approximation equations can be constructed. Let us introduce the variable y1 ¼ y  y∗. Substituting expressions (4.87)–(4.89) into Eq. (4.90) and linearizing the right-hand side of the resulting system of equations yields x€  2ny_  3n2 x ¼ y€1 + 2nx_ ¼  €z + n2 z ¼

0

bFTi x, mB y∗ tan α0

2bcFTi y , mB y∗ tan α0 1

(4.91)

bFTi z, mB y∗ tan α0

where coefficients b and c are determined by the expressions.

b¼

3R2sp y2∗  R2sp

0

1 3R2sp y2∗ A, c ¼  exp @  : y2∗  R2sp y2∗  R2sp tan 2 α0

(4.92)

The system of equations can be rewritten in a dimensionless form after introducing a new independent variable τ and normalized coordinates δ: τ ¼ nt, δx ¼

y1 x z , δy ¼ , δz ¼ : y∗ tan α0 y∗ tan α0 y∗ tan α0

(4.93)

In variables (4.93), Eq. (4.91) takes the form δ00x  2δ0y  ð3 + γ Þδx ¼ 0, δ00y + 2δ0x + 2γcδy ¼ 0,

(4.94)

δ00z + ð1  γ Þδz ¼ 0, where the prime means the derivative with respect to the independent variable τ, and γ is the dimensionless stiffness coefficient of the beam-target interaction γ¼

bFTi : mB n2 y∗ tan α0

(4.95)

In the case when the active spacecraft is located outside the space debris sphere, the inequality y ∗2 > R2sp is satisfied. Analysis of expressions (4.92) and (4.95) allows us to conclude that b, c, and γ are positive.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

The conclusion about the stability of the equilibrium position ρ ∗H ¼ [0, y∗, 0]T can be made using Theorems 5–7 from Section 1.8. Eq. (4.94) can be represented in matrix form (1.146): x0 ¼ Ax,

(4.96)

where x ¼ [δx, δy, δz, δx0 , δy0 , δz0 ]T is the state vector, and the coefficient matrix has the form 2

0

6 0 6 6 6 0 A¼6 63 + γ 6 6 4 0 0

0

0

1

0

0 0

0 0

0 0

1 0

0

0

0

2

2 0

0 0

2γc 0 0 γ1

0

3

07 7 7 17 7: 07 7 7 05

(4.97)

0

The characteristic polynomial det(A  λE) ¼ 0 can be written as 

  λ2 + 1  γ λ4 + ð1 + ð2c  1Þγ Þλ2  2cγ ðγ + 3Þ ¼ 0:

(4.98)

The roots of this equation are λ1,2 ¼  λ3,4

λ5,6

pffiffiffiffiffiffiffiffiffiffiffi γ  1,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 1 2 1+4 c+ ¼  pffiffiffi γ + 28cγ  2γ  ð1 + ð2c  1Þγ Þ, 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 1 2 2 ¼  pffiffiffi  1 + 4 c + γ + 28cγ  2γ  ð1 + ð2c  1Þγ Þ: 2 2

Let us take a closer look at the third root: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 1 2 1+4 c+ λ3 ¼ pffiffiffi γ + 28cγ  2γ  ð1 + ð2c  1Þγ Þ 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 A  B: ¼ pffiffiffi 2 Since b, c, and γ are positive, and 2c  1 ¼

y2∗ + R2sp 2y2∗  1 ¼ > 0, y2∗  R2sp y2∗  R2sp

(4.99)

(4.100)

(4.101)

Relative translation motion of spherical space debris

193

then   1 2 2 A¼1+4 c+ γ + 24cγ + 4γ ð2c  1Þ > 0, 2 B ¼ 1 + ð2c  1Þγ > 0: Taking these estimates into account, it can be concluded that the root λ3 is positive when the inequality A > B2 is satisfied. The last inequality can be reduced to 8cγ ð3 + gÞ > 0,

(4.102)

which always holds. It follows that the root λ3 is always a positive real number. Therefore, the equilibrium position ρ ∗H ¼ [0, y∗, 0]T is unstable according to Theorem 6, which is given in Section 1.8. To confirm the obtained result, let us perform a series of numerical calculations using the equations of motion (4.90). In numerical integration, it is assumed that the system has the following parameters: mB ¼ 500 kg, mB ¼ 5000 kg, α0 ¼ 10°, Rsp ¼ 2 m, y∗ ¼ 10 m, r0 ¼ 7, 371, 000 m, n ¼ 9.9765 s1, and FTi ¼ 0.1 N. In this case, nB0 ¼ 0.9821 and FTc ¼ 0.1098 N. For the equilibrium position ρ∗, Eqs. (4.92) and (4.95) give b ¼ 0.0022, c ¼ 1.0417, and γ ¼ 0.0256. Fig. 4.17 shows the trajectories of the active spacecraft relative to the space debris object obtained for various initial conditions given in Table 4.2. For all cases, x_ 0 ¼ y_0 ¼ z_0 ¼ 0. In addition, in all cases, the active spacecraft descends below the space debris object. This is due to the fact that the decelerating acceleration of the active spacecraft when displaced from the equilibrium position turns out to be greater than the decelerating acceleration transmitted to the space debris object. As a result, the height of the orbit of the active spacecraft falls faster than the height of the orbit of space debris. For cases 1 and 2, at the initial

Fig. 4.17 Trajectories of the active spacecraft relative to the space debris object for different initial conditions given in Table 4.2.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Table 4.2 Initial conditions.

x0, m y0, m z0, m

Case 1

Case 2

Case 3

Case 4

Case 5

0.01 10 0

0 9.99 0

0.01 10 0

0 10.01 0

0 10 0.01

Fig. 4.18 Direction of the ion beam when the active spacecraft is displaced from the equilibrium position.

stage of motion, the active spacecraft rises above the space debris height. In the case when x > 0, the ion force has a downward component (Fig. 4.16), so the space debris itself is displaced downward under the action of this force. However, after some time, the factor of reducing the active spacecraft velocity begins to play a dominant role. Earlier, it was assumed that the axis of the ion beam always remained parallel to its original direction. Let us now consider the case when the control system of the active spacecraft tracks the position of the space debris object and directs the ion beam axis to its geometrical center B (Fig. 4.18). In this case, the acceleration components of the active spacecraft and the space debris object take the form. aAx ¼  aAz ¼  aBx ¼  aBz ¼ 

mB

ηB0 FTi x ηB0 FTi y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , aAy ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , mB x2 + y2 + z2 x 2 + y 2 + z2

mB

ηB0 FTi z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x2 + y 2 + z2

mB

ηB FTi x ηB FTi y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , aBy ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , x 2 + y 2 + z2 mB x2 + y2 + z2

mB

ηB FTi z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x2 + y 2 + z2

(4.103)

(4.104)

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195

The first approximation equations for accelerations (4.103) and (4.104) can be written as x€ 2ny_  3n2 x ¼ 0, y€1 + 2nx_ ¼ 

6FTi y∗ R2sp y1 ,  2 mB y2∗  R2sp tan 2 α0

(4.105)

z€+ n2 z ¼ 0, After passing to dimensionless variables, this system of equations (4.105) takes the form δ00x  2δ0y  3δx ¼ 0, δ00y + 2δ0x + χδy ¼ 0, δ00z

(4.106)

+ δz ¼ 0,

where χ ¼

6FTi y∗ R2sp mB ðy2∗ R2sp Þ n2 tan 2 α0 2

> 0 . The characteristic polynomial for the system of

linear differential equations (4.106) has the form 

  λ2 + 1 λ4 + ðχ + 1Þλ2  3χ ¼ 0:

(4.107)

The roots of this equation are ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 λ1,2 ¼ i, λ3,4 ¼  pffiffiffi 1 + 14χ + χ  ð1 + χ Þ, 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ffiffi ffi  1 + 14χ + χ 2  ð1 + χ Þ: λ5,6 ¼  2 The third root is always positive, since it is easy to prove that 1 + 14χ + χ 2 > ð1 + χ Þ2 : Therefore, the equilibrium position ρ ∗H ¼ [0, y∗, 0]T is unstable according to Theorem 6 of Section 1.8. The results of a numerical simulation of the active spacecraft’s relative motion obtained using Eq. (4.83) with accelerations (4.103) and (4.104) show that its trajectories differ significantly from those shown in Fig. 4.17. The equilibrium position is still unstable, but the active spacecraft is not necessarily below the debris object (Fig. 4.19). In particular, on trajectories 1 and 2, the active spacecraft turns around. Its engines begin to accelerate rather than slow down the spacecraft, which leads to an increase in the height of its orbit. Fig. 4.20 demonstrates the

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.19 Trajectories of the active spacecraft relative to the space debris object for different initial conditions given in Table 4.2.

Fig. 4.20 Dependence of the coordinate z on the angle τ for case 5 from Table 4.2.

change of z coordinate from the angle τ for case 5 (Table 4.2). It can be seen that the out-of-plane oscillations remain stable despite the fact that the in-plane motion is unstable. Obviously, successful implementation of the space debris removal mission requires closed-loop control even in the simplest case of a quasicircular orbit.

4.4.2 Control of the active spacecraft’s relative motion in a quasicircular orbit The simplest and most obvious way to control an active spacecraft is to install thrusters that can generate force in three mutually perpendicular directions. The ideal case is considered, when the active spacecraft’s sensors are able to determine its relative position and velocity without errors and delays. It is assumed that the orientation of the spacecraft does not change during the mission, and the picture shown in Fig. 4.16 is realized. The feedback control law for the engine’s thrust force

Relative translation motion of spherical space debris

197

T FH u ¼ [FTux, FTuy, FTuz] , which keeps the spacecraft in an unstable equilibrium posiT tion ρ∗ ¼ [0, y∗, 0] , can be written in the form

2 _ FH ux ¼ mB n k px x + mB nkVx x, 2 _ FH uy ¼ mB n k py ðy  y∗ Þ + mB nkVy y,

(4.108)

2 _ FH uz ¼ n k pz z + nk Vz z:

where k represents control gains coefficients. The control force projection FTuy can be implemented by the impulse compensation thruster. The equations of the active spacecraft’s relative motion, taking into account the control force (4.108), take the form x€  2ny_  3n2 x ¼  y€ + 2nx_ ¼

FTi f r cos β _ + n2 kpx x + nkVx x, mB

ηB0 FTi FTi f x _  + n2 kpy ðy  y∗ Þ + nkVy y, mB mB

€z + n2 z ¼ 

(4.109)

FTi f r sin β + n2 kpz z + nkVz z: _ mB

Linearization of the right-hand sides of Eq. (4.109) and transition to dimensionless variables (4.93) yield δ00x  2δ0y  ð3 + γ Þδx ¼ kpx δx + kVx δ0x , δ00y + 2δ0x + 2γcδy ¼ kpy δy + kVy δ0y , δ00z

+ ð1  γ Þδz ¼ kpz δz +

(4.110)

kVz δ0z ,

Let us find the conditions for the control gains coefficients that ensure the asymptotic stability of the equilibrium position. These conditions can be formulated on the basis of Theorem 5, given in Section 1.8. Reducing the system of equations (4.110) to the matrix form (4.96) allows us to write the matrix of coefficients as follows: 2

0 0

6 6 6 6 0 A¼6 63 + γ + k px 6 6 4 0

0 0

0 0

1 0

0 1

0 0

0 0

0 kVx

0 2

2γc + kpy

0

2

kVy

0

γ  1 + kpz

0

0

0

3 0 0 7 7 7 1 7 7: 0 7 7 7 0 5

(4.111)

kVz

The characteristic polynomial for this matrix has the form PðλÞ ¼ Pout ðλÞPin ðλÞ,

(4.112)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

where Pin(λ) corresponds to the in-plane motion and Pout(λ) corresponds to the out-ofplane motion: Pout ðλÞ ¼ λ2  kVz λ + 1  γ  kpz ,

(4.113)

    Pin ðλÞ ¼ λ4  kVx + kVy λ3 + 1  γ + 2γc  kpx  kpy + kVx kVy λ2        + 3 + γ + kpx kVy + kVx kpy  2γc λ + kpy  2γc 3 + γ + kpx : (4.114) In order for the equilibrium to be asymptotically stable, all roots of the characteristic polynomial (4.112) must have negative real parts. The roots of polynomial (4.113) are easily found in the form

λ1,2 ¼

kVz  2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2Vz + 4γ + 4kpz  4 2

:

(4.115)

These roots have a negative real part if kVz < 0, k2Vz + 4γ + 4kpz  4 < 0

(4.116)

or kVz

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + k2Vz + 4γ + 4kpz  4 < 0, k2Vz + 4γ + 4kpz  4 0

(4.117)

The combined solution of inequalities (4.116) and (4.117) has the form. kVz < 0, kpz < 1  γ:

(4.118)

The solution of the polynomial (4.114) can be obtained analytically, but in view of the cumbersomeness, the analytical analysis of the roots of the polynomial is not possible. The selection of the gains coefficients for in-plane motion can be carried out numerically. The numerical simulation of the active spacecraft controlled motion with the parameters given in Section 4.4.1 confirms the feasibility of the considered control. Figs. 4.21–4.23 show the trajectory of the spacecraft and its relative coordinates, obtained as a result of numerical integrating the system of equations (4.109). The point ρ0 ¼ [0, 10.01, 0.01]T was chosen as the initial position. The following control coefficients are used in the simulation: kpx ¼  4, kpy ¼  5, kpz ¼ 0.5, kVx ¼  1.5, kVy ¼  3, and kVz ¼  1. The eigenvalues λ1,2 ¼  0.5  i0.9474, λ3,4 ¼  1.852  i 2.776, and λ5,6 ¼  0.398  i 0.5326 correspond to these coefficients. All roots of the characteristic equation have negative real parts, which indicates the asymptotic stability of the equilibrium position. The results shown in Figs. 4.21–4.23 confirm the asymptotic stability of the equilibrium position ρ∗ when using the above control coefficients.

Fig. 4.21 Trajectories of the active spacecraft relative to the space debris object in the BXHYH plane.

Fig. 4.22 Trajectories of the active spacecraft relative to the space debris object in the BYHZH plane.

Fig. 4.23 The dependence of the coordinates of the active spacecraft on time for control force (4.108).

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.24 Trajectory of an active spacecraft and the limit cycle.

In Section 4.4.1, it was shown that in the uncontrolled motion case, oscillations along the z axis can remain limited and behave as stable (Fig. 4.20) with unstable behavior of in-plane oscillations. Numerical simulation showed that if in control (4.108) values kpz ¼ 0 and kVz ¼ 0 are taken, thus turning off the stabilization of out-of-plane oscillations, then the equilibrium position ρ∗ will cease to be asymptotically stable. In its vicinity, a stable limit cycle will arise, to which neighboring phase trajectories will tend. The numerically obtained limit cycle for the system with the above parameters and initial conditions is shown in Fig. 4.24. The appearance of the limit cycle is caused by the fact that out-of-plane oscillations through the normalpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ized forces fr and fx, which include the length ρ ¼ x2 + y2 + z2, will have a periodic perturbing effect on the x and y equations in (4.109). Depending on the parameters of the system, the limit cycle can be quite close to the equilibrium position, which will make it possible to carry out a space debris removal mission successfully even when the active spacecraft moves along this trajectory, and it is not in position ρ∗. In an article by Alpatov et al. (2018) and in Section 6.1.2 of the book by Alpatov et al. (2019), it was noted that due to the fact that the equations for x and y are interconnected, it is possible to carry out the control of the spacecraft in-plane motion using only one thruster. The control force can be given in the form FH ux ¼ 0, 2 2 _ FH uy ¼ mB n k px x + mB nkVx x_ + mB n kpy ðy  y∗ Þ + mB nk Vy y, FH uz

¼ n kpz z + nkVz z: _ 2

(4.119)

Relative translation motion of spherical space debris

201

The equations of the active spacecraft’s relative motion (4.90) in the presence of a control force (4.119) can be rewritten as x€  2ny_  3n2 x ¼  y€ + 2nx_ ¼

FTi f r cos β , mB

ηB0 FTi FTi f x _  + n2 kpx x + nkVx x_ + n2 kpy ðy  y∗ Þ + nkVy y, mB mB

€z + n2 z ¼ 

FTi f r sin β + n2 kpz z + nkVz z: _ mB (4.120)

The equations of the first approximation for system (4.120), taking into account the change of variables (4.93), have the form δ00x  2δ0y  ð3 + γ Þδx ¼ 0, δ00y + 2δ0x + 2γcδy ¼ kpx δx + kVx δ0x + kpy δy + kVy δ0y , δ00z

+ ð1  γ Þδz ¼ kpz δz +

(4.121)

kVz δ0z :

The matrix of coefficients for linear equation (4.121) is 2

0 6 0 6 6 6 0 A¼6 63 + γ 6 6 4 kpx 0

0 0

0 0

1 0

0 1

0

0

0

0

0 2γc + kpy

0 0

0 2 + kVx

2 kVy

0

γ  1 + kpz

0

0

3 0 0 7 7 7 1 7 7: 0 7 7 7 0 5

(4.122)

kVz

The characteristic polynomial for this matrix has the form P(λ) ¼ Pout(λ)Pin(λ), where Pout(λ) is defined by Eq. (4.113), and     Pin ðλÞ ¼ λ4  kVy λ3 + 1  γ + 2γc  kpy  2kVx λ2 + ð3 + γ ÞkVy  2kpx λ   + kpy  2γc ð3 + γ Þ: (4.123) As in the previous case, the analytical solution for the roots of polynomial (4.123) is very cumbersome. The control gain coefficients can be selected numerically from the condition that the real parts of all eigenvalues of the matrix A are negative, which guarantees the asymptotic stability of the equilibrium position according to Theorem 5. Let us carry out a numerical simulation of the motion of the system by choosing the following control gains coefficients: kpx ¼  3, kpy ¼ 1, kpz ¼ 0.5, kVx ¼  4, kVy ¼  1.5, and kVz ¼  1. The eigenvalues λ1,2 ¼  0.5  i0.9474, λ3,4 ¼  0.6875  i 2.6457, and λ5,6 ¼  0.0625  i 0.616 correspond to these coefficients. The dependence of coordinates on time when using control (4.119) is shown in Fig. 4.25.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.25 The dependence of the coordinates of the active spacecraft on time for control force (4.119).

It can be seen from the graphs that the problem of stabilizing the position of the active spacecraft in the orbital plane can be solved using a single thruster. Another thruster is needed to stabilize the out-of-plane motion. In an article by Khoroshylov (2019), the possibility of controlling out-of-plane oscillations of the system without using a thruster directed perpendicular to the ion beam axis was studied. Khoroshylov proposed turning the active spacecraft relative to the XH axis and controlling both the impulse compensation thruster force FTc and the yaw angle ψ, which determines the direction of this force (Fig. 4.26). Thus, both in-plane and out-of-plane motion can be controlled by means of only an impulse compensation thruster. The rotation of the active spacecraft can be carried out by its attitude control system. The results of numerical simulation presented in the paper by Khoroshylov (2019) show that the stabilization of out-of-plane oscillations occurs rather quickly, but yaw angle control leads to a decrease in the efficiency of impulse transfer through the ion beam due to angular deviations. The deterioration of the deorbiting rate due to the yaw attitude deviations does not exceed 1.5% compared Fig. 4.26 Out-of-plane deflection of the active spacecraft.

Relative translation motion of spherical space debris

203

to using control by mutually perpendicular thrusters. To solve this problem, the author proposed turning on the angular control when a certain threshold value of deviations was exceeded, and turning it off when the oscillation amplitude was reduced to acceptable values.

4.4.3 Control of the active spacecraft’s relative motion in an elliptical orbit As in the previous section, it is assumed that the orientation of the spacecraft does not change during the mission. Space debris moves in a Keplerian orbit with a small eccentricity. The motion of an active spacecraft relative to a space debris object in the Hill’s system is described by Eqs. (1.88)–(1.90), which for the considered case of Keplerian orbits take the form x€ ω2Hp x  2ωHp y_  ω_ Hp y  2kx ¼ aAx  aBx , y€ ω2Hp y + 2ωHp x_ + ω_ Hp x + ky ¼ aAy  aBy ,

(4.124)

z€+ ky ¼ aAy  aBy , where k ¼ μr3 and r ¼ p(1 + e cos f)1, and the angular velocity ωHp and the angular acceleration ω_ Hp are defined by the expressions ωHp

¼ f_ ¼

rffiffiffiffiffi μ ð1 + e cos f Þ2 , p3

rffiffiffiffiffi μ ω_ Hp ¼ 2e 3 ð1 + e cos f ÞωHp sin f : p

(4.125)

(4.126)

Since the magnitude of the ion force acting on the space debris object is small, its eccentricity e, semimajor axis a, and semilatus rectum p ¼ a(1  e2) change slowly, which, for example, is confirmed by the results of numerical integration given in Section 4.3. It is assumed that the active spacecraft is equipped with a propulsion system capable of creating thrust in three mutually perpendicular directions FH u ¼ [FTux, FTuy, FTuz]T. One of these engines is the impulse compensation thruster. The value of the thrust force FTc can be represented as the sum of two components: the nominal component FTcn calculated by Eq. (4.61) and the variable component FTcv, which implements the required control law in BYH direction. Let us take as a basis the control force (4.108) that was used for the case of a quasicircular orbit: 2 H FH ux ¼ mB n k px x + mB nkVx x_ + Fux0 , H 2 Fuy ¼ mB n kpy ðy  y∗ Þ + mB nkVy y_ + FH uy0 ,

FH uz

¼ n kpz z + nkVz z_ + 2

(4.127)

FH uz0 :

H H In contrast to Eq. (4.108), terms FH ux0, Fuy0, and Fuz0 are added to the control forces. These force components are functions of the angle of true anomaly f. Equating

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.27 The position of the spacecraft for generating the ion force is opposite to the velocity vector direction.

velocities and accelerations in Eq. (4.124) to zero, it can be shown that the point ρ∗ ¼ [0, y∗, 0]T is not an equilibrium position for the case of an elliptical orbit. The H H components of the control forces FH ux0, Fuy0, and Fuz0 can be chosen to make ρ∗ the equilibrium. It should be noted that when the active spacecraft is located in position ρ∗ ¼ [0, y∗, 0]T, the ion force generated by the ion beam is not always directed opposite to the velocity vector of the space debris object, since the object’s velocity vector has components in the Hill’s coordinate system: T VH B ¼ ½npe sin f , npð1 + e cos f Þ, 0 :

(4.128)

H H The control forces components FH ux0, Fuy0, and Fuz0 can be chosen to make the active spacecraft move along some trajectory ρ ¼ [y∗ sin ψ, y∗ cos ψ, 0]T so that the generated ion force is directed opposite to the velocity vector (Fig. 4.27). In this case, the axis of the ion beam must rotate as the spacecraft moves along the elliptical orbit according to the expression

ψ ¼ arctan

e sin f : 1 + e cos f

(4.129)

Since it is assumed that the axis of the ion beam is always parallel to the BYH axis, it is advisable to place the active spacecraft at the point ρ∗ ¼ [0, y∗, 0]T. At small eccentricities, the angle ψ  0, and at large eccentricities, the ion beam does not hit the space debris object and braking does not occur. Using expression (4.125), let us pass in Eq. (4.124) to a new independent variable, taking into account Eqs. (4.93) and (4.127): δ00x

! 2 γ + kpx 2e sin f + δy + 2δ0y ¼ 1+ δx  1 + ecos f ð1 + ecos f Þ4 1 + e cos f ! 2esin f kVx + + δ0 , 1 + ecos f ð1 + ecos f Þ2 x

Relative translation motion of spherical space debris

! k  2cγ 2esin f 1 py δx + 1  + δ00y ¼ δy  2δ0x 1 + ecos f 1 + e cos f ð1 + e cos f Þ4 ! kVy 2e sin f + + δ0 , 1 + ecos f ð1 + ecos f Þ2 y δ00z

205

(4.130)

! ! 1 2e sin f kVz δz + + ¼  δ0 , 1 + e cos f ð1 + e cos f Þ2 y ð1 + ecos f Þ4 1 + e cos f γ + kpz

H H The components of the control forces FH ux0, Fuy0, and Fuz0 are

3 2 FH ux0 ¼ 2ey∗ n ð1 + e cos f Þ sin f ,   3 4 2 FH uy0 ¼ y∗ n ð1 + e cos f Þ  ð1 + e cos f Þ ,

(4.131)

FH uz0 ¼ 0: In the case of a small eccentricity e ≪ 1, the right-hand sides of Eq. (4.130) can be expanded into a Maclaurin series for e. After discarding the nonlinear terms, we obtain a system of equations describing the relative motion of the active spacecraft in an orbit with a small eccentricity     δ00x ¼ 3 + γ + kpx  4γ + 4kpx + 2 e cos f δx  2e sin f δy + 2δ0y + ðkVx  2ekVx cos f + 2e sin f Þδ0x ,     δ00y ¼ 2e sin f δx + kpy  2γc + 1 + 4kpy + 8cγ e cos f δy  2δ0x   + kVy  2ekVy cos f + 2e sin f δ0y ,

(4.132)

    δ00z ¼ γ + kpz  1  e 4γ + 4kpz  1 cos f δz + ðkVz  2ekVz cos f + 2e sin f Þδ0y : To study the stability of a nonautonomous nonlinear system with periodic coefficients (4.132) using the Lyapunov theorems for the first approximation system, it is necessary to find a fundamental set of solutions for Eq. (4.132). It is very difficult to find these solutions, which complicates the analytical analysis of the limits of change in the coefficients kpx, kpy, kpz, kVx, kVy, and kVz, which ensure the asymptotic stability of the equilibrium position. To study the influence of the orbit eccentricity on the ability to control the relative position of the active spacecraft using the control law (4.127), let us numerically simulate the active spacecraft’s relative motion using Eq. (4.124), with the gains coefficients values obtained for a circular orbit in Section 4.4.2: kpx ¼  4, kpy ¼  5, kpz ¼ 0.5, kVx ¼  1.5, kVy ¼  3, and kVz ¼  1. The point ρ0 ¼ [0, 10.01, 0.01]T is taken as the initial position. Fig. 4.28 shows the dependence of spacecraft coordinates on time for various values of eccentricity. As can be seen from the graphs, the control law developed for a circular orbit can also be successfully applied to

206

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 4.28 The effect of eccentricity on the change in the coordinates of the active spacecraft.

an elliptical orbit with an eccentricity e  0.3. At large values of eccentricity (curves e ¼ 0.4 in Fig. 4.28), the control law with the values of control coefficients chosen for a circular orbit does not allow stabilizing of the active spacecraft in the required relative position. In a study by Khoroshylov (2020), in order to determine the control gains coefficients for elliptical space debris orbit, it was proposed to transfer from a system of differential equations to equations in a discrete form, and to solve the discrete-time linear-quadratic regulator problem. The study uses the methodology detailed by Lewis et al. (2012). The constant control gains coefficients for full-state feedback law are determined to minimize the quadratic cost function J ¼ min

∞  X  QT xj Q + RT uj R , j¼0

where xj is the state vector in the jth step, uj is the control input, and Q and R are weight matrices, which are selected to minimize the position errors and to limit the control forces within the allowable limits set by the active spacecraft’s propulsion system. Since the used methodology involves the construction of a controller for autonomous systems, the parameters of the object are calculated for the orbital position with the maximum angular acceleration. The construction of a periodic discrete linear-quadratic regulator for the considered system was also considered in the paper by Khoroshylov (2020). In this case, the control coefficients are no longer constants, but are periodic functions of the true anomaly angle. To find the matrix of coefficients, it is required to find the

Relative translation motion of spherical space debris

207

solution of the periodic Riccati equation. The algorithm for solving this equation for the discrete case was given by Van Dooren (1981). The numerical algorithm for constructing a solution to the equation for the continuous case was given by Peng et al. (2011). The results of numerical simulations given by Khoroshylov (2020) show that the linear-quadratic regulator and periodic linear-quadratic regulator give close results for orbits with an eccentricity of less than 0.2. For larger eccentricities, a linear-quadratic regulator may not be effective and a periodic regulator should be used. An alternative approach to constructing a controller for a continuous nonautonomous system was considered by Alpatov et al. (2018), who provided a good example of the development of a spacecraft control system capable of operating in the presence of disturbances and inaccuracies in the determination of system parameters and errors in sensor measurements. The key points of the methodology are described below; for a more complete acquaintance with the controller development process, analysis of system robustness, and assessment of the influence of controller parameters on fuel consumption, we refer readers to the study mentioned above or to Chapter 6 of the book by Alpatov et al. (2019). Note that in the mentioned works, the Hill’s reference frame differs from that used in this chapter. Its origin is not in the center of mass of space debris, but in the center of mass of an active spacecraft. The equations of relative active spacecraft’s motion (4.124) in the planer case, by _ y_T , can be represented as introducing the state vector x ¼ ½x, y, x, x_ ¼ Ax + B0 ap + B2 u,

(4.133)

where ap ¼ [aAx  aBx, aAy  aBy]T is the perturbation acceleration and u ¼ [ux, uy]T is the control input vector. The coefficient matrices have the form 2 3 0 0 1 0 0 60 6 0 0 0 1 7 6 7 A¼6 4 ω2Hp + 2k ω_ Hp 0 2ωHp 5, B0 ¼ 4 1 ω_ Hp ω2Hp  k 2ωHp 0 0 2

2 3 3 0 0 0 6 0 7 07 7: 7, B ¼ 6 0 0 5 0 5 2 4 m1 A 0 m1 1 A

The study by Alpatov et al. (2018) suggested using the H∞ methodology (Zhou et al., 1996) to build a controller that can work effectively in the presence of disturbances and various kinds of errors. The general control configuration is determined by a set of matrix equations as follows: x_ ¼ Ax + B1 w + B2 u, z ¼ C1 x + D11 w + D12 ,

(4.134)

v ¼ C2 x + D21 w + D22 u,

ax and ay , xr , yr , Δx, Δy, Δux , Δuy is the exogenous input vector, de where w ¼ de ax , de de ay are the normalized perturbations, xr ¼ 0 and yr ¼ y∗ are the reference active

208

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

spacecraft position, Δx and Δy are the measurement noise, Δux and Δuy are the actuation errors, z ¼ [z1x, z1y, z2x, z2y]T is the minimized output vector, z1 is the control error, z2 is the controller’s output signal, and v ¼ [vx, vy]T is the measured output vector. Matrices of coefficients in Eq. (4.134) have the form 2

0

6 0 6 B1 ¼ 6 max 4 ax 0 2

1 6 0 C2 ¼ 6 4 0 0

D21 ¼

0

0 0 0 0

0

0

0 0 0 0

0

0

0 0 0 0 0 0 0 0 2 0 3T 60 0 6 6 1 7 7 ,D11 ¼ 6 0 60 0 5 6 40 0 0 amax y

0 0 1 0 Δx max 0 0 0 1

0

Δavcx 0 0 0 0 0 0 0

1 0 0 0 0 0

3 1 0 0 0 1 0 T 6 0 1 0 0 0 1 7 0 7 6 7 7 7 , 7, C1 ¼ 6 4 0 0 0 0 0 05 0 5 Δavcy 0 0 0 0 0 0 3 3 2 0 0 0 Δxmax 0 0 60 07 0 Δymax 0 0 7 7 7 6 7 6 0 0 0 07 7, D12 ¼ 6 1 0 7, 60 17 0 0 0 07 7 7 6 40 05 0 0 0 05 0 0 0 0 0 0 0

0 1 0 0 0 0

0 Δy max

3

2

  0 0 , , D22 ¼ 0 0 0 0

0 0

where amax ¼ max(aAx  aBx) and amax ¼ max(aAx  aBx) are the maximum differences x y in the accelerations of the active spacecraft and the space debris object, which, in addition to the accelerations from the thrust forces and the generated ion force, include accelerations from other external perturbations described in Section 1.6. Δxmax and Δymax are the maximum error due to the sensor noise, and Δavcx and Δavcy are actuation errors. If hydrazine thrusters with pulse width modulators are used as the actuators of the control system, then actuation errors can be calculated as Δavcx ¼ Δavcy ¼

min Fth ton , TmA

where Fth is the thruster’s nominal thrust, T is the sampling period of the controller, and tmin on is the minimum pulse width. Thus, the matrices B1, D11, and D21 are selected based on estimates of the maximum input disturbances. The controller design goal is to minimize the relative position errors limiting the effort of the controller. Mathematically, this goal can be written as minimization of the kH∞k norm kFl ðP, KÞk∞ ! min ,

(4.135)

where P is the transfer function of the plant (Alpatov et al., 2018), K is the transfer function of the controller, and Fl(P, K) is the closed-loop transfer function, which

Relative translation motion of spherical space debris

209

describes the relationship between the minimized output vector z and the exogenous input vector w z ¼ Fl ðP, KÞw:

(4.136)

The transfer function can be calculated as PðsÞ ¼ GðsÞWðsÞ,

(4.137)

where G(s) is the transfer matrix of the generalized plant and W(s) is the square diagonal matrix of weight functions 2 6 6 6 Wð sÞ ¼ 6 6 4

W 1x ðsÞ

0

0

0

0

W 1y ðsÞ

0

0

0

0

W 2x ðsÞ

0

0

0

0

W 2y ðsÞ

3 7 7 s=Mij + Ωij 7 : 7, W ij ðsÞ ¼ s + Aij Ωij 7 5

Weight functions contain Wij(s) parameters: the crossover frequencies of the low past filters Ωij, the low frequency gains Aij, and the high frequency gains Mij. The matrix G(s) can be represented as an arrangement of transfer functions: Gij ðsÞ ¼ Ci ðsE  AÞ1 Bj + Dij ,

i, j ¼ 1, 2

(4.138)

A suboptimal controller K(s) can be found from x_ k ¼ AK xk + BK v, u ¼ CK xk + DK v

(4.139)

based on the condition kFl ðP, KÞk∞  γ min : In the study by Alpatov et al. (2019), robustness analysis of the controller was carried out taking into account time-varying and parametric uncertain plant, external perturbations, sensor noise, actuation errors, and limitations on the controller output. It is shown that a rational reduction in the requirements for control accuracy can significantly reduce the required propellant mass for the active spacecraft’s relative position control. The presented results of numerical simulation confirm the effectiveness of the developed controller for controlling the relative position of the active spacecraft during an ion beam-assisted space debris removal mission.

210

4.5

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Space debris removal: Preliminary mission design

This section is devoted to the preliminary design methodology proposed by Urrutxua et al. (2019), which allows us to determine the parameters of an active spacecraft, taking into account the specifics of the contactless ion beam-assisted transportation mission. For the considered mission, several key factors, which largely determine all other parameters of the designed spacecraft, can be distinguished: the separation distance between the spacecraft and space debris, parameters of propulsion, and power subsystems. The problem of electric propulsion subsystem parameter optimization for space debris removal mission was considered by Cichocki et al. (2017). These results can be taken as a basis for further mission development. Preliminary mission design provides initial estimates, which can then be used as a starting point for optimizing the active spacecraft design. The matching chart concept is used to select system configuration based on graphical representation of the operational constraints and technological limitations. The idea of the method is to select two key parameters that affect all other design parameters of the developed system, and construct a region of feasible solutions whose points satisfy all the imposed constraints. To draw this region, operational constraint curves C(i) that correspond to the limitations imposed on the system are plotted. These curves divide the parameter space into regions with acceptable design values and with values that do not satisfy the constraints. Thus, each curve cuts off a part of the space of key parameters. The part of the design space remaining after the construction of all the curves will be the desired area, which is called the design envelope. Based on the analysis of the mutual influence of the parameters of the designed system, Urrutxua et al. (2019) proposed to choose the operational distance along the ion beam axis between the spacecraft and the space debris object ρ and the thrust force of the impulse transfer thruster FTi as the preferred design parameters. The set of operational constraints C(i)(FTi, ρ) that are important for the contactless space debris removal mission is considered. For each constraint, the equation is constructed in such a way that the positive values of the function C(i)(FTi, ρ) correspond to the admissible design values, while the negative ones do not satisfy the constraints. The following constraints are considered. The collision avoidance limitation C(1) is determined by the geometric parameters of the active spacecraft and the space debris object and the minimum allowable distance between them. The sensor sensitivity limitation C(2) is determined by the maximum distance at which the spacecraft’s sensors, which are designed to determine the position and motion parameters of a space debris object, can operate. The momentum transfer efficiency constraint C(3) excludes from the solution space momentum transfer efficiency ηB(ρ) less than some minimum threshold value. The minimum thrust constraint C(4) is the minimum impulse transfer thruster force that it can generate. The signal-to-noise ratio with respect to atmospheric drag constraint C(5) is determined by the allowable ratio between the aerodynamic drag force acting on the space debris object and the generated ion force. With a large drag force, it makes no sense to use an ion beam, since braking is carried out by the atmosphere. The maximum thrust constraint C(6) is the maximum force that the impulse compensation thruster can generate. The maneuverability and control constraint

Relative translation motion of spherical space debris

211

C(7) are determined by the thrust margin of the thrusters, which can be used to correct the relative position of the spacecraft. The onboard power limitation C(8) is that the combined power consumption of the spacecraft’s thrusters shall not exceed the total available power of the propulsion system. The backsputtering contamination constraint C(9) is determined by the minimum distance that the spacecraft can be located from the space debris object at a given impulse transfer thruster thrust level to keep sensors and solar arrays operational throughout the mission. The mission lifetime constraint C(10) defines the maximum allowed time for a mission to complete. Analytic expressions for all the above constraints were given by Urrutxua et al. (2019). The described methodology was implemented by Hodei Urrutxua in the form of an interactive Mathematica document distributed under the GNU GPL v3 license and available at http://sdg.aero.upm.es/ONLINEAPPS/IBS_Design. The document allows us interactively to set some parameters of the active spacecraft and space debris and to draw the constraint curves. Fig. 4.29 shows the result of the program calculations. The left boundary of the design envelope (white region in Fig. 4.29) is of practical interest. The lower left corner of the design envelope provides the best balance between low thrust and high momentum transfer efficiency under given constraints, which allows us to create a lighter spacecraft. In the upper left corner, there is less transfer efficiency at higher thrust, corresponding to a heavier spacecraft but a shorter descent time. It should be noted that the design envelope is built on the basis of the most stringent design constraints, which usually correspond to the start or end of the mission. The reason is that some parameters, such as the mass of the spacecraft or the ratio between aerodynamic drag and ion force, can change during space debris deorbiting, which leads to a change in the constraint curves.

Fig. 4.29 Curves of constraints in the design space.

212

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

References Alpatov, A., Khoroshylov, S., Bombardelli, C., 2018. Relative control of an ion beam shepherd satellite using the impulse compensation thruster. Acta Astronaut. 151, 543–554. https://doi.org/10.1016/j.actaastro.2018.06.056. Alpatov, A.P., Khoroshylov, S.V., Maslova, A.I., 2019. Contactless De-Orbiting of Space Debris by the Ion Beam. Dynamics and Control. Akademperiodyka, Кyiv, https://doi. org/10.15407/akademperiodyka.383.170. Aslanov, V.S., Ledkov, A.S., 2021. Fuel costs estimation for ion beam assisted space debris removal mission with and without attitude control. Acta Astronaut. 187, 123–132. https://doi.org/10.1016/j.actaastro.2021.06.028. Bombardelli, C., Pelaez, J., 2011. Ion beam shepherd for contactless space debris removal. J. Guid. Control. Dyn. 34, 916–920. https://doi.org/10.2514/1.51832. Bombardelli, C., Merino-Martinez, M., Galilea, E., Palaez, J., Urrutxua, H., Herrara-Montojo, J., Iturri-Torrea, A., 2011. Ariadna call for ideas: active removal of space debris. In: Ion Beam Shepherd for Contactless Debris Removal. ESA, Madrid. Bombardelli, C., Urrutxua, H., Merino, M., Ahedo, E., Pela´ez, J., 2012. Relative dynamics and control of an ion beam shepherd satellite. Adv. Astronaut. Sci. 143, 2145–2157. Cichocki, F., Merino, M., Ahedo, E., Smirnova, M., Mingo, A., Dobkevicius, M., 2017. Electric propulsion subsystem optimization for “ion beam shepherd” missions. J. Propuls. Power 33, 370–378. https://doi.org/10.2514/1.B36105. Holste, K., Dietz, P., Scharmann, S., Keil, K., Henning, T., Zsch€atzsch, D., Reitemeyer, M., Nausch€utt, B., Kiefer, F., Kunze, F., Zorn, J., Heiliger, C., Joshi, N., Probst, U., Th€ uringer, R., Volkmar, C., Packan, D., Peterschmitt, S., Brinkmann, K.T., Zaunick, H.G., Thoma, M. H., Kretschmer, M., Leiter, H.J., Schippers, S., Hannemann, K., Klar, P.J., 2020. Ion thrusters for electric propulsion: scientific issues developing a niche technology into a game changer. Rev. Sci. Instrum. 91. https://doi.org/10.1063/5.0010134. Khoroshylov, S., 2019. Out-of-plane relative control of an ion beam shepherd satellite using yaw attitude deviations. Acta Astronaut. 164, 254–261. https://doi.org/10.1016/j. actaastro.2019.08.016. Khoroshylov, S., 2020. Relative control of an ion beam shepherd satellite in eccentric orbits. Acta Astronaut. 176, 89–98. https://doi.org/10.1016/j.actaastro.2020.06.027. Kluever, C.A., 2018. Space Flight Dynamics, second ed. John Wiley and Sons Ltd, Hoboken, NJ. Lev, D., Myers, R.M., Lemmer, K.M., Kolbeck, J., Koizumi, H., Polzin, K., 2019. The technological and commercial expansion of electric propulsion. Acta Astronaut. 159, 213– 227. https://doi.org/10.1016/j.actaastro.2019.03.058. Lewis, F., Vrabie, D., Syrmo, V., 2012. Optimal Control. John Wiley & Sons, New Jersey. Marcucci, M.G., Polk, J.E., 2000. NSTAR xenon ion thruster on deep space 1: ground and flight tests (invited). Rev. Sci. Instrum. 71, 1389–1400. https://doi.org/10.1063/1.1150468. Obukhov, V.A., Kirillov, V.A., Petukhov, V.G., Pokryshkin, A.I., Popov, G.A., Svotina, V.V., Testoyedov, N.A., Usovik, I.V., 2022. Control of a service satellite during its mission on space debris removal from orbits with high inclination by implementation of an ion beam method. Acta Astronaut. 194, 390–400. https://doi.org/10.1016/j.actaastro.2021.09.041. Peng, H., Zhao, J., Wu, Z., Zhong, W., 2011. Optimal periodic controller for formation flying on libration point orbits. Acta Astronaut. 69, 537–550. https://doi.org/10.1016/j. actaastro.2011.04.020.

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Ruiz, M., Urdampilleta, I., Bombardelli, C., Ahedo, E., Merino, M., Cichocki, F., 2014. The FP7 LEOSWEEP project: improving low earth orbit security with enhanced electric propulsion. In: Sp. Propuls. Conf, pp. 35–42. Sengupta, A., Brophy, J., Anderson, J., Garner, C., Banks, B., Groh, K., 2004. An overview of the results from the 30,000 Hr life test of deep space 1 flight spare ion engine. In: 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. American Institute of Aeronautics and Astronautics, Reston, Virigina, pp. 1–21, https://doi.org/10.2514/6. 2004-3608. Stuhlinger, E., 1964. Ion Propulsion for Space Flight. McGraw-Hill, New York. Urrutxua, H., Bombardelli, C., Hedo, J.M., 2019. A preliminary design procedure for an ionbeam shepherd mission. Aerosp. Sci. Technol. 88, 421–435. Van Dooren, P., 1981. A generalized eigenvalue approach for solving Riccati equations. SIAM J. Sci. Stat. Comput. 2, 121–135. https://doi.org/10.1137/0902010. Zhou, K., Doyle, J., Glover, K., 1996. Robust and Optimal Control. Prentice Hall, New Jersey.

Dynamics of passive object attitude motion during ion beam transportation 5.1

5

Mathematical models of a passive space debris object during its contactless transportation by an active spacecraft

The attitude motion of a passive space debris object under the action of ion beam influence is considered in this chapter both in a planar and in a spatial setting. The planar case of motion is not realizable in practice due to the presence of external disturbances; however, the revealed patterns of motion and found control laws can be useful in the analysis of the general case of 3D motion. In this section, besides sufficiently detailed mathematical models of the plane and spatial motion of the considered mechanical system, several simplified models are constructed. These models allow us to perform a qualitative analysis, and are useful for developing control laws.

5.1.1 Mathematical model of the planar motion of space debris Consider the case when a passive space debris object has a plane of symmetry coinciding with the plane of its orbit. It is also assumed that the center of mass of the space debris object is located in this plane. The active spacecraft is considered as a mass point. The planar motion of a mechanical system consisting of the passive object and the active spacecraft can be described by five generalized coordinates (Fig. 5.1): the position vector module r, the true anomaly angle f, the deflection angle of the space object axis from the position vector direction θ, and the coordinates of the active spacecraft in the Hill’s reference frame x and y. The active spacecraft is equipped with an impulse transfer thruster that creates an ion beam, an impulse compensation thruster, of which the main purpose is compensation of the transfer thruster operation, and orientation system thrusters. The resulting thrust of the active spacecraft engines is shown in Fig. 5.1 by projections of the vector PH ¼ [Px, Py, 0]T. The effect of the ion beam on the object can be represented in the form of two mutually perpendicular forces FIx and FIy, which are directed along the axes of the Hill’s coordinate system, and the torque LIz relative to the center of mass B, of which the vector is perpendicular to the plane of Fig. 5.1. It is also assumed that the space debris object is affected by a rarefied atmosphere, the influence of which is determined by the aerodynamic force Fba ¼ [ FaA, FaN, 0]T and torque Lba ¼ [0, 0, Laz]T specified in the bodyfixed coordinate system BXbYbZb. More detailed information about aerodynamic Attitude Dynamics and Control of Space Debris During Ion Beam Transportation. https://doi.org/10.1016/B978-0-323-99299-2.00001-X Copyright © 2023 Elsevier Inc. All rights reserved.

216

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.1 Considered mechanical system in planar case.

forces and torques can be found in Section 1.7.3. The mathematical model below was developed in studies by Aslanov et al. (2021) and Ledkov and Aslanov (2018). Let us use the Lagrange formalism to obtain the equations of the considered mechanical system motion (Section 1.1.2). The kinetic energy of the system is defined as
2 I z f_ + θ_ mA V 2A mB V 2B T¼ + + , 2 2 2

(5.1)

where mA is the mass of the active spacecraft, mB is the mass of the object, Vi is the velocity of the ith point (i ¼ A,B), and Iz is the principle moment of inertia of the object relative BZb axis, which is directed perpendicular to the plane of Fig. 5.1. The first term in Eq. (5.1) defines the kinetic energy of the active spacecraft as a mass point, the second term is the kinetic energy of the center of mass of the space debris object, and the last term is the kinetic energy of the object’s rotation relative to its center of mass. The velocity of point B can be determined through generalized coordinates as follows: V 2B ¼ r 2 f_ + r_2 : 2

(5.2)

To calculate the velocity of the active spacecraft, the projections of point A on the axis of the inertial coordinate system OXpYpZp must first be found: 2

xA

3

2

ðr + xÞ cos f  y sin f

3

6 7 6 7 r pA ¼ 4 yA 5 ¼ 4 ðr + xÞ sin f + y cos f 5, 0 zA

(5.3)

Dynamics of passive object attitude motion during ion beam transportation

217

Finding the derivatives (5.3) and performing simplifications gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 V A ¼ ðr + xÞ2 f_ + y2 f_ + 2ððr + xÞy_  yðr_ + x_ ÞÞf_ + ðr_ + x_ Þ2 + y_2 ,

(5.4)

Substituting expressions (5.2) and (5.4) into Eq. (5.1), we obtain the kinetic energy

T¼

  2 2 2 mA ðr + xÞ2 f_ + y2 f_ + 2ððr + xÞy_  yðr_ + x_ÞÞf_ + ðr_ + x_Þ2 + y_2 + y2 f_  2  mB r 2 f_ + r_2

+

2




I z f_ + θ_ + 2

2

2 :

(5.5) The potential energy of the considered mechanical system P is the sum of the potential energy of the active spacecraft PA and the potential energy of the space debris object PB. The potential energy of the spacecraft as a mass point in the Earth’s gravitational field is determined by the expression PA ¼ 

μmA , rA

(5.6)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where μ is the gravitational constant of the Earth, and r A ¼ xA 2 + yA 2 is the distance between the active spacecraft and the center of the Earth. The potential energy of the passive object as a rigid body can be written in the following form (expression (26) on page 238 in Hughes, 2004):  
 2 2 2 3μ I r + I r + I r x y z x y z μ Ix + Iy + Iz μm PB ¼  B  + , r 2r 3 2r 3

(5.7)

where Ix and Iy are the principle moments of inertia, and r x, r y, and r z are the coordinates of a unit vector directed along the object’s position vector r in the body-fixed frame BXbYbZb. 2

rx

3

2

cos θ

6 7 6 rb ¼ 4 r y 5 ¼ MbH rH ¼ 4  sin θ rz

0

sin θ cos θ 0

32 3 2 3 1 cos θ 76 7 6 7 0 54 0 5 ¼ 4  sin θ 5, 1 0 0 0

(5.8)

where MbH is the transition matrix. The summation of energy in Eqs. (5.6) and (5.7), taking into account Eq. (5.8), gives

 3μ I x cos 2 θ + I y sin 2 θ μmA μmB μ I x + I y + I z P¼   + : rA r 2r 3 2r 3

(5.9)

218

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Since the Lagrangian is defined by the expression L ¼ T  P, Lagrange equations of the second kind (1.4) can be written in the form d ∂T ∂T ∂P  ¼ + Qi , dt ∂q_ i ∂qi ∂qi

(5.10)

where qi is the component of the generalized coordinates vector q ¼ [r, f, θ, x, y], and Qi are the generalized forces Qr ¼ FIx  FaA cos θ  FaN sin θ + Px , Qf ¼ FIy r  FaA r sin θ + FaN r cos θ  Px y + Py ðr + xÞ + Laz + LIz , Qθ ¼ LIz + Laz ,

Qx ¼ Px ,

(5.11)

Qy ¼ P y :

Aerodynamic force and torque can be calculated as FaA ¼

ρa V 2rel ρ V2 ρ V2 SCA ðαÞ, FaN ¼ a rel SCN ðαÞ, Laz ¼ a rel SlCLz ðαÞ 2 2 2

(5.12)

where ρa is the density of the atmosphere and Vrel is the velocity of the space debris relative the atmosphere, which can be calculated using Eq. (1.126). Neglecting the rotation of the Earth, it can be assumed that Vrel ¼ VB. S is the space debris crosssection area, l is the characteristic length of the space debris, CA(α), CN(α), and CLz(α) are dimensionless aerodynamic coefficients of the axial force, normal force, and pitch moment, and α is the angle of attack, which can be calculated as ! p p   V Bx cos ðf þ θÞ þ V By sin ðf þ θÞ p p α ¼ sign V Bx sin ðf þ θÞ  V By cos ðf þ θÞ arccos : VB (5.13)

Here, VpBx and VpBy are projections of the space debris object’s velocity vector on the axis of the inertial reference frame OXpYpZp. Substitution of expressions (5.1), (5.9), and (5.11) into Eq. (5.10) and simplifications give
 2 2 3μ 3I cos θ + 3I sin θ  I  I + I μ F x y x y z x + , r€ ¼ f_ r  2 + mB r 2mB r 4
 3μ I x  I y sin θ cos θ Fy 2f_r_ + f€ ¼   , r mB r mB r 5 2


   3μ I x  I y sin θ cos θ Fy Lz 2f_r_ 1 1 € θ¼  + + + , r Iz Iz mB r mB r 2 r3

(5.14)

(5.15)

(5.16)

Dynamics of passive object attitude motion during ion beam transportation 2 μ ðr + x Þ P , x€ ¼ f€y  r€ + f_ ðr + xÞ + 2f_y_ + x  mA r 3A

y€ ¼ f_ y  f€ðr + xÞ  2f_ðr_ + x_Þ + 2

219

(5.17)

Py μy  ; mA r 3A

(5.18)

where Fx ¼ FIx  FaA cos θ  FaN sin θ and Fy ¼ FIy  FaA sin θ + FaN cos θ are the projections of the resulting force acting on the space debris object on the axes of the Hill’s coordinate system: Lz ¼ LIz + Laz. The obtained system of Eqs. (5.14)–(5.18) describes the plane motion of the considered mechanical system. If it is assumed that the control system of the active spacecraft provides its constant relative position (x ¼ const, y ¼ const), and 1/r ≪ 1, then the object’s equations of motion, after discarding small terms, take the form r€ ¼

h2 μ F  + x, mB r3 r2

(5.19)

rFy , h_ ¼ mB

(5.20)


 Lz 2hr_ 3μ I x  I y sin θ cos θ € + 3 + , θ¼ Iz Iz r3 r

(5.21)

where h ¼ f_r 2 is the angular momentum relative to the center of the Earth per unit mass. Since the magnitude of the force generated by the ion beam is small (it is of the order of 10 mN), it can be assumed that the object’s center of mass moves on the Kepler orbit. In this case, r¼

p , 1 + e cos f

(5.22)

where p ¼ h2 μ1 is the semilatus rectum of the orbit and e is the eccentricity. These values change slowly. Let us fix the values of the semilatus rectum and eccentricity, and choose the new independent variable τ ¼ Ωf , where Ω ¼

(5.23)

qffiffiffiffiffiffiffiffiffiffiffiffi 3|I x I y | 2I z . Time derivatives of some arbitrary function s(t) and derivatives

with respect to the new variable are related by the following equations: s_ ¼

ds dτ c ¼ s0 Ωf_ ¼ s0 Ω 2 ð1 + e cos νÞ2 ¼ s0 Ω dτ t p

rffiffiffiffiffi
2 μ
1 + e cos τ Ω1 , 3 p

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

 rffiffiffiffiffi 0   d 0 μ ds 2 2 0 _ s ð1 + e cos f Þ ¼ Ω ð1 + e cos f Þ  2ð1 + e cos f Þef s sin f ¼ s€ ¼ dt p3 dt  rffiffiffiffiffi rffiffiffiffiffi rffiffiffiffiffi μ 00 μ μ 0 4 3 Ω  2e ð 1 + e cos f Þ sin f ¼ s ð 1 + e cos f Þ s ¼Ω p3 p3 p3
  

 2es0 sin τ Ω1

3 μ 1 + e cos τ Ω1 , ¼ Ω2 3 s00 1 + e cos τ Ω1  Ω p where the prime symbol denotes a derivative with respect to τ. The object’s angular motion Eq. (5.21) can be rewritten as
 2eðΩθ0 + 1Þ sin τ Ω1
 θ ¼

4 + 2
Ω 1 + e cos τ Ω1 μΩ2 I z 1 + e cos τ Ω1 δ sin 2θ
 , +
1 + e cos τ Ω1 p3 Lz

00

(5.24)

where δ ¼ sign(Ix  Iy). The appearance of this coefficient in Eq. (5.24) is due to the presence of the modulus under the root in Ω expression in the change of variable (5.23). For the case of small eccentricity e ≪ 1, Eq. (5.24) can be linearized: θ00 ¼


 Lz p3 e
+ δ sin 2θ + 2 2ðΩθ0 + 1Þ sin τ Ω1 2 Ω Iz μ Ω
1   3
1  4Lz p cos τ Ω 2  δ Ω sin 2θ cos τ Ω  : μI z

(5.25)

For the circular (e ¼ 0) orbit, Eq. (5.24) takes the form θ00 

Lz p3  δ sin 2θ ¼ 0: Ω2 I z μ

(5.26)

Eq. (5.26) can be considered as an equation of unperturbed motion for the studying the attitude motion of the passive object under the action of an external torque Lz.

5.1.2 Mathematical model of the spatial motion of space debris A mathematical model of a mechanical system consisting of a space debris object and an active spacecraft in three-dimensional space is developed in this section. It is assumed that the active spacecraft is a material point A, and the space debris object is a rigid body with a center of mass at point B (Fig. 5.2). The motion of the system occurs only under the influence of the gravitational field of the Earth, the thrust of the active spacecraft’s thrusters, the ion force, and torque generated by the ion beam when interacting with the surface of the space debris. The state of the system can be described by nine generalized coordinates q ¼ [ϑ, ν, r, x, y, z, γ, θ, φ]T. The angles ϑ, ν and the distance r determine the position of

Dynamics of passive object attitude motion during ion beam transportation

221

Fig. 5.2 Considered mechanical system in spatial case.

the center of mass of the space debris object. The coordinates x, y, and z define the position of the active spacecraft relative to space debris in an orbital spherical reference frame BXoYoZo. The three Euler angles γ, θ, and φ define the orientation of the body-fixed reference frame BXbYbZb relative to BXoYoZo. The motion of the centers of mass of a space debris object and the active spacecraft are described by Eqs. (4.28)–(4.33). The equation of motion of a space debris object relative to the center can be obtained using the Euler equations (Schaub and Junkins, 2014): dHbB + ωbbp  HbB ¼ LbG + LbI , dt

(5.27)

where HbB ¼ [I]ωb is the angular momentum vector about the center of mass B, [I] is the space debris inertia matrix, LbG is the gravity gradient torque, LbI ¼ [LIx, LIy, LIz]T is the ion beam torque relative to the space debris object’s center of mass, and ωbbp ¼ [ωx, ωy, ωz]T is the angular velocity of the space debris. All vectors are given by their components in the body reference frame. The angular velocity ωbbp can be represented as the sum (Fig. 1.5) b _ b  ν_ eb + γ_ eb + θe _ b + φe ωbbp ¼ ϑe p32 332 42 513 _ b2 2 3 2 3 2 0 0 0 θ_ 6 7 6 7 6 7 6 7 ¼ Mbp 4 0 5 + Mbo 4 γ_  ν_ 5 + Mb5 4 0 5 + 4 φ_ 5, 0 ϑ_ 0 0

(5.28)

where eij are unit vectors. Using the rotation matrices given in Section 1.2, the coordinates of the angular velocity vector ωbbp can be written as

222

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

 ωx ¼ θ_ cos φ þ ðγ_  ν_ Þ sin θ sin φ þ ϑ_ cos ν ð sin φ cos θ cos γ þ cos φ sin γ Þ  þ sin γ ð cos φ cos γ  sin φ cos θ sin γ Þ , ωy ¼ φ_ þ ðγ_  ν_ Þ cos θ þ ϑ_ ð cos ν sin θ cos γ þ sin ν sin θ sin γ Þ, ωz ¼ θ_ sin φ  ðγ_  ν_ Þ cos φ sin θ þ ϑ_ ð cos νð cos φ cos θ cos γ  sin φ sin γ Þ þ sin νð sin φ cos γ þ cos φ cos θ sin γ ÞÞ: (5.29) The gravity gradient torque LbG acting on the space debris is defined by Eq. (1.115). Eq. (5.27) can be reduced to  Iy  Iz  3μ ωy ωz  3 r y r z + Ix r   Iz  Ix 3μ ωx ωz  3 r x r z + ω_ y ¼ Iy r  Ix  Iy  3μ ω_ z ¼ ωx ωy  3 r x r y + Iz r

ω_ x ¼

LIx , Ix LIy , Iy LIz , Iz

(5.30)

where r i ¼ r i =r are dimensionless coordinates of the space debris object’s position vector rb given in the body-fixed frame. Eqs. (5.29) and (5.30) describe the motion of the space debris object relative to the center of mass under the action of the gravitational and the ion beam torque. Consider the case of motion of a space debris object with small asymmetry. It is assumed that the dimensionless difference of the moments of inertia Δ is a small quantity Δ¼

Ix  Iz , I

(5.31)

where I ¼ (Ix + Iz)/2. In this case, Iz ¼ I(1 + Δ/2) and Ix ¼ I(1  Δ/2). Following the approach described in the book by Aslanov (2017), new variables are introduced based on the classical Lagrange case of motion of a body with a fixed point. In the Lagrange case, the generalized momentum corresponding to rotation and precession angles are integrals of motion. In the case of perturbed motion, these quantities will be slowly changing functions: R ¼ Iy ωy , G ¼ R cos θ + ðωx sin φ  ωz cos φÞ sin θ,

(5.32)

where Iy ¼ I y =I. Expressing the angular velocities from Eqs. (5.29) and (5.32) gives

Dynamics of passive object attitude motion during ion beam transportation

ðG  R cos θÞ sin φ _  ϑ cos φð cos ν sin γ  sin ν cos γ Þ, ωx ¼ θ_ cos φ + sin θ R ωy ¼  , Iy ðG  R cos θÞ cos φ _  ϑ sin φð cos ν sin γ  sin ν cos γ Þ, sin θ ϑ_ cos θð cos ν cos γ + sin ν sin γ Þ G  R cos θ , γ_ ¼ + ν_ + 2 sin θ sin θ

223

(5.33)

ωz ¼ θ_ sin φ 

R ðG  R cos θÞ cos θ ϑ_ ð cos ν cos γ + sin ν sin γ Þ : φ_ ¼    sin θ Iy sin 2 θ

(5.34)

(5.35)

After substituting Eq. (5.33) in Eq. (5.30), expressing the derivatives, we obtain €θ þ ðG  R cos θÞðR  G cos θÞ ¼ sin φ Lz þ cos φ Lx þ θΦ _ 1θ þ ϑΦ _ 2θ þ ϑ_ 2 Φ3θ þ ϑΦ € 4θ Iz Ix sin 3 θ 2 ΔRI ðIx  2IÞð2 cos 2φ  ΔÞðG  R cos θÞ þ , 4I x Iy I z sin θ

Ly ΔðG  R cos θÞ2 sin 2φ _ 2 _ 3R + ϑ_ 2 Φ4R + θ_ ϑΦ _ 5R , + θΦ1R + θ_ Φ2R + ϑΦ R_ ¼  I 2 sin 2 θ    ΔðG  R cos θÞsin 2φ I 2 RðIx  2I Þ cos θ sin φ cos φ G_ ¼ Ly þ L  L sin θ þ 2 I x Iy Iz I Ix x Iz z  ðG  R cos θÞ cos θ _ 2 _ 3G þ ϑ_ 2 Φ4G þ θ_ ϑΦ _ 5G : þ θΦ1G þ θ_ Φ2G þ ϑΦ þ sin 2 θ

(5.36)

(5.37)

(5.38)

where
 3μ I y  I z r y r z 3μðI z  I x Þr x r z Lx ¼ LIx  , Ly ¼ LIy  , r3 r3
 3μ I x  I y r x r y : Lz ¼ LIz  r3
 ΔI 2 R 2I  I y sin 2φ Φ1θ ¼ ,
2I x I y I z 2 ΔI R 2I  I y sin 2φ sin ðν  γ Þ 2ðG  R cos θÞ cos ðν  γ Þ Φ2θ ¼ + , 2I x I y I z sin 2 θ Φ3θ ¼

cos θ cos 2 ðν  γ Þ , Φ4θ ¼ sin ðγ  νÞ, sin θ

(5.39)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Φ1R ¼

ΔðG  R cos θÞ cos 2φ Δ , Φ2R ¼  sin 2φ, sin θ 2

Φ3R ¼

ΔðG  R cos θÞ sin ðν  γ Þ cos 2φ Δ , Φ4R ¼  sin 2 ðν  γ Þ sin 2φ, 2 sin θ

Φ5R ¼ Δ sin 2φ sin ðν  γ Þ, Φ1G ¼

RI sin θ ΔðG  R cos θÞ cos θ cos 2φ ΔRI 2 ðΔ + 2 cos 2φÞ sin θ + + sin θ 4I x I z Iy
 2 3 RI Δ + 4Δ cos 2φ + 4 sin θ  , 4I x I y I z

Φ2G ¼  Φ3G ¼

Φ4G ¼

Δ cos θ sin 2φ, 2

ðΔ cos 2φ  1ÞðG  R cos θÞ sin ðν  γ Þ cos θ sin θ

 I 2 R Δð2I  I x Þ cos 2φ + Δ2 I  2I y sin ðν  γ Þ sin θ  , 2I x I y I z   Δ sin 2φ cos θ 1 cos 2ν cos 2 γ cos 2 ν + sin 2ν sin 2γ 2 2 sin ð2ðν  γ ÞÞ ,  2

Φ5G ¼ Δ sin 2φ cos θ sin ðν  γ Þcos ðν  γ Þ: Eqs. (5.34)–(5.38) can be used to describe the attitude motion of a slightly asymmetrical space debris object.

5.1.3 Mathematical model of the spatial motion of a symmetrical space debris object Consider the case of a symmetrical space debris object motion. It is assumed that Ix ¼ Iz ¼ I and Δ ¼ 0. In this case, equations of a space debris object attitude motion (5.36)–(5.38), taking into account Eq. (5.39), can be written in the form 2 _ €θ + ðG  R cos θÞðR  G cos θÞ ¼ ϑ cos ðν  γ Þ cos θ  ϑ € sin ðν  γ Þ sin θ sin 3 θ
 φ 2 2ϑ_ cos ðν  γ ÞðG  R cos θÞ LIx ðθÞ 3μ sin γ cos θ sin θ I y  I +  , + I r3 I sin 2 θ 2

(5.40) R_ ¼

LφIy ðθÞ , I

(5.41)

Dynamics of passive object attitude motion during ion beam transportation

G_ ¼

LφIy ðθÞ cos θ  LφIz ðθÞ sin θ ϑ_ sin ðv  γ ÞðG cos θ  RÞ  sin θ I
 2 _ϑ2 sin ð2v  2γ Þ 3μ cos γ sin γ sin θ I  I y   θ_ ϑ_ cos ðv  γ Þ  , 2 r3 I

225

(5.42)

where LφIx ¼ LIz sin φ + LIx cos φ, LφIy ¼ LIy, and LφIz ¼ LIz cos φ  LIx sin φ are projections of the ion beam torque on the coordinate system BX4YbZ5, which is rotated relative to the orbital spherical reference frame BXoYoZo at angles γ and θ. To calculate these projections, the technique described in Chapter 3 is used. Calculations show that for a space debris object of cylindrical shape, in which the axis of symmetry is the BYb axis, and the center of mass lies on this axis, projections LφIy and LφIz are equivalent to zero, and the projection LφIx is an odd function of angle θ. The function LφIx(θ) can be represented as a Fourier series: LφIy ¼ LφIz ¼ 0, LφIx ¼ LI ¼ LImax

k X

bj sin jθ,

(5.43)

j

where Lmax is the module of the ion beam torque amplitude value and bj  [1, 1] are I expansion coefficients of the Fourier series. Consider a case when the center of mass of the space debris object moves in the plane BXoYo (ν ¼ 0, ν_ ¼ 0). In reality, when a space debris object is blown with an ion beam, the generated ion force can have a component perpendicular to the plane of the orbit. It is assumed that this component averaged over the period of the object’s oscillations is equal to zero, and therefore the motion of the center of mass can be considered as planar. In this case, the angle ϑ plays the role of the true anomaly angle, so the notation ϑ ¼ f will be used. The space debris object’s center of mass equations takes the form FoIy Fo 2 2f_r_ μ f€ ¼  + , r€ ¼ f_ r  2 + Ix , r rmB mB r

(5.44)

where FoIx and FoIy are the projections of the ion force on the axes of the orbital spherical coordinate system BXoYoZo. Since these forces are small in absolute value, let us consider the orbit as quasi-Keplerian. In this case, the following equations can be used instead of Eq. (5.44): r¼

p , f_ ¼ nð1 + e cos f Þ2 , 1 + e cos f

(5.45)

pffiffiffiffiffiffiffiffiffiffi where n ¼ μp3 , and eccentricity e and semilatus rectum p are slowly varying functions. Let us pass in Eqs. (5.40)–(5.42) from time to a new independent variable f. It also assumed that   G ¼ nk2 G, R ¼ nk2 R,

(5.46)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

where k ¼ 1 + e cos f. Let us calculate the derivative of the function G with respect to time, taking into account Eqs. (5.46) and (5.45): dG dG dG df ¼ 2en2 k3 G sin f + nk2 ¼ 2en2 k3 G sin f + nk2 dt dt df dt 2 3  2 4 0 ¼ 2en k G sin f + n k G ,

(5.47)

0 where the prime means the derivative with respect to the angle f. Expressing G in Eq. (5.47), we obtain

2eG sin f G_ 0 G ¼ 2 4 + : k n k

(5.48)

The second derivative of the angle θ with respect to time is related to the derivative with respect to the true anomaly angle f by the following expression: 2 € θ ¼ θ00 f_ + θ0 f€ ¼ θ00 n2 k4  2en2 k3 θ0 sin f :

(5.49)

After passing to a new independent variable, and using Eqs. (5.43) and (5.48), the object’s attitude motion equations take the form ðG  R cos θÞðR  G cos θÞ L cos 2 γ cos θ + 2I 4 + 3 In k
sin θ sin θ 2 2 cos γ ðG  R cos θÞ 3 sin γ sin 2θ I y  I 2θ0 e sin f , + +  k 2Ik sin 2 θ
 2 sin γ ðG cos θ  RÞ 3 sin 2γ sin θ I y  I 0   G ¼ + cos γ sin γ sin θ 2kI 0  2eG sin f ,  θ0 cos γ + k θ00 ¼ 

2eR sin f 0 R ¼ , k γ0 ¼

cos θ cos γ G  R cos θ : + 2 sin θ sin θ

(5.50)

(5.51) (5.52) (5.53)

In the case of circular orbit, e ¼ 0, k ¼ 1, and Eqs. (5.50)–(5.53) take the form θ00 ¼ 

ðG  R cos θÞðR  G cos θÞ L cos 2 γ cos θ + I2 + 3 sin θ In sin θ
 2   2 cos γ ðG  R cos θÞ 3 sin γ sin 2θ I y  I , +  2I sin 2 θ

(5.54)

Dynamics of passive object attitude motion during ion beam transportation

sin γ ðG cos θ  RÞ + cos γ sin γ  θ0 cos γ G0 ¼ sin θ
 3 sin 2γ sin 2 θ I y  I  , 2I

227

(5.55)

where R ¼ const. The equations obtained in this way describe the attitude motion of the space debris object in the case when its center of mass moves in Keplerian and circular orbits. Consider the motion of a space debris object in a geosynchronous orbit (GEO). A number of assumptions are made to obtain a simplified system of equations: ν_ ¼ 0, ϑ_ ¼

pffiffiffiffiffiffiffiffiffiffi μr 3 ¼ ε, r ¼ const,

(5.56)

where ε ≪ 1 is a small parameter. Consider the case when ε≪

LI ðθÞ : I

(5.57)

For an axisymmetric space debris object, Eqs. (5.34)–(5.38), after discarding small terms, take the form γ_ ¼

G  R cos θ , sin 2 θ

(5.58)

R ðG  R cos θÞ cos θ φ_ ¼   , Iy sin 2 θ

(5.59)

ðG  R cos θÞðR  G cos θÞ LI θ€ + ¼ : I sin 3 θ

(5.60)

where R and G are constants, which are determined by the initial angular velocities and orientation of space debris. Eq. (5.60) can be integrated independently of Eqs. (5.58) and (5.59).

5.1.4 Stationary motions of a symmetrical space debris object In order to find stationary motions, let us consider motion of the space debris object in a circular orbit, which is described by Eqs. (5.53)–(5.55), and equate the derivatives to zero: G0 ¼ 0, θ00 ¼ 0, θ0 ¼ 0, and γ 0 ¼ 0. From Eq. (5.53), it follows that cos γ ∗ ¼

Rcos θ∗  G∗ , cos θ∗ sin θ∗

(5.61)

228

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

ðG∗  R cos θ∗ ÞðR  G∗ cos θ∗ Þ LI ðθ∗ Þ cos 2 γ ∗ cos θ∗ + + sin θ∗ In2 sin 3 θ∗
 2 2cos γ ∗ ðG∗  R cos θ∗ Þ 3 sin γ ∗ sin 2θ∗ I y  I , +  2I sin 2 θ∗
 3 sin 2γ ∗ sin 2 θ∗ I y  I sin γ ∗ ðG∗ cos θ∗  RÞ , 0¼ + cos γ ∗ sin γ ∗  2I sin θ∗ 0¼

(5.62)

(5.63)

where the star index indicates stationary motion and R ¼ const. Eq. (5.63) yields
 3 sin 2 θ∗ cos γ ∗ I y  I G∗ cos θ∗  R ¼ 0: + cos γ ∗  I sin θ∗

(5.64)

Another solution of this equation, sin γ ∗ ¼ 0, contradicts Eq. (5.61). Substitution of Eq. (5.61) into Eq. (5.64) gives
 3 sin θ∗ ðG∗  R cos θ∗ Þ I y  I G sin θ∗  ∗ ¼ 0: I cos θ∗ cos θ∗

(5.65)

From here, we define
 3 cos θ∗ R I y  I  G∗ ¼ : 3I y  4I

(5.66)

From Eqs. (5.61) and (5.66), it follows that cos γ ∗ ¼ 


 RI  , 3I y  4I sin θ∗

(5.67)

Substitution of Eqs. (5.66) and (5.61) into Eq. (5.62) yields
 LI ðθ∗ Þ  3n2 I y  I cos θ∗ sin θ∗ ¼ 0:

(5.68)

The last equation expresses the equality of the gravitational gradient and the ion torques. The solutions of Eq. (5.66) depend on the form of the function LI(θ∗), which in turn depends on the shape of the body and the ion beam parameters. The solution to this nonlinear equation should be searched numerically for a specific space debris object under consideration. Depending on the function LI(θ), Eq. (5.68) can have a different number of roots. Once solutions are found, the corresponding values of G∗ and γ ∗ can be determined from Eqs. (5.66) and (5.67), respectively. Consider the motion of a passive space debris object in GEO, when the dynamics of the object is described by Eqs. (5.58)–(5.60). This case was studied by Aslanov et al. (2020a). To investigate the stationary motions of the system, we equate the derivatives γ_ , €θ to zero in Eqs. (5.58) and (5.60). Two nonlinear equations are obtained:

Dynamics of passive object attitude motion during ion beam transportation

ðG∗  R∗ cos θ∗ ÞðR∗  G∗ cos θ∗ Þ LI ðθ∗ Þ G∗  R∗ cos θ∗ : ¼ 0, ¼ 2 I sin θ∗ sin 3 θ∗

229

(5.69)

The solution of Eq. (5.69) gives the values θ∗, R∗, and G∗, corresponding to the equilibrium position θ∗ ¼ arccos ðG∗ =R∗ Þ,

(5.70)

which must also satisfy the equation LI ðθ∗ Þ ¼ 0:

(5.71)

In addition to the equilibrium position (5.70), the system of Eqs. (5.58) and (5.60) admits a stationary mode of motion, when the angle θ∗ is a solution to Eq. (5.71) but relation (5.70) is not satisfied, and γ_ ¼ ωγ∗ ¼ const. This motion corresponds to the regular precession mode, when the axis of symmetry of the passive body describes a cone around the axis BYo (Fig. 5.3).

5.2

Attitude dynamics of the uncontrolled motion of a passive space debris object at a constant relative position of the active spacecraft

In this section, the motion of a space debris object relative to its center of mass is studied at a constant relative position of an active spacecraft and at constant parameters of the ion beam. In this case, the magnitude and direction of the ion torque acting on the space debris object is determined by its orientation in the beam. In Chapter 3, it was shown that the ion torque depends on the shape of the space debris object, so it is impossible to study the features of the space debris object attitude motion in general Fig. 5.3 Precession of the passive space debris object.

230

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

case without focusing on its shape. In this section, several axisymmetric objects will be considered, the center of mass of which lies on the axis of symmetry. It is assumed that the ion beam axis passes through the center of mass.

5.2.1 Phase portraits and bifurcation diagrams Consider a planar motion of a cylindrical space debris object. In this case, the object’s motion relative to its center of mass is described by Eq. (5.24). The ion beam torque LIz can be approximately represented as a Fourier series: LIz ¼ LImax

ðLiÞ a0

k  X

+

ðLiÞ aj

cos jθ +

ðLiÞ bj

 sin jθ

! ,

(5.72)

j¼1

where Lmax is the maximum absolute value of the ion beam torque, and a(Li) and z j (Li) bj are dimensionless Fourier coefficients. For high orbits, where the effect of the atmosphere is negligible, Eq. (5.25), taking into account Eq. (5.72), can be written as 00

θ  γ IG 0

ðLiÞ a0

þ

k  X

ðLiÞ aj

cos jθ þ

ðLiÞ bj

 sin jθ

!  δ sin 2θ ¼

j¼1

!
 k  
1  ðLiÞ X 2ðΩθ0 þ 1Þ sin τ Ω1 ðLiÞ ðLiÞ @  4γ IG cos τ Ω aj cos jθ þ bj sin jθ a0 þ ¼ e Ω2 j¼1


δ sin 2θ cos τ Ω

 1

1 A:

(5.73) where γ IG is the parameter that determines the ratio of ion beam torque and gravity gradient torque amplitudes: γ IG ¼

LImax p3 2LImax p3 ¼ : 3μ|I x  I y | Ω2 I z μ

(5.74)

As the object’s altitude increases, the influence of the ion beam torque will increase in comparison with the influence of the gravity gradient torque. It can be seen that this parameter is proportional to the object orbit’s semilatus rectum p. The amplitude ratio parameter will change as the altitude of the orbit changes during the object’s transportation mission. To analyze the influence of the γ IG parameter on the space debris object’s attitude motion, let us consider the simplest case, when its center of mass moves in a circular orbit. The motion of the object in a circular orbit will be considered as an unperturbed motion. In this case, e ¼ 0 and Eq. (5.24) takes the form

Dynamics of passive object attitude motion during ion beam transportation

θ00  γ IG

ðLiÞ a0

+

k  X

ðLiÞ aj

cos jθ +

ðLiÞ bj

!  sin jθ

 δ sin 2θ ¼ 0:

231

(5.75)

j¼1

The view of the phase portrait of Eq. (5.75) depends on the amplitude ratio parameter γ IG and the coefficients of the Fourier series expansion a(Li) and b(Li) j j . As mentioned earlier, it is very difficult to analyze the phase portrait of the unperturbed Eq. (5.75) in general cases, since the coefficients of equation depend on many parameters, including the moments of inertia, shape and layout of the object, the parameters of the ion beam, its direction, and the distance between the active spacecraft and the object. Let us perform further analysis for a specific passive space debris object shown in Fig. 5.4. The object has a cylindrical shape. It is characterized by the following parameters: the mass of the passive object is mB ¼ 2000 kg, the radius of the cylinder is Rc ¼ 1.5 m, and its length is Lc ¼ 6 m. It is supposed that the active spacecraft creates the ion beam with parameters given in Table 3.2. The distance between the active spacecraft and the space debris object is AB ¼ 15 m. Figs. 5.5–5.7 show the dependence of the ion beam torque and forces on the object’s deflection angle for various positions of the center of mass Δ ¼ xc/Lc. The shift of the center of mass to the lower end of the cylinder leads to the fact that two points of intersection of the curve LIz(θ) with the abscissa remain on the graph. From a physical point of view, this means that when only the torque of the ion beam acts on the body, two equilibrium states will exist: the stable position θ ¼ 3π/2 and the unstable position θ ¼ π/2. It should be noted that a change in the position of the center of mass leads to a change in the ion beam forces, as shown in Figs. 5.6 and 5.7, since within the framework of the considered assumptions, the axis of the ion beam is directed to the center of mass. Thus, as the center of mass shifts, the direction of the ion beam also changes, which in turn leads to a change in resultant force. The dependences of the ion beam torque on the angle shown in Fig. 5.5 have kinks. They are explained by the transition of the end surfaces of the cylinder to the shadow region and the nonparallel propagation of the ion beam particles. As an example, consider points A and B on the graph corresponding to the center of mass position Δ ¼ 0 in Fig. 5.5. A sharp change in the direction of the curves occurs at points θA ¼ 11.5° and θB ¼ 169°. In the case θ < θA, the ion beam only affects the part of the cylindrical surface facing it (shown in green in Fig. 5.8A). The end surfaces and the rear of the

Fig. 5.4 The space debris object’s geometry.

232

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.5 Dependence of the ion beam torque LIz on the object’s deflection angle for various displacements of the center of mass (Δ ¼ xc/Lc).

Fig. 5.6 Dependence of the ion beam force FIx on the object’s deflection angle for various displacements of the center of mass (Δ ¼ xc/Lc).

Fig. 5.7 Dependence of the ion beam force FIy on the object’s deflection angle for various displacements of the center of mass (Δ ¼ xc/Lc).

Dynamics of passive object attitude motion during ion beam transportation

233

Fig. 5.8 Face transition to a shaded area.

cylinder are in shadow, and they are not affected by the ion beam particles. Position θ ¼ θA corresponds to the situation when the upper end plane of the cylinder becomes parallel to the trajectories of the flow particles flying nearby—that is, it is located on the border of the shadow. For θ > θA, the upper end plane leaves the shade and is affected by the ion beam (Fig. 5.8B). Since the area of this surface is comparable with the area of the blown surface of the cylinder, this exit from the shadow significantly affects the curve of the Lz(θ) graph. A similar pattern is observed near point B, with the only difference being that at first, the lower surface is blown by the ion beam (Fig. 5.8C), and at θ > θB, it turns into a shadow (Fig. 5.8D). According to Eq. (5.75), the attitude motion of the passive object in a circular orbit depends on the ion beam torque view (through the coefficients a(Li) and b(Li) j j ), and γ IG max parameter (5.74) which include ion beam torque amplitude LI , the moments of inertia Ix and Iy, and the semilatus rectum of the orbit p. We study the influence of the parameter γ IG and the object’s center of mass position on the location and type of equilibrium states of the unperturbed system. Two fundamentally different cases will be considered: a prolate (Ix > Iy) and an oblate (Ix < Iy) space object. Case 1: prolate object, of which the principle moments of inertia are Ix ¼ 2250 kg m2 and Iy ¼ Iz ¼ 6000 kg m2. The influence of the position of the center of mass of the object on the position of singular points are demonstrated by bifurcation diagrams shown in Figs. 5.9 and 5.10. Unstable saddle-type equilibrium positions are shown by solid lines in the diagrams. Stable center-type equilibriums are shown by dashed lines. Analyzing the bifurcation diagrams shown in the figures, we can conclude that during the transportation of the prolate cylinder, a phase portrait of one of four types can be observed. The greatest diversity is observed when the center of mass is located near the geometric center of the cylinder (Fig. 5.9C). In the case Δ ¼ 0 (Fig. 5.9A) there are two bifurcation points (γ ∗1 ¼ 2.18, γ ∗2 ¼ 2.34) that allow one to distinguish three different variants of phase portraits (Fig. 5.11). The differences in the topology of phase spaces are observed in the vicinity of points θ ¼ πn. For small γ IG, the main influence on the objects exerts a gravity gradient torque, and θ ¼ π is a singular point of the center type. With increasing γ IG the height of the orbit decreases and, at the same time, the influence of the gravity gradient decreases in comparison with the ion beam torque. Bifurcation occurs at the point

234

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.9 Bifurcation diagram for the case Ix < Iy.

Fig. 5.10 Bifurcation diagram for the case Ix < Iy.

Dynamics of passive object attitude motion during ion beam transportation

235

Fig. 5.11 Phase portraits for Δ ¼ 0.

γ IG ¼ 2.18. Two new centers and two saddles appear in the phase portrait. In the vicinity of the point θ ¼ π, three centers are observed, around which lie the oscillation areas separated by separatrices (Fig. 5.11). With a further increase in the γ IG parameter, the next bifurcation occurs at the point γ IG ¼ 2.34. The center located at point θ ¼ π disappears and a saddle point is observed in its place (Fig. 5.11). In the manufacture of a real spacecraft, the center of mass is displaced from the plane of symmetry. Even a small displacement leads to a qualitative change in the shape of the bifurcation diagram. Figs. 5.9 and 5.10 show points A, B, and C corresponding to different bifurcations. In the case Δ ¼ 0, points A and C occur at the same value of the bifurcation parameter γ IG ¼ γ ∗1 ¼ γ ∗3. With increasing displacement of the center of mass, points B and C are shifted to the left, and point A is shifted to the right (Fig. 5.10). In addition, points B and C approach each other (Fig. 5.10B and C) and disappear as the displacement of the center of mass increases (Fig. 5.9B). In the case when the center of mass is at a distance Δ > 0.0006, points B and C merge and disappear and only one bifurcation is observed. At small values of γ IG, one center is observed in the vicinity of θ ¼ πn points (γ IG ¼ 2 on Fig. 5.12). As the parameter increases, bifurcation occurs. Another center and a saddle point

236

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.12 Phase portraits for Δ ¼ 0.0015 various parameter γ IG.

appear (γ IG ¼ 6 on Fig. 5.12). The separatrix passing through this saddle point encompasses the centers, forming two oscillation areas. As can be seen from Fig. 5.9, in all considered cases, positions π/2 and 3π/2 are unstable. With increasing γ IG and distance Δ, stable positions in the vicinity of the unstable position 3π/2 approach it. In all cases, two large oscillation areas separated by a separatrix can be distinguished in the phase portrait. Inside these oscillation areas, small internal oscillation areas bounded by separatrices can be observed. The described bifurcations affect these small areas. Case 2: oblate object, of which the principle moments of inertia are Ix ¼ 6000 kg m2 and Iy ¼ Iz ¼ 2250 kg m2. In the case of an oblate object, a much simpler behavior is observed. When the center of mass lies in the geometric center of the cylinder (Δ ¼ 0), the bifurcation diagram does not contain bifurcation points (Fig. 5.13A). The corresponding phase portrait is shown in Fig. 5.14A. In the case of a slight displacement of the center of mass, bifurcations are possible (Fig. 5.13B). As the parameter γ IG increases, the center located in the interval (π/2; π) shifts to the right, closer to the saddle point, until it disappears. The center on the interval (0; π/2) behaves similarly, shifting to the left (Fig. 5.14B and C). This moment corresponds to bifurcation γ ∗1 in Fig. 5.13B. With increasing displacement of the center of mass, this bifurcation value decreases, and point A in Fig. 5.13 is shifted to the left. At large displacements, a new bifurcation γ ∗2 arises, which shifts to the right with increasing Δ (point B in Fig. 5.13C). With small γ IG, one center is observed (Fig. 5.14D). After bifurcation γ ∗2, two new centers are born from it, and a saddle point appears in its place (Fig. 5.14C). In contrast to the case of a prolate object, in this case

Dynamics of passive object attitude motion during ion beam transportation

237

Fig. 5.13 Bifurcation diagram for the case Ix > Iy.

there can exist both two (Fig. 5.14A) and one (Fig. 5.14B–D) external oscillation areas. Let us study an influence of the object oscillation on average ion force. It was demonstrated above that even in a circular orbit, a space object can demonstrate the complex dynamics of attitude motion. As shown in Figs. 5.6 and 5.7, the magnitude of the force transmitted to the object by the ion beam depends on the orientation of the object. In order to determine the most favorable motion modes, the averaged values of the forces during the oscillation period for various phase trajectories were calculated. Fig. 5.15 shows the various phase trajectories for prolate space object (case 1). The corresponding averaged forces are shown in Table 5.1. The data presented in Table 5.1 shows that the most effective from the point of view of transportation are trajectories number 8 and 9, which correspond to oscillations near stable equilibrium positions of the center type. For these trajectories, the generated

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.14 Phase portraits of oblate space object.

Fig. 5.15 Phase portraits for prolate space object, Δ ¼ 0.025, γ IG ¼ 6.

Dynamics of passive object attitude motion during ion beam transportation

239

Table 5.1 Ion beam forces for prolate space object. Trajectory

1

2 7

FIx, N FIy, 103 N

1.03  10 7.8

Trajectory

7

FIx, N FIy, 103 N

1.40  105 8.96

8.91  10 7.7

3 8

4 7

5 11

6 6

1.10  105 8.87

5.57  10 7.21

1.12  10 8.01

3.32  10 7.53

8

9

10

11

12

3.47  105 29.05

3.47  105 29.05

3.31  106 7.53

8.95  108 7.7

1.04  107 7.8

average force FIy is maximum in absolute value (shown in bold in Table 5.1). It should be noted that the magnitude of the force FIx reaches a relatively large value. This ion beam force component will lead the object away from the ion beam axis, and additional fuel costs of active spacecraft will be required to compensate for this removal. The smallest magnitude of FIx force is observed on the fourth trajectory, where oscillations with a large amplitude occur. In the case of an oblate space object, a similar picture is observed: the most favorable from the point of view of the transmission of force FIy is transportation in which the object oscillates around stable equilibrium positions (Fig. 5.16, trajectories 8 and 9). The average values of forces for this case are shown in Table 5.2. Bold font indicates the maximum absolute value of force FIy.

Fig. 5.16 Phase portraits for oblate space object, Δ ¼ 0.025, γ IG ¼ 2. Table 5.2 Ion beam forces for oblate space object. Trajectory

1

2

3

4

5

6

FIx, N

4.47  108

1.48  109

1.59  1010

1.29  109

1.35  108

1.5  104

7.85

7.35

8.20

8.5

8.42

8.42

Trajectory

7

8

9

10

11

12

FIx, N

1.49  104

7.86  105

8.43  105

8.48  105

8.48  105

4.54  108

FIy, 103 N

8.4

29.11

29.11

7.17

7.17

7.85

3

FIy, 10

N

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

The data given in Tables 5.1 and 5.2 shows that the region of the phase space in which the relative motion occurs has a significant effect on the magnitude of the force transmitted by the ion beam. The difference in ion beam force is about 18%. Thus, a space object exhibits complex behavior that can change as the altitude of the orbit changes, affecting the γ IG parameter.

5.2.2 Chaotic motion of the object relative to its center of mass in an elliptical orbit in the planar case of motion When a space object moves in an elliptical orbit, the distance from its center of mass to the center of mass of the Earth is periodically changed. This change in altitude leads to a periodic change in the magnitude of the gravitational gradient torque, which can be considered as a periodic disturbance. The presence of saddle points in the phase portrait of an unperturbed system and periodic disturbances creates the prerequisites for chaos. The appearance of chaos can lead to an uncontrolled transition of the object’s oscillations to the rotation mode and vice versa. For the considered contactless space debris removal mission, it can lead to a decrease in the average ion force generated by the ion beam and, as a result, to a decrease in the efficiency of the contactless transportation system, which ultimately translates into an increase in the time required to complete the removal mission. Therefore, the identification of possible chaos is an important stage in mission design. The data presented in this section is based on studies by Aslanov et al. (2020b, 2021) and Aslanov and Ledkov (2020a). One of the methods for detecting chaos is the construction of the Poincare section. Regular trajectories on Poincare sections look like smooth lines. Chaotic trajectories are displayed as a cloud of points. More detailed information about the construction of Poincare sections and their use can be found in Section 1.9. In order to construct the Poincare section, several initial points in the phase portrait are given. For each such point, the numerical integration of Eq. (5.73) is performed. The points of the obtained phase trajectory corresponding to the moments τ ¼ 2π Ωj (j  ℕ) are plotted on the Poincare section. Since the construction of Poincare sections is based on the numerical integration of the equation of motion, it is necessary to choose the mass-geometric parameters of the equation. This section uses the system parameters given in Section 5.2.1 (Figs. 5.17 and 5.18). It is assumed that the eccentricity of the space Fig. 5.17 Poincare cross section for prolate object, Δ ¼ 0.025, γ IG ¼ 6, e ¼ 0.01.

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241

Fig. 5.18 Poincare cross section for oblate object, Δ ¼ 0.025, γ IG ¼ 2, e ¼ 0.01.

debris orbit is e ¼ 0.01. A comparison of the Poincare sections with the phase trajectories of the unperturbed system shown in Figs. 5.15 and 5.16 allows us to conclude that in the vicinity of the unperturbed separatrices, a chaotic layer is observed in the form of a point cloud, and in the distance from the separatrices, the motion remains regular. Calculations show that the area of the region in the phase portrait, which is occupied by the chaotic layer, increases with increasing eccentricity (Fig. 5.19). Fig. 5.18 is of particular interest. Phase trajectories favorable from the point of view of average force FIy (trajectories 8 and 9 in Fig. 5.16) appear inside the chaotic layer in the case of the elliptical orbit with nonzero eccentricity. In this case, the image point can move inside the entire chaotic layer, which corresponds to vibrations of a space object with a large amplitude. As a result, the average force will change compared with the case of unperturbed motion in a circular orbit. Table 5.3 shows the averaged values of the ion beam forces calculated for the initial conditions corresponding to the phase trajectories shown in Fig. 5.16. Averaging was carried out at intervals equal to 10 periods. As can be seen from Table 5.3, the average force FIy dropped significantly compared with the unperturbed case, which is conditionalized by another mode of oscillation of the object. For trajectories 8 and 9, the average ion force drops are 9.8% and 5.6%, respectively. Despite the deterioration, the phase trajectory 9

Fig. 5.19 Change in the thickness of the chaotic layer with the eccentricity change for oblate object, Δ ¼ 0.025, γ IG ¼ 2.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Table 5.3 Ion beam forces for oblate space object (e ¼ 0.01). Trajectory

1

2

3

4

5

6

FIx, N FIy, 103 N

1.21  107 7.86

3.95  109 7.66

6.45  107 8.16

2.40  107 8.5

3.50  106 8.56

1.5  104 8.43

Trajectory

7

8

9

10

11

12

FIx, N FIy, 103 N

1.48  104 8.39

4.72  106 8.22

3.68  106 28.6

8.97  105 7.28

9.00  105 7.28

1.24  107 7.85

remains the most promising in comparison with other trajectories (shown in bold in Table 5.3). In addition to constructing Poincare sections, the presence of chaos in the system can be confirmed by calculating the Lyapunov spectrum for individual trajectories. The Lyapunov exponents making up this spectrum characterize the evolution of trajectories in a certain volume near the trajectory under consideration in different directions of the phase volume. A numerical algorithm for calculating Lyapunov exponents is described in Section 1.9. Let us consider perturbed motion of the system in an extended phase space. Eq. (5.73) can be written in the vector form x0 ¼ FðxÞ,

(5.76)

where the prime denotes the derivative with respect to τ given by expression (5.23), x ¼ [x1, x2, x3]T ¼ [θ0 , θ, τ]T is the vector of variables, and F(x) ¼ [f1, f2, f3]T is a set of functions ðLiÞ a0

f 1 ¼ γ IG

+



k  X

ðLiÞ aj

cos jx2 +

ðLiÞ bj

 sin jx2

! + δ sin 2x2

j¼1



 2ðΩx1 + 1Þ sin x3 Ω1  δ sin ð2x2 Þ cos x3 Ω1 2 Ω !! k   X
 ðLiÞ ðLiÞ ðLiÞ 1 a0 + 4γ IG cos x3 Ω aj cos jx2 + bj sin jx2 , +e

j¼1

f 2 ¼ x1 , f 3 ¼ 1: This system belongs to the class of autonomous systems of dimension N ¼ 3. Lyapunov characteristic exponents λi are constructed for the trajectory starting at the point x0. In ε-vicinity of this point (ε ¼ 105), three starting points x01, x02, and x03 are taken, such that the vectors x0j  x0 (j ¼ 1, 2, 3) are mutually orthogonal. A time interval T ¼ 0.5 is chosen for calculations. Using Eq. (5.76), we carry out numerical integration up to the moment T. The resulting points xk1, xk2, and xk3 are used to calculate the sums inside the Lyapunov exponents:

Dynamics of passive object attitude motion during ion beam transportation

243

Fig. 5.20 Lyapunov characteristic exponents for the case e ¼ 0.01, Δ ¼ 0.025, and γ IG ¼ 2, x0 ¼ [0, 3.2, 0]T.

M 1 X xekj , λj ¼ ln ε M T k¼1

j ¼ 1, 2, 3;

(5.77)

where M is the number of iterations and the symbol “” means orthogonalization of the vector in accordance with the Gram-Schmidt method, described in Section 1.9. Fig. 5.20 shows the Lyapunov exponents calculated for a chaotic trajectory starting near the saddle point x0 ¼ [0, 3.2, 0]T. Fig. 5.21 shows the Lyapunov exponents for a trajectory starting at point x0 ¼ [0, 3.7, 0]T. These graphs were obtained for e ¼ 0.01 and γ ¼ 2, for which Poincare cross sections are shown in Fig. 5.18. For trajectories starting inside the chaotic layer, the spectrum contains positive, zero, and negative Lyapunov exponents λ  [0.510, 0,  0.561]. Such a set indicates that the trajectory is a strange attractor, and dynamic chaos is observed in the system. For a regular trajectory, with the characteristic Lyapunov exponents shown in Fig. 5.21, it is impossible to draw any conclusion about the type and stability, since all exponents tend to zero. Along with Lyapunov exponents and Poincare cross sections, the Melnikov criterion is often used to determine the presence of chaos in a system. Melnikov’s theory allows us to write the necessary condition for chaos (Melnikov, 1963; Wiggins, 2003). However, the construction of the Melnikov criterion is difficult for the considered

Fig. 5.21 Lyapunov characteristic exponents for the case e ¼ 0.01, Δ ¼ 0.025, and γ IG ¼ 2, x0 ¼ [0, 3.7, 0]T.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

unperturbed Eq. (5.75) due to the lack of analytical expressions for its heteroclinic trajectories.

5.3

Dynamics of the controlled motion of a passive object

This section is devoted to the development of various ion beam control laws that provide the transfer of the space debris object attitude motion to the required angular mode of motion. It is assumed that the control system of the active spacecraft keeps it in a constant position in relation to the space debris object.

5.3.1 Control approaches: Control of beam rate and direction In Section 5.2.1, it was demonstrated that the attitude motion of a passive object during its contactless transportation by an ion beam influences the magnitude of the average ion force generated by the flow. Therefore, the transfer of the object to a favorable angular motion mode from the point of view of maximizing the average force leads to an increase in the efficiency of the system and a decrease in the mission time for deorbiting the object. When developing the control law for the angular motion of a space debris object by an ion beam, two approaches can be proposed: the first one involves changing the thrust of the impulse transfer thruster, which leads to a change in the velocity of the particles of the beam, and the second is based on the change of the ion beam axis direction (Fig. 5.22). A change in thrust leads to stretching of the ion torque function LIz(θ) along the ordinate axis, without changing its nature and the

Fig. 5.22 Approaches to controlling the angular position of a passive object.

Dynamics of passive object attitude motion during ion beam transportation

245

location of the points of intersection with the abscissa axis. In the case of thrust control, the ion torque (5.72) can be rewritten as LIz ¼

u LImax

ðLiÞ a0

+

k  X

ðLiÞ aj

cos jθ +

ðLiÞ bj

 sin jθ

! ,

(5.78)

j¼1

where u ¼ [0, 1] is the dimensionless control parameter. The value u ¼ 0 corresponds to the off thruster, and the value u ¼ 1 corresponds to the case when it is running at full power. From a physical point of view, the control parameter u is proportional to the square 2 rate of discharge of ions and is determined by the voltage in the ion engine u ¼ (V0/Vmax 0 ) . Since the ion engine cannot inhale the ions inwards, this control parameter is positive. Changing the direction of the flow axis leads to a complete change in the graph of the function LIz(θ).

5.3.2 Control of the space debris attitude motion in a planar case This section is devoted to the development of control laws, which provide the stabilization of a space debris object in a stable equilibrium position. The data presented in Tables 5.1 and 5.2 indicates that for cylindrical objects, contactless transportation in the position of angular equilibrium is effective in terms of maximizing the average generated ion force. Consider the case when the space debris object moves in a circular orbit and its motion is described by Eq. (5.26) with torque (5.78): ! k   0 max 3 X u ð θ, θ Þ L p ð Li Þ ð Li Þ ð Li Þ I θ00  δ sin 2θ  a0 + aj cos jθ + bj sin jθ ¼ 0: Ω2 I z μ j¼1 (5.79) Equating the acceleration in this equation to zero, the equation for determining the equilibrium position is LIz ðθ∗ Þp3 + δ sin 2θ∗ ¼ 0: Ω2 I z μ

(5.80)

For simplicity, it is assumed that θ∗ ¼ 0 is the equilibrium position. If this is not the case, and it is necessary to stabilize the object in the angular position θ∗ 6¼ 0, then a change of variable e θ ¼ θ  θ∗ can be used, after which all the arguments below e are valid for θ . According to Eq. (5.80), position θ∗ ¼ 0 is an equilibrium if θ∗ is the root of the equation LIz(θ∗) ¼ 0. Consider a situation of uncontrolled motion, when the engine is turned on at full power (u ¼ 1). In the particular case when the space debris center of mass lies in the plane of symmetry (Fig. 5.23, dash-dotted line), its torque Lu¼1 Iz (θ) is an odd function of the angle θ for β ¼ 0, where β is the deviation angle of the ion beam axis from the line connecting the centers of mass of the active spacecraft and the

246

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.23 The dependences of the ion beam torque on the angle of the space debris deflection θ for various positions of the center of mass and the angles of the ion beam axis deviation β.

space debris object (Fig. 5.22). Figures in this section are plotted for cylindrical space debris with a length of 6.5 m, and radius 1.2 m, and the center of mass is shifted 0.2 m to the bottom. Eq. (5.79) has an equilibrium position θ ¼ 0, θ0 ¼ 0. For this case, the following expression for the Fourier series coefficients (5.78) can be written: a(Li) ¼ 0. In the general case, when the center of mass does not j lie in the symmetry plane, the value θ ¼ 0 is not the root of the function Lu¼1 Iz (θ) for β ¼ 0 (Fig. 5.23, dotted line), and θ ¼ 0, θ0 ¼ 0 is not the equilibrium position. Changing the angle β leads to a change in the graph of the function Lu¼1 Iz (θ). A series of calculations showed that for a cylindrical body, the value β ¼ β∗ can be found such that Lu¼1 Iz (0) ¼ 0 (Fig. 5.23, solid line). In this case, θ ¼ 0 is the equilibrium position of Eq. (5.79), and the following relation can be written for the P∞ coefficients of the Fourier series (5.78): j¼0 aj ¼ 0. It is assumed below that β ¼ β∗, and θ∗ ¼ 0 is the equilibrium position. Cases when the ion beam axis is deflected by an angle βmax or βmin, providing a maximum or minimum of ion beam torque at the point θ ¼ 0 should also be considered (Fig. 5.24), since a larger modulus of torque allows us to expect faster stabilization. In

Fig. 5.24 The dependences of the ion beam torque on the angle of the space debris deflection θ for β ¼ βmin and β ¼ βmax.

Dynamics of passive object attitude motion during ion beam transportation

247

the cases where β ¼ βmin and β ¼ βmax, function Lu¼1 Iz (θ) does not equal to zero at the point θ ¼ 0 and its neighborhood. Ion engine thrust control for two cases will be considered below. In the first case (β ¼ β∗), the function Lu¼1 Iz (θ) becomes zero at the point θ ¼ 0, and the function changes sign when passing through this point. In the second case (β ¼ βmin or β ¼ βmax), θ ¼ 0 is not the root of the function Lu¼1 Iz (θ). The laws of the ion engine thrust control that are being developed are based on the idea that the torque created by the engine should be directed against the angular velocity of rotation of the space debris. ∗ 5.3.2.1 Thrust control for the case when Lu¼1 Iz (0) ¼ 0 and β ¼ β

It is proposed to use the control in the form 8 2 <  kΩ I z μ θθ0 , p3 u¼ : 0,

when θθ0 < 0,

(5.81)

when θθ0  0;

where k is the constant control parameter, which is chosen from the condition u < 1. In order to maximize control impact, the control parameter should be chosen as k¼

p3 : Ω2 I z μ max ðjθθ0 jÞ

(5.82)

Since the amplitude of the angle oscillations and its angular velocity decrease as the equilibrium position is approached, it is advisable to recalculate this coefficient periodically. For example, at the moment when θ ¼ 0 and θ0 ¼ 0, the coefficient k could be recalculated on the basis of data for the previous period of the angle θ oscillation. Consider the basic idea of the control law u ¼  kΩ2Izμp3θθ0 in the absence of the constraint u > 0. Consider the ion beam torque Lu¼1 Iz of a full-powered engine acting on the space debris object. The period of its oscillations can be divided into four zones (Fig. 5.25). The dashed line shows oscillations without control, when u ¼ 1 and Lz ¼ Lu¼1 Iz . The solid line demonstrates controlled motion. At zones I and III, the direction of the torque Lu¼1 coincides with the direction of the space debris rotation Iz

Fig. 5.25 Space debris oscillation.

248

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.26 The ion beam torque.

(Fig. 5.26). This torque tends to increase the angular velocity of rotation. In order to slow down the rotation of the space debris, the direction of the torque must be changed to the opposite. Therefore, the control parameter u must be negative. In this case, sign(Lz) ¼  sign (Lu¼1 Iz ). At zones II and IV, the directions of the space debris rotation and the ion beam torque Lu¼1 do not coincide. In this case, the torque slows down the Iz rotation, and the control coefficient u should be positive. In this case, sign(Lz) ¼ sign (Lu¼1 Iz ). Thus, the idea is that the control parameter u is chosen so that the product uLu¼1 has a sign opposite to the angular velocity θ0 . Fig. 5.27 demonstrates schematz ically the position and direction of the space debris rotation. In a real situation, when control u < 0 is not physically feasible, the ion engine is turned off (in zones I and III), in order not to accelerate the rotation of the space debris.

Fig. 5.27 The ion beam torque for different angular states of space debris.

Dynamics of passive object attitude motion during ion beam transportation

249

Two cases will be considered separately when control u ¼  kΩ2Izμp3θθ0 is implemented and when u ¼ 0. We first reject the constraint u > 0 and show that the control u ¼  kΩ2Izμp3θθ0 can provide the asymptotic stability of the equilibrium position θ ¼ 0, θ0 ¼ 0. To study the stability, we use the first approximation approach described in Section 1.8 and by Hahn et al. (1963). It is assumed that θ and θ0 are small values of ε order. The expansion of trigonometric functions in Eq. (5.79) in series gives θ00  2δθ +


 4δ 3 4δ 5 θ  θ + … ¼ kθθ0 c0 + c1 θ + c2 θ2 + c3 θ3 + … : 3 15

(5.83)

Transferring the nonlinear terms in the right-hand side of the equation gives
 4δ
 4δ 5 θ00  2δθ ¼ kθθ0 c0 + c1 θ + c2 θ2 + c3 θ3  θ3 + θ + O ε6 , 3 15

(5.84)

P P L max Pn ðLiÞ ðLiÞ 2 ðLiÞ where c0 ¼ LImax nj¼0 aj , c1 ¼ LImax nj¼1 jbj , c2 ¼  I2 , and j¼1 ðjÞ aj 4LImax Pn
j 3 ðLiÞ c3 ¼  3 are the coefficients of the ion beam torque decomposition. j¼1 2 bj In the case Lu¼1 Iz (0) ¼ 0, the coefficient c0 ¼ 0. Eq. (5.84) can be written in the matrix form x0 ¼ Ax + gðxÞ,

(5.85)

where x ¼ [x1, x2]T is the vector, x1 ¼ θ0 , x2 ¼ θ, and

A¼

0 1

 2δ , 0

(5.86) Pn

ðkÞ (k) k¼1 gj , gi is the function, which ð3Þ (2) (3) 2 g(2) 1 ¼ g2 ¼ g2 ¼ 0, and g1 ¼ kc1 x1 x2 

g ¼ [g1, g2]T is the nonlinear terms vector, gj ¼

contains the variables of the kth order, pffiffiffiffiffi The matrix (5.86) has two eigenvalues: λ1,2 ¼  2δ. In the case δ ¼ 1 (when Ix > Iy), the considered equilibrium position is unstable, since among the roots of the characteristic polynomial of the first approximation Eq. (5.85), there are roots with a positive real part (Theorem 6 from Section 1.8). In this case, the proposed control cannot stabilize the object in this equilibrium position. In the case δ ¼  1 (when pffiffiffi Ix < Iy), the matrix A has two purely imaginary eigenvalues: λ1,2 ¼ i 2 . This is the simplest critical case described in Section 29.b of the book by Hahn et al. (1963). The conclusion about the stability of the equilibrium position cannot be made on the basis of the analysis of the linear system x0 ¼ Ax. For stability analysis, it is necessary to take into account the contribution of nonlinear expansion terms. In order to use the methodology described by Hahn et al. (1963), let us make the change of the independent variable

4δ 3 3 x2 .

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

τ1 ¼

pffiffiffi 2τ:

(5.87)

 0 1 ð3Þ , g1 ¼ k1 c1 x1 x22 + 23 x32 , and the prime in 1 0 Eq. (5.85) means the derivative with respect to the new variable τ1. The Lyapunov function for the equation containing terms of order ε3 will be searched in the form In this case, k1 ¼ 2k, and A ¼

V ¼ x21 + x22 +


 1 4 2 2 2 x1 + x1 x2 + x1 x2 x21  x22 γ 4 , 3 3

(5.88)

dV for Eq. (5.85) is where γ 4 ¼  k14c1 . The total derivative V 0 ¼ dτ 1


2
 V 0 ¼ γ 4 x21 + x22 + O ε7 :

(5.89)

Taking into account the smallness of the values of xj, it can be concluded that the Lyapunov function V given by Eq. (5.88) is positive definite (Fig. 5.28A) at small values x1 and x2, and its derivative V0 given by Eq. (5.89) is positive definite when γ 4 > 0, and negative definite when γ 4 < 0 (Fig. 5.28B). In other words, the equilibrium is asymptotically stable in the case when γ 4 < 0, and unstable in the case when γ 4 > 0. Thus, the condition for the asymptotic stability of the equilibrium state can be written as k1 c1 > 0:

(5.90)

Consider now the case when u ¼ 0, and δ ¼  1. The Lyapunov function for Eq. (5.79) in this case can be written in the form
 2 V ¼ θ0 + 2 1  cos 2 θ :

Fig. 5.28 Lyapunov function (5.88) and its derivative (5.89) for γ 4 ¼  0.2.

(5.91)

Dynamics of passive object attitude motion during ion beam transportation

251

The derivative of function (5.91), taking into account Eq. (5.79), with u ¼ 0 is identically zero: V 0 ¼ 0:

(5.92)

According to the Lyapunov stability theorem, the equilibrium state θ ¼ 0, θ0 ¼ 0 is stable. The use of control (5.81) allows transferring of the phase trajectory to a neighborhood of the equilibrium position θ ¼ 0, θ0 ¼ 0. When θθ0 < 0, the trajectory approaches the equilibrium position due to its asymptotic stability, and when θθ0 > 0, the trajectory does not move away from the equilibrium position due to its stability. Thus, control (5.81) consistently approximates the phase trajectory to the equilibrium position.

5.3.2.2 Thrust control for the case when Lu¼1 Iz (0) 6¼ 0 and β ¼ βmin or β ¼ βmax Consider the case when the ion flux torque does not become zero in the vicinity of the point θ ¼ 0. The following control law is proposed: uðθ0 Þ ¼ kΩ2 I z μp3 θ0 :

(5.93)

u¼1 where sign(k) ¼ sign (Lu¼1 Iz (0)). Since in the considered case the function LIz does not change its sign, the torque when using the control (5.93) is always directed opposite to the direction of rotation and slows it down. In this case, the equation of motion (5.79) takes the form

θ00  δ sin 2θ + kθ0 LImax

ðLiÞ a0

k   X ðLiÞ ðLiÞ + aj cos jθ + bj sin jθ

! ¼ 0:

(5.94)

j¼1

To prove that the equilibrium position θ ¼ 0, θ0 ¼ 0 is asymptotically stable, we use the first approximation approach. After expanding Eq. (5.94) in a series in the point θ ¼ 0, and passing to the independent variable (5.87), the result can be written in matrix form (5.85), where.

A¼

k1 c0

δ

1

0

"

 , g¼

#
 2 2 k1 x1 c1 x2 + c2 x22 + c3 x32 + … + x32  x52 : 3 15 0

The eigenvalues of matrix A have the form λ1,2

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k c k 1 c0 ¼ 1 0 + δ: 2 2

(5.95)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

The equilibrium position θ ¼ 0, θ0 ¼ 0 is asymptotically stable regardless of nonlinear terms g(x), if the real parts of all eigenvalues (5.95) are less than zero. In the case δ ¼ 1 (Ix > Iy), then one of the λ roots is positive, so the equilibrium position is unstable. In the case δ ¼  1 (Ix < Iy), both roots are less than zero when the following is satisfied: k1 c0 > 0:

(5.96)

Condition (5.96) is satisfied when the above ratio sign(k1) ¼ sign (Lu¼1 Iz (0)) ¼ sign (c0) is true.

5.3.2.3 Ion beam direction control As mentioned in Chapter 3 and shown in Fig. 5.24, the ion beam torque LIz(θ) depends on the angle of deflection of the ion beam axis β. In this section, the task of stabilizing space debris attitude motion in a circular orbit by changing the β angle is considered. The task of finding the time-optimal control that transfers the system to the equilibrium position is posed to determine control structure. After transition to a new independent variable (5.87), the controlled motion Eq. (5.79) can be written as θ00 

δ sin 2θ ¼ wðθ, βÞ, 2

(5.97)

where wðθ, βÞ ¼ LIzΩðθ,2 IβμÞp is the control function. The cost function is 3

z

ðτk J¼

dτ:

(5.98)

0

For Eq. (5.97) and cost function (5.98), the Hamilton-Jacobi-Bellman equation can be written in the following form (Aslanov and Ledkov, 2020b): 2

0 0 B B 4 ∂φ ðτ1 , θ, θ Þ þ ∂φ ðτ1 , θ, θ Þ θ0 ∂τ1 ∂θ w½w min ; w max 

min

3   ∂φ ðτ1 , θ, θ Þ 1 þ  sin 2θ þ w þ 15 2 ∂θ0 B

0

(5.99)

¼ 0, where φB(τ1, θ, θ0 ) is an unknown function. Since all the trajectories must reach the point θ ¼ 0, θ0 ¼ 0 at the moment τ1 ¼ τk, the boundary condition is defined only at this point φB(T, θ, θ0 ) ¼ 0. In accordance with the Bellman method, the structure of the optimal control can be found from the condition of the minimum of the expression in brackets in Eq. (5.99):

Dynamics of passive object attitude motion during ion beam transportation

w¼

8 > > < w max , > > : w min ,

∂φB ðτ1 , θ, θ0 Þ 0; ∂θ0 ∂φB ðτ1 , θ, θ0 Þ when > 0: ∂θ0

253

when

(5.100)

Eq. (5.100) shows that optimal control is relay. An attempt to solve Eq. (5.99) makes it necessary to calculate the integral 0

ðθ

dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2wðx  θÞ + 4ð cos ðx=2Þ  cos ðθ=2Þ θ0  2 + F 2wθ + θ0  4 cos ðθ=2Þ ,

φ ðτ1 , θ, θ Þ ¼  B

which is impossible to do analytically. Therefore, we focus on building simplified control. Consider a relay control when the angle β changes instantly. This assumption is fully justified, since the period of oscillations of space debris without a control is about an hour and the engine axis can be rotated in seconds. If the space debris is in fast rotation mode, and this assumption is not fulfilled, then it should be firstly transferred into oscillation mode. To slow down the rotation angular velocity, it is necessary to turn the axis of the ion beam to obtain the maximum torque that will slow down the rotation (Aslanov and Ledkov, 2017). After the angular velocity decreases and the space debris goes into oscillation mode, relay control (5.100) can be used to stabilize it. Fig. 5.24 demonstrates the dependence of the LIz on the angle of deviation of space debris θ for various values of β. An analysis of Fig. 5.29 shows that the maximum values of the moment LIz corresponds to an angle βmax ¼  10°, and the minimum values corresponds to an angle βmin ¼ 9°. With a further increase in the angle modulus, the moments decrease as part of the passive object goes beyond the flow.

Fig. 5.29 Dependence of torque LIz on angle θ for various β.

254

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.30 Phase portrait. Solid lines correspond to β ¼ βmin and dashed lines correspond to β ¼ βmax.

Using the dependences of the moment LIz on the angle θ shown in Fig. 5.24, the phase portrait of Eq. (5.79) for β ¼ βmin and β ¼ βmax was constructed (Fig. 5.30). Solid lines show phase trajectories corresponding to β ¼ βmin, and dashed lines demonstrate phase trajectories for the case β ¼ βmax. Analysis of the phase portrait shows that the transition to the equilibrium position θ ¼ 0, θ0 ¼ 0 is possible either by phase trajectory I or by trajectory II (Fig. 5.30). Thus, the transition to the equilibrium position can be divided into two stages. First, it is necessary to transfer the phase point to the trajectory I or II, and then the point moves along this trajectory to the equilibrium position. Phase trajectories I and II divide the phase space into two areas. In the white zone, to go to trajectory II we need to use control β ¼ βmax, and in the gray area, we need to use control β ¼ βmin to go to trajectory I. For example, if we are at point A in Fig. 5.30, then we need to rotate the ion beam axis by an angleβ ¼ βmin; when we are in point B, we need to rotate the axis by angle β ¼ βmax. This will allow us to get to the point θ ¼ 0, θ0 ¼ 0 along trajectory I. It should be noted that if at the initial moment of time the body oscillates at a relatively small amplitude around the equilibrium position  π/2, then it is necessary to turn off the engine and wait until the imaging point leaves the vicinity of this equilibrium position. The described control requires obtaining an analytical equation for the phase trajectories I and II (Fig. 5.30). The solution of this problem causes difficulties in connection with complex nature of the moment Lz. It is proposed to use an approximate equation of the trajectory to determine the moment of control switching (for δ ¼  1): 1 θ ¼ Ω 0

rffiffiffiffiffiffiffiffiffiffiffiffi L0 p3 θ , μI z

(5.101)

where L0 ¼ Lz(0) is the ion beam torque value corresponding to the equilibrium position θ ¼ 0. Taking into account Eq. (5.101), the following control can be offered:

β¼

8 > > > < β min , > > > : β max ,

μI z θ0 Ω2 signðθ0 Þ, L0 p3 2

when θ > 

μI θ0 Ω2 when θ <  z 3 signðθ0 Þ: L0 p 2

(5.102)

Dynamics of passive object attitude motion during ion beam transportation

255

Fig. 5.31 Phase portrait.

When using the control (5.102), the switching moment does not coincide with the optimal one; therefore, the transition to the equilibrium position is performed in more than one switching. Fig. 5.31 shows schematically the phase trajectory and positions of space debris and active spacecraft at several points. At points D and H, control is switched. If we knew the analytical equation of the boundary, which is shown by a dashed line I, the transition from point A to the origin of coordinates could be done in one switch (near point D).

5.3.2.4 Comparison of the effectiveness of control methods To compare the effectiveness of the control laws described above, let us carry out numerical simulation using Eq. (5.79). As an example, an attitude motion of a rocket stage in a circular orbit of 700 km altitude is considered. The stage has a mass mB ¼ 1400 kg, its length is l ¼ 6.5 m, and its radius is 1.2 m. The center of mass is shifted 0.2 m to the bottom of the rocket. The moments of inertia are Ix ¼ 1300 kg m2 and Iy ¼ Iz ¼ 6800 kg m2. Table 5.4 contains the values of the Fourier coefficients for various values of the deviation angle of the flow axis. Value β∗ ¼ 0.2036° corresponds to the case Lu¼1 Iz (0) ¼ 0, value β ¼  10° corresponds to the maximum torque LIz, and value β ¼ 9° corresponds to the minimum torque LIz. Numerical simulation of attitude motion stabilization will be considered below: when the space debris is in oscillation mode, and when the space debris is in rotation mode. The case when space debris is in oscillation mode is considered here. It is assumed that in the initial moment the following initial conditions are given: θ0 ¼ 0.5 rad and θ00 ¼ 0. The numerical integration of differential Eq. (5.79) was carried out using various ion engine thrust control laws. In the first case (a thin solid line in Fig. 5.32), the ion beam axis was deflected by an angle β ¼ β∗ ¼ 0.2036°, and control law (5.81) was used. After each half-cycle of oscillations, the control coefficient k was recalculated according to Eq. (5.82). The dependence of control u(θ, θ0 ) on an independent variable τ is shown in Fig. 5.33. The control does not reach the maximum value of 1, since it is

Table 5.4 Fourier coefficients for various values of the ion flow axis deviation angle. β 5 β∗ 5 0.2036°

β 5 βmax 5 2 10°

β 5 βmin 5 9°

j

a(Li) j , Nm

b(Li) j , Nm

a(Li) j , Nm

b(Li) j , Nm

a(Li) j , Nm

b(Li) j , Nm

0 1 2 3 4 5 6

1.10158  103 2.22337  103 8.89402  104 7.20263  104 2.83589  104 4.03508  104 8.73372  105

0 1.37822  104 9.95015  103 1.85450  104 5.53070  103 1.52168  104 1.97856  103

0.0368453 6.24380  103 0.0242794 1.54778  104 8.98374  103 3.60726  104 2.95004  104

0 5.95066  104 7.52650  103 5.23968  104 1.07517  103 5.47992  104 1.71206  103

0.0370000 6.01756  103 0.0237260 4.99732  104 8.94706  103 2.14785  104 8.13403  104

0 1.11596  103 7.71685  103 1.57499  104 1.79172  103 5.89558  104 1.32353  103

Dynamics of passive object attitude motion during ion beam transportation

257

Fig. 5.32 Changing the space debris deflection angle θ using ion engine thrust control.

Fig. 5.33 Control law for the first case when β ¼ β∗ and u(θ, θ0 ) is given by Eq. (5.81).

proportional to the amplitude of angle θ oscillations, which decreases as a result of the control. In the second case (dashed line in Fig. 5.32), the beam axis is deflected by an angle βmax ¼  10°, which provides the maximum modulus of ion beam torque Lu¼1 Iz at θ ¼ 0, and the control law (5.81) was used. Fig. 5.34 shows that the implementation of this law (dashed curve) requires significantly less engine thrust than in the first case. In the third case (a thick solid line in Fig. 5.32), the axis is deflected by an angle βmax ¼  10° and the control law is given by Eq. (5.93). The control law is shown in Fig. 5.34 by a solid line (curve 3). The results show that the third control law is most effective in terms of stabilization rates, while the first law is the least effective. In the first case, the angle passes into the ε-neighborhood of the equilibrium position θ ¼ 0 in 163,658 s (45 h, 27 min, and 38 s). In the second case, this process takes Fig. 5.34 Control law for the cases when β ¼ βmax and u(θ, θ0 ) is given by Eq. (5.81) (curve 2) or by Eq. (5.93) (curve 3).

258

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.35 Changing the space debris deflection angle θ using ion beam axis deflection control.

101,572 s (28 h, 12 min, and 52 s). In the third, it takes just 51,508 s (14 h, 18 min, and 28 s). In the calculations, it was chosen that ε ¼ 106. Consider the stabilization of space debris by changing the angle of the ion beam axis deviation β. The result of integrating Eq. (5.97) with the control (5.102) is shown in Fig. 5.35. The law of the ion beam axis deviation angle is shown in Fig. 5.36. Calculations show that this law makes it possible to stabilize space debris by transferring it to the ε-neighborhood of the point θ ¼ 0, θ0 ¼ 0 in 524 s, which is 8 min and 44 s. Thus, controlling the orientation of the ion beam axis allows stabilization of the oscillations of space debris almost 100 times faster than in the case of ion engine thrust control. Consider the case when the space debris object rotates relative to its center of mass. It is assumed that at the initial time, the following initial conditions are satisfied: θ0 ¼ 0 and θ00 ¼ 105.5 (this value corresponds to angular velocity θ_ 0 ¼ 10 degrees=s ). Research by Albuja et al. (2015) showed that space debris can move at such angular velocities. As in Section 5.3.2.3, two methods of space debris attitude motion control are considered: by changing the ion engine thrust and by controlling the inclination of the ion beam axis. Consider space debris detumbling by ion engine thrust control. The simulations show that the use of control law (5.81) does not allow stopping the rotation of space debris, while control (5.93) copes well with this task. Since the study by Aslanov and Ledkov (2017) showed that in order to stop the rotation of a body by an ion beam, the Fig. 5.36 Ion beam axis deflection angle control.

Dynamics of passive object attitude motion during ion beam transportation

259

Fig. 5.37 Space debris detumbling using controls (5.93) and (5.102).

Fig. 5.38 Final stage of the space debris detumbling using controls (5.93) and (5.102).

beam axis should be turned in the same direction as the direction of the body rotation, in the considered case, β ¼ βmin ¼ 9° should be given. Results of integration of Eq. (5.79) using the control law (5.93) are shown in Figs. 5.37 and 5.38 by a dashed line. The constant control parameter was given as k ¼  100. Since the control u(θ0 ) is physically bounded above, it was assumed that if u > 1, then u ¼ 1. Calculations show that the phase trajectory goes into the ε-neighborhood of equilibrium position θk ¼ 891π in 76,766 s, which equates to 21 h, 19 min, 26 s. The dependence of the control law u(θ0 ) on an independent variable τ1 is shown in Fig. 5.39. Fig. 5.39 Control during space debris detumbling by thrust.

260

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Control of ion beam axis direction can also be used to space debris detumbling. Results of numerical integration of Eq. (5.97) using the control law (5.102) are shown in Figs. 5.37 and 5.38 by solid lines. Unlike the case of engine thrust control, stabilization occurs at a different equilibrium position θk ¼ 890π (Fig. 5.38) and takes less time: 33,663 s, which equates to 9 h, 21 min, and 3 s. In both cases, the detumbling process can be divided into two stages. The first stage is the reduction of the angular velocity of the space debris rotation. The second stage is a decrease in the oscillation amplitude. The first stage is the same in both cases. The ion beam axis is deflected by an angle β ¼ βmin and the ion engine is turned on at full power. This stage ends at the moment τ1 ¼ 52.75 (Fig. 5.38), after which control can be carried out in different ways. Calculations showed that the second control method is preferable from the viewpoint of minimizing stabilization time. Analysis of the control laws given by Eqs. (5.100) and (5.102) shows that the implementation of the combined control, when simultaneously controlling the ion beam axis deflection angle and the ion engine, is inexpedient since these types of control fundamentally contradict each other. Relay angle control requires the inclusion of thrust at full power, but the thrust control implies varying thrust and even intervals with the engine off. Comparing Figs. 5.32 and 5.35 shows that the angle control solves the problem of stabilization for substantially less time compared with the control of the ion engine thrust. The advantage of relay angle control is the possibility of transferring space debris to the equilibrium position for a finite time, while engine thrust control allows it to approach this position asymptotically, and it takes infinite time to reach the equilibrium. In addition, the rotation of a light active spacecraft by its orientation engines is considered by the authors to be a simpler task than controlling the engine thrust in a wide range.

5.3.2.5 Ion beam control based on energy estimation In the previous subsections, control laws that ensure the transfer of the angular motion of a space debris object to an equilibrium position were considered. However, it is possible that the shape and layout of the space debris object is such that the equilibrium position is not the optimal angular position in terms of the generated ion force. There may exist a phase trajectory in space (θ, θ0 ) for which the averaged ion force calculated on this trajectory turns out to be greater than the force in the equilibrium position. In this case, the goal of the developed control law should be the transition of the object oscillations to this phase trajectory. When developing the control law, control of the direction of the ion beam axis is used because, as shown in Section 5.3.2.4, this approach is more efficient than thrust control. A control based on an estimate of the energy of the angular motion of a space debris object is being developed. It is assumed that the attitude motion of the object is described by Eq. (5.79), which can be rewritten as θ00 ¼ δ sin 2θ + ξLIz ðθ, βÞ,

(5.103)

Dynamics of passive object attitude motion during ion beam transportation

261

where ξ ¼ p3(Ω2Izμ)1, and LIz(θ, β) is the ion beam torque that depends on the angle of deflection of the ion beam axis β, which is a control parameter. For this equation, the energy integral can be written in the form ð 2 θ0 δ Eðθ, θ , βÞ ¼ + cos 2θ  ξ LIz ðθ, βÞdθ: 2 2 0

(5.104)

The space debris oscillation energy E retains its value along each phase trajectory. Let E∗ denote the energy corresponding to the motion along the target trajectory to which the transition should be made. It should be noted that if there are several regions of oscillations in the phase space, which are separated by separatrices, then one energy level can correspond to several trajectories in different regions. In this case, in addition to the energy E∗, the target trajectory must be identified by the boundary values θ1 and θ2 corresponding to the target oscillation region. The idea of the proposed control method is to deviate the ion beam axis by the angle βmin or βmax so that the work performed by the ion toque brings the current energy E(θ, θ0 , 0), calculated by Eq. (5.104), closer to the target value E∗. If the current energy level is greater than the target energy E(θ, θ0 , 0) > E∗, current angle θ  [θ1, θ2], and θ0 < 0, then the ion beam axis must be deflected by the angle β ¼ βmax. If E(θ, θ0 , 0) > E∗, θ  [θ1, θ2], and θ0 > 0, then the ion beam axis must be deflected by the angle β ¼ βmin. If the current energy level is less than target energy E(θ, θ0 , 0) < E∗ and θ0 < 0, then the ion beam axis must be deflected by the angle β ¼ βmin. If E(θ, θ0 , 0) < E∗ and θ0 > 0, then the ion beam axis must be deflected by the angle β ¼ βmax. The described control methodology is used in Section 5.4. Fig. 5.40 schematically shows the proposed control method. Let the motion start at point A1. The energy level at this point in the phase space exceeds the energy on the required phase trajectory E∗, so the ion beam axis, in accordance with the proposed method, deviates by an angle β ¼ βmin. At point A2, the phase trajectory leaves the region θ  [θ1, θ2] and the axis returns to the position β ¼ 0. At point A3, the phase trajectory returns to the region θ  [θ1, θ2] and, since θ0 < 0, the ion beam axis deviates by

Fig. 5.40 Proposed control scheme.

262

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

an angle β ¼ βmax. At point A4, the current energy level coincides with E∗, and the axis is directed to the center of mass of the object β ¼ 0. If the phase trajectory moves in one of the inner oscillation regions and it is necessary to transfer the phase trajectory to the neighboring inner region, then this maneuver is carried out in two steps. First, an intermediate target phase trajectory is selected in the outer region. This trajectory should cover the target oscillation region. The representing point is translated first into this trajectory. At the second step, the representative point is transferred from the intermediate trajectory to the target one.

5.3.3 Control of the space debris attitude motion in a spatial case in GEO Consider the three-dimensional motion of a mechanical system consisting of a symmetrical space debris object and the active spacecraft in GEO. It is assumed that the active spacecraft maintains a constant position relative the space debris object, and the ion beam axis is directed to the center of mass of the object. Let us construct the thrust control of the active spacecraft’s impulse transfer thruster, which provides the translation of the angle θ in the equilibrium position θ∗, which is calculated as the root of the nonlinear Eq. (5.69). The ion torque generated by the ion beam can be written as LI ¼ uLImax

k X

ðLiÞ

bj

sin jθ∗ ,

(5.105)

j

where u¼

8
 h
 i < 1 + k θ  θ θ_ Η θ  θ θ_ , ∗ ∗ :

1,


 when k θ  θ∗ θ_ > 1;
 when k θ  θ∗ θ_ 1;

(5.106)

where Η is the Heaviside theta function and k is the constant control parameter, which is chosen from the condition u < 1. The case u < 0 is not physically feasible, because it means an ion beam that is drawn into the engine of the spacecraft. It is proposed to take k ¼ 1 initially, and then to choose this parameter based on data from the previous oscillation period T: k¼

1 
 : _ max j θ  θ∗ θj

(5.107)

½tT, t

The results of numerical simulation presented in the next section confirm the possibility of using this control law to transfer the angle θ to the equilibrium position θ∗. To study the stability of the controlled motion, let us make the change of variable x ¼ θ  θ∗ in Eq. (5.60), taking into account Eq. (5.105):

Dynamics of passive object attitude motion during ion beam transportation

263

     k R  G cos x + θ∗ G  R cos x + θ∗ uL max X ðLiÞ   x€ ¼  bj sin jðx + θ∗ Þ: + I I sin 3 x + θ∗ j

(5.108)

Two cases of the system motion will be considered below: when control u ¼ 1 +
 k θ  θ∗ θ_ ¼ ð1 + kxx_ Þ is used, and when u ¼ 1. Let investigate the stability of motion in the first case u ¼ ð1 + kxx_ Þusing the apparatus of stability according to the first approximation. The right-hand side of the equation can be expanded into a Maclaurin series: x€ ¼ a01 x + a11 xx_ + a02 x2 + a12 x2 x_ + a03 x3 , where a01 ¼ c1  a02 ¼  a03 ¼  c1 ¼

3b2 cos θ∗ sin θ∗

3c2 ð1 + 3 cos 2 θ∗ Þ

+

2 sin 2 θ∗

c1 ð8 + 19 cos 2 θ∗ Þ 6 sin 2 θ∗

G2 + R2 2GR cos θ∗ sin 2 θ∗



¼



LImax I

k P j

5c1 cos θ∗ 2 sin θ∗



ðLiÞ

bj j cos jθ∗, a11 ¼ kc2, a12 ¼ RG sin θ∗



c2 cos θ∗ ð9 sin 2 θ∗ 20Þ

LImax I

2 sin 3 θ∗

k P j

ðLiÞ

bj

(5.109)

LImax 2I

+

k P j

kLImax I

k P j

ðLiÞ

bj j cos jθ∗ ,

ðLiÞ

bj j2 sin jθ∗ ,

2RG cos θ∗ sin 2 θ∗

sin jθ∗ , and c2 ¼



LImax 6I

k P j

ðLiÞ

bj j3 cos jθ∗ ,

ðGR cos θ∗ ÞðRG cos θ∗ Þ sin 3 θ∗

.

Eq. (5.109) can be written in the matrix form x_ ¼ Ax + gðxÞ,

(5.110) ð2Þ

(3) (2) (3) T 2 _ x2 ¼ x, g ¼ [g(2) where x ¼ [x1, x2]T, x1 ¼ x, 1 + g1 , g2 + g2 ] , g1 ¼ a11 xx_ + a02 x ,

 0 a01 pffiffiffiffiffiffiffiffiffiffi ð3Þ g1 ¼ a12 x2 x_ + a03 x3 , and A ¼ . The eigenvalues λ ¼  a01 of the 1 0 matrix A determine the stability of the equilibrium position θ∗. If a01 < 0, then one of the eigenvalues is a positive real number and the motion is unstable. In this case, the proposed control (5.106) does not allow transferring the system to the required equilibrium position. When a01 > 0, the real part of the eigenvalues λ is zero, and in order to make a conclusion about the stability of the motion, it is necessary to study the nonlinear terms of Eq. (5.110). The methodology for obtaining the Lyapunov function for this critical case is described in detail in Section 30 of the book by Hahn et al. (1963) and in more detail in Section 37 of the work by Malkin (1959). For the considered nonlinear Eq. (5.110), the Lyapunov function can be written in the form


2  a11 þ a03 x41 x21 2a02 x32 2a11 x31 2 V¼ þ x2  þ pffiffiffiffiffiffi þ a01 3a01 2a01 3 a01
3  ða01 a12 þ a02 a11 Þ x1 x2  x1 x32 a03 x21 x22 pffiffiffiffiffiffi þ : þ a01 a3 01

(5.111)

264

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

For the case of small values of xj, the sign of the function V is determined by the first two terms of Eq. (5.111). It is assumed that xj has order of smallness ε. In a small neighborhood of zero, the Lyapunov function is positive definite. The total derivative _ Vcalculated by virtue of Eq. (5.110), 2
 ða a + a a Þ
V_ ¼ 01 12 3 02 11 x21 + a01 x22 + O ε5 , 4a01

(5.112)

for small values of xj is positive defined when (a01a12 + a02a11) > 0, and negative defined when a01a12 + a02a11 < 0. Thus, the condition for asymptotic stability of the equilibrium position can be written in the form a01 a12 + a02 a11 < 0:

(5.113)

In the second case, u ¼ 1, the energy integral can be written for Eq. (5.109): E¼

x21 + W ðx2 + θ∗ Þ + E0 , 2

(5.114)

where x1 ¼ x,_ x2 ¼ x, E0 is the integration constant, and W is reduced potential energy: ðLiÞ k G2 + R2  2GR cos θ LImax X bj W ðθ Þ ¼ cos jθ: + I j 2 sin 2 θ j

(5.115)

Eq. (5.109) describes the motion of a conservative mechanical system. For the study of the stability of such systems, the direct Lyapunov method has proven itself well. The energy integral (5.114) can be taken as the Lyapunov function ðLiÞ

k b x2 G2 + R2  2GR cos ðx2 + θ∗ Þ LImax X j V ðx1 , x2 Þ ¼ 1 + + 2 I j 2 sin 2 ðx2 + θ∗ Þ j

cos jðx2 + θ∗ Þ

ðLiÞ

k b G2 + R2  2GR cos θ∗ LImax X j   2 I j 2 sin θ∗ j

(5.116)

cos ðjθ∗ Þ,

where the constant of integration E0 is chosen so that V(0, 0) ¼ 0. Since function (5.116) is an integral of the equation of motion, its derivative is equal to zero due to this equation, V_ ðx1 , x2 Þ≡0. According to Lyapunov’s theorem, the equilibrium is stable if the function V(x1, x2) is a positive definite function. This condition can be written as
 k bðLiÞ

  G2 + R2  2GR cos x2 + θ∗ LImax X j
 + cos j x2 + θ∗  cos jθ∗ 2 I j 2 sin x2 + θ∗ j 

G2 + R2  2GR cos θ∗ >0 2 sin 2 θ∗ (5.117)

Dynamics of passive object attitude motion during ion beam transportation

265

for all x2 6¼ 0. For motion in a small neighborhood of the equilibrium position, let us expand the left-hand side of inequality (5.116) into a Maclaurin series:
 3 G2 + R2  2GR cos θ∗ 2G2 + 2R2  GR cos θ∗  4 2 sin 2 θ∗ 2 sin θ∗ 

LImax

k X

2I

(5.118)

! ðLiÞ

bj j cos jθ∗


 x22 + O x32 > 0:

j

For small x2, the sign of the series is determined by the sign of the coefficient at x22. To prove this statement, it is enough to show that point x2 ¼ 0 is a local minimum point of the series in the left-hand part of Eq. (5.118) by calculating the first and second derivatives of the series at this point. In view of the above, Eq. (5.118) can be rewritten as
 3 G2 + R2  2GR cos θ∗ 2 sin 4 θ∗



2G2 + 2R2  GR cos θ∗ 2 sin 2 θ∗

k L max X ðLiÞ  I bj j cos jθ∗ > 0: 2I

(5.119)

j

It should be noted that inequality (5.119) coincides with the sufficient condition for the minimum reduced potential energy of Eq. (5.109). This condition, up to a factor, coincides with the condition a01 > 0, which provides a pair of imaginary eigenvalues λ of the linearized Eq. (5.110) matrix. Thus, in the case of controlled contactless transportation with the control (5.106), the following behavior is observed when conditions (5.113) and (5.119) are met. If kðθ  θ∗ Þθ_ > 1, then the equilibrium position θ ¼ θ∗ is asymptotically stable and the phase trajectories approach it. If kðθ  θ∗ Þθ_ 1, then this equilibrium position is stable, and phase trajectories do not move away from it. These phases alternate, and the proposed control consistently approximates the phase trajectories to the equilibrium state. The obtained Lyapunov functions prove the stability of the system in each of the considered cases. However, since during the motion there is a switching between these cases, the obtained conditions (5.113) and (5.119) are necessary, but are not sufficient conditions for stability. As an example, the motion of a space debris object close in its mass-geometric parameters to the Meteosat-8 (Schmetz et al., 2002) satellite in a GEO is studied below. The controlled motion of the object with law (5.106) is considered. It is assumed that the object’s orbit is specified by the following parameters: ν ¼ 0, ϑ ¼ 0, ν_ ¼ 0, ϑ_ ¼ 0, r ¼ 42, 164, 000 m, and r_ ¼ 0. Motion simulation requires a preliminary calculation of ion beam forces and torques depending on the orientation of the object in the ion beam. The geometric parameters of the considered object and an example of the computational mesh are shown in Fig. 5.41. The mass of the object is mB ¼ 1100 kg; principal moments of inertia are Ix ¼ Iz ¼ 2100 kg m2 and Iy ¼ 1400 kg m2. The parameters of the ion beam correspond to the parameters given in Table 3.2. They were chosen based on an analysis of existing ion engines. The

266

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.41 Considered space debris object and calculation mesh.

Table 5.5 Fourier series coefficients for ion beam torque. j

b(Li) j

j

b(Li) j

j

b(Li) j

j

b(Li) j

1 2 3 4

1 0.4482 0.0002 0.8870

5 6 7 8

0.0378 0.0394 0.0304 0.2792

9 10 11 12

0.0109 0.0076 0.0083 0.1466

13 14 15 16

0.0040 0.0066 0.0013 0.0800

calculated Fourier series expansion coefficients b(Li) are given in Table 5.5, and j Lmax ¼ 3.706  103 Nm. I As an example, consider the stabilization of the spatial motion of space debris using control (5.106). Before proceeding to the simulation of controlled motion, let us draw up a diagram illustrating the possibility of applying control (5.106) using inequalities (5.113) and (5.119). For each pair of R and G values, the equilibrium position θ∗ is calculated using Eq. (5.69). Then, for these values of R, G, and θ∗, the fulfillment of conditions (5.113) and (5.119) is checked. If both conditions are met, then the point on the diagram is shaded gray, otherwise the point is shaded white. Fig. 5.42 shows a diagram obtained in the described manner. Region I corresponds to a pair of values R and G, where the control (5.106) does not provide an asymptotic stability of the equilibrium position. In region II, both inequalities (5.113) and (5.119) are satisfied, and the equilibrium position is asymptotically stable. Numerical calculations have shown that for the considered object, inequality (5.119) is satisfied in the entire region where inequality (5.113) is satisfied. Fig. 5.42B shows an enlarged view of the central area of the diagram. It can be seen from the figure that at small values of R and G, the regions have a more complex form. For small absolute values of R and G, which can be observed at low angular velocities of the object (according to Eq. (5.32)), region II

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267

Fig. 5.42 Diagram illustrating the asymptotic stability of an equilibrium position.

does not have holes, which indicates that the proposed control can be used for stabilizing the oscillations of slowly rotating objects (ω0 < 0.002 rad/s). For the case of fast rotating objects, before applying control (5.106), it is necessary to make sure that the object is in region II. Figs. 5.43 and 5.44 show the graph of the Lyapunov function for cases u ¼ 1 + kðθ  θ∗ Þθ_ and u ¼ 1, which were calculated for values R ¼ 0.002 rad/s, G ¼  0.003 rad/s from region II of the stability diagram. In both cases, the Lyapunov functions are _ the derivative of this function is positive definite. In the case where u ¼ 1 + kðθ  θ∗ Þθ, negative definite (Fig. 5.45). In the second case, the derivative is identically zero. Let us simulate the controlled motion of the system using Eqs. (5.34), (5.35), and (5.40)–(5.42) and the initial conditions given in Table 5.6. In all cases, γ 0 ¼ φ0 ¼ 0 and ωz0 ¼ 0. It follows from Eq. (5.29) that θ_0 ¼ ωz0 ¼ 0 . In the first four cases in Table 5.6, space debris is not twisted about the axis of symmetry. Fig. 5.46 shows the dependence of the angle θ on time when using control for the first four cases in Table 5.6. Curve 3 shows a very slow approach to the equilibrium position. This curve is shown on a larger scale in Fig. 5.47.

Fig. 5.43 Lyapunov function for u ¼ 1 + kðθ  θ∗ Þθ_ and R ¼ 0.002 rad/s, G ¼  0.003 rad/s.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.44 Lyapunov function for u ¼ 1 and R ¼ 0.002 rad/s, G ¼  0.003 rad/s.

Fig. 5.45 Lyapunov function derivative for u ¼ 1 + kðθ  θ∗ Þθ_ and R ¼ 0.002 rad/s, G ¼  0.003 rad/s.

Table 5.6 Initial conditions. No

ωx0, rad/s

ωy0, rad/s

θ0, rad

G, rad/s

R, rad/s

θ∗, rad

Region

1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0.001 0.001 0.001 0.001 0.001 0.1 0.1

0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.15 0.15

0.01 0.3 1 2.5 0.01 0.3 1 2.5 0.302 0.3 1.1

4.9999  105 1.4776  103 4.2074  103 2.9924  103 7.1663  104 2.1145  103 4.5676  103 2.4583  103 2.1236  103 0.10802 0.16392

0 0 0 0 6.6667  104 6.6667  104 6.6667  104 6.6667  104 6.6667  104 0.06667 0.06667

2.4715 2.1185 1.7414 1.8498 2.1234 1.8158 1.444 1.7642 1.8145 0.9058 1.152

II II II II II II I II II II II

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269

Fig. 5.46 Dependences of the angle θ on time for initial conditions 1–4 given in Table 5.6.

Fig. 5.47 Dependences of the angle θ on time for curve 3 from Table 5.6.

In cases 5–8, space debris is slightly twisted around its longitudinal axis, but the angular rate of its rotation is small. Fig. 5.48 shows graphs for these cases. In contrast to Fig. 5.46, here one of the trajectories is unstable, and an increase in the amplitude is observed (curve 7 in Fig. 5.48). The second significant difference is that curves 6 and 8 are stabilized not in the global equilibrium position θ∗, which is indicated in Table 5.6, but in the local equilibrium. Fig. 5.49 shows graphs of reduced potential energy (5.115) for cases 2 and 6. In case 2, the reduced potential energy has one minimum, which corresponds to a stable equilibrium position θ∗. In case 6, there are two minimum points on the graph. The global minimum is shown by a black point and it is Fig. 5.48 Dependences of the angle θ on time for initial conditions 5–8 given in Table 5.6.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.49 Reduced potential energy (5.115) for cases 2 and 6.

given in Table 5.6. The second minimum θ∗2 ¼ 1.3387rad is shown by a white point. Curve 6, shown in Fig. 5.48, tends to θ∗2. It is impossible to say in advance to which equilibrium position the phase trajectory will be attracted. For example, in case 9, the initial angle θ0 differs by 0.002 rad from case 6, but the phase trajectory is attracted to the global minimum θ∗ (Fig. 5.50). Getting into one or another area of attraction is very sensitive to the initial conditions, and given the presence of external disturbances—for example, gravitational or solar wind—we can say that it is random. In the numerical examples discussed above, space debris had a very low angular velocity. Calculations show that with an increase in the angular velocity, the control becomes less efficient and takes more time. Figs. 5.51 and 5.52 show the graphs plotted for cases 10 and 11 from Table 5.6. The frequency of angle θ oscillations has increased significantly in comparison with the cases considered above. Although case 10 corresponds to the stable region II in the diagram (Fig. 5.42), it can be seen that the motion is unstable and the amplitude of oscillations increases (Fig. 5.51A). The reason is that the proposed control law provides local asymptotic stability of equilibrium positions θ∗. Numerical calculations show that when the angular velocity of rotation of the body increases, the region of the equilibrium position attraction decreases. In the case shown in Fig. 5.51, the initial angle θ0 ¼ 0.3rad is far from the equilibrium position θ∗ ¼ 0.9058rad (case 10). If we take the initial position closer to the equilibrium θ0 ¼ 1.1rad (case 11), the phase trajectory will approach it. Fig. 5.52 shows the angle oscillations in case 11. As can be seen, the amplitude of angle θ decreases rather slowly. Fig. 5.50 Dependences of the angle θ on time for initial conditions 6 and 9.

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271

Fig. 5.51 Dependences of the angle θ on time for initial conditions 10 given in Table 5.6.

Fig. 5.52 Dependences of the angle θ on time for initial conditions 11 given in Table 5.6.

Thus, the proposed control law makes it possible to stabilize the space debris oscillations and transfer the motion of the axis of the space debris in the regular precession mode. This law can be used when transporting slowly rotating axisymmetric objects in orbit.

5.3.4 Detumbling of axisymmetric space debris in a spatial case This section is devoted to the development of a control law for an active spacecraft engine thrust, which ensures the stabilization of spatial oscillations of cylindrical space debris in a stationary angular position. A simplified mathematical model (5.50)–(5.53) describing the motion of a dynamically symmetric rigid body in a Keplerian orbit is used to describe the dynamics of a space debris object. It is assumed that during the motion, the spacecraft’s control system keeps it in a constant position

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

relative to the space debris x ¼ d ¼ const, y ¼ 0, and z ¼ 0. The axis of the ion flow is always directed to the space debris object’s center of mass. To control the attitude motion of space debris, we will control the thrust of the engine that creates the ion beam. In this case, the ion torque can be defined as LI ¼ uðθ, θ0 , G, γ ÞLu¼1 ðθÞ, I

(5.120)

where u  [0, 1] is a dimensionless control parameter, which is proportional to the square rate of the ions velocity and can be changed by the voltage in the thruster. The value u ¼ 0 corresponds to the off engine, and the value u ¼ 1 corresponds to (θ). To calculate the dependence of the the engine turned on at full power LI ¼ Lu¼1 I ion torque Lu¼1 on the angle θ, an author’s MATLAB implementation of the compuI tational procedure described in detail in Chapter 3 is used. To control the space debris attitude motion, it is proposed to use the following feedback control law, which puts the system in a stationary motion mode (Aslanov and Ledkov, 2021): 8 0, > > < u ¼ u, > > : 1,

when u 0; when 0 < u < 1;

(5.121)

when u  1;

where

  In2 k4 u ¼ 1 + kθ ðθ∗  θÞ  kΩ θ0 + kγ γ ∗  γ + kG ðG∗  GÞ u¼1 , LI ðθÞ

(5.122)

where kj are control law parameters. To determine the control law parameters, the cost function is introduced in the form

2 2 F kθ , kΩ , kγ , kG ¼ ðθ∗  θTav Þ2 + θ0Tav + γ ∗  γ Tav 2 + ðG∗  GTav Þ ,

(5.123)

0 where θTav, θTav , γ Tav, and GTav are average values of the variables calculated for the last period of oscillations, and θ∗, γ ∗, and G∗ are values corresponding to the equilibrium position and described in Section 5.1.4. To calculate this function, the system of Eqs. (5.53)–(5.55) is numerically integrated with the specific values kθ, kΩ, kγ, and kG transferred as parameters. Integration is performed on an independent variable interval of 10 periods. Then the last period is taken, and the amplitude values of the variables for this period are found. The arithmetic mean values for these amplitude values 0 are substituted into the formula (5.123) as θTav, θTav , γ Tav, and GTav. To find a set of control parameters kθ, kΩ, kγ , and kG that ensure the minimum cost function (5.123), the Nelder-Mead method (Lagarias et al., 1998), which is implemented in Matlab as FMINSEARCH, is used.

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273

Fig. 5.53 Dependence of ion beam torque LImax on angle θ.

As an example, let us consider the deorbiting of a hypothetical rigid body. Cases of uncontrolled and controlled motion will be compared below. It is assumed that space debris mass mB ¼ 1400 kg, moments of inertia Ix ¼ 1300 kg m2, Iy ¼ Iz ¼ I ¼ 6800 kg m2, the length of the debris object is 6 m, and its diameter is 2.4 m. The center of mass of the space debris lies on the axis of symmetry and is shifted to the lower end by 0.5 m. The distance between the spacecraft and space debris center of mass d ¼ 15 m. The ion beam parameters are given in Table 3.2. Fig. 5.53 shows the dependence of the ion torque on the angle θ, which was obtained for body and ion beam with the above parameters. At the initial moment of time, the space debris center of mass has the following motion parameters: r0 ¼ 6, 671, 000 m, r_0 ¼ 0, and f_0 ¼ 1:1587 103 rad=s. These values correspond to n ¼ 1.1587 103rad/s and e ¼ 0. Let us take ωx ¼ 0.03 rad/s, ωy ¼ 0.0005 rad/s, and ωz ¼ 0.02 rad/s, for which according to Eqs. (5.32) and (5.46) G ¼ 0:0803, and R ¼ 0:0825. Let us find stationary values. Numerical solution of Eq. (5.68) gives the following roots: θ∗1 ¼ 0, θ∗2 ¼ 1.9324 rad, θ∗3 ¼ π, θ∗4 ¼ 4.3508 rad, and θ∗5 ¼ 2π. Modeling shows that roots with odd indices correspond to unstable equilibrium, and roots with even indices are stable equilibrium. Let us choose the second root as a stationary position to which we will stabilize the oscillations: θ∗ ¼ θ∗2. Eqs. (5.66) and (5.67) give the following results for this value: G∗ ¼ 2:0668  102 and γ ∗ ¼ 1.5451 rad. It is assumed that at the initial moment of time, θ0 ¼ 2.2 rad, γ 0 ¼ 1.7 rad, and G ¼ 0:0803 . Let us simulate the motion of the system using the control law (5.121). The minimization of the cost function (5.123) using the FMINSEARCH function in MATLAB gives the following values of the control law parameters: kθ ¼ 16.7469, kΩ ¼ 8.3178, kG ¼ 12.3538, and kγ ¼ 43.9145. Figs. 5.54–5.56 show the graphs obtained in the case of uncontrolled motion and when using the proposed control law. The dependence of the dimensionless function of the control law u on the angle f is shown in Fig. 5.57. The figure shows that at the initial stage, the control has a close to relay character, but as the system parameters approach the stationary

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.54 Dependence of angle θ on angle f.

Fig. 5.55 Dependence of angle γ on angle f.

Fig. 5.56 Dependence of G on angle f.

position, the control takes the form of a nonlinear function bounded from above (when f > 66.14 rad). At u ¼ 1 and u ¼ 0, the control remains continuous, but not smooth. Analysis of the graphs of controlled motion allows us to hypothesize that the found stationary position θ∗, G∗, and γ ∗ is asymptotically stable. For a strictly mathematical

Dynamics of passive object attitude motion during ion beam transportation

275

Fig. 5.57 Dependence of control function (5.121) on angle f.

proof of this hypothesis, a rougher study using Lyapunov’s theory is required. Since the control (5.121) is a continuous piecewise nonsmooth function, the derivative of which is not defined at the switching points, the direct use of classical Lyapunov’s theorems is impossible, and the use of the theory of differential inclusions is required (Leine and Nijmeijer, 2013). This issue will be the topic of our future research. Here we will focus on the numerical analysis of the mechanical system behavior. A series of numerical calculations with different initial conditions was carried out to determine the region of attraction of this equilibrium. It is difficult to visualize the surface that bounds the region of attraction in four-dimensional space (θ, γ, θ0 , and  Figs. 5.58 and 5.59 show two cross sections. Fig. 5.58 was built for constant G). values θ00 ¼ 0 and G0 ¼ G∗, and Fig. 5.59 was built for θ0 ¼ θ∗ and γ 0 ¼ γ ∗. The gray points in Figs. 5.58 and 5.59 correspond to the initial conditions for which the phase trajectory passes into the vicinity of the stationary position at an interval of 500 rad. According to the data given in the study by Sˇilha et al. (2018), the angular velocities of the rocket stages can reach values of 409.6 deg/s. Using Eq. (5.33), it can be shown that the modulus of angular velocity inside the region of attraction, which is shown in Fig. 5.59, does not exceed 0.0034 rad/s. Thus, the proposed control can be used to stabilize slowly rotating objects. Fig. 5.58 The region of attraction of stationary position in (θ, γ) space.

276

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.59 The region of attraction of stationary position in (G, θ0 ) space.

Fig. 5.60 Dependence of angle θ on angle f for various eccentricities.

Calculations show that the proposed control law can be used for weakly elliptical orbits. Fig. 5.60 shows the dependence of angle θ on angle f for various values of the eccentricity. Figs. 5.61 and 5.62 show the change in angle γ and dimensionless var respectively. iable G, It can be seen that at the eccentricity value of 0.04, the control law copes with the task, and the variables approach the stationary values. With an eccentricity value of Fig. 5.61 Dependence of angle γ on angle f for various eccentricities.

Dynamics of passive object attitude motion during ion beam transportation

277

Fig. 5.62 Dependence of G on angle f for various eccentricities.

0.05, the control is ineffective and rather leads to a buildup of the system. This is due to the narrowing of the region of attraction of an asymptotically stable equilibrium position as a result of an increase in perturbations associated with a change in the gravitational moment in an elliptical orbit. Moreover, the equilibrium position itself is impossible in an elliptical orbit. Instead of a position of equilibrium, the phase trajectories are attracted to a stable limit cycle, which is closed trajectory in phase space. Limit cycle projections on planes for various eccentricities are shown in Figs. 5.63 and 5.64. To obtain these graphs, the numerical integration of the equations of motion (5.50)–(5.53) over a large interval (f  [0, 2500] rad) was performed, and then the last period was plotted. Calculations have shown that in the case of nonzero eccentricity, the control parameters kθ, kΩ, kγ , and kG, which provide a minimum to cost function (5.123), change, but insignificantly. This refinement does not lead to a qualitative change in the observed behavior, but requires significant computational costs. Therefore, for the purpose of optimization, it was decided to carry out calculations with the values of the control coefficients obtained for a circular orbit. When preparing real missions, more accurate and resource-intensive calculations, without this simplification, must be performed.

Fig. 5.63 Stable limit cycles for various eccentricities in (θ, θ0 ) space.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.64 Stable limit cycles for various eccentricities in (γ, G) space.

5.4

Fuel costs estimation for ion beam-assisted space debris removal mission with and without attitude control

The aim of this section is to evaluate the effect of controlling the attitude motion of space debris on the amount of fuel consumption during the ion beam contactless transportation of space debris. Several strategies for controlling the space debris object will be proposed and compared.

5.4.1 Equations of motion The planar motion of the system is considered. The space debris object is a rigid body of cylindrical shape. The motion of a system consisting of an active spacecraft and a space debris object is described by Eqs. (5.14)–(5.18). For the purposes of further analysis, let us write the equation of the unperturbed motion of a space debris object. Unperturbed motion is understood as the motion of space debris in a circular orbit when the spacecraft retains its relative position and angular orientation unchanged. In this case, r ¼ const, f_ ¼ ω ¼ const, x ¼ const, y ¼ const, β ¼ const, and the space debris object oscillations can be described by the equation
 3μ I y  I x sin θ cos θ LIz ðθ, α, x, yÞ € θ+ ¼ , Iz r3 Iz

(5.124)

where α ¼ β + γ is the angle between the ion beam axis and the line connecting the centers of mass of the spacecraft and the space debris object (Fig. 5.65), γ ¼ arctan(x/y). Eq. (5.124) can be integrated and reduced to the form ð


 LIz ðα, x, yÞdθ  _2 3μ I y  I x θ _ E θ, θ, α, x, y ¼   cos 2θ, Iz 2 4r 3 I z 

(5.125)

Dynamics of passive object attitude motion during ion beam transportation

279

Fig. 5.65 Contactless space debris removal scheme.

where E is the space debris oscillation energy, which retains its value along each phase trajectory. The equilibrium positions θ∗ of Eq. (5.124) are the roots of nonlinear equation
 3μ I y  I x sin θ∗ cos θ∗  r 3 Lz ðθ∗ , α, x, yÞ ¼ 0:

(5.126)

Since the orbital radius r is included in the equation, the equilibrium positions change during the descent of the space debris.

5.4.2 Control strategies Several control strategies for the active spacecraft will be proposed and analyzed in this section. The first strategy does not take into account the motion of the space debris object relative to its center of mass. The spacecraft’s control system holds it at a point with coordinates x0 ¼ 0 and y0 ¼ d, and the ion beam axis is directed to the object’s center of mass. The distance d is mainly determined based on the condition of minimizing the effect of backsputtering contamination due to the sputtering phenomenon (Urrutxua et al., 2019). This distance is of the order of tens of meters. This strategy will be called “transport without attitude control” (Fig. 5.66). All other control methods will be compared with this strategy. This control strategy was used in many studies (Alpatov et al., 2018; Bombardelli et al., 2012; Khoroshylov, 2020; Kulkov et al., 2020). In order to move the spacecraft to the required relative position and to translate it into the required angular position, it is proposed to use the following linear thrusters control laws: _ Px ¼ kx ðx0  xÞ  kdx x, _ Mz ¼ kβ ðβ0  βÞ  kdβ β,

_ Py ¼ ky0 + ky ðy0  yÞ  kdy y,

(5.127) (5.128)

where kj are gain coefficients, x0 and y0 are the coordinates corresponding to the required relative position of the spacecraft, and β0 is the angle corresponding to the required orientation of the spacecraft. Numerical calculations have shown that reorienting the spacecraft to change the direction of the ion beam axis using control

280

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.66 Control strategies for an active spacecraft.

(5.128) is an ineffective solution, since it is time-consuming and significantly increases fuel costs. It is proposed to mount the impulse transfer thruster and compensation thruster on a movable platform and change the ion beam direction using an electric motor. Further, it is assumed that the electric motor turns the platform according to the following law: 8 > 1 α > >  f < 2 α¼ > 1 αf > > :2 

   α0 ðt  t0 Þπ cos + α0 , tf  t0 2    α0 ðt  t0 Þπ cos + αf , tf  t0 2

if αf > α0 ; (5.129) if αf < α0 ;

where t0 and tf are initial and final time corresponding to the moments when the electric motor is turned on and turned off, respectively, and α0 and αf are initial and final ion beam axis angle, respectively. Since the force created by an ion thruster is limited, it is proposed in the simulation to smooth out the thrust force according to the law

P lim i

8 Pi , if Pi p1 ; > >   > > < Pi  p1 πp p sin + p1 , if p1 < Pi < p1 + 2 ; 2 ¼ p 2 2 > > > πp > 2 : p1 + p2 ; if Pi  p1 + ; 2

(5.130)

where Pi is the value of the thrust force calculated according to Eq. (5.127), Plimi is the value of the thrust force bounded from above, and p1 and p1 are coefficients defining smoothing (Fig. 5.67).

Dynamics of passive object attitude motion during ion beam transportation

281

Fig. 5.67 Limiting thrust force.

The second strategy involves the stabilization of the space debris object in an equilibrium position θc1 and its subsequent transportation with small angular oscillations. Stabilization can be carried out both by changing the thrust of the impulse transfer thruster and by changing the direction of the ion beam axis. The second method is much more efficient in terms of speed of the stabilization (Aslanov and Ledkov, 2020b). The relative position of the active spacecraft is chosen so that the ion force lies on the BYH axis. This strategy will be referred to below as the “transport in equilibrium state” (Fig. 5.66). To control the ion beam axis direction, it is proposed to use the approach described in Section 5.3.2.5. The third strategy proposes the transportation of a space debris object in an oscillation mode. In this case, the space debris is transferred to the oscillation mode for which the ion force averaged over the period of angular oscillations is maximum. As in the previous case, the relative position of the spacecraft is chosen so that the averaged force lies on the axis BYH (Fig. 5.68). This strategy will be called “transport with the maximum average ion force.” The third strategy involves calculating the force averaged over the oscillation period. The average value of the ion force can be assigned to each phase trajectory of the unperturbed Eq. (5.124). For each initial point on the phase portrait, a phase trajectory can be constructed by numerically integrating the equation of unperturbed motion. As a result of integration, the dependence θ(t) can be obtained, and the oscillation period T for a given phase trajectory can be calculated. This dependence can be used to calculate the projections of average ion forces on orbital reference frame axes: 1 FIx ¼ T

ðT 0

1 FIx ðθðtÞÞdt, FIy ¼ T

ðT

FIy ðθðtÞÞdt, FI ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 FIx + FIy ,

(5.131)

0

where FIx and FIy are the projections of the ion force on the axes of the Hill’s coordinate system BXHYH (Fig. 5.65). In order to the averaged ion force vector to lie on the axis BYo, it is necessary to move the active spacecraft to point with coordinates x ¼ d sin γ and y ¼ d cos γ, for which FIx ¼ 0. As in the previous case, the energy of the unperturbed system is used to build a control strategy. The target trajectory with maximum average ion force FI can be found by performing a series of calculations with different initial conditions.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.68 Control strategies for an active spacecraft taking into account the angular motion of the transported object.

The fourth strategy involves the stabilization of space debris in an angular position corresponding to the maximum module of the ion force. Since this position is not the equilibrium position of the system, it will be necessary to return the space debris object periodically into this angular position by changing the direction of the ion beam axis. Again, the relative coordinates of the spacecraft are chosen so that the maximum ion force lies on the axis BYH (Fig. 5.68). This strategy will be called “transport with the maximum ion force.” For the fourth strategy, the following approach to controlling the α angle is proposed. The strategy of “transport with the maximum ion force” allows us to transfer the representing point to the phase trajectory passing through the point θ ¼ θ4 and θ_ ¼ 0 of the phase space (point 4 in Fig. 5.69). Since θ ¼ θ4 is not an equilibrium position, then after some time the representing point in the phase portrait of the unperturbed Eq. (5.124) will leave the vicinity of this position and the angle θ will change by Δθ. At this point in time, control should be turned on again and the representing point should be transferred to the phase trajectory passing through point

Fig. 5.69 Phase portrait of the unperturbed Eq. (5.124) for fourth control strategies.

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283

θ ¼ θ4 and θ_ ¼ 0. This process is repeated to keep space debris in Δθ vicinity of angular position θ ¼ θ4.

5.4.3 Comparison of fuel costs for different control strategies As an example, consider the deorbiting of Russian SL-8 Kosmos upper stage (Table B.9 in Pardini and Anselmo, 2018) having the parameters given in Table 5.7. The methodology described in Section 4.2 was used to calculate fuel costs. Since there are no detailed data on rocker stages of this class in the public domain, the mass-geometric parameters given in the table are very approximate and are suitable only for a rough estimate. The active spacecraft has mass mA ¼ 450 kg and moments of inertia IAx ¼ 1000 kg m2, IAy ¼ 2000 kg m2, and IAz ¼ 2500 kg m2. The spacecraft’s thrusters and plume have following parameters (Patterson et al., 2003): thrust force Fmin ¼ 0.0496 N, Fmax ¼ 0.209 N, flow rate m_ min ¼ 2:29 106 kg=s, m_ max ¼ 5:21 106 kg=s , electrical power pmin ¼ 1080 W, and pmax ¼ 6075 W. Substitution of these values in Eqs. (4.42) and (4.43) allows calculation of the coefficients: a ¼ 1:5135 103 N, b ¼ 3:1912 105 NW1 , a0 ¼ 1:6586 106 kg=s, b0 ¼ 5:8458 1010 kg s1 W 1 , c0 ¼ 0: The ion beam divergence angle is α0 ¼ 15°, and the distance between the spacecraft and the space debris object centers of mass is d ¼ 15 m. It is assumed that the impulse transfer thruster creates constant thrust FT ¼ 0.2N. In this case, Eqs. (4.55) and (4.34) give the axial velocity as u0 ¼ 39642 m/s. The ion beam density, according to Eq. (4.58), for RT ¼ 0.2 m and ion mass mi ¼ 2.18  1025 kg is n0 ¼ 4.6457  1015 m3. Control law (5.127)coefficients are kx ¼ ky ¼ 1000 kg/s2, kdx ¼ kdy ¼ 100 kg/s, and ky0 ¼ 0.28mA/mB. Smoothing law (5.130) coefficients are p1 ¼ 0.03 N and p2 ¼ 0.01 N. The calculation of the ion force and torque for the case x ¼ 0, y ¼ d was carried out using the method described in Section 3.4. Fig. 5.70 shows graphs of the dependences of the ion force projections on the axis of the body FbIx, FbIy and orbital FIx, FIy reference Table 5.7 SL-8 Kosmos upper stage parameters. Parameter

Value

Parameter

Value

USSTRATCOM catalog number Perigee altitude Apogee altitude Eccentricity

1575

Dry mass mA

1435 kg

1,368,200 m 1,517,000 m 0.009522

6m 2.4 m 0.5 m

Orbit’s parameter

7,812,892 m

Inclination

56.1 degrees

Length Diameter Center of mass displacement Longitudinal moment of inertia Ix Transverse moment of inertia Iy, Iz

1300 kg m2 6800 kg m2

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.70 Dependence of ion force projections on the deflection angle θ.

frames, as well as the magnitude of the ion force FI ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F2Ix + F2Iy . Fig. 5.70 also

shows curve 10FIx in order to demonstrate more clearly the dependence FIx(θ) at the selected scale. The maximum ion force is FImax ¼ 0.0329 N. This force corresponds to the angular position θ4 ¼ 0.4363 rad. For the force to be directed along the axis BYH, the spacecraft must be at the point x ¼ 14.9971 m, y ¼ 0.2941 m. In this case, γ ¼ arctan( FIx/FIy) ¼ 0.0196 rad. The dependence of the ion torque on the angle θ for cases β ¼ 0, α ¼ αmin, and α ¼ αmax is shown on Fig. 5.71. For the considered space debris object, the following angles are obtained: αmin ¼ 11° and αmax ¼  11°. Graphs of the dependence of the equilibrium positions θc1 and θc2 and the angles θ31 and θ32 corresponding to the maximum average force on the orbit height are shown in Fig. 5.72, where RE is the Earth radius. When calculating these angles, the γ angle, which specifies the relative position of the spacecraft, was chosen so that the ion force lies on the BXo axis. The corresponding graphs for γ angles are shown in Fig. 5.73. The phase portrait of the unperturbed Eq. (5.124) for the case r ¼ 7, 812, 900 m and γ ¼ 0 is shown in Fig. 5.74. The corresponding averaged force is shown in Fig. 5.75. To construct this graph, the initial point θ(0) ¼ θ0, θ_ð0Þ ¼ 0 was taken and the numerical integration of the unperturbed Eq. (5.124) was carried out until the moment when the Fig. 5.71 Dependence of ion torque on the deflection angle θ.

Dynamics of passive object attitude motion during ion beam transportation

285

Fig. 5.72 Dependence of a stable equilibrium position θ∗ on the altitude.

Fig. 5.73 Dependence of the angle γ on the altitude. Fig. 5.74 Phase portrait for r ¼ 7, 812, 900 m.

phase trajectory returned to the initial point. Then the averaged force was calculated using Eq. (5.131). The results show that for the body under consideration, the point corresponding to the maximum averaged force FI ¼ 0:0323N and θ3 ¼ 3.5966 rad coincides with the center θc1. The second and third control strategies involve the displacement of the spacecraft in a point for which the ion force or the averaged ion force

286

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 5.75 The dependence of the average ion force FI on initial angle θ 0.

is directed along the BXH axis. Comparison of equilibrium positions and points corresponding to the maximum average force is shown in Fig. 5.72. Let us simulate the descent of the stage using various control strategies given in Section 5.4.2 with the help of the system of Eqs. (5.14)–(5.18). The orbital perigee altitude is tracked during the simulation. We will restrict ourselves to considering the descent of the perigee by 50 km. The choice of such a small decrease in height for analysis is due to the high computational costs required for the calculations. The following initial conditions are used: r ð0Þ ¼ 7739199m, r_ ð0Þ ¼ 0, f ð0Þ ¼ 0, f_ð0Þ ¼ 9:229 104 rad=s, θð0Þ ¼ 3:8442rad, θ_ ð0Þ ¼ 0, xð0Þ ¼ 0, x_ ð0Þ, yð0Þ ¼ 15m, y_ ð0Þ ¼ 0, βð0Þ ¼ 0, β_ ð0Þ ¼ 0: In addition, consider the descent using the first control strategy, when the initial deflection angle corresponds to the trajectory with the minimum average ion force θ(0) ¼ 3π/2  4.7124 rad. This is expected to be the worst case in terms of time and fuel consumption. The simulation results show that the first strategy is most effective if the stage is initially stabilized in the equilibrium position (row 3 in Table 5.8). However, as the orbital radius gradually decreases, the phase trajectory will move away from the equilibrium position (Fig. 5.73), which will lead to an increase in fuel consumption at large time intervals. The second strategy, which implies the transportation of the space debris object in an equilibrium angular position, is effective when the stage is not initially stabilized in the equilibrium position. Since the rotation of the ion beam axis is carried out by electric motors, the necessary permanent correction of the angular position of the stage does not lead to an increase in fuel consumption. Fuel consumption in the case of the second and third control strategies coincides, since the phase trajectory on which the maximum averaged force is very close to the equilibrium position θc1 (Fig. 5.72). Active stabilization of the transported object in the equilibrium position can significantly reduce fuel consumption compared to using the first strategy (up to 11.6% according to data given in rows 1 and 4 in Table 5.8), which does not imply control of the angular movement of the object.

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Table 5.8 Comparison of fuel costs for different control strategies to reduce of the perigee by 50 km. Case 1 2 3 4 5 6 7 8 9 10 11 12

Strategy Strategy Strategy Strategy Strategy Strategy Strategy Strategy Strategy Strategy Strategy Strategy

1 1 1 2 2 2 3 3 3 4 4 4

Initial angle θ(0), rad

Descent time, tk, h

Total propellant mass mprop, kg

4.7124 3.8442 3.6442 4.7124 3.8442 3.6442 4.7124 3.8442 3.6442 4.7124 3.8442 3.6442

332.5624 293.6431 292.1181 293.3749 293.1618 293.1517 293.3877 293.1765 293.1601 467.0538 466.9007 466.8918

13.9339 12.3247 12.2605 12.3131 12.3044 12.3040 12.3131 12.3044 12.3040 20.3431 20.0813 20.0805

However, if at the beginning of transportation the object is near the equilibrium position, the gain is not so obvious. A comparison of rows 2 and 5 of the table shows that the fuel economy is only 0.2%. Therefore, it is advisable to stabilize the space debris object in a stable equilibrium position before the start of transportation mission, and then intermittently to correct the angular oscillations of the stage, returning it to the equilibrium position. Calculations have shown that the fourth strategy is ineffective (rows 10–12 in Table 5.8). Although the angle θ4 corresponds to the maximum ion force, for about half of the transportation time the system moves in a mode when it returns to the vicinity of the point θ4. In this case, the axis of the ion beam is deflected by an angle of αmin or αmax, which leads to a significant decrease in the ion force generated on the object. The time and fuel consumptions for different initial angular positions of space debris within the same control strategy (except for strategy 1) turn out to be close, since the transition from the initial to the required angular position occurs in a time that is significantly less than the entire subsequent transportation operation. For example, for the second strategy, with the initial position, shown in row 4 of Table 5.8, the transition to the vicinity of the equilibrium position takes 42 min. Fig. 5.76 shows the phase trajectory at the initial stage of motion in the case of using different control strategies. It can be seen that the trajectories are not periodic. This is due to the fact that the orbit of the center of mass is not circular, but its radius changes during transportation. Fig. 5.77 demonstrates the change in the orbital pericenter radius during the descent. The first and second strategies turn out to be very close when the rocket stage is initially stabilized. Fig. 5.78 shows the change in the thrust force of the spacecraft thrusters at the initial stage of transportation for strategy 2. Periodic bursts of force are caused by the rotation of the ion beam axis (Fig. 5.79) to stabilize the stage in the equilibrium position. It can be seen that the control strategy presupposes a fairly frequent adjustment of the angular position of the transported object.

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Fig. 5.76 Phase trajectories at the initial stage of ion transportation for θ0 ¼ 3.8442 rad.

Fig. 5.77 Perigee radius in case of using different control strategies.

Fig. 5.78 Spacecraft’s thrusters force for strategy 2.

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Fig. 5.79 Ion beam axis deflection angle for strategy 2.

Thus, the calculations showed that under unfavorable initial conditions, the control of the angular motion can have a noticeable effect on the fuel consumption. For the case when space debris object is not in the position of angular equilibrium, the most preferable from the point of view of minimizing fuel consumption is the second strategy. The fourth strategy turned out to be ineffective because it is necessary constantly to return the transported object to the angular position corresponding to the maximum ion force. It should be noted that even when using the most efficient control strategies, fuel consumption is quite high. This is primarily due to the low value of momentum transfer efficiency coefficient ηB. For given engine parameters and the relative position of the active spacecraft, most of the ion beam particles pass by the space debris object and do not participate in the generation of ion force. Decreasing the distance between the active spacecraft and the target space debris object and decreasing the beam divergence angle α0 allow us to increase momentum transfer efficiency.

References Albuja, A.A., Scheeres, D.J., McMahon, J.W., 2015. Evolution of angular velocity for defunct satellites as a result of YORP: an initial study. Adv. Space Res 56, 237–251. https://doi.org/ 10.1016/j.asr.2015.04.013. Alpatov, A., Khoroshylov, S., Bombardelli, C., 2018. Relative control of an ion beam shepherd satellite using the impulse compensation thruster. Acta Astronaut. 151, 543–554. https:// doi.org/10.1016/j.actaastro.2018.06.056. Aslanov, V.S., 2017. Rigid Body Dynamics for Space Applications. Elsevier, https://doi.org/ 10.1016/C2016-0-01051-3. Aslanov, V.S., Ledkov, A.S., 2017. Attitude motion of cylindrical space debris during its removal by ion beam. Math. Probl. Eng. 2017. https://doi.org/10.1155/2017/1986374. Aslanov, V., Ledkov, A., 2020a. Chaotic motion of a passive space object during its contactless ion beam transportation. In: 2020 International Conference on Information Technology and Nanotechnology (ITNT). IEEE, pp. 1–6, https://doi.org/10.1109/ ITNT49337.2020.9253185. Aslanov, V.S., Ledkov, A.S., 2020b. Space debris attitude control during contactless transportation in planar case. J. Guid. Control. Dyn. 43, 451–461. https://doi.org/10.2514/1. G004686.

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Aslanov, V., Ledkov, A., 2021. Detumbling of axisymmetric space debris during transportation by ion beam shepherd in 3D case. Adv. Space Res. https://doi.org/10.1016/j. asr.2021.10.002. Aslanov, V., Ledkov, A., Konstantinov, M., 2020a. Attitude dynamics and control of space object during contactless transportation by ion beam. In: Proc. Int. Astronaut. Congr. IAC 2020-Octob, 1–7. Aslanov, V., Ledkov, A., Konstantinov, M., 2020b. Chaotic motion of a cylindrical body during contactless transportation from MEO to LEO by ion beam. Nonlinear Dyn. 101, 1221– 1231. https://doi.org/10.1007/s11071-020-05822-0. Aslanov, V.S., Ledkov, A.S., Konstantinov, M.S., 2021. Attitude motion of a space object during its contactless ion beam transportation. Acta Astronaut. 179, 359–370. https://doi.org/ 10.1016/j.actaastro.2020.11.017. Bombardelli, C., Urrutxua, H., Merino, M., Ahedo, E., Pela´ez, J., 2012. Relative dynamics and control of an ion beam shepherd satellite. Adv. Astronaut. Sci. 143, 2145–2157. Hahn, W., Hosenthien, H.H., Lehnigk, H., 1963. Theory and Application of Liapunov’s Direct Method. Prentice-Hall Englewood Cliffs. Hughes, P.C., 2004. Spacecraft attitude dynamics. In: Orbital Mechanics for Engineering Students. Dover Publications, New York. Khoroshylov, S., 2020. Relative control of an ion beam shepherd satellite in eccentric orbits. Acta Astronaut. 176, 89–98. https://doi.org/10.1016/j.actaastro.2020.06.027. Kulkov, V.M., Markin, N.N., Egorov, Y.G., 2020. Issues of controlling the motion of a space object by the impact of the ion beam. IOP Conf. Ser. Mater. Sci. Eng. 927. https://doi.org/ 10.1088/1757-899X/927/1/012051. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E., 1998. Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9, 112–147. Ledkov, A.S., Aslanov, V.S., 2018. Attitude motion of space debris during its removal by ion beam taking into account atmospheric disturbance. J. Phys. Conf. Ser. 1050. https://doi. org/10.1088/1742-6596/1050/1/012041, 012041. Leine, R.I., Nijmeijer, H., 2013. Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer Science & Business Media. Malkin, I.G., 1959. Theory of Stability of Motion. US Atomic Energy Commission, Office of Technical Information. Melnikov, V.K., 1963. On the stability of a center for time-periodic perturbations. Tr. Mosk. Mat. Obs. 12, 3–52. Pardini, C., Anselmo, L., 2018. Evaluating the environmental criticality of massive objects in LEO for debris mitigation and remediation. Acta Astronaut. 145, 51–75. https://doi.org/ 10.1016/j.actaastro.2018.01.028. Patterson, M., Domonkos, M., Foster, J., Haag, T., Soulas, G., Kovaleski, S., 2003. NEXT: NASA’s evolutionary xenon thruster development status. In: 39th AIAA/ASME/SAE/ ASEE Joint Propulsion Conference and Exhibit. American Institute of Aeronautics and Astronautics, Reston, VA, pp. 1–10, https://doi.org/10.2514/6.2003-4862. Schaub, H., Junkins, J.L., 2014. Analytical Mechanics of Space Systems, third ed. American Institute of Aeronautics and Astronautics, Inc, Washington, DC, https://doi.org/10.2514/ 4.102400. Schmetz, J., Pili, P., Tjemkes, S., Just, D., Kerkmann, J., Rota, S., Ratier, A., 2002. An introduction to meteosat second generation (MSG). Bull. Am. Meteorol. Soc. 83, 977– 992. https://doi.org/10.1175/1520-0477(2002)0832.3.CO;2.

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Sˇilha, J., Pittet, J.-N., Hamara, M., Schildknecht, T., 2018. Apparent rotation properties of space debris extracted from photometric measurements. Adv. Space Res 61, 844–861. https:// doi.org/10.1016/j.asr.2017.10.048. Urrutxua, H., Bombardelli, C., Hedo, J.M., 2019. A preliminary design procedure for an ionbeam shepherd mission. Aerosp. Sci. Technol. 88, 421–435. Wiggins, S., 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, Springer-Verlag, New York, https://doi.org/10.1007/b97481.

The use of contactless ion beam technology 6.1

6

Orbital flight in one plane

The purpose of this chapter is to demonstrate the possibility of using technology of contactless ion beam-assisted transportation to solve the problems of a passive space object transfer. This task is similar to low-thrust orbit transfers, but it has a significant difference. Since the engine generating thrust is located not on the object itself, but on the external active spacecraft, it is necessary to move the active spacecraft to a new point relative to the object to change the direction of generated thrust. These active spacecraft’s flights require additional fuel consumptions, which should be taken into account when developing optimal control laws. The development of control laws for low-thrust engines is a difficult task itself, to which many studies have been devoted (Conway, 2010; Morante et al., 2021; Petropoulos, 2003). Some basic points primarily related to the missions of active space debris removal and disposal are considered below. Only the plane case of motion is considered. The orbit of an object under the action of a perturbing acceleration aH B ¼ [aBx, aBy, aBz]T can be described by Gauss planetary equations (4.9), which for the considered planar case takes the form  2a2
aBx e sin f + ð1 + e cos f ÞaBy , h   að 1  e 2 Þ e cos 2 f + 2 cos f + e aBx sin f + e_ ¼ aBy , h 1 + e cos f a_ ¼

að1  e2 Þ ω_ ¼ he



  2 + e cos f a sin f  aBx cos f , 1 + e cos f By

(6.1) (6.2)

(6.3)

pffiffiffiffiffi where h ¼ pμ and p ¼ a(1  e)2. It is assumed that the acceleration vector aH B has a constant magnitude and can change its direction: aBx ¼ aB sin ϕ, aBy ¼ aB cos ϕ, aBz ¼ 0,

(6.4)

where aB ¼ const is the acceleration magnitude and ϕ is the angle defining the direction of the acceleration vector (Fig. 6.1). For the active space debris removal mission, the periapsis radius rp is of great importance, since the goal of the active spacecraft control can be to minimize this

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation. https://doi.org/10.1016/B978-0-323-99299-2.00008-2 Copyright © 2023 Elsevier Inc. All rights reserved.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 6.1 Orbit of a passive object and acceleration vector direction.

parameter, which allows the atmosphere to be used faster to decelerate a space debris object. Let us write an expression for rp: rp ¼

p ¼ að 1  e Þ 1+e

(6.5)

and find its derivative by virtue of Eqs. (6.1) and (6.2): r_p ¼ a_ ð1  eÞ  ae_ ¼ 

  r 2p e cos 2 f + 2 cos f  e  2 aBy : aBx sin f + 1 + e cos f h

(6.6)

Consider the case when the vector aH B retains its coordinates in the Hill’s reference frame. Since the accelerations are small, the parameters of the orbit change insignificantly per revolution. Let us average the right-hand parts of Eq. (6.1) over the angle of the true anomaly. 1 a_ ¼ 2π

Z2π

 2a2
2a2 aBx e sin f + ð1 + e cos f ÞaBy df ¼ a , h h By

(6.7)

0

The analysis of Eq. (6.7) shows that with a constant magnitude and relative direction of acceleration aH B , an increase in the average value of semimajor axis of the elliptical orbit a is observed when aBy > 0, and a decrease occurs when aBy < 0. Averaging Eq. (6.6) gives r_ p ¼

 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1  e2 ð1  eÞ ð1 + eÞ2  1  e2 heð1 + eÞ

aBy :

(6.8)

As for the semimajor axis a, the average value of the periapsis radius r p increases with positive aBy and decreases with negative. Thus, the problem of deorbiting a space debris object with an ion beam or rising the object into a disposal orbit can be solved by placing an active spacecraft on the axis BYH.

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Consider the case when the direction of the acceleration vector aB, given by the angle ϕ, can change during the motion of an object in orbit. Eqs. (6.1)–(6.3), (6.6) can be rewritten as a_ ¼

2a2 aB Fa ðe, f , ϕÞ, h

e_ ¼

að1  e2 ÞaB Fe ðe, f , ϕÞ, h

(6.10)

ω_ ¼

að1  e2 ÞaB Fω ðe, f , ϕÞ, he

(6.11)

r_p ¼

r 2p aB F ðe, f , ϕÞ: h r

(6.12)

(6.9)

where functions Fj(e, f, ϕ) are defined as Fa ðe, f , ϕÞ ¼ e sin f sin ϕ + ð1 + e cos f Þ cos ϕ, Fe ðe, f , ϕÞ ¼ sin f sin ϕ +  Fω ðe, f , ϕÞ ¼

2 + e cos f 1 + e cos f

e cos 2 f + 2 cos f + e cos ϕ, 1 + e cos f

(6.13)

(6.14)

 sin f cos ϕ  sin ϕ cos f ,

(6.15)

e cos 2 f + 2 cos f  e  2 cos ϕ: 1 + e cos f

(6.16)

Fω ðe, f , ϕÞ ¼  sin f sin ϕ 

Factors at functions Fj(e, f, ϕ) in Eqs. (6.9)–(6.12) are positive. Therefore, the sign of the right-hand sides of the equations is determined by the functions Fj(e, f, ϕ). Let us find the dependence ϕ(e, f) that provides the maximization of the right-hand sides of Eqs. (6.9)–(6.12). To do this, we find the derivative of the functions (6.13)–(6.16) with respect to the angle ϕ and equate it to zero, then we express the dependence ϕ(e, f). For Eq. (6.9), we have ∂Fa ¼ e sin f cos ϕ  ð1 + e cos f Þ sin ϕ ¼ 0: ∂ϕ

(6.17)

It follows from the last equation that 

 e sin f ϕa ¼ arctan : 1 + e cos ðf Þ

(6.18)

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

This angle coincides with the angle between the velocity vector VH ¼ [VBx, VBy, VBz]T and the BYH axis, where V Bx

rffiffiffi rffiffiffi μ μ e sin f , V By ¼ ð1 + e cos f Þ, V Bz ¼ 0: ¼ p p

(6.19)

That is, the orientation of the acceleration vector along the velocity provides the greatest impact on a. It is interesting that the greatest impact on the periapsis radius rp is carried out with a different angle: 

 sin f ð1 + e cos ðf ÞÞ ϕr ¼ arctan , 2 cos ðf Þ  2  e sin 2 f

(6.20)

∂Fe r which is the root of the equation ∂F ∂ϕ ¼ 0. The solution of the equation ∂ϕ ¼ 0 can be written as

1 0 8 > 2 > > e cos f + 2 cos f + e C B > > when f < π; arccos @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, > > 2 > < ðe cos 2 f + 2 cos f + eÞ + sin 2 f ð1 + e cos f Þ2 1 0 ϕe ¼ > > > 2 > e cos f + 2 cos f + e C B > > 2π  arccos @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, when f  π: > > : 2 2 2 ðe cos 2 f + 2 cos f + eÞ + sin f ð1 + e cos f Þ (6.21) ω This function provides maximum F(e, f, ϕ). The solution of the equation ∂F ∂ϕ ¼ 0 has the form

8   cos f ð1 + e cos f Þ > > > <  arctan sin f ð2 + e cos f Þ ,   ϕω ¼ > cos f ð1 + e cos f Þ > > , : π  arccos sin f ð2 + e cos f Þ

when f < π; (6.22) when f  π:

Dependences of the angles ϕj on the true anomaly angle are shown in Fig. 6.2. To implement the dependences shown in the figure using an ion beam generated by an active spacecraft, the spacecraft must be located at a point with the following coordinates in the Hill’s reference frame BXHYHZH: x ¼ d sin ϕj , y ¼ d cos ϕj , z ¼ 0,

(6.23)

where d is the distance between the active spacecraft and the object. It is assumed that the generated ion force is directed along the line connecting the center of mass of the active spacecraft and the transported object. In the case of maximum impact on the

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Fig. 6.2 Dependence of the angle ϕj, corresponding to the maximum of the function Fj, on the angle of the true anomaly f for e ¼ 0.3.

orbital element a, the active spacecraft is located near the BYH axis. In the case of maximum impact on the periapsis radius, the active spacecraft moves along an arc ϕr  [ π/2, π/2]. It should be noted that a jump in the angle ϕr is observed during the passage of the periapsis, which cannot be physically realized (Fig. 6.2). To implement dependences ϕe(f) and ϕω(f), the active spacecraft must make a complete revolution around the transported object during the period of the object’s revolution around the Earth. Obviously, the implementation of these spacecraft’s flights require fuel costs, which should be taken into account when developing optimal control laws. Figs. 6.3–6.6 show the dependences of Fj(e, f, ϕ) functions on the angle of the true anomaly f when implementing the laws of changing the acceleration direction ϕi(f). Fig. 6.3 Dependence of the function Fa on the angle of the true anomaly f for various ϕi(f) and e ¼ 0.3.

Fig. 6.4 Dependence of the function Fr on the angle of the true anomaly f for various ϕi(f) and e ¼ 0.3.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 6.5 Dependence of the function Fe on the angle of the true anomaly f for various ϕi(f) and e ¼ 0.3.

Fig. 6.6 Dependence of the function Fω on the angle of the true anomaly f for various ϕi(f) and e ¼ 0.3.

An analysis of the figures shows that a change in one of the elements of the orbit causes a change in other elements as well. Moreover, there are decays of the orbit, where the directions of the change of elements can coincide (for example, the section f  [0, π] for the curves Fa(e, f, ϕa) and Fω(e, f, ϕa)), and can be opposite (the section f  (π, 2π) for the curves Fa(e, f, ϕa) and Fω(e, f, ϕa)). This circumstance should be taken into account when developing schemes and laws for controlling an object using low thrust. For example, the paper by Petropoulos (2003) proposed the time-to-go and proximity-quotient control concepts, which allow the required change in the orbit elements to be carried out.

6.2

Deorbiting of space debris object into the atmosphere

Consider the problem of deorbiting a space debris object. For numerical simulation, the system of Eqs. (5.14)–(5.18) is used. The relative position of the active spacecraft is controlled using the law (5.127). It is assumed that the space debris object has the parameters given in Section 5.4.3. The object moves in an elliptical orbit with parameters e ¼ 0.1, p ¼ 8108100 m, a ¼ 8190000 m, and b ¼ 8148947 m, and has the initial conditions θ0 ¼ 2.5 rad, θ_ 0 ¼ 0, x0 ¼ 0, y0 ¼ 15 m, x_0 ¼ y_0 ¼ r_0 ¼ 0, f0 ¼ 0, f_0 ¼ 0, and r0 ¼ 7, 371, 000 m. The goal of control is to reduce the radius of the periapsis of the orbit to the atmosphere border. Let us consider several control scenarios. In case 1,

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Table 6.1 Results of calculations for various control methods.

Spacecraft location Control strategy (Section 5.4.2)

Maneuver time Fuel mass

Case 1

Case 2

Case 3

Case 4

x¼0 y¼d Strategy 1 (without attitude control) 291.61 h 11.86 kg

x¼0 y¼d Strategy 2 (in equilibrium)

x ¼ d sin ϕr y ¼ d cos ϕr Strategy 1 (without attitude control) 273.63 h 208.59 kg

x ¼ d sin ϕr y ¼ d cos ϕr Strategy 2 (in equilibrium)

273.26 h 11.179 kg

255.26 h 195.53 kg

the active spacecraft is located at the point with coordinates x ¼ 0, y ¼ d, and the motion of the space debris object relative to its center of mass is not controlled. In case 2, the active spacecraft is located at the point with coordinates x ¼ 0, y ¼ d, and the angular position of the transported object is controlled in accordance with scenario 2 (transportation in equilibrium position) described in Section 5.4.2. In case 3, the active spacecraft is located at the point with coordinates x ¼ d sin ϕr, y ¼ d cos ϕr, where ϕr is defined by Eq. (6.20); the attitude motion of the space debris object is not controlled. In case 4, the active spacecraft is located at the point with coordinates x ¼ d sin ϕr, y ¼ d cos ϕr, and the attitude motion of the space debris object is controlled in accordance with scenario 2 described in Section 5.4.2. Since the calculation of the entire descent mission to the atmospheric boundary takes quite a long time, let us carry out an estimated calculation, when the spacecraft descends 50 km, in order to choose the most economical control method. Table 6.1 contains some results of the calculations. Fig. 6.7 shows dependencies for the periapsis radius obtained for different control cases given in Table 6.1. It can be seen that case 4 copes with the task most quickly, which involves controlling the position of the active spacecraft that provides the greatest impact on the object’s periapsis radius. However, in this case, the fuel costs are very high. This is due to the fact that the active spacecraft has to be actively moved Fig. 6.7 Periapsis radius for various cases given in Table 6.1.

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Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

Fig. 6.8 Trajectory of an active spacecraft relative to a space debris object in the Hill’s reference frame for two revolutions in orbit (Δy ¼ y  d).

relative to the space debris object (Fig. 6.8). In cases 1 and 2, fuel costs are an order of magnitude lower. In case 3, the descent time is comparable to case 2. This is because in case 3, the object will make uncontrolled oscillations and rotations, as a result of which the direction of the generated ion force does not coincide with the required direction corresponding to the angle ϕr. Figs. 6.9 and 6.10 show phase portraits for cases 1 and 3, when the angular motion of the object is not controlled.

Fig. 6.9 Phase portrait for case 1 for the first eight orbits.

Fig. 6.10 Phase portrait for case 3 for the first eight orbits.

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Fig. 6.11 The dependence of the angle θ on time during deorbiting.

Let us simulate the descent of a space debris object to the boundary of the atmosphere (100 km above the Earth’s surface) using the control described in case 2. Calculations show that the descent takes 5338.94 h, while the fuel costs are 204.11 kg. Fig. 6.11 shows the dependence of the angle θ change during transportation, which is provided by the control strategy 2 described in Section 5.4.2. The dependence of the periapsis radius rp on time is shown in Fig. 6.12. As the height of the orbit decreases, an increase in eccentricity is observed (Fig. 6.13). The results of the numerical simulations showed that the need to relocate the active spacecraft relative to the space debris object in order to implement a control law can Fig. 6.12 The dependence of the periapsis radius on time during deorbiting.

Fig. 6.13 The dependence of the eccentricity on time during deorbiting.

302

Attitude Dynamics and Control of Space Debris During Ion Beam Transportation

lead to a significant increase in fuel costs. In this regard, the existing optimal solutions for spacecraft low-thrust transfers cannot be directly used for the contactless ion beam-assisted transportation problem.

6.3

Transportation of space debris into disposal orbit

Consider the transporting a space debris object from a GEO to a disposal orbit located 250 km above. To carry out this maneuver, the active spacecraft is held by its control system at a point with coordinates x ¼ 0, y ¼ 15 m relative the space debris object. The control law (5.127) is used. It is assumed that the spacecraft’s impulse transfer thruster is placed on a movable platform that allows the ion beam axis direction to be changed without turning the active spacecraft. A control strategy for the attitude motion of the space debris object assuming transport in a stable equilibrium position is used (strategy 2 in Section 5.4.2). Numerical simulation is carried out for the system with the parameters specified in Section 5.4.3. Calculations show that this maneuver can be carried out in 144.34 h, and 5.53 kg of fuel is required. Such low fuel consumption is due to the fact that the spacecraft changes its relative position very little during the mission. Fig. 6.14 shows the dependence of the orbit radius on time. The change in the angle θ provided by the ion beam direction control system is shown in Fig. 6.15.

Fig. 6.14 The dependence of the object’s radius on time.

Fig. 6.15 The dependence of the angle θ on time.

The use of contactless ion beam technology

303

Simulation results show that contactless ion beam-assisted transportation can be used to solve the problem of transfer a space debris object from GEO to a disposal orbit.

References Conway, B.A., 2010. Spacecraft Trajectory Optimization. Cambridge University Press, Cambridge. Morante, D., Sanjurjo Rivo, M., Soler, M., 2021. A survey on low-thrust trajectory optimization approaches. Aerospace 8, 88. https://doi.org/10.3390/aerospace8030088. Petropoulos, A.E., 2003. Simple control laws for low-thrust orbit transfers. In: AAS/AIAA Astrodyn. Spec. Conf. 1–19.

Index Note: Page numbers followed by f indicate figures and t indicate tables. A Active space constant relative position, uncontrolled motion of, 229–244, 231–232f, 235f Active spacecraft, 298–299 Active spacecraft relative position, 188–209 quasicircular orbit, control of, 196–203, 199f, 202f quasicircular orbit, stability of, 189–196, 189f Active space debris removal, 62–63, 62f contact methods of, 71–80 contact space debris removal methods (see Contact space debris removal methods) space debris capturing (see Space debris capturing) target selection for, 68–71, 70t Active space debris removal mission, 293–294 Auxiliary reference frames, 138f Axisymmetric space debris detumbling, spatial case, 271–277, 273–274f, 276f B Blanket thermal insulation (BTI) materials, 130–132, 131f Body center of mass motion, 34 Body electrostatic charging modeling, 87, 87f Body surface, ion beam interaction with, 123–135, 124f auxiliary reference frames, 138f forces and torque calculating, 135–141 ion beam parameter, 133t linear cascade regime, 125 single knock-on regime, 125 spike regime, 125 C Chaotic dynamics, 47 nonlinear system, 46–50, 48–49f

Chaser’s mass estimation, 97–98 Circular orbit, 181–187 Classical mechanics, 3 Clohessy-Wiltshire equations, 30 Complex physical phenomena, 114 Contactless ion beam technology transportation disposal orbit, space debris into, 302–303, 302f orbital flight in one plane, 293–298, 294f, 297–298f space debris object deorbiting, atmosphere, 298–302, 299f, 299t, 301f Contactless space debris removal approaches Eddy currents, contactless detumbling using, 98–99, 99f electrostatic interaction, transportation by, 85–90 gravity fields transportation, 96–98 ion beam-assisted transportation (see Ion beam-assisted transportation) lasers transportation, 90–96 Contactless transportation, 114–115 Contact space debris removal methods drag augmentation devices, 79 electrodynamic tethers, 76–77, 77f momentum exchange, 77–78, 78f solar sails, 79–80, 80f tethered towing, 75–76, 75f Controlled motion dynamics, 244–277 beam rate and direction, 244–245 control approaches, 244–245 control methods effectiveness, 255–260, 256t, 257–259f energy estimation, ion beam control based on, 260–262, 261f ion beam direction control, 252–255, 253f planar case, space debris attitude motion in, 245–262, 246f thrust control, 247–252, 247–248f, 250f

306

Coordinate systems, 4–10 earth-centered-inertial (ECI) frame, 4–5, 5f Cylinder, 148–157, 149–150f, 152–153f, 155f, 157f D Diagonal matrix, 31 Disposal orbit, space debris into, 302–303, 302f Drag augmentation devices, 79 E Earth-centered-inertial (ECI) frame, 4–5, 5f Eccentricity vector, 13 Electric propulsion subsystem, 82 Electrodynamic tethers, 76–77, 77f Electromagnetic field, 60 Electro-optical sensors, 58f, 59 Electrostatic interaction, transportation by, 85–90 body electrostatic charging modeling, 87, 87f plasma interactions, 86, 86f space debris dynamics and control, 89 Emitted atoms, 132 EP2PLUS software, 129–130 Euler’s angles, 31–32 Euler’s equations, 30–31 F Fuel consumption calculation, 174–181 G Gauss planetary equations, 23, 36–37, 167–170 Gravity fields transportation Chaser’s mass estimation, 97–98 space debris removal scenario, 98 Ground-based observations, 58–59 H Harpoon, 74–75, 74f Hill’s reference frame, 294 I IADC guidelines, 56 Ideal central gravitational field, 34–35 Inertial parameters estimation, 57–61

Index

Inertia matrix, 31 International legal aspects, 54–57 Inverse matrix, 1 Ion beam-assisted space debris removal mission, fuel costs estimation for, 278–289 control strategies, 279–283, 280–282f different control strategies, fuel costs for, 283–289, 283t, 285f, 287t, 288–289f equations of motion, 278–279, 279f Ion beam-assisted transportation, 80–84, 81f electric propulsion subsystem, 82 ion beam force, 81–82 space debris dynamics and control, 82–83 torque calculation, 81–82 Ion beam force, 81–82 Ion beam parameter, 133t Ion beam physics body surface, ion beam interaction with auxiliary reference frames, 138f forces and torque calculating, 135–141 ion beam parameter, 133t linear cascade regime, 125 single knock-on regime, 125 spike regime, 125 body surface, ion beam interaction with a, 123–135, 124f ion force/torque calculations, 141–163 calculation assumptions and methodology, 141–144, 142–143f cylinder in, 148–157, 149–150f, 152–153f, 155f, 157f rectangular prism in, 157–159, 158–159f rocket stage and cylinder, comparison of, 159–162, 160–161f solar panels, cylindrical satellite with, 162–163, 162–163f sphere in, 145–148, 145–148f mathematical modeling of, 113–116, 114f simplified ion beam models, 116–123 Ion force/torque calculations, 141–163 calculation assumptions and methodology, 141–144, 142–143f cylinder in, 148–157, 149–150f, 152–153f, 155f, 157f rectangular prism in, 157–159, 158–159f rocket stage and cylinder, comparison of, 159–162, 160–161f

Index

solar panels, cylindrical satellite with, 162–163, 162–163f sphere in, 145–148, 145–148f J Just-in-time collision avoidance, 63–68, 64–67f K Kepler’s equation, 16 Kinematic motion parameters, 59–60 L Lagrange equations, 3, 24 Laser ablation, 92 Lasers transportation, 90–96 laser ablation, 92 photon momentum transfer, 90 space debris removal projects, 93, 94f Linear cascade regime, 125 Local vertical, local horizontal (LVLH), 7 Low Earth Orbit Security With Enhanced Electric Propulsion (LEOSWEEP), 80–81 Low-thrust orbit transfers, 293 Lyapunov exponents, 48 Lyapunov stability theory, 42–46, 43f M Massless angular momentum vector, direct calculation of, 11 Mathematical/mechanical preliminaries, 1–4 classical mechanics fundamentals, 3–4 vector and matrix operations, 1–3, 2f, 5f MATLAB, 141 Momentum exchange, 77–78, 78f N Net capturing, 72, 73f Newton’s second law, 25–26 O Object relative to its center of mass, chaotic motion of, 240–244, 240–241f, 242t, 243f Object using low thrust, 297–298 Optimal control laws, 293 Orbital and attitude perturbations, 34–42

307

atmospheric influence, 38–40 earth’s magnetic field torques, 41–42, 41f moon and sun gravity, 37–38, 37f nonspherical central body gravitational field, 34–37, 35f solar radiation pressure, 40–41 Orbital eccentricity, 6 Orbital elements, 18 Orbital flight in one plane, 293–298, 294f, 297–298f P Passive object attitude motion, ion beam transportation active space constant relative position, uncontrolled motion of, 229–244, 231–232f, 235f controlled motion dynamics, 244–277 beam rate and direction, 244–245 control approaches, 244–245 control methods effectiveness, 255–260, 256t, 257–259f energy estimation, ion beam control based on, 260–262, 261f ion beam direction control, 252–255, 253f planar case, space debris attitude motion in, 245–262, 246f thrust control, 247–252, 247–248f, 250f detumbling of axisymmetric space debris, spatial case, 271–277, 273–274f, 276f ion beam-assisted space debris removal mission, fuel costs estimation for, 278–289 control strategies, 279–283, 280–282f different control strategies, fuel costs for, 283–289, 283t, 285f, 287t, 288–289f equations of motion, 278–279, 279f mathematical models of, 215–229 planar motion of space debris, 215–220, 216f spatial motion of a symmetrical space debris object, 224–227 spatial motion of space debris, 220–224, 221f symmetrical space debris object, stationary motions of, 227–229

308

Passive object attitude motion, ion beam transportation (Continued) object relative to its center of mass, chaotic motion of, 240–244, 240–241f, 242t, 243f space debris attitude motion, spatial case, 262–271, 266t, 266–270f, 268t Perturbed motion planetary equations, 18–24 Photon momentum transfer, 90 Planar motion of space debris, 215–220, 216f Plasma interactions, 86, 86f Poincare section, 47 Position vector modulus, 22 Postmission disposal, 61–62 Preliminary mission design, 210–211 Propellant mass, 181–187 R Rectangular prism, 157–159, 158–159f Relative orbital motion, 24–30, 25f Rigid body attitude motion in orbit, 30–34, 33f Robotic arms/tentacles, 71, 72f Rocket stage and cylinder, comparison of, 159–162, 160–161f Rotation matrices, 6 Rotation matrix, 7 S Second-order vector differential equation, 16–17 Self-similar models, 115–116 Simplest plasma propagation models, 115 Simplest spacecraft formation flight geometry, 25 Simple trigonometric transformations, 21 Simplified ion beam models, 116–123 Single knock-on regime, 125 Solar panels, cylindrical satellite with, 162–163, 162–163f Solar sails, 79–80, 80f Spacecraft low-thrust transfers, 301–302 Space debris attitude motion, spatial case, 262–271, 266t, 266–270f, 268t Space debris capturing harpoon, 74–75, 74f net capturing, 72, 73f robotic arms/tentacles, 71, 72f tethered space robot, 73–74, 74f

Index

Space debris dynamics and control, 82–83, 89 Space debris object deorbiting, atmosphere, 298–302, 299f, 299t, 301f Space debris prevention guidelines, 54–57 Space debris problem active space debris removal, 62–63, 62f contact methods of, 71–80 contact space debris removal methods (see Contact space debris removal methods) space debris capturing (see Space debris capturing) target selection for, 68–71, 70t contactless space debris removal approaches Eddy currents, contactless detumbling using, 98–99, 99f electrostatic interaction, transportation by, 85–90 gravity fields transportation, 96–98 ion beam-assisted transportation (see Ion beam-assisted transportation) lasers transportation, 90–96 electromagnetic field, 60 electro-optical sensors, 58f, 59 ground-based observations, 58–59 IADC guidelines, 56 inertial parameters estimation, 57–61 international legal aspects, 54–57 just-in-time collision avoidance, 63–68, 64–67f kinematic motion parameters, 59–60 postmission disposal, 61–62 review of, 53–71 rotation properties, 57–61 space debris prevention guidelines, 54–57 space debris threat, 53–54 stereovision-based algorithm, 59 Space debris removal projects, 93, 94f, 98, 210–211 Space debris threat, 53–54 Space flight mechanics and control chaotic dynamics of nonlinear system, 46–50, 48–49f coordinate systems, 4–10 earth-centered-inertial (ECI) frame, 4–5, 5f Kepler’s equation, 16 Lyapunov stability theory, 42–46, 43f

Index

mathematical/mechanical preliminaries, 1–4 classical mechanics fundamentals, 3–4 vector and matrix operations, 1–3, 2f, 5f orbital and attitude perturbations, 34–42 atmospheric influence, 38–40 earth’s magnetic field torques, 41–42, 41f moon and sun gravity, 37–38, 37f nonspherical central body gravitational field, 34–37, 35f solar radiation pressure, 40–41 perturbed motion planetary equations, 18–24 relative orbital motion, 24–30, 25f rigid body attitude motion in orbit, 30–34, 33f two-body problem, 10–17, 10f orbital plane motion, 10–17 Spatial motion space debris, 220–224, 221f symmetrical space debris object, 224–227 Sphere, 145–148, 145–148f Spherical orbital reference frame, 9 Spherical reference frame, equations of motion in, 170–174 Spherical space debris, relative translation motion of active spacecraft relative position, 188–209 quasicircular orbit, control of, 196–203, 199f, 202f quasicircular orbit, stability of, 189–196, 189f

309

circular orbit, 181–187 fuel consumption calculation, 174–181 Gauss planetary equations of, 167–170 general assumptions, 167 preliminary mission design, 210–211 propellant mass for, 181–187 space debris removal, 210–211 spherical reference frame, equations of motion in, 170–174 thruster exhaust velocity, 181–187 thruster plume parameters, 174–181, 176–177f, 179–180f Spike regime, 125 Sputtering, 124 STL format, 141 Symmetrical space debris object, stationary motions of, 227–229 T Tethered space robot, 73–74, 74f Tethered towing, 75–76, 75f Thruster exhaust velocity, 181–187 Thruster plume parameters, 174–181, 176–177f, 179–180f Time-to-go and proximity-quotient control concepts, 297–298 Torque calculation, 81–82 Transport theorem, 27 Two-body problem, 10–17, 10f orbital plane motion, 10–17