Cable-Driven Parallel Robots: Proceedings of the 4th International Conference on Cable-Driven Parallel Robots [1st ed.] 978-3-030-20750-2;978-3-030-20751-9

Table of contents :
Front Matter ....Pages i-xii
Front Matter ....Pages 1-1
Planar Cable-Driven Robots with Enhanced Orientability (M. Vikranth Reddy, N. C. Praneet, G. K. Ananthasuresh)....Pages 3-12
Chain Driven Robots: An Industrial Application Opportunity. A Planar Case Approach (Guillermo Rubio-Gómez, David Rodríguez-Rosa, Jorge A. García-Vanegas, Antonio Gonzalez-Rodríguez, Fernando J. Castillo-García, Erika Ottaviano)....Pages 13-22
Non-slipping Conditions of Endless-Cable Driven Parallel Robot by New Interpretations of the Euler-Eytelwein’s Formula (Takashi Harada, Koki Hirosato)....Pages 23-34
Analysis of Cable-Configurations of Kinematic Redundant Planar Cable-Driven Parallel Robot (Koki Hirosato, Takashi Harada)....Pages 35-46
Improving cable length measurements for large CDPR using the Vernier principle (Jean-Pierre Merlet)....Pages 47-58
Front Matter ....Pages 59-59
Stiffness of Planar 2-DOF 3-Differential Cable-Driven Parallel Robots (Lionel Birglen, Marc Gouttefarde)....Pages 61-71
Stability Analysis of Pose-Based Visual Servoing Control of Cable-Driven Parallel Robots (Zane Zake, Stéphane Caro, Adolfo Suarez Roos, François Chaumette, Nicolò Pedemonte)....Pages 73-84
Practical Stability of Under-Constrained Cable-Suspended Parallel Robots (Dragoljub Surdilovic, Jelena Radojicic)....Pages 85-98
Singularity Characteristics of a Class of Spatial Redundantly actuated Cable-suspended Parallel Robots and Completely actuated ones (Lewei Tang, Xiaoyu Wu, Xiaoqiang Tang, Li Wu)....Pages 99-107
Kinetostatic Modeling of a Cable-Driven Parallel Robot Using a Tilt-Roll Wrist (Saman Lessanibahri, Philippe Cardou, Stéphane Caro)....Pages 109-120
Static Analysis of a Two-Platform Planar Cable-Driven Parallel Robot with Unlimited Rotation (Thomas Reichenbach, Philipp Tempel, Alexander Verl, Andreas Pott)....Pages 121-133
Front Matter ....Pages 135-135
Calculation of the cable-platform collision-free total orientation workspace of cable-driven parallel robots (Marc Fabritius, Christoph Martin, Andreas Pott)....Pages 137-148
Workspace Analysis of Cable Parallel Manipulator for Side Net Cleaning of Deep Sea Fishing Ground (Liping Wang, Haisheng Li, Zhufeng Shao, Zhaokun Zhang, Fazhong Peng)....Pages 149-160
Identifying the largest sphere inscribed in the constant orientation wrench-closure workspace of a spatial parallel manipulator driven by seven cables (Ambuj Shahi, Sandipan Bandyopadhyay)....Pages 161-172
A Bounding Volume of the Cable Span for Fast Collision Avoidance Verification (M. Lesellier, M. Gouttefarde)....Pages 173-183
Computation of the interference-free wrench feasible workspace of a 3-DoF translational tensegrity robot (Marc Arsenault)....Pages 185-196
Antipodal Criteria for Workspace Characterization of Spatial Cable-Driven Robots (Leila Notash)....Pages 197-206
Front Matter ....Pages 207-207
Robust Adaptive Control of Over-Constrained Actuated Cable-Driven Parallel Robots (Alireza Izadbakhsh, Hamed Jabbari Asl, Tatsuo Narikiyo)....Pages 209-220
Model Predictive Control of Large-Dimension Cable-Driven Parallel Robots (João Cavalcanti Santos, Ahmed Chemori, Marc Gouttefarde)....Pages 221-232
Linearised Feedforward Control of a Four-Chain Crane Manipulator (Michael Stoltmann, Pascal Froitzheim, Normen Fuchs, Wilko Flügge, Christoph Woernle)....Pages 233-244
An experimental study on control accuracy of FAST cable robot following zigzag astronomical trajectory (Hui Li, Mingzhe Li)....Pages 245-253
Front Matter ....Pages 255-255
Path Planning of a Mobile Cable-Driven Parallel Robot in a Constrained Environment (Tahir Rasheed, Philip Long, David Marquez-Gamez, Stèphane Caro)....Pages 257-268
Development of Emergency Strategies for Cable-Driven Parallel Robots after a Cable Break (Roland Boumann, Tobias Bruckmann)....Pages 269-280
A Conditional Stop Capable Trajectory Planner for Cable-Driven Parallel Robots (Patrik Lemmen, Robin Heidel, Tobias Bruckmann)....Pages 281-292
Front Matter ....Pages 293-293
Modeling of Elastic-Flexible Cables with Time-Varying Length for Cable-Driven Parallel Robots (Philipp Tempel, Dongwon Lee, Felix Trautwein, Andreas Pott)....Pages 295-306
Static and dynamic analysis of a 6 DoF totally constrained cable robot with 8 preloaded cables (Damien Gueners, B. Chedli Bouzgarrou, Hélène Chanal)....Pages 307-318
Slackening Effects in 2D Exact Positioning in Cable-Driven Parallel Manipulators (Erika Ottaviano, Andrea Arena, Vincenzo Gattulli, Francesco Potenza)....Pages 319-330
Front Matter ....Pages 331-331
Automatic Self-Calibration of Suspended Under-Actuated Cable-Driven Parallel Robot using Incremental Measurements (Edoardo Idá, Jean-Pierre Merlet, Marco Carricato)....Pages 333-344
Eye-on-Hand Calibration Method for Cable-Driven Parallel Robots (Nicolas Tremblay, Kaveh Kamali, Philippe Cardou, Christian Desrosiers, Marc Gouttefarde, Martin J.-D. Otis)....Pages 345-356
On the automatic calibration of redundantly actuated cable-driven parallel robots (Han Yuan, Yongqing Zhang, Wenfu Xu)....Pages 357-366
Towards a Precise Cable-Driven Parallel Robot - A Model-Driven Parameter Identification Enhanced by Data-Driven Position Correction (Marcus Hamann, Pauline Marie Nüsse, David Winter, Christoph Ament)....Pages 367-376
Front Matter ....Pages 377-377
Design, Implementation and Long-Term Running Experiences of the Cable-Driven Parallel Robot CaRo Printer (Andreas Pott, Philipp Tempel, Alexander Verl, Frederik Wulle)....Pages 379-390
A Dual Joystick-Trackball Interface for Accurate and Time-Efficient Teleoperation of Cable-Driven Parallel Robots within Large Workspaces (Kwun Wang Ng, Robert Mahony, Darwin Lau)....Pages 391-402
Active Vibration Damping of a Cable-Driven Parallel Manipulator Using a Multirotor System (Yue Sun, Matthew Newman, Arthur Zygielbaum, Benjamin Terry)....Pages 403-414
Reproduction of Long-Period Ground Motion by Cable Driven Earthquake Simulator Based on Computed Torque Method (Daisuke Matsuura, Taishu Ueki, Yusuke Sugahara, Minoru Yoshida, Yukio Takeda)....Pages 415-424
Back Matter ....Pages 425-426

Citation preview

Mechanisms and Machine Science 74

Andreas Pott Tobias Bruckmann Editors

Cable-Driven Parallel Robots Proceedings of the 4th International Conference on Cable-Driven Parallel Robots

Mechanisms and Machine Science Volume 74

Series Editor Marco Ceccarelli, Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy Editorial Board Alfonso Hernandez, Mechanical Engineering, University of the Basque Country, Bilbao, Vizcaya, Spain Tian Huang, Department of Mechatronical Engineering, Tianjin University, Tianjin, China Yukio Takeda, Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan Burkhard Corves, Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany Sunil Agrawal, Department of Mechanical Engineering, Columbia University, New York, NY, USA

This book series establishes a well-defined forum for monographs, edited Books, and proceedings on mechanical engineering with particular emphasis on MMS (Mechanism and Machine Science). The final goal is the publication of research that shows the development of mechanical engineering and particularly MMS in all technical aspects, even in very recent assessments. Published works share an approach by which technical details and formulation are discussed, and discuss modern formalisms with the aim to circulate research and technical achievements for use in professional, research, academic, and teaching activities. This technical approach is an essential characteristic of the series. By discussing technical details and formulations in terms of modern formalisms, the possibility is created not only to show technical developments but also to explain achievements for technical teaching and research activity today and for the future. The book series is intended to collect technical views on developments of the broad field of MMS in a unique frame that can be seen in its totality as an Encyclopaedia of MMS but with the additional purpose of archiving and teaching MMS achievements. Therefore, the book series will be of use not only for researchers and teachers in Mechanical Engineering but also for professionals and students for their formation and future work. The series is promoted under the auspices of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). Prospective authors and editors can contact Mr. Pierpaolo Riva (publishing editor, Springer) at: [email protected]. Indexed by SCOPUS and Google Scholar.

More information about this series at http://www.springer.com/series/8779

Andreas Pott Tobias Bruckmann •

Editors

Cable-Driven Parallel Robots Proceedings of the 4th International Conference on Cable-Driven Parallel Robots

123

Editors Andreas Pott University of Stuttgart Stuttgart, Germany

Tobias Bruckmann University of Duisburg-Essen Duisburg, Germany

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-030-20750-2 ISBN 978-3-030-20751-9 (eBook) https://doi.org/10.1007/978-3-030-20751-9 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

International Scientific Committee

S. Agrawal, Columbia University M. Arsenault, Laurentian University Ph. Cardou, Lavel University S. Caro, CNRS, LS2N, IRT Jules Verne M. Carricato, University of Bologna C. Gosselin, Laval University M. Gouttefarde, LIRMM, CNRS, Univ. Montpellier J.-P. Merlet, INRIA L. Notash, Queen’s University D. Schramm, University of Duisburg-Essen A. Verl, University of Stuttgart

v

Preface

In 2012, leading experts from three continents gathered during the “First International Conference on Cable-Driven Parallel Robots (CableCon 2012)” in Stuttgart, Germany. This conference initiated a forum for the cable robot community. Due to the great success, the event was continued by the “Second International Conference on Cable-Driven Parallel Robots (CableCon 2014)” at the University Duisburg-Essen in 2014, establishing CableCon as a home for cable robot researchers. In 2017, the “Third International Conference on Cable-Driven Parallel Robots (CableCon 2017)” was organized by the Université Laval in Québec, Canada. This time, the conference went across the Atlantic Ocean and emphasized its role as an international event, where researchers from all over the world find a place to connect with the most recognized experts from the field. Meanwhile, practical investigations on cable robots are attracting the focus of the research teams around the world. At the same time, fundamental research still continues to provide new insights to the physical understanding of cable-driven parallel robots. This broad variety of research activities is reflected by the content of this book. The editors would like to thank the authors for their valuable contributions. Noteworthy, the strict schedule in the preparation of this book would not have been feasible without the support of the reviewers and the scientific committee. All three conferences in the past were organized under the patronage of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). This time, the “Fourth International Conference on Cable-Driven Parallel Robots” takes even place as part of the “15th IFToMM World Congress” in Krakow, Poland. We are proud that IFToMM decided to welcome the fourth edition of CableCon at this very special event, which celebrates the 50th anniversary of IFToMM at the same time. CableCon 2019 was organized by the University of Duisburg-Essen and the University Stuttgart together with the organizers of the “15th IFToMM World

vii

viii

Preface

Congress”. We would like to express our gratefulness to Tadeusz Uhl and his supporting team – namely Anna Inglot and Marta Polak – for their continuous assistance. Stuttgart, Germany Duisburg, Germany

Andreas Pott Tobias Bruckmann Editors

Contents

Part I

Design

Planar Cable-Driven Robots with Enhanced Orientability . . . . . . . . . . . M. Vikranth Reddy, N. C. Praneet and G. K. Ananthasuresh Chain Driven Robots: An Industrial Application Opportunity. A Planar Case Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guillermo Rubio-Gómez, David Rodríguez-Rosa, Jorge A. García-Vanegas, Antonio Gonzalez-Rodríguez, Fernando J. Castillo-García and Erika Ottaviano

3

13

Non-slipping Conditions of Endless-Cable Driven Parallel Robot by New Interpretations of the Euler-Eytelwein’s Formula . . . . . . . . . . . Takashi Harada and Koki Hirosato

23

Analysis of Cable-Configurations of Kinematic Redundant Planar Cable-Driven Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koki Hirosato and Takashi Harada

35

Improving cable length measurements for large CDPR using the Vernier principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Pierre Merlet

47

Part II

Kinematics and Static

Stiffness of Planar 2-DOF 3-Differential Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lionel Birglen and Marc Gouttefarde Stability Analysis of Pose-Based Visual Servoing Control of Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zane Zake, Stéphane Caro, Adolfo Suarez Roos, François Chaumette and Nicolò Pedemonte

61

73

ix

x

Contents

Practical Stability of Under-Constrained Cable-Suspended Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dragoljub Surdilovic and Jelena Radojicic Singularity Characteristics of a Class of Spatial Redundantly actuated Cable-suspended Parallel Robots and Completely actuated ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lewei Tang, Xiaoyu Wu, Xiaoqiang Tang and Li Wu

85

99

Kinetostatic Modeling of a Cable-Driven Parallel Robot using a Tilt-Roll Wrist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Saman Lessanibahri, Philippe Cardou and Stéphane Caro Static Analysis of a Two-Platform Planar Cable-Driven Parallel Robot with Unlimited Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Thomas Reichenbach, Philipp Tempel, Alexander Verl and Andreas Pott Part III

Workspace

Calculation of the cable-platform collision-free total orientation workspace of cable-driven parallel robots . . . . . . . . . . . . . . . . . . . . . . . 137 Marc Fabritius, Christoph Martin and Andreas Pott Workspace Analysis of Cable Parallel Manipulator for Side Net Cleaning of Deep Sea Fishing Ground . . . . . . . . . . . . . . . . . . . . . . . 149 Liping Wang, Haisheng Li, Zhufeng Shao, Zhaokun Zhang and Fazhong Peng Identifying the largest sphere inscribed in the constant orientation wrench-closure workspace of a spatial parallel manipulator driven by seven cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Ambuj Shahi and Sandipan Bandyopadhyay A Bounding Volume of the Cable Span for Fast Collision Avoidance Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 M. Lesellier and M. Gouttefarde Computation of the interference-free wrench feasible workspace of a 3-DoF translational tensegrity robot . . . . . . . . . . . . . . . . . . . . . . . . 185 Marc Arsenault Antipodal Criteria for Workspace Characterization of Spatial Cable-Driven Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Leila Notash

Contents

Part IV

xi

Control

Robust Adaptive Control of Over-Constrained Actuated Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Alireza Izadbakhsh, Hamed Jabbari Asl and Tatsuo Narikiyo Model Predictive Control of Large-Dimension Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 João Cavalcanti Santos, Ahmed Chemori and Marc Gouttefarde Linearised Feedforward Control of a Four-Chain Crane Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Michael Stoltmann, Pascal Froitzheim, Normen Fuchs, Wilko Flügge and Christoph Woernle An experimental study on control accuracy of FAST cable robot following zigzag astronomical trajectory . . . . . . . . . . . . . . . . . . . . . . . . 245 Hui Li and Mingzhe Li Part V

Motion Planning

Path Planning of a Mobile Cable-Driven Parallel Robot in a Constrained Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Tahir Rasheed, Philip Long, David Marquez-Gamez and Stèphane Caro Development of Emergency Strategies for Cable-Driven Parallel Robots after a Cable Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Roland Boumann and Tobias Bruckmann A Conditional Stop Capable Trajectory Planner for Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Patrik Lemmen, Robin Heidel and Tobias Bruckmann Part VI

Advanced Cable Modeling

Modeling of Elastic-Flexible Cables with Time-Varying Length for Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Philipp Tempel, Dongwon Lee, Felix Trautwein and Andreas Pott Static and dynamic analysis of a 6 DoF totally constrained cable robot with 8 preloaded cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Damien Gueners, B. Chedli Bouzgarrou and Hélène Chanal Slackening Effects in 2D Exact Positioning in Cable-Driven Parallel Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Erika Ottaviano, Andrea Arena, Vincenzo Gattulli and Francesco Potenza

xii

Part VII

Contents

Calibration and Identification

Automatic Self-Calibration of Suspended Under-Actuated Cable-Driven Parallel Robot using Incremental Measurements . . . . . . . 333 Edoardo Idá, Jean-Pierre Merlet and Marco Carricato Eye-on-Hand Calibration Method for Cable-Driven Parallel Robots . . . 345 Nicolas Tremblay, Kaveh Kamali, Philippe Cardou, Christian Desrosiers, Marc Gouttefarde and Martin J.-D. Otis On the automatic calibration of redundantly actuated cable-driven parallel robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Han Yuan, Yongqing Zhang and Wenfu Xu Towards a Precise Cable-Driven Parallel Robot - A Model-Driven Parameter Identification Enhanced by Data-Driven Position Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Marcus Hamann, Pauline Marie Nüsse, David Winter and Christoph Ament Part VIII

Applications

Design, Implementation and Long-Term Running Experiences of the Cable-Driven Parallel Robot CaRo Printer . . . . . . . . . . . . . . . . . 379 Andreas Pott, Philipp Tempel, Alexander Verl and Frederik Wulle A Dual Joystick-Trackball Interface for Accurate and Time-Efficient Teleoperation of Cable-Driven Parallel Robots within Large Workspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Kwun Wang Ng, Robert Mahony and Darwin Lau Active Vibration Damping of a Cable-Driven Parallel Manipulator Using a Multirotor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Yue Sun, Matthew Newman, Arthur Zygielbaum and Benjamin Terry Reproduction of Long-Period Ground Motion by Cable Driven Earthquake Simulator Based on Computed Torque Method . . . . . . . . . 415 Daisuke Matsuura, Taishu Ueki, Yusuke Sugahara, Minoru Yoshida and Yukio Takeda Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Part I

Design

Planar Cable-Driven Robots with Enhanced Orientability M. Vikranth Reddy1, N. C. Praneet2, and G. K. Ananthasuresh3 [1-2-3] Indian Institute of Science, Bengaluru-560012, India [email protected], [email protected], [email protected]

Abstract. Cable-driven robots consist of a moving platform to which cables are attached and pulled using ground-mounted motors. Even though there are many advantages of these over conventional robots, there is a serious limitation of restricted orientations of the platform. Furthermore, the orientation depends on the external load. In this work, the inherent limited orientation attribute of planar cable robots is demonstrated with analysis and simulations. Also presented is a concept and implementation of complete orientability of the end-effector attached to the moving platform of a planar cable-driven robot by using only one additional cable. The new concept, by design, gives rise to a moment load on the moving platform, which does not yet appear to have been considered in cable-driven robots. Implications in this analysis and path planning are discussed. Working prototype of a planar three-cable robot with an extra cable to enhance the orientability was built and tested. Additionally, for a given configuration and loading, cable tensions were pre-calculated and stored in a structured database for expeditious execution of the path and orientation of the robot. Keywords: Planar cable-driven robot, orientations.

1

Introduction

Cable-driven robots are structurally similar to parallel robots with the fundamental difference that cables can only pull the moving platform but cannot push it. Due to this feature, well-known results in robotics for trajectory planning and control must be modified to reflect the constraints of non-negative cable tensions [1]. The limitations of these robots are restricted orientations of the moving platform and a constraint of nonnegative tensions. The past two decades have seen rapid growth in this area, which gave rise to a variety of applications. Few applications of these include automated cranes [24], ultra-high speed robots [5], and virtual reality interfaces [6]. Numerous research studies have been reported for determining the conditions for obtaining the workspace, stiffness, singularities, etc. They include NIST Robocrane [2], workspace analysis methodology that can be applied for optimal design of the planar cable robots [7], controller designs [8], and workspace analysis [9-11], and constraints on tensions are used to check the feasible orientations of a rectangular platform suspended by cables in a square workspace [12]. To the best knowledge of the authors, limited work has been done on the orientation aspect of the moving platform of the cable-driven robots. In this work, limited orientations of the moving platform of the cable-driven robots are discussed and a concept to obtain the complete orientability of the end effector mounted on the moving platform with an extra cable and a linear spring is presented. The new concept, by design, gives rise to a moment load on the moving platform. To confirm

© Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_1

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M. V. Reddy et al.

the theory with experiments, an experimental test bed of a planar cable-driven robot with three-cables and an extra cable to enhance the orientability of the end effector is fabricated. The organization of this paper is as follows: Section 2 highlights the restricted orientations of the moving platform in workspace and a procedure is described to check the feasible orientations of a planar cable robot. A concept for providing complete orientability of the end effector mounted on a moving platform is presented in Section 3. Implementation of this concept in a prototype along with results are presented in Section 4. Discussion and closing remarks are in Section 5.

2

Limited orientability of the moving platform in cabledriven robots

In this section, a model of a three cable and a four cable robot is discussed to highlight the limited orientations of the platform. 2.1

Three-cable planar robot

A three-cable planar robot consists of a triangular platform constrained to move in a plane that is connected by three cables to a base platform at three fixed points. Let us consider a reference frame (O, X , Y ) fixed to the centroid of the base called the base frame. Coordinates of the fixed points on the base frame and the contact points on the moving platform are denoted by ( x fj , y fj ) and ( xmj , ymj ) respectively (where j =1, 2, 3). Selecting a reference point P ( x, y ) on the moving platform and a direction Y c fixed relative to the moving platform, we obtain the moving frame ( P, X c, Y c) . Its orientation, denoted by I , is the relative angle between the fixed Y axis and the moving Y c axis with the centroid of the base frame as the reference. T j is the angle between the j th cable and the X axis of the base frame, measured counter-clockwise. Fig. 1 shows the representation of such a model. The procedure that is used to check the non-negative tensions at a given point ( x, y ) for a given orientation I is taken from [12] and is briefly presented below. For the system to be in static equilibrium, we need to balance cable forces in X and ª¬ Fx

Y directions with the given external load f

Fy

T

M z º¼ , and also balance mo-

ment about the Z axis. n

Fx

¦T

j

cos T j

¦T

j

sin T j

(1a)

j 1 n

Fy

j 1

§ xmj · § T j cos T j · R¨ ¸u¨ ¸ ¦ j 1 © ymj ¹ © T j sin T j ¹ n

Mz

(1b)

Planar Cable-Driven Robots with Enhanced Orientability

5

Fig. 1. Model of the three-cable planar robot

 sin I · ¸ is the rotation matrix; ( xmj , ymj ) are the initial coordinates cos I ¹ of the moving platform (where j = 1, 2, 3); T j is the tension in j th cable; where R

§ cos I ¨ © sin I

c · c · § y fj  ymj § xmj tan 1 ¨ ; and ¨ ¨ x  xc ¸¸ c ¸¹ mj ¹ © ymj © fj To determine the tensions t [T1 trix form as follows.

Tj

§ xmj · § x· ¨ ¸  R¨ y ¸ . © y¹ © mj ¹ T2

T3 ]T , Eqs. 1(a)-1(b), can be rewritten in ma-

Lt

f

§ · ¨ ¸ cos T1 cos T 2 cos T3 ¨ ¸ ¸ where sin T1 sin T 2 sin T 3 L ¨ ¨ ¸ ¨ § xm1 · § cos T1 · § xm 2 · § cos T 2 · § xm 3 · § cos T 3 · ¸ ¸u¨ ¸u¨ ¸u¨ ¸¸ ¸ R¨ ¸ R¨ ¨R¨ © ym 2 ¹ © sin T 2 ¹ © ym 3 ¹ © sin T3 ¹ ¹ © © ym1 ¹ © sin T1 ¹ When rank (L) =3, we calculate the tensions as

(2)

(3)

t = L-1f (4) If all the elements in t are non-negative, then the considered orientation at the given point is feasible. If L is rank deficient, Eq. 4 is not applicable. It is possible have a solution to Eq. (2) only if f is in the left nullspace of L and its components are all non-negative [12]. The procedure to obtain non-negative tensions is described next. L_aug = > L f @ (5)

6

M. V. Reddy et al.

rank (L) rank (L_aug) t p = LT (LLT )-1 (-f) N nullmatrix  L t = t p + ND

(6)

t p is the particular solution and its elements can be any real value; N is the null matrix and its elements are real values; D is an arbitrary scalar when rank (L) =2 and is a vector of 2 elements when rank (L) =1. For example, consider the configuration as shown in Fig. 2. The coordinates of the base frame and the platform (initially) are (0, 5.77), (5, -2.89), (-5, -2.89) and (0, 1.15), (1, -0.58), (-1, -0.58). For an external force f

I

1q at a point ( x, y )

>0

T

1 0@ and for an orientation

(2, 2) , tensions are computed using the Eqs. 1(a)-3, and 4(or

5 and 6). For this case, rank (L) =3. Using Eq. 4, the calculated values of tensions in the cables are t [3.126 9.687 8.673]T . As the tensions in the cables are all non-negative for I 1q , it is considered as a feasible orientation. We get a range of such feasible orientations for this point by varying I from > 90q,90q@ with an increment of 1º while ensuring that there is no interference between the cables. The feasible orientations for this example range from 1º-14º and are indicated by the yellow sector shown in Fig. 2. For I 15q , the calculated tensions in the cables are t [1.11 0.25 0.03]T . Hence, I 15q is not a feasible orientation. The plot illustrated in Fig. 3 indicates the region of feasibility in orientations where tensions in the cables are non-negative. For chosen limits of the workspace, we exhaustively search and determine the feasible orientations at each point within the limits following the aforementioned procedure. Feasible orientations at various points for two different configurations of the planar three-cable robot are shown in Fig. 4, indicated by green sectors in the circles. The center line in the circle indicates 0º w.r.t the Y axis. Green sectors depicted on the left and right to the center line indicate positive and negative orientations respectively.

Fig. 2. Tensions in the cables and orientation of the platform at a point ( x, y ) rotation of 1º are depicted.

(2, 2) for a

Planar Cable-Driven Robots with Enhanced Orientability

7

Fig. 3. Orientations feasible at (2, 2)

Fig. 4. Orientations are depicted when a force F = [0 -1 0]T (left) and F = [0 1 0]T (right) is applied on the moving platform. Coordinates of the base frame are indicated in red.

2.2

Four-cable robot

A four-cable robot model is developed similar to the three-cable model. The platform here is a quadrilateral suspended at four points by the cables. The notations used for this model hold the same meaning as they did in the three-cable model. In this case, L is (3 u 4) matrix and t is (4 u 1) vector. The procedure stated in Section 2.1 is used to determine the feasible orientations with the modification that only Eqs. 5-6 are used to compute the values of tensions in the cables for any rank (L). Results of simulations for two such configurations are presented in Fig. 5. From Fig. 5, it is observed that the range of orientations of the moving platform can be improved by varying the geometries of the platform for the same workspace. However, it is still restricted. We propose a concept to achieve complete orientability with the help of one extra cable and a translational spring which is detailed in the next section.

3

Complete orientability in cable robots

We now introduce a concept to achieve complete orientability of the end effector, using an extra cable and a translational spring.

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Fig. 5. A Force F = [0 -1 0]T is applied on the moving platform. Orientations are depicted for (i) (left) configuration with square platform in a square workspace and (ii) (right) rectangular platform in a square workspace. Note that for the same workspace, the rectangular platform has more orientability.

3.1

An extra cable to enhance orientation

We propose to fit the end-effector to the moving frame using a revolute joint. The endeffector is modelled as a spool with a helical groove. An extra cable is wound over the spool along its groove and it is connected to a rotary actuator on one end and to a linear spring on the other. Fig. 6 illustrates such a model, where the end-effector is mounted on a triangular platform which is connected by three cables to the base frame at three fixed points. The cable is kept in pre-tension by elongating the spring to a set limit. This is done to reduce the backlash error arising at the interface of the cable and the endeffector. When actuated, the difference in tensions between the two ends of the cable results in the rotation of the winch at any given point in the workspace. The pre-loaded spring acts as an additional actuator and assists in rotating the end effector back to its initial orientation when the cable is released by the rotary actuator. As a result, complete orientability of the end effector is realized. Furthermore, the design of the revolute joint includes the flexibility of arresting it to the moving platform thereby giving rise to an external moment load on the platform.

Fig. 6. (Left)Model for enhancing orientation. (Right) Spool with a helical groove with the cable wound over it.

Planar Cable-Driven Robots with Enhanced Orientability

3.2

9

Re-calculated tensions for orienting in a given location

Ts1 and Ts 2 are the tensions in the cables connected to the actuator and the spring respectively, as shown in Fig. 6. The resolved components of these tensions act along the X and Y axes, which contribute to the external forces. Thus, tensions in the cables have to be re-calculated taking these new external forces into count. Within the feasible range of orientations at a point, there exists a unique orientation satisfying the given objective function such as minimizing or maximizing sum of tensions. Hence, for a given objective function and chosen path (finite number of points), there is a unique locus of such orientations. Path planning of the planar cable robot is done by commanding the platform to move along such a locus. The procedure for doing this is briefly presented next. We consider the centroid of the moving platform to be the reference point for any specified path. First, for a given path, when there is no change in the orientation of the end effector, Ts1 and Ts 2 are equal in magnitude and opposite in di-

rection. L is calculated at every point in the path as presented in Section 2.1. Ts1 and Ts 2 are determined from the following equation: Ts1 Ts 2 k ('x ) (7) where 'x is the pre-set elongation of the spring to provide the required pre-tension in the cable and k is the stiffness of the spring. Next, forces on the moving platform are re-calculated using: Fx _ new Fx  Ts1 cos(T s1 )  Ts 2 cos(T s 2 ) (8) Fy _ new Fy  Ts1 sin(T s1 )  Ts 2 sin(T s 2 )

T s1 and T s 2 are the angles made by the cables w.r.t the X axis on the actuator and spring sides respectively. Lastly, quantities in Eq. 4 (or 5 and 6) are re-calculated. When the end effector is given certain rotation, we re-calculate the tensions with the following modifications: i. When the rotary actuator is winding the cable, quantities in Eq. 4 (or 5 and 6) and 8 are re-calculated using: Ts k ('x c) Ts1 Ts 2 (e PT w ) Ts 2 Ts

(9)

where P is the coefficient of friction between the end-effector and the cable; T w the wrapping angle of the cable on the end-effector; and 'x c is the elongation of the spring. ii. When the rotary actuator is releasing the cable, the quantities in Eq. 4 (or 5 and 6) and 8 are re-calculated using: Ts k ( 'x c) (10) Ts1 Ts 2 / (e PT w ) Ts 2 Ts Path Planning. Fig. 7 presents the consolidation of three frames of simulation. The platform is programmed to go along a vertically linear path from (0, -4) to (0, 4) w.r.t. the base frame. The first frame depicts the end effector at (0, 0) oriented at 0º. The second frame depicts the end effector undergoing a rotation of 30º at (0, -4). With the

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orientation fixed at 30º, the platform moves vertically upward to (0, 4). In the third frame, the end effector is rotated by -30º at (0, 4). The platform is then made to return to its initial position as shown in the first frame.

Fig. 7. A Force of [0 5.59 0]T is acting on the triangular platform.

4

Prototype

A prototype of a planar cable robot was developed to confirm the theory by experimenting with three and four cables for various geometries of the moving platform. The cable robot is an open-loop system containing electronic and mechanical modules. The electronic module consists of an Apple MacBook Pro computer, an Arduino MEGA 2560 microcontroller board, micro-stepping motor drivers, and stepper motors. The simulation code is written in MATLAB 2018b and the results are downloaded on to the microcontroller board. The results of the simulations are encoded by the microcontroller board before being fed to micro-stepping motor drivers which then convert the input command signals into the power necessary for energizing the stepper motor windings. The mechanical module consists of a base plate, a movable platform suspended by the cables, a spring, end effector, and pulleys secured to the shaft of the motors (Refer Table 1 for specifications of the setup). Cable stoppers are used as the fixed points on the base frame through which the cables are passed from pulleys mounted on the motors to the platform. These fixed points are connected by the lines depicted in Fig. 8(a). A photograph of the prototype is shown in Fig. 8. Table 1. Specifications of the prototype Component/ Part Computer Microcontroller board Micro-stepping motor driver DC stepper motor Power supply Base plate Triangular platform Rectangular platform Cables

Specifications Apple MacBook Pro ± i5 (2016) Arduino MEGA 2560 board TB6560 NEMA ± 23 , holding torque = 1.89Nm at 2.8A/ per phase 12V 5A 60W SMPS unit Mild steel ( length = 1m, breadth = 1m, thickness = 2.2mm) Aluminum alloy ( base = 180mm, height = 180mm, thickness = 8mm) Alloy steel ( length = 121.24 mm, breadth = 70mm, thickness = 15mm) Synthetic tennis gut (nylon) , diameter =1.56mm

Planar Cable-Driven Robots with Enhanced Orientability

Pulley End effector (spool with helical groove) Spring

11

Aluminum (Outer diameter = 42.60mm , width = 32mm, pitch = 4mm) Aluminium ( pitch = 4mm, outer diameter = 42.60mm, width = 40mm) Spring steel, D2 grade, stiffness = 3.2517N/cm

Fig. 8. a) Prototype of the three-cable robot b) Pre-loaded spring c) End effector with cable wound d) Triangular platform e) Pulley mountable on motor shaft f) Cable stopper g) Motor driver

4.1

Three-cable robot with complete orientability

An end-effector modelled as a spool with a helical groove that can freely rotate about its axis is mounted on a triangular platform. For our experiments, we considered two windings of the cable over the end effector with one end of the cable wound on a pulley mounted on the shaft of a DC stepper motor, and the other end hooked to a translational spring. Using the method presented in Section 3, the end effector can be rotated by any angle at any point in the workspace. The path considered in Section 3.2 is depicted in Fig. 9 where the end-effector is rotated by an angle of 30°when it arrives at the end points of a vertically linear path (from (0, -4) to (0, 4)). A 10% error was observed in the rotation of the end effector. We attribute this error to the lack of sufficient friction between the cable and the smooth surface of the pulley.

5

Closure

Inherent restricted orientation attribute of the moving platform in planar cable robots is explained with the aid of simulations results for three- and four-cable robots. A concept, involving the use of one extra cable and a translational spring, to achieve complete orientability of the end effector is explained in Section 3. Experimental values of orientations obtained deviated from the orientation values considered in Section 3.2 by 10%. This may be attributed to slip of the cable on the spool and other reasons that will

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be considered in our future work. Closed-loop control is also likely to improve the accuracy of position and orientation.

Fig. 9. Enhanced orientability of the end effector. Note that in image 2 and 3 there is a rotation of about 27º (indicated by the black pointer attached to the end-effector).

References 1. Oh, S. R., Agrawal, S. K.: Cable suspended planar robots with redundant cables: Controllers with positive tensions. IEEE Transactions on Robotics, 21(3), 457-465. 2. Albus, J., Bostelman, R., Dagalakis, N.: The NIST robocrane. Journal of robotic systems, 10(5), 709-724. 3. Yang, L. F., Mikulas, M. Jr.: Mechanism synthesis and 2-D control designs of an active three cable crane. In 33rd Structures, Structural Dynamics and Materials Conference (p. 2342). 4. Shiang, W. J., Cannon, D., Gorman, J.: Optimal force distribution applied to a robotic crane with flexible cables. In Robotics and Automation, 2000. Proceedings. ICRA'00. IEEE International Conference on (Vol. 2, pp. 1948-1954). IEEE. 5. Kawamura, S.: Development of an ultrahigh speed robot FALCON using wire drive system. Robotics and Automation, 215-220. 6. Eichstadt, F., Campbell, P., Haskins, T.: Tendon suspended robots: Virtual reality and terrestrial applications (No. 951571). SAE Technical Paper. 7. Fattah, A., Agrawal, S. K.: On the design of cable-suspended planar parallel robots. Journal of mechanical design, 127(5), 1021-1028. 8. Alp, A. B., Agrawal, S. K.: Cable suspended robots: design, planning and control. In Robotics and Automation, 2002. Proceedings. ICRA'02. IEEE International Conference on (Vol. 4, pp. 4275-4280). IEEE. 9. Barette, G., Gosselin, C. M.: Kinematic analysis and design of planar parallel mechanisms actuated with cables. In Proceedings of ASME Design Engineering Technical Conference (pp. 391-399). 10. Fattah, A., Agrawal, S. K.: Workspace and design analysis of cable-suspended planar parallel robots. In ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (pp. 1095-1103). American Society of Mechanical Engineers. 11. Fattah, A., Agrawal, S. K.: Design of cable-suspended planar parallel robots for an optimal workspace. In Proceedings of the workshop on fundamental issues and future research directions for parallel mechanisms and manipulators (pp. 195-202). 12. Roberts, R. G., Graham, T., Lippitt, T.: On the inverse kinematics, statics, and fault tolerance RIFDEOHဨVXVSHQGHGURERWV Journal of Robotic Systems, 15(10), 581-597.

Chain Driven Robots: An Industrial Application Opportunity. A Planar Case Approach Guillermo Rubio-Gómez1, David Rodríguez-Rosa1, Jorge A. García-Vanegas2, Antonio Gonzalez-Rodríguez1, Fernando J. Castillo-García1, and Erika Ottaviano3[0000-00027903-155X] 1

University of Castilla-La Mancha. Avda. Carlos III. Real Fábrica de Armas, TO 45071, Spain 2 University of Ibagué. Carrera 22 Calle 67 B/Ambalá. Ibagué, Tolima, Colombia 3 University of Cassino and Southern Lazio, Viale dell'Università, 03043 Cassino, Italy [email protected]

Abstract. This work presents Chain-Driven Parallel Robots replacing cables by chains. The use of conventional sprockets adds some important advantages with regards to Cable-Driven Parallel Robots. The most important ones are: a) no drum is required; b) no cable plasticity limitation must be imposed; c) using counterweights the manipulator can move the required payload with low motorization. In this paper some design considerations for allowing an accurate positioning and maximizing the robot workspace are presented. As example, a 2 Degrees-ofFreedom planar manipulator has been designed and built. The robot can command a 60 kg payload into a 0.8m ൈ 1.8m workspace using only two 150W DC motor. Keywords: Chain-Driven Robot, Parallel Robot, Industrial Applications.

1

Introduction

The use and applications of Cable Driven Parallel Manipulators (CDPMs) are increasing in the last decades thanks to the main advantages with respect their classical counterparts. These advantages refer to the very large workspace, relatively lightweight structure, overall reduction of the costs related to the main components of the system. Application of CPDMs for high-speed pick and place manipulators taking advantage of the low-inertia are reported in [1, 2]. Applications related to the large workspace include displacement of materials over large areas, positioning systems for heavy objects, rescue operations, service, assistance, rehabilitation, [3, 4, 5, 6]. Main components of the CDPMs are the fixed frame (or structure), the end-effector, the cables, the actuation and transmission system. Each of these elements should be correctly modelled in order to get a good mechanical design and realistic simulations RI WKH FDEOH V\VWHP¶V EHKDYLRU, [7, 8, 2]. One of the challenges in the study of the CDPMs is the cable and its correct modeling, since it is the main tool for operating payloads and influences the kinematic, static, dynamic analyses and control.

© Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_2

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Several studies on cable-driven robots deal with their kinematics [9] under static or quasi-static conditions [10, 1]. Even in this condition is immediately unclear that assuring cables to be in tension affects the workspace and the available operational wrench of such manipulators, which turns out to be quite different from that of conventional parallel mechanism. As clearly reported in the relevant literature, cables cannot be modeled as rigid, contrary to conventional limbs in parallel manipulators. This feature has led to model the stiffness of cable-actuated mechanisms. Models including sagging cables have been discussed in [11, 12, 13, 14], all of those models have advantages but possess drawbacks. The main issue is computational complexity and computational time, vs accurate modeling [14]. CDPMs are often classified referring to the end-effector DOFs that can be controlled. When the CDPR is in a crane configuration, and gravity acts like an additional cable [7] not all the end-effector DOFS can be controlled. Underconstrained cable robots offer several advantages with respect to fully constrained ones. A smaller number of cables means reduced number of actuators, overall costs and setup time, improved ease of assembly and a lower likelihood of cable interference. These features make underconstrained CDPRs well suited for all those applications in which not all the end-effector DOFs can be controlled but having the advantage of decreasing complexity, such as rehabilitation tasks, rescue operations, positioning of objects over large workspaces, [15]. Besides the interest towards fully constrained robots the suspension of the endeffector has recently drawn attention in under-constrained CDPRs [16, 2]. Since cables can exert forces only when they are taut, additional actuators can be used with respect to the number of degrees of freedom (DOFs) to be controlled to prevent cables from becoming slack. This is often the case for redundantly actuated cabledriven robots. One of the main difficulties in the modelling and control of suspended manipulators is the cable behavior that influences correct positioning, kinematics and dynamics. The kinematic problem has been addressed solving simultaneously kinematic compatibility and equilibrium equations, as proposed in [7, 17, 13, 18]. Conventional displacement analysis problem was defined as geometrico-static as in [7] also expressing some connections between stability and energy [10]. As a completely new approach, in this paper we propose the use of chain instead cables to drive and locate the end-HIIHFWRU7RWKHDXWKRUV¶NQRZOHGJHWKHXVHRIFKDLQ instead cables in CDPMs has never been proposed before, therefore, we are addressing the study of Chain Driven Robot (CDRs). The use of chain instead cables may be effective for a number of advantages that make them suitable for displacing large payloads and gaining accurate positioning. In traditional CDPMs, in order to modify the cable length, cables are usually wound around a spool and the spool is rotated to release or retract the cable through suitable actuation system mounted on the same axis of the spool. The anchor position is the point at which the cable leaves the spool. The change in diameter of the spool due to the cable being wound results in the change in location of anchor position. When the cable is retracted the diameter of the spool gets larger and when the cable is let out the diameter gets smaller. In addition, as the cable travels along the axis of spool, while it is retracted or released, it moves out of the plane of mobile platform. These two issues are usually avoided in practical mechanical design of a cable-based manipulator by

Chain Driven Robots: An Industrial Application…

15

placing a pulley between the spool and the cable attachment point on the mobile platform, which gives uncertainties in the determination of the cables length and positioning, as reported in [2]. Additional problems are related to the axial and transversal flexibility of the cable, as previously reported. The use of chain prevents from problems related to the drum and pulleys. Roller chain drives are traditionally and widely used in various high-speed, high-load and power transmission applications, because their main characteristics of high reliability and durability, on the contrary of cables they are inextensible, there is no slippery. In this paper, we evaluate the use of chain driven robot for industrial application and detail the design requirements for a proper design.

2

Chain-Driven Robot vs Cable-Driven Robot

As introduced in Section 1, chain-driven robots present several important advantages in comparison to cable-driven ones. The main advantages, which have been considering for the design, are: x No drum is required: Using chain and sprockets, we avoid the use of any mechanism for roll in and out cable (like drums, see Fig. 1a and 1b).

a)



b)



c)



Fig. 1. Drum vs. Sprocket

x

x

Cable plasticity limitation: When parallel robots are commanded by means of cable all transmission elements (pulleys or drums) must be designed considering the minimum radius to avoid cable plasticity. In the case of using chains the restriction is regarded to the size of the chain link. In this sense with cables, when payload increase, cable size also increases, and this requires that all drums and pulleys must be larger for avoiding cable plasticity. With chain, any compatible sprocket can be applied without plasticity limitation (see Fig. 1c, where both, small and large sprockets can couple a heavy chain). Counterweight for decreasing torque requirements: As chain and sprocket are coupled, counterweight can be used, and motor torque needed, ߬௜ , notoriously decreases (see Fig. 2).

On the other hand, using chains for commanded the end-effector, has several problems, still opened: x Dynamics model implications of using chains owing to the non-negligible weight of chains.

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x x x

Minimum tension of chains for ensuring an accurate positioning (sagged chains). Using chains for commanded low payloads. Spatial designing for connecting frame to end-effector by means of chains and swivel sprockets. Wi

counterweight

end-effector

Fig. 2. Counterweight for decreasing needed torque.

3

Preliminary Prototype

3.1

Prototype design

In this paper a 2 degrees-of-freedom planar chain-driven robot has been designed and built. Reflexive pulleys have been included in end-effector for an accurate kinematic model [2]. Figure 3 shows represent the robot scheme where ܹ and ‫ ܪ‬are the width and height of the frame and, ‫ ݓ‬and ݄ the width and height of the end-effector, which planar pose is ሾ‫ݔ‬௘ ǡ ‫ݕ‬௘ ሿ் . The mass of both counterweights has been noted as ݉ଵ and ݉ଶ , respectively, since the mass of end-effector is ݉௘ . Finally, all sprockets radius is ‫ݎ‬, since the motors angular positions are ߙଵ and ߙଶ . Note that motor 1 (ߙଵ ) commands the pair of chains on the left so they are parallel and the same with motor 2 (ߙଶ ) and the pair of chains on the right. This scheme maintains constant the angle of end-effector [1]. t

r

Ś

T1

T2

T2

T1 ǁ

D2

D1

me Ś

m1

,

m2 [x e , ye ]

Fig. 3. Scheme of 2DOF planar chain-driven robot

Chain Driven Robots: An Industrial Application…

3.2

17

Experimental platform

The experimental parameters are summarized in Table 1. Table 1. Parameters of experimental platform Parameter

Value

ܹ (m)

1.680

‫( ܪ‬m)

2.310

‫( ݓ‬m)

0.320

݄ (m)

0.140

‫( ݎ‬m)

0.050

The values of end-effector mass, ݉௘ , and counterweights, ݉ଵ and ݉ଶ will be configured for maximizing workspace ensuring an accurate positioning. Figure 4 shows the final aspect of the experimental platform.

a)

c)



 Fig. 4. Experimental platform

 b)

d)



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The robot is motorized by two 150W DC motor RE40 Maxon Motor coupled to a 74:1 gearbox. The nominal values of the motor are 0.177 Nm as nominal torque (13.098 Nm at the gearbox output) and 6940 rpm as nominal speed (93.78 rpm at the gearbox). 3.3

Tension Analysis

In this section we analyze the minimum tension of chain for an accurate positioning. Figure 5 represents different end-effector pose for a 2 kg payload. Note that sagging chains are present is different end-effector position when chain tension is lower than a limit which depends of the chain properties.



















Fig. 5. End-effector poses for a 2kg payload

For determining the minimum tension allowed for accurate positioning we have blocked the sprockets at a position corresponding to a centered end-effector pose with known chains angle (ߠଵ and ߠଶ in Fig. 3). For ensuring non-sagging chains, we have started the experiments with a very-large payload, ݉௘ ൌ30 kg, and we have decreased it until we note that sagged effect is present. This lower limit payload corresponds to the lower limit tension (ܶଵ and ܶଶ in Fig. 3): ܶ௜ǡ௠௜௡ ൌ

௠೐ ௚ ଶୱ୧୬ሺఏ೔ ሻ

(1)

Chain Driven Robots: An Industrial Application…

19

where ܶ௜ǡ௠௜௡ is the minimum tension of chain ݅. The sagged effect has been detected by measuring the variation of the end-effector pose allowing a maximum distance variation of 1 mm. Figure 6 summarizes the experiments, where the end-effector pose has been established for ߠଵ ൌ ͳ͵ͷι and ߠଶ ൌ Ͷͷι. The resulting payload for non-sagged effect was ݉௘ ൌ10.20 kg, which correspond to ܶଵǡ௠௜௡ ൌ ܶଶǡ௠௜௡ =70.81 N.



a)

b)



c)





d)

Fig. 6. Minimum tension determination

3.4

Workspace Analysis

Once, minimum tension for accurate positioning has been determined, the end-effector mass, ݉௘ , together with the mass of counterweights, ݉ଵ and ݉ଶ , can be obtained for maximizing the workspace for a given maximum torque of motor/gearbox set (߬௠௔௫ ൌ13.098 Nm). The torque of motor will be exerted for moving end-effector. On the other hand, motors should move the end-effector in the opposite direction of counterweight effect. These both conditions can be written as: ȁ߬௜ ȁ ൌ ȁ‫ ݎ‬൉ ሺܶ௜ െ ݉௜ ݃ሻȁ ൏ ߬௠௔௫

(2)

being ߬௜ the torque exerted by motor ݅. The optimal workspace is obtained with ݉௘ ൌ ͸ͶǤͺ Kg and ݉ଵ ൌ ݉ଶ ൌ ͶͲǤ͹ Kg. (see Fig. 7a). If counterweights are not properly designed and are lower than the optimal one, workspace is reduced as in Fig. 7b. If end-effector mass is not properly designed and is lower than the optimal one, workspace is reduced as in Fig. 7c.

a)



b)



Fig. 7. Workspace analysis

c)



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Finally, Fig. 8 represents the maximum rectangular workspace which is 0.810 m ൈ 1.86 m. Note that the designed robot can manipulate a payload of 64.8 Kg into a 0.810 m ൈ 1.86 m workspace only with 2 motors of 150W.

Fig. 8. Maximum rectangular workspace

4

Results

4.1

Experiments setup

A Graphical User Interface (GUI) has been designed on MATLAB. This GUI allows establishing the manoeuvres parameters such as initial and final pose or time profile of the trajectory. The GUI send messages to a CAN-BUS node used as a bridge between computer and the controller of each motor (Maxon Motor EPOS 2 70/10). Finally, the controllers execute the motor orders and, when the manoeuvre finishes, return the control to the GUI for following movements (see Fig. 9).

Fig. 9. Hardware architecture of the robot

Chain Driven Robots: An Industrial Application…

4.2

21

First movements

For the first experiments, a 30 Kg payload (2 ൈ 15 Kg) has been placed at the endeffector, since both counterweights have been set to 15 Kg. Using the hardware architecture shown in Fig. 9, some controlled movements have been executed. To achieve the desired end-effector movements the Inverse Kinematic Transform of the robot has been developed as in [2]. The experimental results shown as the 2 ൈ 150W motorization is able to move the required payload into the workspace although no positioning accuracy has been still evaluated.

Fig. 2. First experiments setup.

5

Conclusions

In this paper a chain-driven parallel manipulator has been presented, replacing cables by chains for commanding the movements of a payload. This new kind of manipulator can be used for industrial applications, for example, pick and place. The main advantages of these manipulators are: x No drum is required to be designed. x The size of the pulleys, now sprockets, can be reduced because no plasticity limitation should be considered. x The use of counterweights reduces the needed torque for moving the payload. On the other hand, the implications of using chains over the dynamics model should be explored. In this paper several design considerations have been presented, such as the transmission design, based on commercial sprockets, or the end-effector and counterweight design for maximizing the feasible workspace of the robot. As example, a 2 Degreesof-Freedom planar manipulator has been designed and built. The robot can command a 60 kg payload into a 0.8m × 1.8m workspace using only two 150W DC motor.

Acknowledgements This work was partially supported by EU Call RFCS-2017 through the research project DESDEMONA (grant agreement number 800687).

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References 1. Castelli, G., Ottaviano, E., Gonzalez, A.: Analysis and simulation of a new Cartesian cablesuspended robot. Proc. IMechE Part C: J. Mechanical Engineering Science 224: 1717{1726 (2010). doi:10.1243/09544062JMES1976 2. Gonzalez-Rodriguez, A., Castillo-Garcia, F.J., Ottaviano, E., Rea, P., Gonzalez-Rodriguez, A.G.: On the effects of the design of cable-Driven robots on kinematics and dynamics models accuracy. Mechatronics 43: 18{27 (2017). doi:10.1016/j. mechatronics.2017.02.002 3. Albus, J., Bostelman, Dagalakis N.: The nist robocrane. Journal of Robotic Systems10 (5), 709{724 (1993). 4. Castelli, G., Ottaviano, E., Rea, P.: A Cartesian Cable-Suspended Robot for improving endusers' mobility in an urban environment. Robotics and Computer-Integrated Manufacturing 30(3): 335{343 (2014). doi:10.1016/j.rcim.2013.11.001 5. Havlik, S.: A cable-suspended robotic manipulator for large workspace operations. Comput. Aided Civil Infrastruct. Eng. 15(6): 56{68 (2000). 6. Nan, R., Peng, B.: A chinese concept for 1km radio telescope. Acta Astronautica 46 (1012): 667{675 (2000). 7. Abbasnejad, G., Carricato, M.: Direct geometrico-static problem of underconstrained cabledriven parallel robots with n cables. IEEE Transactions on Robotics 31 (2): 468{478 (2015). 8. Du, J., Agrawal, SK.: Dynamic Modeling of Cable-Driven Parallel Manipulators with Distributed Mass Flexible Cables. J. Vib. Acoust. 137(2): 1{8 (2015). doi:10.1115/1.4029486 9. Bosscher, P., Ebert-Uphoff, I.: Disturbance robustness measures for underconstrained cabledriven robots. Proc. IEEE Int. Conf. Robot. Autom., Orlando, FL: 4205{4212 (2006). 10. Carricato, M., Merlet, J-P.: Stability analysis of underconstrained cable-driven parallel robots. IEEE Trans. on Robotics 29(1):288{296 (2013). 11. Kozak et al.: Static Analysis of Cable-Driven Manipulators with Non-Negligible Cable Mass. IEEE Transactions on Robotics 22(3):425{433 (2006). doi:10.1109/TRO.2006.870659 12. Merlet, J.-P.: A generic numerical continuation scheme for solving the direct kinematics of cable-driven parallel robot with deformable cables. In IEEE International Conference on Intelligent Robots and Systems: 4337{4343 (2016). 13. Ottaviano, E., Gattulli, V., Potenza, F.: Elasto-Static Model for Point Mass Sagged CableSuspended Robots. Advances in Robot Kinematics 2016. Springer Proceedings in Advanced Robotics, vol 4. Springer, Cham (2018). 14. Pott, A, Tempel, P.: A Uni_ed Approach to Forward Kinematics for Cable-Driven Parallel Robots Based on Energy. Advances in Robot Kinematics (ARK): 401{409 2018. doi:10.1007/978-3-319-93188-3 46. 15. Merlet, J.-P., Daney, D.: A portable, modular parallel wire crane for rescue operations. IEEE International conference on robotics and automation, Anchorage, Alaska, 3{8 May 2010: 2834{2839 (2010). 16. Collard, J.F., Cardou, P.: Computing the lowest equilibrium pose of a cable-suspended rigid body. Optimization and Engineering 14(3): 457{476 (2013). 17. Merlet, J.-P., dit Sandretto, A.: The Forward Kinematics of Cable-Driven Parallel Robots with Sagging Cables. In Cable-Driven Parallel Robots: 3{15 (2014) Springer. 18. Pott, A.: An algorithm for real-time forward kinematics of cable-driven parallel robots. In Advances in Robot Kinematics: 529{538 (2010), Springer.

Non-slipping Conditions of Endless-Cable Driven Parallel Robot by New Interpretations of the Euler-Eytelwein’s Formula Takashi Harada1 and Koki Hirosato2 1

2

Kindai University, 3-4-1 Kowakae Higashiosaka Osaka 577-8502, Japan, [email protected], WWW home page: https://sites.google.com/site/parallelmech/ Kindai University, Graduate School of Science and Engineering Research

Abstract. Non-slipping conditions of endless-cable driven parallel robot (E-CDPR) which enables unlimited rotation of the hand are discussed in this paper. Instead of fixing the end of the cable to the pulley and the winch, endless-cable (loop-cable) is turned around the endless-pulley and the endless-winch. Because friction forces between the cable and the drum transfer the cable tension, slipping of the cable which is dominated by the well-known Euler-Eytelwein’s formula is taking into consideration of the statics of the E-CDPR. In this paper, a new interpretation of the Euler-Eytelwein’s formula is proposed by using a graph that the nonslipping condition is expressed as an area in the cable tensions. Equations of the statics of the endless-pulley and the endless-winch are superimposed on the graph, then the non-slipping conditions of the E-CDPR are derived. Keywords: endless-Cable, friction drive, the Euler-Eytelwein’s formula, statics

1

Introduction

Cable-driven parallel robots (CDPRs) have excellent characteristics such as high speed and large translational workspace [1]. There are a lot of researches and applications of the CDPSs using these characteristics [2][3]. However, the rotational workspace of the CDPR is limited by the collisions of the cables. Additional rotational DOF is one of the most promising ideas for solving this problem[4] [5][6][7]. In this paper, a novel CDPR which enables unlimited rotation about one axis is proposed. Our idea is similar to the drum-embedded CDPR in [4], but be different in that the drum is rotated unlimitedly by a novel endless-cable driven mechanism. Instead of fixing each end of the cable to the pulley and the winch, looped endless-cable is turned around the endless-pulley and the endlesswinch. Because friction forces between the cable and the drum transfer the cable tension, slipping of the cable is taking into consideration of the statics of the endless-cable driven parallel robot (E-CDPR). The Euler-Eytelwein’s formula [8][9] is well known as when the ratio of the cable tensions Tt /Ts between the © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_3

23

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tight side Tt and the slack side Ts is less than eμθ , then the cable does not slip on the drum. Where, μ is the coefficient of friction between the cable and drum materials, and θ is the total contact angle swept by all turns of the cable, measured in radians. In this paper, a new interpretation of the Euler-Eytelwein’s formula is proposed by using a graph that the non-slipping condition is expressed as an area in the cable tensions. Equations of the statics of the endless-pulley and the endless-winch are superimposed on the graph, then the non-slipping conditions of the E-CDPR are derived.

2

Endless-cable driven parallel robot

Fig. 1. Schematic image of Endless-cable driven parallel robot: OCTABE

Figure 1 represents six-dof eight-cables driven parallel robot, dubbed OCTAVE [7]. The moving part of OCTAVE generates six-dof motions, i.e., three translational motions along the xyz axis and three rotational motions around the xyz axis. An endless-pulley is embedded inside the moving part for expanding the rotational workspace around the z axis. Instead of fixing the end of the cable to the pulley or the moving part, two of eight-cables, C1 -C2 ,. . . ,C7 -C8 are looped then be turned around the endless-pulley. Thus OCTAVE is driven by four loop-cables (endless-cables). All endless-cables are wraped around the identical endless-pulley. The driving side of the loop-cable is comprised of two endless-winches and a tensioner by constant pressure spring for the purpose winding the cable extra length when the moving part moves. These mechanisms enable the endless-pulley to unlimited rotation. Force of the tensioner acts as a bias torque for the motors driving the winch, and it effectively utilizes the motor torque range on the negative side to the cable tension on the side from the winch to the pulley. In addition, the bias force makes the cable wrapped around

Non-slipping Conditions of Endless-Cable Driven Parallel Robot…

25

the winch less slippery. Thus, OCTAVE totally has seven-dof mechanism, which means that OCTAVE has one kinematic redundancy for six-dof hand attached at the tip of the endless-pulley. The seven-dof mechanism of Octave is controlled by eight-cables, which means that OCTAVE has one actuation redundancy like as the conventional CDPRs.

3 3.1

Statics of the endless-pulley single-drum and double-drums

Fig. 2. Options of the endless-pulley

There are two typical types of the endless-pulley; i.e., the single-drum (nongrooved) and the double-drums (grooved) as shown in Fig. 2 (a) and (b). Figure 2 (a) illustrates an endless-pulley of single-drum is driven by a cable by the frictions between the drum and the cable. The cable is several turned around the drum such as U-shaped capstan drum. The generated torque τp of the pulley is given by the cable tensions ta and ta as τp = rp (ta − tb )

(1)

rp represents the radius of the drum. In the case of the double-drums as shown in Fig. 2 (b), the cable is alternately several turned around and between the main grooved-drum and the idle grooveddrum. When the winding number of the cable is large, the double-drum is suitable because the cable is stably guided by the grooves. Assuming that the generated torque of the idle pulley is zero, one obtains, ta1 = ta , ta2 = tb1 , ...., ta(i) = tb(i−1) , ...., tb(n) = tb Thus, the generated torque of the pulley is given as  n  n   tai − tbi = rp (ta − tb ) τp = rp i=1

(2)

(3)

i=1

The relation between the tensions of the cable and the generated torque of the pulley becomes same as a single-drum pulley of Eq. (1).

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Fig. 3. Schematic images of cable-driven pulleys

3.2

A new interpretation of the Euler-Eytelwein’s formula

The Euler-Eytelwein’s formula [8][9] is well known with Fig.3 (a) as when the ratio of the cable tensions Tt /Ts between the tight side Tt and the slack side Ts (Tt > Ts ) is less than eμθ , then the cable does not slip on the drum. The non-slipping condition of the cable on the drum is given by Tt < eμθ Ts

(4)

Where, μ is the coefficient of friction between the cable and drum materials, and θ is the total contact angle swept by all turns of the cable, measured in radians. When the pulley rotates at high speed, the centrifugal force of mv 2 acts on the cable then the frictional force decreases. In this research, it is assumed that a nylon cable with a diameter of 3 mm (7 g/m) is operated at a maximum speed of 30 m/s. At that time, bias tension of approximately mv 2 = 60 N is applied for compensating the influence of the centrifugal force. For convenience, the influence of the centrifugal force is omitted in this paper. The contact angle θ of the double-grooved-drums of Fig. 2 (b) can be made larger, thus be grippy than the single-drum. In this subsection, we expand the understanding of the Euler-Eytelwein’s formula and propose a new interpretation of the formula for the E-CDPR. In the case of the E-CDPR, the endless-pulley must generate not only positive (CCW direction) torque τp  0 but also negative (CW direction) torque τp < 0 while changing the tight and slack sides of the cables. Thus the non-slipping condition of the cable on the drum for the E-CDPR is given as ⎧ ta ⎪ ⎪ ⎨ < eμθ (ta ≥ tb ) tb (5) 1 t ⎪ a ⎪ ⎩ > μθ (ta < tb ) tb e In the case of the double-drums pulley, the total contact angle θ is given by the sum of the contact angle swept by all turns of the cable on the driven pulley. Rewite Eqs.(1) and (3) in vector form as 

ta = JT t τp = rp −rp tb

(6)

Non-slipping Conditions of Endless-Cable Driven Parallel Robot…

27

where JT represents the transposed Jacobian. The cable tension t is derived by inverting Eq. (6) as

 

t = JT + τp + hσ =

1 2rp − 2r1p

τp +

√1 2 √1 2

σ

(7)

JT + represents the pseudo-inverse of JT , which gives the minimum norm solution of Eq. (6). h represents the vector spanning the null space of JT , and σ is an arbitrary scholar. This situation is illustrated in Fig. 3 (b). Relations of the non-slipping conditions of Eq. (5), generated torque of the pulley by the cable tensions of Eq. (6), and the cable tensions for desired torque of Eq. (7) are newly interpreted by the graph in Fig.4. The vertical axis and the horizontal axis represent cable tensions ta and tb , respectively. The nonslipping conditions of Eq. (5) is represented by the light gray area in Fig. 4. Note here that the area includes both (ta ≥ tb ) and (ta < tb ) in Eq. (5). The inclinations of the upper and lower border lines lu and lb of the non-slipping area are represented by eμθ and 1/eμθ , respectively. If μ, the coefficient of friction, gets larger, i.e., materials or surface texture of the pulley and/or cable are selected less slippy, or θ, the total contact angle between the cable and drum, gets larger, i.e., the numbers of turns of the cable around the drum are increased, then the inclinations of the upper borderline lu of eμθ is increased and that of lower borderline lb of 1/eμθ is decreased, thus the borders of the non-slipping area is expanded from lu to lux and from lb to lbx as shown in the figure.

Fig. 4. Statics of the endless-pulley on the new interpretation of the E-E’s formula

Here and the former part of the following subsection, we focus on the area of τp ≥ 0 (ta ≥ tb ). Line l represents the constant torque line for the cable tensions of Eq. (7). Cable tensions at the point (tb , ta ) on the line generate the same

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torque of τp at the pulley. Cable tension at the point A on the line corresponds to the minimum-norm solution of Eq. (6), namely, t = JT + τp . As shown in Fig.4, cable tensions at point A are not only out of the non-slipping area but also one of the cable tension ta is not a positive value. Unit vector h and scholar σ in Eq. (7) correspond to the unit direction vector of the line of the constant torque and the distance from the minimum-norm solution along the line, respectively. In the physical meanings, hσ corresponds to the bias tensions which do not affect the output torque of τp . In the case of the E-CDPR, the bias tensions of cables can be controlled so as not only the cable tensions are not negative values, but also the cable satisfies the non-slipping condition of Eq. (5). When the proportionality constant σ of the bias tensions is gradually increased from zero to the positive value, the corresponding point of the tensions on the line moves from the point A(σ = 0) to the right upper direction. When the point crosses over the point √ B(σ = 2r2p ), both cable tensions ta and tb become positive but still be out of the non-slipping area. 3.3

Non-slipping condition of the endless-pulley

When the point crosses over the point C in Fig. 4, cable tensions enter the nonslipping area. After the point C, cable tensions keep in the non-slipping area even if the proportionality constant σ of the bias tensions is set larger value. The non-slipping condition of σ in τp ≥ 0 (ta ≥ tb ) is given as, √   2τp eμθ + 1 σ> (8) 2rp eμθ − 1 In the case of τp < 0 (ta ≥ tb ), the constant torque line, the minimum-norm solution, the border of the positive tensions, and the border of the non-slipping conditions are represented by l , A , B  , and C  in Fig. 4. The non-slipping condition of σ is given as, √   2τp eμθ + 1 (9) σ>− 2rp eμθ − 1 The new interpretation of the Euler-Eytelwein’s formula using Fig. 4 is convenient not only for understanding the non-slipping conditions of the endlesspulley of the E-CDPR but also for designing the non-slipping condition of the endless-winch which is discussed in the following section.

4 4.1

Statics of the endless-winch Endless-winch system in the E-CDRP

Figure 5(a) represents the endless-winch system in the E-CDPR, which is also represented in Fig 1. The endless-winch system is comprised of two endlesswinches A and B, an endless-cable and a tensioner. In Fig 5, endless-winch is

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29

comprised of the double-drums in which one drum is driven by a motor. The tensioner is comprised of two fixed pulleys Pa and Pb , a moving pulley Pm guided by a slider and pulled by a constant force spring. ts represents the constant pulling force by the spring. Two motors and the tensioner generates the cable tensions ta and tb which drive the endless-pulley of the previous section.

Fig. 5. Endless-winch system and its simplified model

Figure 5(b) represents the simplified model of an endless-winch for the static analysis. tw represents the tension to the endless-pulley which corresponds to the tensions ta or tb in Fig. 5(a). The double-drums is replaced by a single-drum of both values of coefficient of friction μ between the cable and drum materials, the total contact angle θ swept by all turns of the cable, and the radius rp of the drum are equivalent. tb represents the bias tension by the tensioner. In the case of Fig 5(a) in which the endless-cable is pulled by one moving pulley, tb equals to fs /2. τw represents the torque of the motor. 4.2

Statics and non-slipping condition of the endless-winch

The goal of this section is to derive the relationships between the range of the cable tension tw , the range of the motor torque τw and bias tension tb . These relationships are useful for designing the endless-winch system. The simplified model of the endless-winch in Fig. 5 (b) looks similar to the model of the endless-pulley in Fig. 3. Indeed, both model are dominated by the Euler-Eytelwein’s formula. However, the input and the output of each model are different. In the case of the endless-pulley of Eq. (1), two cable tensions are the input and the generated torque of the endless-pulley is the output. On the other hand, in the case of the endless-winch, bias tension tb by the tensioner and the

30

T. Harada and K. Hirosato

motor torque τw are the input and the cable tension tw to the endless-pulley is the output which is given by τw tw = + tb (10) rw In general, motor can generate both positive (CCW) and negative (CW) torque. The range of the motor torque τw of the winch is defined as −τw0 ≤ τw ≤ τw0

(11)

Negative torque of the motor generates negative (pushing) cable tension to the endless-pulley, which cannot be supported by the flexible body of the cable. It means that the negative range in Eq. (11) is not valid for the CDPR. In order to effectively use of the motor torque for CDPRs, bias tensions for the winch are commonly used in the previous researches [10]. The tensioner of our research acts not only as the bias force for the effective use of the motor but also as an additional force for the non-slipping between the endless-cable and the endlesswinch. The cable slipping phenomena of the endless-winch can be expressed by

Fig. 6. Statics of the endless-winch on the new interpretation of the E-E’s formula

the new interpretation of the Euler-Eytelwein’s formula as shown in Fig. 6. In the figure, the horizontal axis and the vertical axis represent bias tension tb and tension tw of the cable exported from the endless-winch, respectively. Hereafter, tw is called as winch tension. The non-slipping area of the cable tensions is indicated by light gray same as Fig. 4. When the bias tension tb = tb0 (= 0), range of the winch tension is expressed by the hollow arrow on the line of tb = tb0 in the figure. Not only the winch tension below zero is not only valid but also the hollow arrow exists out of the non-slipping area. When the bias tension tb is increased, the hollow arrow which expresses the range of the winch tension moves

Non-slipping Conditions of Endless-Cable Driven Parallel Robot…

31

to the right upper side as shown in the figure. The upper and the bottom edge of the hollow allow lying on the line l and l , respectively. These lines correspond to the constant torque lines in Fig. 4. When the bias tension tb = tb1 (= τw0 /rw ), the minimum and maximum value of the winch tensions become tw1n = 0 and tw1x = 2τw0 /rw , respectively. In this case, the range of the winch motor seems to be effectively used for the winch tension. However, as shown in Fig. 6, the lower part of the hollow arrow is out of the non-slipping area. Under this condition, 1 τw0 if the winch tension below tw1b = eμθ rw is applied, the endless-cable slips on the endless-winch. The part of the hollow arrow in the non-slipping area is filled with dark gray as shown in the figure. When the bias tension is given as tb = tb2 , the range of the winch tension becomes inside the non-slipping area as shown in the figure. At that time, the minimum and maximum value of the winch tensions become tw2n and tw2x , respectively. tb2 , tw2n and tw2x are given as,   τw0 eμθ (12) tb2 = rw eμθ − 1 ⎧   ⎪ τw0 1 ⎪ ⎪ ⎨ tw2n = rw eμθ − 1 ⎪ τw0 ⎪ ⎪ ⎩ tw2x = tw2n + 2 rw

(13)

Note here that in the case of the non-slipping condition of the endless-winch, minimum winch tension greater than zero. As shown in Fig. 6, the bias tension tb is set as greater than tb2 , the range of the winch tension keeps inside the nonslipping area, but the minimum and the maximum value of the winch tension are got larger. At that time, the non-slipping condition of the bias tension tb of the endless-winch and range of the winch tension tw are given as   τw0 eμθ (14) tb ≥ rw eμθ − 1 tb −

5

τw0 τw0 ≤ tw ≤ tb + rw rw

(15)

Statics of general E-CDPR

In this section, the non-slipping condition of the endless-pulley is extended for general E-CDPRs such as OCTAVE in Fig. 1, the endless-pulley is driven by more than two endless-cables as shown in Fig. 2 (c). The inverse statics equation of the E-CDPR from the generalized force f of m dof mechanism to the generalized cable tension t of (m + 1) cables comprised of n = (m + 1)/2 endless-cables is given as t = JT + f + hσ

(16)

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T. Harada and K. Hirosato

Where, JT + , h and σ represent pseudo-inverse of the transposed Jacobian matrix, (m + 1) × 1 vector which spans null space of JT and an an arbitrary scholar, respectively. Equation (16) is divided into each endless-cable as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ta1 ha1 ⎢ tb1 ⎥ ⎢ A1 ⎥ ⎢ hb1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ (17) ⎢ . ⎥ = ⎢ . ⎥ f + ⎢ .. ⎥ σ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ tan ⎦ ⎣ ⎦ ⎣ han ⎦ An tbn hbn Statics of the ith (i = 1, . . . , n) endless-cable is given as ti = Ai f + hi σ   tai hai , hi ≡ ti ≡ hbi tbi

(18)

Fig. 7. Non-slipping conditions of endless-pulley of the E-CDPR

The non-slipping condition of each endless-cable of the E-CDPR can be explained by the new interpretation of the Euler-Eytelwein’s formula as shown in Fig. 7. The horizontal and the vertical axis of the figure represent each tension of ith endless-cable of tai and tbi , respectively. The minimum-norm solution t = JT + f of Eq. (16) is projected to each endless-cable tension’s space as ti = Ai f of Eq. (18), which corresponds to the tensions on the point Ai in Fig. 7. In the simple case of a single endless-pulley with a single endless-cable of Eq. (7), angle ϕhi of the unit vector h with respect to the horizontal axis equals π/4. In general, μθ in the Euler-Eytelwein’s formula greater than zero, then eμθ > 0, thus one obtaines 1 tan−1 μθ < ϕhi < tan−1 eμθ (19) e Under the condition of Eq. (19), it is obvious from Fig. 4 that the cable tensions

Non-slipping Conditions of Endless-Cable Driven Parallel Robot…

33

Fig. 8. Slipping conditions of endless-pulley of the E-CDPR

can be shifted on the constant torque line then into the non-slipping area with appropriate bias tensions. After the tensions in the non-slipping area, they keep inside the area when the bias tensions are set larger value. Same situations may occur in the case of the general E-CDPR, but there exist some differences. Here, the E-CDPR satisfies the wrench-closure condition [11][12], all elements of the vector h in of Eq. (16) share the same signum. In convenient, the signum of each element of h is set as positive. At that time, the bias vector hi σ in Eq.(18) can be set as its direction be going to right upper side. However, vector hi is not an unit vector nor its direction angle ϕhi = tan−1 (hai /hbi ) is not equals to π/4 as illustrates in Fig. 7. If, ϕhi < tan−1 (1/eμθi ) or ϕhi > tan−1 eμθi , then the cable tensions cannot be shifted in the non-slipping area as shown (a) and (b) in Fig. 8, or cable tenshions is shifted outside of the non-slipping area when the bias tenshion hi σ is set as large value as shown (c) in Fig. 8. θi is the total contact angle swept by all turns of the ith endless-cable. From the above discussions, the sufficient condition of the endless-cables’ non-slipping on the endless-pulley is given as tan−1

1 < ϕhi < tan−1 eμθi , (i = 1, · · · , n) eμθi

(20)

At the end of this section, σ in Eq. (17) for the non-slipping condition is derived. In the case of the E-CDPR, σ is a common value for all endless-cables. Firstly, as shown in Fig. 7, each σic at the point Ci that the cable tensions come into the non-slipping area is derived. If the cable tensions at the point Ai is in the non-slipping area, σic is set as zero. Then, σx , the minimum value of σ that satisfies the non-slipping condition for all endless-cables is given as σx = max{σci , i = 1, · · · , n}

6

(21)

Conclusions

Non-slipping conditions of E-CDPR which enables unlimited rotation of the hand were discussed in this paper. A new interpretation of the Euler-Eytelwein’s for-

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T. Harada and K. Hirosato

mula was proposed by using a graph that the non-slipping condition is expressed as an area in the cable tensions. Equations of the statics of the endless-pulley and the endless-winch are superimposed on the graph, then, the non-slipping conditions of general E-CDPR were derived.

Acknowledgement This work was supported by JSPS KAKENHI Grant Number 18K04068.

References 1. Pott, A.: Cable-Driven Parallel Robots, Theory and Application. Springer, (2018). doi:10.1007/978-3-319-76138-1 2. Cable-Driven Parallel Robots, Mechanisms and Machine Science 13, Editors, Bruckmann, T., Pott, A., Springer (2013). doi:10.1007/978-3-642-31988-4 3. Cable-Driven Parallel Robots, Mechanisms and Machine Science 53, Editors, Goselin, C., Cardou, P., Bruckmann, T, Springer (2017). doi:10.1007/ 978-3-319-61431-1 4. Fortin-Cˆ ot´e, A., Faure, C., Bouyer, L., McFadyen, B.J., Mercier, C., Bonenfant, M., Laurendeau, D., Cardou, P., Gosselin, C.: On the Design of a Novel CableDriven Parallel Robot Capable of Large Rotation About One Axis. in Cable-Driven Parallel Robots, Mechanisms and Machine Science 53, pp. 390-401, Springer (2017). doi:10.1007/978-3-319-61431-133 5. Miermeister, P., Pott, A.: Design of Cable-Driven Parallel Robots with Multiple Platforms and Endless Rotating Axes. Interdisciplinary Applications of Kinematics, Mechanisms and Machine Science 26, pp. 21-29 Springer, (2015). doi: 10.1007/978-3-319-10723-3 3 6. Pott, A., Miermeister, P: Workspace and Interference Analysis of Cable-Driven Parallel Robots with an Unlimited Rotation Axis. in Advances in Robot Kinematics, pp. 341-350 Springer, (2018). doi:10.1007/978-3-319-56802-7 36 7. Makino, T., Harada, T.: Cable Collision Avoidance of a Pulley Embedded CableDriven Parallel Robot by Kinematic Redundancy. 4th International Conference on Control, Mechatronics and Automation, pp. 117-120, Barcerona (2016). doi: 10.1145/3029610.3029620 8. Meriam, J.J, Kraige, L.G.: Engineering Mechanics, Statics (Seventh Edition). pp. 377-378, John Wiley & Sons (2006). ISBN:978-0-470-61473-0 9. Popov, V.L.: Contact Mechanics and Friction, Physical Principles and Applications. pp. 148-149, Springer (2010). doi:10.1007/978-3-642-10803-7 10. Zitzewitz, J.V., Fehlberg, L., Bruckmann, T., Vallery, H.: Use of Passively Guided Deflection Units and Energy-Storing Elements to Increase the Application Range of Wire Robots. Cable-Driven Parallel Robots, Mechanisms and Machine Science 12, pp. 167-184, Springer (2013). doi:10.1007/978-3-642-31988-4 11 11. Gouttefarde, M., Gosselin, C.: L., Analysis of the Wrench-Closure Workspace of Planar Parallel Cable-Driven Mechanisms. IEEE Transactions on Robotics, VOL. 22, No. 3, pp. 434-445 (2006). doi:10.1109/TRO.2006.870638 12. Loloei, A.Z., Taghirad, H.D.: Controllable Workspace of Cable-Driven Redundant Parallel Manipulators by Fundamental Wrench Analysis. Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 3, pp. 297-314 (2012)

Analysis of Cable-Configurations of Kinematic Redundant Planar Cable-Driven Parallel Robot Koki Hirosato1 and Takashi Harada2 1

Kindai University, Graduate School of Science and Engineering Research 3-4-1, Kowakae Higashiosaka Osaka 577-8502, Japan, [email protected] 2 Kindai University

Abstract. A redundant planar three-dof cable-driven parallel robot is proposed and analysis of its cable-configuration is discussed in this paper. Loop cables and constant force springs unlimitedly rotate the endlesspulley which is embedded inside the moving part. The angle of the hand is redundantly given by the sum of the angles of the moving part frame and the endless-pulley. Three-dof hand is controlled by four-dof mechanism using five-cables. This means that the proposed CDPR is a novel cable-driven parallel robot which simultaneously has the kinematic and actuation redundancies. Tactical design and control for the cableconfigurations are proposed as the robot satisfies the wrench-closure condition. Keywords: Cable-driven, Redundantly Acutuated, Cable configurations, Wrench-closure condition, Kinematic redundancy

1

Introduction

Cable-Driven parallel robot (CDPR) has a large translational workspace when using long cables. However, collisions between cables and singular configurations restrict its rotational workspace [7]. In order to enlarge the rotational workspace of CDPRs, additional rotational mechanism such as crank [6] or pulley [1] [3] is embedded inside the moving part. In this paper, a novel planar CDPR with a pulley is embeded inside the moving part as an additional rotational degrees of freedom (dof) is proposed. The pulley embedded mechanism of the proposed CDPR is similar to [1], but be different in that; – The pulley embedded in the moving part is driven by loop-cables which are tensioned by constant springs. This mechanism enables unlimited rotation of the hand which is attached at the end of the pulley. Hereinafter, this mechanism is called endless-pulley. – Not only the angle of the endless-pulley, but also the angle of the moving part frame are actively controlled by the cables. Angle of the hand is given by the sum of the angle of the moving part frame and the endless-pulley, thus, the proposed CDPR has kinematic redundancy. © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_4

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Three-dof hand is controlled by four-dof mechanism (translations along x and y direction, rotation of the moving part φm and additional rotation of the endless-pulley φp ) which is driven by five-cables. The five-cables are spanned as a star-hexagon which looks like the family crest of the Japanese diviner Abe no Seimei. So the proposed CDPR is dubbed SEIMEI. The kinematic redundancy is applied not only for enlarging the rotational range of the hand, but also for a tactical design of the cable configurations and control for the wrench-closure condition of SEIMEI. For the n-dof CDPR driven by n + 1 cables, if and only if the transposed Jacobian matrix (TJM) has full rank and all elements of the null space vector of the TJM share the same signum, then the CDPR satisfy the wrench-closure condition [2] [4] [10]. At that time, tensions of all cables can be positive by adding positive bias tensions, which are projected to the null space of the TJM. The remainder of the paper proceeds as follows. In Sect. 2, design and modeling of SEIMEI are introduced. In Sect. 3, equations of kinetostatics of SEIMEI are derived. In Sect. 4, tactical designs of cable-configurations and control for the wrench-closure condition of SEIMEI is discussed. In Sect. 5, derived cableconfigurations in Sect. 4 were confirmed the wrench-closure condition by numerical tests.

2 2.1

Mechanical design of unlimited rotatable CDPR Planar three-dof by five cables CDPR “SEIMEI”

Schematic image of SEIMEI is shown in Fig. 1. In convenient, base frame and the moving part (MP) frame are formed of regular pentagon. Pbi and Pmi at each vertex of these pentagons represent points of ith cable port on the base frame and the MP frame, respectively. An endless-pulley is embedded into the MP frame as its rotational axis be fixed to the MP frame via a rotational pair. Five cables control position x, y, angle φm of the MP frame and angle φp of the endless-pulley. Σb represents the coordinate frame fixed on the base, Σm and Σp are those on the MP frame, and on the endless-pulley respectively. The length of the ith cable is defined as li , which corresponds to the distance between Pbi and Pmi . As shown Fig.1(b), a hand is attached to the endless-pulley. Angle θ of the hand is redundantly given by sum of the angle φm of the MP frame and the angle φp of the endless-pulley as θ = φ m + φp

(1)

Thus, SEIMEI is categorized into the kinematic redundant parallel robot. Even if collisions between cables and singular configuration occur in a specific configuration, the kinematic redundancy contributes to avoid them by changing angle φm of the MP frame, then changing the direction of the cables without changing position x, y and angle θ of the hand. Instead of fixing each end of the cable to the winch and the endless-pulley, cable-loop mechanism is applied to SEIMEI, which

Analysis of Cable-Configurations of Kinematic Redundant Planar…

37

Fig. 1. Planar three-dof CDPR “SEIMEI”

enables unlimited rotation of the hand. The cable-loop mechanism is comprised of loop cable, winches, constant force spring and slider. Figure 2 illustrates 3D sketch of the cable-loop mechanism. The loop cable from the winch j via the port Pbi is imported inside the MP via the port Pmj , next the cable is wound several turns around the endless-pulley, then the cable is exported from the MP via the port Pmi , then imported in the frame via the port Pbi , wound several turns around the other winch i. After that the loop cable is guided by the two fixed pulleys and tensioned by constant force spring via a moving pulley on the slider. Finally, the loop cable is wound several turnes around the winch j. While fixing the position and angle of the MP frame, the cable can run then turn the endless-pulley. When the MP moves as shown by dashed line in Fig.2, excessive length of the loop cable is pulling by the constant force spring. Note here that the cable is driven by frictions between the cable and the winch or the endless-pulley, that may occurs slip when the tension of the cable is getting to be small. Bias tension for the cable by the constant force spring contributes the cable to non-slipping. The neighboring two cables C1-C2 and C3-C4 are set as loop-cables as shown in Fig.1 (a). The rest cable C5 in Fig.1 (a) is jointed to center of the MP frame by using a rotational pair. This makes specific elements of the TJM equal 0, thus the analysis of the cable configurations can be simple.

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Fig. 2. Cable-loop mechanism

2.2

Modeling

Fig. 3. Kinematic model of SEIMEI

Fig. 4. Relation of cable length

Kinematic model of SEIMEI around the MP is schematically shown in Fig. 3. p is position vector of the MP in the base coordinate frame Σb . bi and vi represent the position vector of ith cable port of the frame and unit direction vector of the ith cable in Σb , respectively. si is the position vector of the cable port Pmi in Σb . li represents the length of the ith cable from the port Pbi to the port Pmi . The vector loop equation of the ith cable is given as p + s i = bi + li v i

(2)

Differentiate both sides of Eq.(2) with respect to time, one obtains l˙i = (vi × si )φ˙ m + vi T p˙

(3)

As mentioned in Sect.2.1, while the position and angle of the MP are fixed, the loop cable can be run then turns endless-pulley as shown in Fig.4. di represents the running length of the loop cable on the ith winch.

Analysis of Cable-Configurations of Kinematic Redundant Planar…

39

Relations between the cable length li , running length di and angle of the endless-pulley φp , and their differential with respect to time are given as Eq.(4) and Eq.(5), respectively. d i = l i + ri φ p

(4)

d˙i = l˙i + ri φ˙ p

(5)

where ri = ±rp (rp : radius of the endless-pulley), manner of the signum is given as Fig.5. The signum is determined by the winding direction of the cable to the pulley, which is named the pulley winding direction in this paper.

Fig. 5. Signum of ri

By substituting Eq.(3) into Eq.(5), one obtains d˙i = ri φ˙ p + (vi × si )φ˙ m + vi T p˙ Kinematics and statics of SEIMEI are given as ⎡ ⎤ ⎤ ⎡ d˙1 φ˙ p ⎢ .. ⎥ ⎣ . ⎦ = Jdx ⎣ φ˙ m ⎦ p˙ d˙5 f x = Jdx T f d

(6)

(7)

(8)

f x and f d represent the wrench of the hand and cable tensions, respectively. Jdx T represents the transposed Jacobian matrix (TJM) of SEIMEI which is given as ⎡ ⎤ r1 . . . r 5 T Jdx = ⎣ a1 . . . a5 ⎦ (9) v1 . . . v5 where ai = |vi ||si |sinθi , θi is the angle from vi to si .

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K. Hirosato and T. Harada

Expand the null space vector of the transposed Jacobian matrix wrench-closure condition

Inverse statics from the wrench f x to the cable tensions f d is given by inverting Eq.(8) as +

f d = Jdx T f x + hdx σdx

(10)

T+

Jdx , hdx and σdx represent the pseudo-inverse of the TJM, 5 × 1 vector spans the null space of the TJM and an arbitrary scholar, respectively. hdx σdx corresponds to the bias tension on the cable. Because Jdx T (hdx σdx ) = 0, the bias tension doesn’t influence to the wrench f x of the hand. In order to the following analysis, rewriting the part of each column of the TJM as ⎡ ⎤  r1 . . . r 5

r . . . r5 = w1 . . . w5 (11) Jdx T = ⎣ a1 . . . a5 ⎦ = 1 t1 . . . t 5 v1 . . . v5 The null space vector hdx of the TJM is derived by the Cramer’s rule [5] as ⎤ ⎡ ⎤ ⎡ det([ w5 w2 w3 w4 ]) det([B1 ]) ⎢ det([ w1 w5 w3 w4 ]) ⎥ ⎢ det([B2 ]) ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ hdx = ⎢ (12) ⎢ det([ w1 w2 w5 w4 ]) ⎥ = ⎢ det([B3 ]) ⎥ ⎣ det([ w1 w2 w3 w5 ]) ⎦ ⎣ det([B4 ]) ⎦ −det([C]) −det([ w1 w2 w3 w4 ]) The wrench-closure condition for the n-dof CDPR driven by n + 1 cables is given as follows; i) rank (Jdx T ) = n(full-rank) ii) all elements of hdx share the same signum When a CDPR satisfies the wrench-closure condition, all cable tensions are controlled to be positive value by applying an appropriate bias tension hdx σdx . 3.2

Cable configurations

A lot of researchers can be found related to the wrench-closure condition of CDPRs, e.g., wrench-closure configuration, wrench-closure workspace [8] [10], and cable-configurations for satisfying the wrench-closure condition [2] [9]. It is difficult to find these conditions analytically, so one-by-one numerical search used to be applied in the previous researcher [11]. In this paper, an analytical approach for cable-configurations that satisfy the wrench-closure condition are proposed. Each element of the null space vector of the TJM is given by corresponding minor of the TJM. Each minor is converted to polynomial by the cofactor expansion, then be related to three kinematic parameters about cable-configurations of the robot. Strategies of design and control for the three kinematic parameters of SEIMEI are proposed so as all polynomials, i.e., all elements of hdx , share the same signum.

Analysis of Cable-Configurations of Kinematic Redundant Planar…

3.3

41

Cofactor expansion of the null space vector of the transposed Jacobian matrix

Each elements of the null space vector hdx of Eq.(12) is given by the corresponding minor of the TJM of Eq.(11). As an example, row 5 of hdx is expanded about ri as   r1 r2 r3 r4 det([ w1 w2 w3 w4 ]) = det t1 t2 t3 t 4 = r1 det([ t2 t3 t4 ]) − r2 det([ t1 t3 t4 ]) + r3 det([ t1 t2 t4 ]) − r4 det([ t1 t3 t3 ])

(13)

Furthermore, det([ t2 t3 t4 ]) of the first term is expanded as   a2 a3 a 4 det([ t2 t3 t4 ]) = det v2 v3 v4 = a2 det([ v3 v4 ]) − a3 det([ v2 v4 ]) + a4 det([ v2 v3 ]) (14) Each element of hdx is converted to polynomial of twelve terms which are composed by ri , ai , det([ vi vj ]). These valuables are related to the kinematic parameters about the cable-configurations of SEIMEI.

4 4.1

Determine cable configurations Three kinematic parameters

In this section, we evaluate strategies of design and control for the three kinematic parameters so as SEIMEI satisfy the wrench-closure condition. Basic idea of the strategy is to find kinematic conditions that the signums of specific combinations of the twelve terms keep their signums during SEIMEI moves in the specific area. As mentioned in Sect.2.1, four of five cables form a pair, not formed the cable is jointed to center of the MP. This makes specific elements r5 and a5 of the TJM equal 0, that enables the analysis to be simple. Cross product by directions vectors of two cables ; det([vi vj ]) For the simplicity, size of the MP is assumed to be zero in this subsection. Det([ vi vj ]) denotes cross product of unit direction vectors of cables, vi and vj . As shown in Fig.6, this corresponds to sinθij (θij is the angle from vi to vj ) because vi and vj are unit direction vectors. Signum of det([ vi vj ]) changes when the MP across the line which connect cable port Pbi and Pbj of the base frame. The following strategy a) is applied. a) A specific limited workspace is determined so as the cross products of two cable’s direction vectors are not changed when the moving part is in this area. By doing this, signums of the valuables are fixed. This area is surrounded by diagonal lines of the base frame, and it is in the shape of the inverse pentagon as shown in Fig.7. Signums of all det([ vi vj ]) do not change in the limited workspace.

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Fig. 6. The angle from vi to vj

Fig. 7. Limited workspace

The moving part pulling direction; ai

ai = |vi ||si |sinθi is given by the cross product of the cable unit direction vector vi and position vector si of the cable ports Pmi in the Σb of the base coordinate frame. θi corresponds to the angle from vi to si . Signum of ai are determined by value of angle θi as shown in Fig.8. Possible initial configurations of all cables


 








Fig. 8. Signum of ai

of SEIMEI about the MP pulling direction are represented in Fig.9. Each cable pair C1-C2 and C3-C4 is consisted by a loop cable. Remained cable C5 is jointed to the center of the MP via a rotational pair. At that time, s5 equals zero vector, thus a5 equals zero, then the analysis becomes simpler. The signum is determined by the pulling direction of the cable to the MP, which is named the MP pulling direction in this paper. The MP pulling direction changes when the MP moves as shown in Fig.10. Note here that SEIMEI has kinematic redundancy as Eq.(1). The hand of SEIMEI can be rotated by the endless-pulley while keeping the

Analysis of Cable-Configurations of Kinematic Redundant Planar…

43

MP pulling direction. Namely, the following strategy is applied for control of SEIMEI. b) During the motions of SEIMEI, the MP pulling direction of each cable to the moving part frame is actively controlled not to be changed by using the kinematic redundancy. By doing this, signums of ai is fixed.

Fig. 10. Changes the MP pulling direction Fig. 9. Cable configurations about ai

The pulley winding direction; ri As mentioned in Fig.5, the signum of ri is determined by the pulley winding direction. Remained cable C5 is jointed to the center of the MP and not to wind to the endless-pulley, thus ri equals zero, then the analysis becomes simpler. Possible configurations about the pulley winding direction are showed in Fig.11. Signum of ri is only affected by winding direction, therefore it is fixed even if the MP moves anywhere in the limited workspace and the MP frame and the endlesspulley has angles. Cable configurations which make all terms of polynomial which are composed by kinematic parameters share same signum are derived. This is the strategy c). c) Under the conditions applied strategies a) and b), the pulley winding direction is determined so as to satisfy the wrench-closure condition.

4.2

Verification of cable configurations of SEIMEI

In this subsection, cable configurations which all elements of the null space vector of TJM share the same signum are derived. For example, the cable configuration whose signums of kinematic parameters is based on Table 1 is verified.

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Fig. 11. Cable configurations about ri Table 1. Combination of signum r1 +

r2 −

r3 −

r4 +

r5 0

a1 −

a2 +

a3 −

a4 +

a5 0

One obtains cofactor expand of det([B1 ]) of hdx as det ([B1 ]) = det([ w5 w2 w3 w4 ]) = r5 det([ t2 t3 t4 ]) − r2 det([ t5 t3 t4 ]) + r3 det([ t5 t2 t4 ]) − r4 det([ t5 t2 t3 ]) = r5 (a2 sinθ34 − a3 sinθ24 + a4 sinθ23 ) − r2 (a5 sinθ34 − a3 sinθ54 + a4 sinθ53 ) + r3 (a5 sinθ24 − a2 sinθ54 + a4 sinθ52 ) − r4 (a5 sinθ23 − a2 sinθ53 + a3 sinθ52 ) (15) Taking r5 = 0 and a5 = 0 into account, Equation (15) is simplified as (−) + sinθ54 (−) ) det([ w5 w2 w3 w4 ]) = rp a2 (+) (sinθ53

+ rp a4 (+) (sinθ53 (−) − sinθ52 (+) ) − rp a3 (−) (sinθ54 (−) + sinθ52 (+) )

(16)

It is evident from substituting signums of Table.1 for ri and ai that 1st and 2nd terms, rp a2 (sinθ53 + sinθ54 ) and rp a4 (sinθ53 − sinθ52 ) are negative regardless of the value of elements. The formula in the parenthesis of 3rd term is rearranged as Eq.(17). sinθ54 + sinθ52 = sinθ54 + sinθ(54−24) = sinθ54 + sinθ54 cosθ24 − cosθ54 sinθ24 = sinθ54 (−) (1 + cosθ24 ) − cosθ54 sinθ24 (+)

(17)

Signum of cosθij is changed when the MP across the circle line whose diameter is given by the line from Pbi to Pbj . Thus additionally limited workspace is defined. This area is showed the hatched part in Fig.12, and be similar to the

Analysis of Cable-Configurations of Kinematic Redundant Planar…

45

star-shaped which is given by trimming the circle off the limited workspace in Fig.7. Signums of all cosθij is fixed when the MP is in this area. Thus signums of cosθ24 and cosθ54 are positive and 3rd term has negative value. Value of three terms are negative, therefore signum of row 1 of hdx is negative. Signum of det([Bi ]) (i = 2 . . . 4) and −det([C]) are regarded as negative in the same way. All elements of hdx have negative value, cable configuration showed in Table 1 satisfies the wrench-closure condition.

Fig. 12. Additionally limited workspace

5

Fig. 13. Wrench-closure workspace

Simulation

This section investigates whether derived cable configuration in Sect.4 satisfies the wrench-closure condition. Various parameters are given later, radius of the pulley rp equals 5mm, distance between center of the MP and cable port Pmi equals 10mm, distance between center of the base frame and cable port Pbi equals 100mm. Figure 13 shows the wrench-closure workspace. The black dot line represents the boundary of the wrench-closure workspace when the angle φm of the MP frame equals 0◦ , the hatched part represents the limited workspace. Proposed additionally limited workspace is conservative, and actual workspace of SEIMEI is more larger in that additionally limited workspace is in the boundary of the wrench closure workspace. The cable-configuration derived in Sect.4.2 satisfies the wrench-closure condition, and proposed tactical design and control for cable configuration are valid.

6

Conclusion

A redundant planar three-dof cable driven parallel robot “SEIMEI” was proposed. SEIMEI is different from conventional CDPR in that, having cable loop mechanism and kinematic redundancy. Cable-loop mechanism rotates the endlesspulley embedded in the MP unlimitedly. SEIMEI can control the angle of the

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endless-pulley and the MP frame actively by having kinematic redundancy. Tactical design of cable-configurations and control for the wrench-closure condition were proposed. The cable-configuration which satisfies the wrench-closure condition is derived by using the “strategy” and kinematic redundancy. Derived cable configuration were confirmed the validity by numerical tests. Acknowledgement This research was supported by JSPS KAKENHI Grant Number 18K04068.

References 1. Fortin-Cˆ ot´e, A., C´eline, F., Bouyer, L., J.McFadyen, B., Mercier, C., Bonenfant, M., Laurendeau, D., Cardou, P., Gosselin, C.: On the Design of a Novel CableDriven Parallel Robot Capable of Large Rotation About One Axis. Cable-Driven Parallel Robots, pp.390-401. Springer (2017) 2. Gouttefarde, M., Gosselin, C.: Analysis of the Wrench-Closure Workspace of Planar Parallel Cable-Driven Mechanisms. IEEE TRANSACTIONS ON ROBOTICS, vol.22, pp. 434-445 (2006) 3. Makino, T., Harada, T.: Cable collison avoidance of a pulley embedded cable-driven parallel robot by kinematic redundancy. 4th International Conference on Control, Nechatronics and Automation, pp. 117-120, Barcelona (2016) 4. Marc Arsenault.: Optimization of the prestress stable wrench closure workspace of planar parallel three-degree-of-freedom cable-driven mechanisms with four cables. 2010 IEEE International Conference on Robotics and Automation, pp. 1182-1187, Alaska (2010) 5. McColl, D., Notash, L.:WORKSPACE ENVELOPE FORMULATION OF PLANAR WIRE-ACTUATED PARALLEL MANIPULATORS. Transactions of the Canadian Society for Mechanical Engineering, vol.33, No.4, pp. 547-560 (2009) 6. Miermeister, P., Pott, A.: Design of Cable-Driven Parallel Robots with Multiple Platforms and Endless Rotating Axes. Interdisciolinary Applications of Kinematics, pp. 21-29. Springer (2014) 7. Pott, A.: Cable-Driven Parallel Robots. Springer (2017). 8. Pott, A., Kraus, W.: Determination of the Wrench-Closure Translational Workspace in Closed-Form for Cable-Driven Parallel Robots. IEEE International Conference on Robotics and Automation, pp. 882-887, Stockholm (2016) 9. Seon, J.A., Park, S., Ko, S.Y., Park, J.O.: Cable Configuration Analysis to Increase the Rotational Range of Suspended 6-DOF Cable Driven Parallel Robots. 2016 16th International Conference on Control, Automation and Systems, pp. 10471052, Korea (2016) 10. Sheng, Z., Park, J.P., Stegall, P., Arrawal, S.K.: ANALYTIC DETERMINATION OF WRENCH CLOSURE WORKSPACE OF SPATIAL CABLE DRIVEN PARALLEL MECHANISMS. ASME 2015 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, DETC201547976, Boston (2015) 11. Tadokoro, S., Nishioka, S., Kimura, T., Hattori, M., Takamori, T., Maeda, K.: On fundamental Design of Cable Configurations of Cable-Driven Parallel Manipulators with Redundancy. Transactions of the Japan Society of Mechanical Engineers, series C, vol.66, No.647, pp.2247-2254 (2000)

Improving cable length measurements for large CDPR using the Vernier principle Jean-Pierre Merlet J-P. Merlet, HEPHAISTOS project, Universit´e Cˆote d’Azur, Inria, France [email protected]

Abstract. Cable lengths is an important input for determining the state of a CDPR. Using drum with helical guide may be appropriate for small or mediumsized CDPR but are problematic for large one. Another issue is the initialization of the cable length as measurements based on drum rotation are incremental. We propose to address automatic initialization and improvement of cable length measurements by using regularly spaced color marks on the cable combined with color sensors in the mast of the CDPR. We show that this disposition allows one to automate the initialization issue and then how it allows to get regularly accurate estimation of the cable length by using the Vernier principle. Keywords: cable length, accuracy, initialization, calibration

1

Introduction

Measuring the cable lengths of CDPR with rotary winch is usually done by measuring the rotation of the drum with an encoder. Coiling the cable on the drum is usually done in two manners: the drum may have a spiral guide and a guiding mechanism moves synchronously the cable in front of the free part of the spiral or the cable is just coiled on the fly on the drum. An alternate to using rotary actuators is the use of linear actuator with a pulley mechanism that amplifies the stroke of the actuator [Merlet(2008)] but we will not consider this case in this paper. The spiral-guided mechanism is the one used in many cases for small to medium sized CDPR such as IPANEMA [Pott et al(2012)] or COGIRO [Gouttefarde et al(2012)]. It leads to a one-to-one and fixed relationship between the cable length and the drum rotation and if elasticity can be neglected it provides an accurate measurement of the cable length but it has drawbacks: – the drum may accept only a single layer. This limits the available total cable length as increasing the drum radius will both increases the length measurement inaccuracy due to error in the drum rotation measurement and the necessary motor torque for a given cable tension. Hence such a mechanism may not be appropriate for very large CDPR that involves coiling several dizains or hundreds meters of cable – the friction of the cable on the spiral guide increases the cable wear and cable elasticity is not taken into account – according to the cable tension the cable may jump to another part of the guide © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_5

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J.-P. Merlet

Another issue for such a winch system is that the drum encoder usually provides only a relative measurement. When starting the CDPR it is therefore necessary either to calibrate the platform’s pose [Alexandre(2012)], [Alexandre et al(2013)], [Chen(2004)], [Baczynski(2010)], [Miermeister and Pott(2012)] or to measure the initial length of the cable, both tasks being tedious. To the best of the author knowledge an automatic determination of the initial cable lengths has not yet been presented. We propose a method that allows both to obtain the initial cable lengths and provides regularly information on the cable length.

2

Approach

Our method is directly inspired by a method we have implemented successfully on our MARIONET-ASSIST CDPR [Merlet(2010)]. This CDPR uses synthetic cables and we have glued several aluminium foils on the cables at known distance from the platform cable attachment point B. At the winch level the cable goes through two electrically isolated Delrin guides that have been covered with aluminium foils, the mid-point between the guides being the output point A of the winch. Each time a cable foil goes through the guides an electric contact is established providing a boolean information for the control computer. Using this event a semi-automated procedure may be used for initialization (the robot stops when detecting a foil and the operator input manually the corresponding cable length) and, in operation, for updating the current estimation of the cable length. In this paper we propose to improve this method by using colored marks on the cable and several color sensors in the mast that constitute the support structure of the CDPR (figure 1) for benefiting from a Vernier scale. Color sensor consists in leds that provides a constant illumination and receptors that are sensitive to a particular color (figure 1). Such sensors are inexpensive and our tests have shown that they can reliably detect at least the three RGB colors. They can easily be integrated in the support mast in small non transparent boxes with a circular opening for the cable (figure 1), the color sensor being protected from external illumination.

support

color sensor

Fig. 1. A color sensor and the principle of color marks on the cable and color sensor in the mast

Improving cable length measurements for large CDPR…

3

49

The initialization problem: a first approach

A major problem for CDPR is to determine automatically what are the cable lengths at the start of the operation. The basic idea of our approach for automatizing this initialization process is the detection by a color sensor of n successive color marks when coiling (or uncoiling) the cable, while having disposed the colors marks on the cable in such a way that there is only a single occurrence of any n successive colors on the cable. For example if we use 3 possible colors R, G, B for the mark, then there will be, for example, a single GBR sequence on the whole cable, the other sequences being any triplet among the set (R, B, G), possibly with repetition (for example GGG or GGB). Let us assume that a given color sequence (n1 , n2 , n3 ) has been detected and that dl is the known distance between A and the color sensor. As a given color sequence (n1 , n2 , n3 ) is unique on the cable the detection of n3 gives us the corresponding mark number on the cable and consequently the distance d between B and the mark. The cable length ρ = ||AB|| is therefore obtained as ρ = d − dl . Hence there is a one-to-one relationship between all possible sequences (n1 , n2 , n3 ) and ρ. Therefore the initialization method consists simply in uncoiling the cable until we have at least n marks between a color sensor and the B point and then coiling the cable until the color sensor has detected n marks, such a process being easy to automatize. A faster initialization process will be presented in section 5.3.

4

Number of marks

As will be presented in the next sections we also intend to use the marks to update the current cable length. Intuitively it makes sense to have a large number of marks on the cable for that purpose, but our initialization process imposes the constraint of having a single occurrence of n successive marks, whatever is the color combination. We will now investigate the influence of this constraint on the maximal number of marks on a cable. Assume that we use marks with k different colors: our problem is to determine the maximal number of marks m that we may have on the cable so that all subsets of n successive marks are different in term of color value. In other words we have to determine the largest color sequence that satisfy this constraint. This problem is well known in combinatorial theory and such a sequence is called a De Bruijn sequence [Bruijn(1975),Bruijn(1946)]. For a cable with k possible mark values that has all n combination of the mark values a single time in the sequence, the sequence length (i.e. the number of marks) is kn for a cyclic sequence. So for n = k = 3 we have a sequence of length 27. But as our sequence is not cyclic we may add 2 additional marks (but not 3, which will be contradictory) so we may have up to 29 marks on the cable. An example of such sequence (the colors are indicated by the number 1, 2, 3) is: [12312112213222323313111333212]. Using this coding the detection of 3 successive colors by a color sensor allows one to determine the location of the last mark on the cable and consequently the cable length. For n = 4 the De Bruijn sequence has a length of 34 = 81. and we may add 3 additional marks while keeping the property so that we will end up with a total of 84 marks on the cable. For k = 4 colors and a sequence of n = 3 marks the De Bruijn

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sequence has a length of 64 to which we may add 2 marks for a total of 66 marks while for a sequence of n = 4 marks we will have a total of 259 marks on the cables. We have now to examine how the marks may be used to determine the cable length.

5

Measuring cable lengths

The height of the tower will be denoted h (in the examples we will assume h = 15) and we assume that there is a color sensor every dm meters starting from the base of the tower so that we have kmax sensors in the tower with kmax = f loor(h/dm ), where f loor is the largest integer lower than h/dm . The cable sensor will be numbered from 1 marks on the 1 to kmax , the sensor kmax being the highest in the tower. We have kmax cable that are assumed to be distributed regularly on the cable, the distance between two successive marks being denoted by dc meters.The marks are numbered from 1 to 1 , mark k1 1 kmax max being the one the closest to B. The distance between mark kmax and the attachment point B, called the dead length, will be denoted by b. When the cable is completely uncoiled we assume that the first mark on the cable is located at the kmax sensor. The total length Lt of the cable between the base of the tower and B is 1 Lt = kmax dm + (kmax − 1)dc + b

(1)

If we assume that the k1 -th mark is located at the k-th color sensor, then the cable length L between the color sensor and B is: 1 L = (kmax − k1 )dc + b

(2)

and the length ρ of the cable between A, B is then obtained from L by subtracting the distance h − kdm between the k-th sensor and the top of the tower: 1 ρ = (kmax − k1 )dc + b − (h − kdm )

(3)

1 Consequently the lowest value ρmin for ρ is obtained for k1 = kmax and k = 1 with the value ρmin = b − h + dm We will choose b = h−dm so that this minimum is 0 in order to have always positive ρ when a mark lies in front of a color sensor. Consequently ρ is obtained as: 1 ρ = (kmax − k1 )dc + (k − 1)dm

(4)

By taking all possible values for k, k1 we get all ρ that will correspond to a mark detection by a color sensor, that we will call a sensor event, and is defined by a triplet (k1 , k, ρ) corresponding respectively to the mark number, to the sensor number and the 1 ] there will be corresponding cable length.. As k, k1 lie respectively in [1, kmax ], [1, kmax 1 at most kmax kmax different ρ. This is an upper bound as the same ρ may possibly be obtained for different pairs (k1 , k). For these values of ρ we will get a sensor event that will allow us to determine the cable length between A, B. Sorting by increasing value all possible values of ρ and taking the differences between successive values will provide the various cable length changes Δ ρ between two sensor events. Note that Δ ρ cannot exceed dm which correspond to the case where a

Improving cable length measurements for large CDPR…

51

given mark is seen successively by the same color sensor. Between two sensor events the cable length will be interpolated from the measurement of the drum rotation with some uncertainty because of modeling error on the coiling process. Clearly there is an interest of having the largest possible number of significant sensor event (i.e. one that provides an update on the cable length), a relatively flat distribution of the Δ ρ with a low average value. Another interest of a flat distribution is that we may relate two successive sensor events (which will provide a change Δ ρ in the cable length) to the corresponding drum rotation Δ θ in order to obtain an estimate of the mean drum radius that will be updated at each sensor event. This estimate will then be used to determine the cable length between two sensor events. 5.1

Significant sensor events

Clearly we are interested in having the largest number of different ρ. Consequently we should avoid having two sensor events for the same ρ. In other words we have to determine if we may have Δ ρ equal to 0 being given dm , dc . Consider now 2 sensor events defined by the triplets (k1 , k, ρ), (k1 , k , ρ  ). Using equation (4) we may calculate the Δ ρ = |ρ − ρ  | as Δ ρ = |(k1 − k1 )dc + (k − k )dm | There is a symmetry in this relation as, being given the events (k1 , k), (k1 , k ) we will get 1 ,k  1  the same Δ ρ for the events (k1 − kmax max − k ,), (kmax − k1 , kmax − k ). Let us assume that the ratio dm /dc is a rational number p/q and consider the rational number k1 − k1 p1 = q1 (k − k ) which is such that |p1 | is the lowest possible value for |k1 − k1 | while |q1 | is the minimal value of |k − k |. We have Δ ρ = dc |(k − k )(

−p1 p + )| q1 q

(5)

As dc is a known positive constant the minimum of Δ ρ will be obtained when |(k − p  1 k )( −p q1 + q )| is minimal. As |(k − k )| ≥ |q1 | the minimum of Δ ρ is obtained for the rational p1 /q1 that is the closest to p/q. 1 = Equation (5) is essential to assert the distribution of the Δ ρ. For example for kmax 29 (29 marks on the cable) and dm = 2.4 (which leads to kmax = 6) and dc = 2 we get dm /dc = 1.2 = 6/5. If we set p1 = 6, q1 = 5 we get k1 − k1 = 6 leading to k1 = k1 + 6 ∈ [7, 29] and k1 ∈ [1, 23]. We have also k − k = 5 and as k cannot exceed 6 we get k = 1, k = 6. The ρ obtained for the pair (k1 ∈ [7, 29], 6) will be the same than for the pair (k1 − 6, 1) so that we will get 23 identical ρ. As the maximum number of ρ is 29 × 6 = 174 there will be 174-23=151 different ρ available. . However we are not only interested in discarding the sensor events that will give the same ρ but also in the set of Δ ρ that are lower than dm . Hence we have to find the positive rationals p1 /q1 with a denominator at most equal to kmax − 1 and lower or equal 1 that will lead to |Δ ρ| that are lower of equal to dm . For that purpose we consider to kmax

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the following theorem that is used for studying the Farey sequences1 : Theorem 1: Let s = a/b and t = c/d > s and if there is no rational between s and t with a denominator that is lower than the largest b or d, then bc − ad = 1 and the a+c rational with smallest denominator between s and t is b+d . Let us illustrate the application of this theorem in our previous example for dm = 2.4 and dc = 2. We have t = p/q = 6/5 and we are looking for s that is smaller than t so that there is no rational between s,t with a denominator that is smaller than a, 5. According to the theorem we must have 6b − 5a = 1 that is clearly satisfied for a = b = 1. Then according to the theorem the next rationals having the lowest denominator are 7/6, 13/11 and so on with increasing denominator. As the denominator are larger than 5, then the closest valid rational p1 /q1 to p/q, that is lower than p/q, is 1/1. As b = k − k we get the minimal Δ ρ as dc × (−1 + 6/5) = dc /5 = 0.4. For any ρ obtained for the pair (k1 , k), then the ρ obtained for the pair (k1 + 1, k + 1) will differ from the previous one by 0.4. We may now examine the closest valid rational p1 /q1 to p/q, that is larger than p/q. Using the theorem we get the condition 5c − 6d = 1 that is fulfilled by c = 5, d = 4 leading also to Δ ρ = 0.4. For being more systematic we may use the following theorem Theorem 2: let two rationals a/b and c/d and let u = p/q the rational closest to c/d and larger than c/d with denominator lower or equal to n. Let k be the largest integer such that k ≤ (n + b)/d. The value of p, q are given by p = kc − a

q = kd − b

We will use this theorem by starting from the lowest possible successive value for a/b, c/d which are 1/(kmax − 1), 1/(kmax − 2) and construct the full Farey sequence or order 5 corresponding to .dm = 2.4 and dc = 2. We get the following pairs (p/q, |Δ ρ|): (1/1 or 5/4, 0.4), (4/3, 0.8), (3/2, 1.2), (2/1, 1.6), (7/5, 2). Note that Δ ρ = dm is always possible by setting k1 = 1, k = kmax , k = kmax − 1. But although we have obtained what are the possible values of Δ ρ we have not established their frequency for a given configuration. The dead length is b = 12.6 and measurements for ρ between 2 and 68 will be obtained. We get 142 successive ρ that differs by 0.4, 2 that differs by 0.8, 1.2, 1.6 and 1 that differ by 2. The Δ ρ distribution as a function of the cable length ρ is provided in figure 2. As may be seen, apart at the extremity, the Δ ρ is everywhere equal to 0.4 with a mean value of 0.442 and a variance of 0.04571. Let us now consider that dm = 1.3, dc = 2 so that dm /dc = 13/20 and kmax = 11. We get the following pair (p/q, |Δ ρ|): (2/3, 0.1), (5/8, 0.4), (1/2, 0.6), (1/1, 0.7), (3/4, 0.8), (5/7, 0.9), (7/10, 1), (4/7, 1.1) and Δ ρ = 0.1 may always be obtained . This is confirmed by the calculation with 216 measurement differing by 0.1, 72 by 0.4, 12 by 0.5, 12 by 0.6, 4 by 0.7 and 1 for 1.3. The value of b is 13.7 and we get measurements for ρ between 1.3 and 69 (figure 2). The mean value of the Δ ρ is 0.213 and its variance is 0.0322. It may be observed that although we have almost doubled the number of 1 A Farey sequence of order n are the irreducible rationals between 0 and 1 whose denominator is lower or equal to n

Improving cable length measurements for large CDPR…

53

Fig. 2. On the left the distribution of Δ ρ as a function of ρ for dm = 2.4, dc = 2 with 29 marks and 6 color sensors. On the right the distribution for dm = 1.3, dc = 2 with 29 marks and 11 color sensors.

color sensors compared to the previous case we still get a large number of cases with Δ ρ = 0.4 so that the choice of dm = 1.3 may not be optimal. Optimal configuration is addressed in the next section. 5.2

Optimal configuration

Optimal choice of sensor distance dm for a given dc A key point for realizing such a measurement system is first to determine what could be the number kmax of color sensors and then select the dm for the best performance. For a given kmax , dm should lie in the range ]h/(kmax +1), h/kmax ]. Consider for example that h = 15, kmax = 6, dc = 2. We may draw the curve of the mean and variance values of the Δ ρ as a function of dm , figure 3. As may be seen the mean value of Δ ρ is an increasing function of dm while there is a discontinuity point for dm = 2.4 with a sudden increase of the mean value but also a sudden decrease of the variance. To understand this behavior we have to consider how many minimal Δ ρ we will obtained for a given rational ratio dm /dc = p/q. For that purpose we have to look at the rational p1 /q1 which has a denominator q1 lower or equal to kmax and that is the closest to p/q. The larger the sets of possible k1 , k are the larger will be the set of 1 − minimal Δ ρ. As the values of k1 , k are restricted to lie respectively in the range [1, kmax p1 ],[1, kmax − q1 ] the larger the sets will be if p1 , q1 are close to 1. Let us assume that q is lower than kmax . According to theorem 1 the closest rational p1 /q1 that is lower than p/q should satisfy pp1 − q1 q = 1. If we set p1 = q1 = 1 we get the constraint (A) p − q = 1 so that dm = pdc /(p − 1) Hence dm is a decreasing function of p that cannot be greater than kmax = 6. If we look at all possible value for p between 1 and 6, then we found out that (A) may be satisfied only for p = 6. The corresponding dm /dc ratio is 6/5 = 2.4 and for this ratio we will get the maximal number of Δ ρ that are at the minimum (in this case Δ ρ = 0.4 as seen on figure 2). However for this value

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Fig. 3. The mean value and variance of Δ ρ as a function of the distance dm between the color sensor for dc = 2 and 29 color marks

the mean value is not that good: other value of dm may lead to a sequence of .Δ ρ whose minimum is lower than 0.4. Let us look now at dm = 2.35, a value that presents the second minimal value for the variance. For this value we get 23 Δ ρ = 0.25, 140 Δ ρ = 0.35, 2 Δ ρ = 0.6, 0.95, 1.3, 1.65 and 1 Δ ρ = 2. Here we have a larger number of Δ ρ equal to 0.25 or 0.35 than compared to the value 0.4 obtained for dm = 2.4. Hence apparently for a given dc we shall consider as possible optimal values of dm the one having the lowest variances. We may also consider increasing dc to a larger value for CDPR having large cable length. For example if we have 29 marks with dc = 5, dm = 2.2 (6 sensors) we get a measuring range of 151 meters with the following pairs of Δ ρ and their number: (0.6, 112), (1, 27), (1.6, 30). If we move to dc = 10 and the same number of sensors and dm , then the measuring range increases to 291 meters with (1, 28), (1.2, 56), (2.2, 88).

Influence of the number of marks Although the number of sensors in a system is important, these sensors are inexpensive and require a low level of maintenance while the marks will require more attention. We may thus consider having less than the maximum number of marks imposed by the the initialization process, but this will impose to have larger dc in order to have a sufficient total cable length. As seen in the previous section the ratio dm /dc = p/q that has the maximum of lowest Δ ρ is such the closest rational with a denominator lower or equal to q should be p1 /q1 = 1/1. Using theorem 1 we get p − q = 1 if p1 /q1 is lower than p/q so that dm = dc p/(p − 1) provided that p − 1 is lower or equal to the number of sensors f loor(h/dm ). If p1 /q1 is greater than p/q we get q − p = 1 so that dm = dc p/(p + 1) Therefore being given dc we are able to find all the valid dm .

Improving cable length measurements for large CDPR…

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Assume that we will use only 20 marks instead of 29 but set dc to 3 in order to still have a large total cable length. For dc = 3 table 1 provides some examples of dm leading to interesting set of Δ ρ with their distribution.

dm 4.5 4 3.75 2.5 2.4 2.15 1.9 1.1

ρmax number of sensors Δρ number 66 3 1.5-3 40 1 65 3 1-2-3 55 2 1 68.25 4 0.75 -1.5-2.25-3 73 2 2 1 69.5 6 0.5-1-1.5-2-2.5 111 2 2 2 1 69 6 0.6-1.2-1.8-2.4 97 2 2 1 67.75 6 0.4-0.45-0.85 34 54 27 68.4 7 0.3-0.5-0.8-1.1-1,9 72 34 26 4 1 70.2 13 0.1-0.2-0.3-0.5-0.8-1.1 32 85 126 6 6 3

Table 1. Minimal Δ ρ measurement and their number for various dm being given dc = 3 and 20 marks on the cable

It may be seen that the best compromises for 3 sensors is dm = 4 with an increase of accuracy for 4 sensors with dm = 3.75, while dm = 2.5 is optimal for 6 sensors. Note that we have added a line with dm = 2.15 that correspond to the lowest variance beside the one leading to the maximal number of minimal Δ ρ as it offers interesting performances. As may be seen from this table increasing the number of sensors may have a large benefit on the measurement. For 13 sensors we get 243 changes for Δ ρ in the range [0.1, 0.3]. We may now consider going into the opposite direction by increasing the number of marks in order to increase the accuracy of the system, at the price of requiring the reading of 4 marks to initialize the cable length (although this argument will be invalidated if we use the initialization procedure described in section 5.3). Consider first that dc = 1.5 so that we have 40 marks. We consider the case where the variance is minimal and table 2 summarizes the result. As it seems difficult to position the sensor with an

dm 2.4 2.25 1.6875 1.8 1.83 1.66 1.35 1.27

ρmax number of sensors Δρ number 69.75 6 0.3-0.6-0.9-1.5 189 12 4 1 70.5 6 0.75-1.5 89 1 70.3125 8 0.1875 . . . 305 71.1 8 0.3-0.6-0.9-1.2-1.5 209 2 2 2 1 71.31 8 0.15-0.18-0.33 . . . 102 140 69 71.78 9 0.16-0.22-. . . 312 31 72 11 0.15-. . . 392 71.2 11 0.11-0.12-0.23-. . . 136 175 118

Table 2. Minimal Δ ρ measurement and their number for various dm being given dc = 1.5 and 40 marks on the cable

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accuracy of half a centimeter, the optimal solution for dm seem to be 2.4, 1.8, 1.83, 1.66, 1.35. Table 3 summarizes good design solutions for cable length in the range of 60-70 meters. number of marks 12 15 20 20 20 29 29 40 40 40 40 60 60 60 84

dc 5 4 3 3 3 2 2 1.5 1.5 1.5 1.5 1 1 1 0.7

dm number of sensors Δρ number 3.75 4 1.25 43 3 5 1 58 1 3 1 55 3.75 4 0.75 73 2.5 6 0.5 111 2.4 6 0.4 142 2.35 6 0.25-0.35 23-140 2.4 6 0.3-0.6 189-12 1.83 8 0.15-0.18-0.33 102-140-69 1.66 9 0.16-0.22 312-31 1.35 11 0.15 392 2.4 6 0.3 289 1.75 8 0.25 363 1.3 11 0.1 0.2 375 272 1 15 0.1 639

Table 3. Best system arrangement for various dc (distance between marks) and number of sensors (dm = distance between color sensors)

5.3

The initialization problem: a second approach

We have proposed in section 3 a first approach to determine the initial cable length based on the detection of n successive marks by one sensor, the coding of the cable being such that there is a single occurrence of a given sequence of n colors on the cable. For the initialization process we therefore need to coil the cable by ndc . However we may get a faster initialization process by taking into account that we have several color sensors in the mast so that we may use all event detection event for performing the initialization process. Clearly we aim at determining the cable length by coiling the cable by less than ndc . If we have n different colors on the cable and m marks on the cable so that m is maximal, then we will have roughly m/n marks of the same color. For example for n = 3, m = 29 and the marking presented in section 3 we have 10 marks of color 1, 2 and 9 marks of color 3. Remind that if the color sensor ns detects mark ms , then the cable length ρ is given by ρ = ns dm − h + L0 − (ms − 1)dc

(6)

where L0 is the total length of the cable. After starting the coiling there will be a first sensor detection event that provide ns while ms is not known. However the detected

Improving cable length measurements for large CDPR…

57

color provides a limited choice for ms (at most 10 in our example) and consequently a limited set {ρ1 , . . . , ρl } of possible ρ. For each of the ρi we consider the set of all possible values of ρ that are lower than ρi obtained for all possible values of ns , ms and we then sort this set by decreasing value of ρ. The first element of the ordered set provides what will be the next sensor detection event if the current value of the ρ is ρi . Hence we get a list L of (ρi , nis , ci ) where nis is the sensor number and ci is the mark color for the next event. When a new sensor event occurs we check the coherence of the ns and the color with each nis , ci : each element of L that is not coherent with the sensor number or color is removed from L . This leads to a new list that contains all possible values of the current value of ρ and prediction about the next sensor event and we repeat the process. As the coherence test allows one to reduce the size of L , we shall end up with a list with a single element that is the current value of ρ. For example we consider the case where we have 29 marks separated by 2 meters with 3 different colors and 6 color sensors separated by 1.6 meter. If the cable length is ρ = 21.5 when starting the initialization process, then the first detection event is color 3 on sensor 4. Using color 3 we obtain the list L ={(45.4, 5, 1), (29.4, 5, 2), (21.4, 5, 2), (17.4, 5, 3), (15.4, 5, 1), (11.4, 5, 1), (3.4, 5, 3), (1.4, 5, 3)}. The second sensor event is color 2 for sensor 5 meaning that among the previous list only the value 21.4 and 29.4 are coherent, leading to L = {(29, 6, 2), (21, 1, 2)}. As the next event has color 2 and sensor 1 we may discard 29 and we will have determined that ρ = 20.2, after coiling only 1.3 meter of cable. More generally figures 4 shows how much cable should be coiled before getting the current ρ value as a function of the initial value of ρ and the number of detection events that are necessary to identify this value.

Fig. 4. On the left the coiling amount that is necessary to identify the current ρ as a function of the ρ value. On the right the number of sensor events that are necessary to identify the current ρ as a function of the ρ value (29 marks, 6 sensors, dc = 1.6, dm = 2).

As may be seen on these figures the average amount of coiling distance for the identification of the current ρ is 1.069 meter (variance: 0.054, min: 0.4, max: 1.95) for an initial ρ between 10 and 50 meter, much better than the value of 6 meters required

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for the method proposed in section 3. If we use dm = 1.83, dc = 1.5, 8 color sensors and 40 marks, then in average we need only 0.622 meter of coiling (variance: 0.03, min: 0.34, max: 1.29) for determining the current ρ.

6

Conclusion

In this paper we propose a setup for both allowing an automated determination of the initial cable lengths and improving the cable length measurements. The method is simple and allows one to obtain very good estimation of the current cable lengths especially for large CDPR (for example being able to provide the cable length for any change of 0.4 meter for a 60 meters cable). It allows also to get an estimate of the drum radius at each sensor event, thereby allowing to improve the estimation of the cable length between two sensor events. The method is robust: being given the sensor price it is possible to use sever color sensors within a given sensor box. Furthermore it is possible to detect sensor failure based on the absence of a color signal during a significant change of cable length. We intend also to explore if this system may not allow to estimate the cable elastic deformation by comparing the difference between the distance between marks (that is known at rest) and the distance observed between sensor events. On-line calibration may also benefit from being able to fix the cable lengths at known values.

References [Baczynski(2010)] Baczynski J, Baczynski M (2010) Simple system for determining starting position of cable-driven manipulator. In: IEEE Int. Conf. on Computer Information Systems and Industrial Management Applications (CISIM), Cracow, pp 102–106 [Bruijn(1946)] Bruijn N (1946) A combinatorial problem. Indagationes Math 8:461–467 [Bruijn(1975)] Bruijn N (1975) Acknowledgment of priority to c. flye sainte-marie on the counting of circular arrangements of 2n zeros and ones that show each n-letter word exactly once. Indhove: Technische Hogeschool Eidenhoven [Chen(2004)] Chen Q, et al (2004) An integrated two-level self-calibration method for cabledriven manipulator. IEEE Trans on Robotics 10(2):380–391 [Gouttefarde et al(2012)] Gouttefarde M, et al (2012) Simplified static analysis of largedimension parallel cable-driven robots. In: IEEE Int. Conf. on Robotics and Automation, Saint Paul, pp 2299–2305 [Merlet(2008)] Merlet JP (2008) Kinematics of the wire-driven parallel robot MARIONET using linear actuators. In: IEEE Int. Conf. on Robotics and Automation, Pasadena [Merlet(2010)] Merlet JP (2010) MARIONET, a family of modular wire-driven parallel robots. In: ARK, Piran, pp 53–62 [Miermeister and Pott(2012)] Miermeister P, Pott A (2012) Auto calibration method for cabledriven parallel robot using force sensors. In: ARK, Innsbruck, pp 269–276 [Pott et al(2012)] Pott A, et al (2012) IPAnema: a family of cable-driven parallel robots for industrial applications. In: 1st Int. Conf. on cable-driven parallel robots (CableCon), Stuttgart, pp 119–134 [Alexandre(2012)] Alexandre dit Sandretto J, Daney D, Gouttefarde M (2012) Calibration of a fully-constrained parallel cable-driven robot. In: RoManSy, Paris, pp 12–41 [Alexandre et al(2013)] Alexandre dit Sandretto J, et al (2013) Certified calibration of a cable-driven robot using interval contractor programming. In: Computational Kinematics, Barcelona

Part II

Kinematics and Static

Stiffness of Planar 2-DOF 3-Differential Cable-Driven Parallel Robots Lionel Birglen1 and Marc Gouttefarde2 1

Department of Mechanical Engineering, Polytechnique Montr´eal, QC, Canada 2 LIRMM, University of Montpellier, CNRS, Montpellier, France [email protected], [email protected]

Abstract. Planar 2-degree-of-freedom (DOF) 3-differential Cable-Driven Parallel Robots (CDPRs) consist of a point-mass end-effector driven by a number of cables. Each cable is divided into four segments, three of them being connected to the point-mass end-effector by means of routing pulleys. This paper deals with the stiffness analysis of such planar 2-DOF 3-differential CDPRs. Based on the usual linear spring cable elongation model, the expression of the stiffness matrix is derived. The stiffness and workspace of several examples of planar 2-DOF 3-differential CDPRs are then compared. The results of these comparisons illustrate that the stiffness of planar CDPRs can be significantly improved by means of pulley differentials. Keywords: Cable-driven parallel robots, differential pulley actuation, stiffness analysis

1

Introduction

This paper deals with cable-driven parallel robots (CDPRs) whose cable segments are not each driven by a single actuator but are coupled through differentials. Coupling and transmitting the actuation torque of a single actuator to several cable segments has been shown in previous works to be a possible solution to extend the workspace of CDPRs by purely passive mechanical means [1–3], i.e. without relying on additional actuators or relocating the latter on a structural frame. Compared to conventional CDPRs, more cable segments then cluttered the workspace but this should not be a critical issue for planar and suspended CDPRs. In the specific example studied here, a single actuator simultaneously controls the lengths of three cable segments going from the ground to the point-mass end-effector of a planar robot and the coupling between these cable segments is implemented through pulley differentials, similarly in principle to the well-known block and tackle. The present paper presents and quantifies the stiffness of such planar 2-DOF 3-differential CDPRs. Such differential couplings will be shown to provide a noticeable improvement in the stiffness of the CDPR. As shown in Fig. 1, each cable of the CDPR consists of four successive segments Bi P , P Ei , Ei Mi , and Mi P . Three of these cable segments link the point © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_6

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Fig. 1. A planar 2-DOF 3-Differential CDPR where each cable consists of four segments Bi Pi , Pi Ei , Ei Mi , and Mi Pi , three of these segments connecting the end-effector P to the ground points Bi , Mi and Ei .

mass P , which plays the role of the CDPR 2-DOF end-effector, to the ground. In the remainder of this paper, these three segments are referred to as the active cable segments. The other one, Ei Mi , extends between points Ei and Mi which are both fixed to the ground. In other words, cable i exits the base first at point Bi , and then goes in this order through points P , Ei , Mi to finally be attached to point P . Hence, these pulley drives are referred to as 3-differential since each one has three “active” outputs. Figure 2 shows this cable routing. In order to implement such a mechanism, pulleys need to be used at points P , Ei , and Mi . However, as a first approximation, the diameters of these pulleys are neglected (considered to be null) and pulley friction is ignored (no friction). The latter assumption implies that the cable tension is constant all along the cable, i.e., the four cable segments have the same tension and each active segment applies a force of identical magnitude on the end-effector. Let n be the number of cables used to drive point P (n = 3 in the case of Fig. 1). This paper deals with the case of planar 2-DOF 3-differential CDPRs with n ≥ 2 cables. The mass of the cables is neglected and all the cable segments are considered to be straight line segments. Stiffness and vibration analyses of usual CDPRs, where the driving cables are directly attached to the end-effector, have been proposed in a number of works, e.g. [4–11]. As previously mentioned, the main difference with these previous works is that the present paper deals with the stiffness analysis of planar point-

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Fig. 2. Cable routing in a single differential: The cable starts at point Bi where the actuated drum is located, goes to P (the end-effector), then to Ei where it is redirected to point Mi in order to be finally attached to point P .

mass CDPRs with 3-differential pulley drive. To the best of our knowledge, the only previous work on stiffness analysis of differential-pulley planar CDPRs is [12] where the case of cables with two active segments is considered. Moreover, the stiffness analysis in [12] only considered stiffness in a single direction and did not mention the trade-off between the size of the workspace and the improvement in stiffness inherent to pulley differentials as will be shown here. This paper is organized as follows: First, Section 2 presents the Jacobian and wrench matrices of differential-pulley-driven planar point-mass CDPRs with three active cable segments. Then, in Section 3, the stiffness matrix of these mechanisms is derived based on the usual linear spring cable elongation model. Finally, Section 4 reports simulations of the resulting stiffness of several examples of planar 2-DOF 3-differential CDPRs.

2

Wrench and Jacobian Matrices

The active length of the cable i in each differential routing is defined as li = −−→ −−→ −−→ lBi P +lEi P +lMi P where lBi P = Bi P , lEi P = Ei P  and lMi P = Mi P  are the strained lengths of the active cable segments Bi P , Ei P and Mi P , respectively. Length li is the sum of the cable segment lengths that change when the position of point P changes. It should not be confused with the total length of the cable which includes the length of the cable segment between points Ei and Mi . Let us T define the vector l of the active cable lengths li as l = [l1 , l2 , . . . , ln ] . Moreover, T ˙ let l be the time derivative of l, p = [px , py ] be the position vector of point P in the fixed reference frame (0, x, y), and p˙ the velocity of point P . A Jacobian matrix J then maps p˙ (resp. dp) into l˙ (resp. dl) ⎡˙ ⎤ l1 ⎢ l˙2 ⎥ ⎢ ⎥ l˙ = ⎢ . ⎥ = Jp˙ ⎣ .. ⎦ l˙n

⇐⇒

dl = Jdp

(1)

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with

⎤ uTB1 + uTE1 + uTM 1 ⎢ ⎥ .. J=⎣ ⎦ . uTBn + uTEn + uTM n n×2 ⎡

(2)

where uBi , uEi and uM i are the unit vectors directed along the cable segments Bi P , Ei P , and Mi P , pointing from the base points Bi , Ei , and Mi , to point T P , respectively. The vector of cable tensions ti is denoted t = [t1 , t2 , . . . , tn ] . Pulley friction being neglected, the cable tension ti is constant all along the cable. The force f applied by the n cables to the point-mass P is then given by f = Wt

(3)

where the 2 × n wrench matrix W is defined as W = −J . T

3

Stiffness Analysis

The elastic potential energy of cable i is given by 1 2 ki (lti − l0i ) 2 where lti is the total length of cable i Vi =

(4)

lti = li + lEi Mi

(5)

with lEi Mi the length of the passive cable segment Ei Mi and li = lBi P + lEi P + lMi P the active length of cable i, as defined in Section 2. l0i is the corresponding unstrained length, l0i = l0Bi P +l0Ei P +l0Ei Mi +l0Mi P . The elongation is supposed to be uniform along the cable length and the cable is assumed to be a linear spring with ti = ki (lti − l0i ), ki = AE l0i , where A and E are the cable cross-sectional area and elastic modulus, respectively. The total elastic potential energy V of the 3-differential CDPR is then the sum of the elastic potential energies of all its n cables V =

n  i=1

n

Vi =

1 2 ki (lti − l0i ) 2 i=1

(6)

Computing the derivative of Equation (6) yields dV =

n  i=1

dVi =

n  i=1

ki (lti − l0i ) dlti =

n 

ki (lti − l0i ) dli

(7)

i=1

where dlti = dli according to Eq. (5) and to the fact that dlEi Mi = 0 (the distance between the ground points Ei and Mi is constant). Then, defining the diagonal matrix D and vectors lt and l0 as follows ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ lt1 l01 k1 ⎢ lt2 ⎥ ⎢ l02 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ D = ⎣ . . . ⎦ , lt = ⎢ . ⎥ and l0 = ⎢ . ⎥ (8) ⎣ .. ⎦ ⎣ .. ⎦ kn ltn l0n

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Eq. (7) can be written T

dV = (lt − l0 ) Ddl

(9)

According to Eq. (1), dl is equal to Jdp so that according to Eq. (9) ⎡ ∂V ⎢ ∂px dV =⎢ ⎣ ∂V dp ∂py

⎤

⎥ ⎥ = JT D (lt − l0 ) ⎦

(10)

where it can be noted that D (lt − l0 ) = t since ti = ki (lti − l0i ). The stiffness matrix K of the 3-differential CDPR is defined as the second derivative of the elastic potential energy, namely ⎡ ⎤ ∂2V ∂2V ⎢ ∂p2 ∂px ∂py ⎥ d2 V x ⎢ ⎥ K= = (11) ⎢ ⎥ dp2 ⎣ ∂2V ∂2V ⎦ ∂py ∂px ∂p2y Taking the derivative of Eq. (10) with respect to p, the following classic expression of the stiffness matrix is obtained K = JT D

dJT dJT dl + D (lt − l0 ) = JT DJ + t dp dp dp

(12)

where, once again, the equality dlt = dl has been used. It is worth noting that this stiffness matrix K can only be used in local stiffness analyses because Eq. (12) is obtained from Eq. (10) by assuming that l0 is constant. In other words, the matrix K is defined at an equilibrium, defined by both the position p and the cable tensions t, and it can be used to analyze the stiffness of the robot in the vicinity of this equilibrium. In this local analysis, the actuators are assumed to be locked or their position controlled to a constant value. Note also that, according dV to Eq. (3) and (10), = JT t = −f , and thus dp   d dV d(−f ) K= = ⇐⇒ Kdp = d(−f ) (13) dp dp dp Hence, in the vicinity of an equilibrium defined by p and t, K maps an infinitesimal displacement dp in the position of point P to the corresponding infinitesimal change in the force −f , where f is the force applied by the cables to point P (Eq. (3)). The approach used above to derive the stiffness matrix is a rather classic one, but care must be taken to use in Eq. (12) the Jacobian matrix defined in Eq. (1), where each row of this matrix is equal to the sum uTBi + uTEi + uTM i of the unit vectors directed along the three straight line segments delineated by the active cable segments.

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Referring to the right-hand side of Eq. (12), the stiffness matrix is seen to be the sum of two matrices. In the next section of this paper, for simplicity and due to space limitations, only the influence on stiffness of the matrix JT DJ will be considered.

4

Results

Fig. 3. Example planar 2-DOF 3-differential CDPR with 3 cables (Case 1)

This section reports simulations of the stiffness of six planar 2-DOF 3differential CDPRs with n = 3 cables, across their respective workspaces. These six examples are defined as Cases 1 to 6 in Tables 1 and 2. Case 1 is shown in Fig. 3 together with its Wrench-Closure Workspace (WCW) [13] and WrenchFeasible Workspace (WFW) [14]. Here, the WFW is defined as the set of positions of point P such that any force f of magnitude less or equal to 0.1 N can be generated by the cables at P with tensions ti , i = 1 . . . n, satisfying tmin ≤ ti ≤ tmax where tmin = 0.1 N and tmax = 1 N. Case 2 defined in Table 1 is a planar 2-DOF 3-differential CDPR where the three active cable segments are superposed. It is obtained from Case 1 by taking Bi ≡ Ei ≡ Mi for i = 1, 2 and 3. Case 3 in Table 1 defines a CDPR without pulley differential where each cable consists of only one segment from Bi to P . It is obtained from Case 1 by keeping segment Bi P and removing the other cable segments P Ei , Ei Mi , and Mi P . Case 4 defined in Table 2 is another planar 2-DOF 3-differential CDPR. It is shown in Fig. 4 where its WFW is defined as above for Case 1. Case 5 is a planar 2-DOF 3-differential CDPR obtained from Case 4 by superposing the 3

Stiffness of Planar 2-DOF 3-Differential Cable-Driven Parallel Robots

B1 E1 M1 B2 E2 M2 B3 E3 M3

Case 1 x (m) y (m) cos(π/3) sin(π/3) cos(2π/3) sin(2π/3) 0 1 −1 0 cos(−2π/3) sin(−2π/3) cos(−5π/6) sin(−5π/6) cos(−π/3) sin(−π/3) 1 0 cos(−π/6) sin(−π/6)

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Case 2 Case 3 x (m) y (m) x (m) y (m) cos(π/3) sin(π/3) cos(π/3) sin(π/3) cos(π/3) sin(π/3) × × cos(π/3) sin(π/3) × × −1 0 −1 0 −1 0 × × −1 0 × × cos(−π/3) sin(−π/3) cos(−π/3) sin(−π/3) cos(−π/3) sin(−π/3) × × cos(−π/3) sin(−π/3) × ×

Table 1. Geometries of the first three examples of planar 2-DOF 3-differential CDPRs. The crosses × indicate that the corresponding point Ei or Mi does not exist, i.e., Case 3 defines a CDPR without pulley differential where each cable consists of only one segment from Bi to P .

B1 E1 M1 B2 E2 M2 B3 E3 M3

Case 4 x (m) y (m) 1 0 cos(2π/3) sin(2π/3) cos(π/3) sin(π/3) cos(2π/3) sin(2π/3) cos(−2π/3) sin(−2π/3) −1 0 cos(−2π/3) sin(−2π/3) 1 0 cos(−π/3) sin(−π/3)

Case 5 Case 6 x (m) y (m) x (m) y (m) 1 0 1 0 1 0 × × 1 0 × × cos(2π/3) sin(2π/3) cos(2π/3) sin(2π/3) cos(2π/3) sin(2π/3) × × cos(2π/3) sin(2π/3) × × cos(−2π/3) sin(−2π/3) cos(−2π/3) sin(−2π/3) cos(−2π/3) sin(−2π/3) × × cos(−2π/3) sin(−2π/3) × ×

Table 2. Geometries of the last three examples of planar 2-DOF 3-differential CDPRs. Case 6 defines a CDPR without pulley differential where each cable consists of only one segment from Bi to P .

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Fig. 4. Example planar 2-DOF 3-differential CDPR with 3 cables (Case 4)

active segments of each cable and Case 6 is a CDPR without pulley differential obtained from Case 4 by removing, for each cable, all segments but Bi P . As mentioned at the end of Section 3, the subsequent stiffness analysis is based on considering that the stiffness matrix is K = JT DJ. To “normalize” the stiffness coefficient with respect to specific values, we use E = 1 Pa and A = 0.001 m2 . Note that these values, together with the tmin and tmax values defined above, do not correspond to actual physical values since, for instance, steel wire ropes typically have elastic modulus E of tens of GPa. However, it is not important for the purposes of this section where stiffness comparisons between several 2-DOF 3-differential CDPRs are made. Moreover, in the calculation of D, for simplicity, we assume that the unstrained length of each cable segment is equal to its strained length. For a given position of the end-effector P , the two eigenvalues of the 2 × 2 stiffness matrix K can then be calculated. The minimum (resp. maximum) stiffness values shown in the left part (resp. right part) of Fig. 5 are defined as the minimum (resp. maximum) of these two eigenvalues. The values of the ratios of the minimum to the maximum eigenvalues over the WCW of Case 1 are shown in Fig. 6. The means of the minimum stiffness values, maximum stiffness values and ratio of minimum to maximum stiffness values over the WCW of the planar 2-DOF 3-differential CDPR examples defined in Tables 1 and 2 are given in Table 3. The mean of the minimal stiffness in Case 1 is 2.4 times larger than in Case 3, where Case 3 corresponds to a classic CDPR with no pulley differential. The means of the maximum stiffness and stiffness ratio in Case 1 are both 1.4 times larger than in Case 3. Comparing Case 2 to Case 3, the minimum and maximum stiffness values are 3 times larger for Case 2 while the stiffness ratios are identical. Hence, planar 2-DOF 3-differential CDPRs (Cases 1 and 2) present a larger stiffness over their workspace than the corresponding CDPR without

Stiffness of Planar 2-DOF 3-Differential Cable-Driven Parallel Robots

Fig. 5. Minimum and maximum stiffness values over the WCW of Case 1 Case min (N/m) max (N/m) 1 0.0018 0.0041 2 0.0022 0.0086 3 0.00074 0.0029 4 0.0012 0.0016 5 0.0022 0.0086 6 0.00074 0.0029

ratio 0.43 0.31 0.31 0.79 0.31 0.31

Table 3. Means of the minimum stiffness values (min), maximum stiffness values (max) and ratio of minimum to maximum stiffness values (ratio) over the WCW of the 2-DOF 3-differential CDPR examples

pulley differential (Case 3). Similar results are obtained by comparing Cases 4 and 5 to Case 6. The cases with superposed cable segments (Cases 2 and 5) have larger minimum and maximum stiffness values than the ones with distinct cable segments (Cases 1 and 4). Note that Case 4 has the best stiffness ratio. Finally, Table 4 shows a comparison of the size of the WCW and WFW in each case. The WFW is defined for all the six cases as it has been defined above for Case 1. From these examples, it can be concluded that the cases of 2-DOF 3-differential CDPRs with superposed cable segments (Cases 2 and 5) lead to the largest stiffness improvements but with a marginal gain on the sizes of the WCW and WFW, as compared to the cases without pulley differentials (Cases 3 and 6). In comparison, the stiffness improvement is less significant in the cases of 2-DOF 3-differential CDPRs with distinct cable segments (Cases 1 and 4) with, however, a slightly better improvement in the WCW and WFW sizes.

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Fig. 6. Ratio of minimal to the maximal stiffness values over the WCW of Case 1

5

Conclusion

This paper presented the stiffness analysis of planar 2-DOF CDPRs with pulley differentials where a single actuator simultaneously controls the lengths of three cable segments going from the ground to the point-mass end-effector. Based on the usual linear spring cable elongation model, the expression of the stiffness matrix has been derived. The stiffness and workspace of several examples of planar 2-DOF 3-differential CDPRs have then been compared. These comparisons show that the stiffness of such planar CDPRs can be significantly improved compared to more common CDPRs without pulley differentials, without decreasing the size of the CDPR workspace. Additionally, the presented example comparisons Case 1 2 3 4 5 6

WCW 37.6% 32.4% 32.4% 33.6% 32.4% 32.4%

WFW 29.5% 27.3% 24.7% 22.6% 27.3% 24.7%

Table 4. WCW and WFW coverage of the total area occupied by the CDPR examples. This total area is the one delimited by the circles shown in Fig. 3 and 4.

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also showed that a trade-off may exist between designing a CDPR with pulley differentials to improve stiffness and designing one to increase the workspace size.

References 1. H. Khakpour, L. Birglen, and S. Tahan, “Synthesis of differentially driven planar cable parallel manipulators,” IEEE Transactions on Robotics, vol. 30, no. 3, pp. 619–630, June 2014. 2. H. Khakpour and L. Birglen, “Workspace augmentation of spatial 3-dof cable parallel robots using differential actuation,” in 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, Sept 2014, pp. 3880–3885. 3. H. Khakpour, L. Birglen, and S.-A. Tahan, “Analysis and optimization of a new differentially driven cable parallel robot,” Journal of Mechanisms and Robotics, vol. 7, no. 3, pp. 034 503–034 503–6, Aug 2015. [Online]. Available: http://dx.doi.org/10.1115/1.4028931 4. S. Behzadipour and A. Khajepour, “Stiffness of cable-based parallel manipulators with application to stability analysis,” ASME J. Mech. Des., vol. 128, pp. 303–310, jan 2006. 5. X. Diao and O. Ma, “Vibration analysis of cable-driven parallel manipulators,” Multibody System Dynamics, vol. 21, pp. 347–360, 2009. 6. M. Arsenault, “Workspace and stiffness analysis of a three-degree-of-freedom spatial cable-suspended parallel mechanism while considering cable mass,” Mechanism and Machine Theory, vol. 66, pp. 1–13, 2013. 7. H.Yuan, E. Courteille, and D. Deblaise, “Static and dynamic stiffness analyses of cable-driven parallel robots with non-negligible cable mass and elasticity,” Mechanism and Machine Theory, vol. 85, pp. 64–81, 2015. 8. H.Yuan, E. Courteille, M. Gouttefarde, and P.-E. Herv´e, “Vibration analysis of cable-driven parallel robots based on the dynamic stiffness matrix method,” Journal of Sound and Vibration, vol. 394, pp. 527–544, 2017. 9. M. Anson, A. Alamdari, and V. Krovi, “Orientation workspace and stiffness optimization of cable-driven parallel manipulators with base mobility,” ASME Journal of Mechanisms and Robotics, vol. 9, 2017. 10. S. Abdolshah, D. Zanotto, G. Rosati, and S. K. Agrawal, “Optimizing stiffness and dexterity of planar adaptive cable-driven parallel robots,” ASME Journal of Mechanisms and Robotics, vol. 9, 2017. 11. H. Jamshidifar, A. Khajepour, B. Fidan, and M. Rushton, “Kinematicallyconstrained redundant cable-driven parallel robots: Modeling, redundancy analysis, and stiffness optimization,” IEEE/ASME Transactions on Mechatronics, vol. 22, no. 2, pp. 921–930, April 2017. 12. C. Nelson, “On improving stiffness of cable robots,” in Cable-Driven Parallel Robots - Proceedings of the 3rd International Conference on Cable-Driven Parallel Robots, ser. Mechanisms and Machine Science, vol. 53. Netherlands: Springer Netherlands, 2018, pp. 331–339. 13. M. Gouttefarde and C. M. Gosselin, “Analysis of the wrench-closure workspace of planar parallel cable-driven mechanisms,” IEEE Transactions on Robotics, vol. 22, no. 3, pp. 434–445, June 2006. 14. P. Bosscher, A. T. Riechel, and I. Ebert-Uphoff, “Wrench-feasible workspace generation for cable-driven robots,” IEEE Transactions on Robotics, vol. 22, no. 5, pp. 890–902, Oct 2006.

Stability Analysis of Pose-Based Visual Servoing Control of Cable-Driven Parallel Robots Zane Zake12 , St´ephane Caro13 , Adolfo Suarez Roos2 , Fran¸cois Chaumette4 , and Nicol`o Pedemonte2 1

Laboratoire des Sciences du Num´erique de Nantes, UMR CNRS 6004, 1, rue de la No¨e, 44321 Nantes, France, [email protected], 2 IRT Jules Verne, Chemin du Chaffault, 44340, Bouguenais, France [email protected], 3 Centre National de la Recherche Scientifique (CNRS), 1, rue de la No¨e, 44321 Nantes, France, [email protected] 4 Inria, Univ Rennes, CNRS, IRISA, Rennes, France, [email protected],

Abstract. Cable-driven parallel robots are robots with cables instead of rigid links. The use of cables introduces advantages such as high payload to weight ratio, large workspaces, high velocity capacity. Cables also bring drawbacks such as bad accuracy when the robot model is not accurate. In this paper, a visual servoing control is proposed in order to achieve high accuracy no matter the robot model precision. The stability of the solution is analyzed to determine the tolerable perturbation limits. Experimental validation is performed both in simulation and on a real robot to highlight the differences. Keywords: cable-driven parallel robots, visual servoing, stability

1

Introduction

In cable-driven parallel robots (CDPRs) rigid links are substituted by flexible cables. This substitution leads to advantages such as: (i) large workspace (WS), (ii) reconfigurability [1], and (iii) high payload to weight ratio. However, the accuracy of CDPRs is to be improved. Previously, the following methods to improve accuracy have been studied: (i) increasing the complexity of the CDPR model in model-based control [2]; using proprioceptive sensors, such as (ii) force sensors to measure cable tensions [3] or (iii) angular position sensors to measure cable angles [4]; using exteroceptive sensors, namely vision sensors [5] [6] [7]. The increasing popularity of vision-based control is due to its high robustness to unexpected change in the environment. The two main approaches of such control are: image-based visual servoing (IBVS) and pose-based visual servoing (PBVS) [8]. The latter is used when information from the image and some additional knowledge about the object (usually its model) allows us to estimate © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_7

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the 3D pose of the object in the camera frame Fc . The control consists of minimizing the difference between the acquired object pose and the desired one at each iteration. IBVS is used when estimation of the 3D pose is not possible. Here, visual features such as 2D image coordinates or image moments are used instead. The control then consists of minimizing the error in the image space by comparing the desired and the current visual features. Vision-based control on CDPRs is not yet well researched. It is surprising given the challenge to achieve accurate model-based control, since it requires to predict complex aspects like cable elongation and sagging, and pulley effect. Furthermore, to find the solutions to the Forward Kinematic problem for CDPRs is a tedious task that requires a good knowledge of actual cable lengths and tensions, acquired by proprioceptive sensors. Although the addition of proprioceptive sensors can increase the robot accuracy, vision-based control is inherently robust to modeling errors and uncertainties, and avoids the need of computing the Forward Kinematic problem. This is attained by actively perceiving either the MP with a static camera in the so-called eye-to-hand configuration [5] [6], or the object of interest with a moving camera mounted on MP in the eye-in-hand configuration [7]. The robot is then actuated according to what is perceived. Dallej et al. used multiple static cameras facing the MP as well as observing cable exit points to determine their sag [5]. Multiple control schemes were proposed. Similarly, Chellal et al. used 6 infra-red sensors to determine the MP pose with high accuracy [6]. Remy et al. used the eye-in-hand configuration with a single camera [7]. Their task was to control a spatial CDPR with three translational degrees of freedom. Finally, a control scheme was proposed in [9] to control all six degrees of freedom while still using only one camera mounted on the MP. One of the main characteristics of any control scheme is its stability. It was found in [9] that the PBVS control of a CDPR is highly robust to modeling errors. However due to the wide range of such errors and their combined effect on the system it was not possible to determine their maximum values. In this paper, possible sources of errors are studied separately to find their maximum values for different MP working range. Such a separated analysis does not allow us to evaluate the interaction between different errors. For this reason, a second evaluation is done, assuming a constant non-negligible perturbation. These values are experimentally validated both in simulation and on a CDPR prototype. This paper is organized as follows. Section 2 presents the control scheme. Stability condition is established in Section 3. A case study is shown in Section 4. Finally, the conclusions are drawn in Section 5.

2 2.1

Vision-Based Control of a CDPR CDPR Kinematics

The schematic of a spatial CDPR in a suspended configuration is shown in Fig. 1. Given that the camera is mounted on the MP, the transformation matrix p Tc between the camera frame Fc and the MP frame Fp is constant. On the contrary, the transformation matrices b Tp and c To change with time.

Stability Analysis of Pose-Based Visual Servoing Control…

cble

i li

Bi

75

moving pltform

p

bi

Op

ai

p b

b

Ob

bse cmer

c

c

o

p

c Oc

o

object

Oo

Fig. 1. Schematic of a spatial CDPR with eight cables, a camera mounted on its MP and an object in the WS.

# » The length li of the ith cable is the 2-norm of the vector Ai Bi pointing from cable exit point Ai to cable anchor point Bi , namely,  # »   (1) li = Ai Bi  2

with

# » li b ui = b Ai Bi = b bi − b ai = bRp p bi + b tp − b ai # » where b ui is the unit vector of b Ai Bi that is expressed as: b

» b# b b Ai Bi b − ba R p b − b a + b tp   i# »i = p  i # »i ui =  # » =  b  b  b Ai Bi   Ai Bi   Ai Bi  2

2

(2)

(3)

2

The cable velocities l˙i are obtained upon differentiation of Eq. (2) with respect to (w.r.t.) time: (4) l˙ = A b vp where b vp is the Cartesian velocity of the MP expressed in Fb , l˙ is the cable velocity vector, and A is the Forward Jacobian matrix of the CDPR, defined as [10]: ⎤ uT1 (bRp p b1 × b u1 )T ⎥ ⎢ .. A = ⎣ ... ⎦ . p b T b b T u m ( Rp b m × u m ) ⎡b

2.2

(5)

Pose-Based Visual Servoing

The control scheme proposed in this paper is shown in Fig. 2. An image is retrieved from the camera and processed with a computer vision algorithm, that returns the current pose of the object s. It is compared to a previously

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known desired pose s∗5 and an error vector e is defined as e = [eTt eTω ]T , where ∗ et = c to − c to∗ = [ex ey ez ]T and eω = uθ, u being the axis and θ the angle of the rotation matrix cRc∗ . s

e +

 
 L s

c

b

vc

d 

s

vp

 

l

Computer Vision algorithm

1=r!

qm

CDPR Camera

image

Fig. 2. Control scheme for pose-based visual servoing of a CDPR

To decrease the error e, an exponential decoupled form is selected e˙ = −λe with a positive adaptive gain λ, that is computed at each iteration, depending on the current ||e||2 [7]. The relationship between e˙ and the Cartesian velocity of the camera c vc , expressed in Fc , is defined as: e˙ = Ls c vc

(6)

where Ls is the interaction matrix and it is defined in [8]: Finally, the instantaneous velocity of the camera in its own frame is expressed as a function of the pose error as follows: c

−1 e vc = −λ L s

(7)

s . Note −1 is the inverse of the estimation of the interaction matrix L where L s that the inverse is directly used, because for PBVS Ls is a (6 × 6)–matrix that is of full rank [8]. 2.3

Kinematics and Vision

To combine the modeling shown in Sects. 2.1 and 2.2, the MP twist b vp is expressed as a function of camera velocity c vc : b

vp = Ad c vc

where Ad is the adjoint matrix that takes the following form [11]:

b  b b R c tc × R c Ad = b 03 Rc

(8)

(9)

where b tc = bRp p tc and p tc is the vector pointing from Op to Oc . 5

In this paper, a superscript ∗ denotes the desired value, e.g. desired object pose s∗ ∗ and c∗ in c to∗ refers to desired camera frame Fc∗ in which the object is in the desired pose s∗

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Stability Condition

One of the main characteristics of any system is its stability. It is a measure to assess the effects of estimation quality. That is, how coarse can the estimation be for the system to still converge to its goal [12]. Lyapunov analysis is used to determine the stability of the closed-loop visual servoing system. From Eqs. (4), (6) and (8) the model is the following: † ˙ e˙ = Ls A−1 d A l

(10)

Upon injecting (8) and (7) into (4), the output of the control scheme, that is, the cable velocity vector takes the form: A d L −1 e l˙ = −λ A s

(11)

and A d are the estimations of A and Ad , respectively. where A The following closed-loop equation is obtained from (10) and (11): † −1 e˙ = −λ Ls A−1 d A A Ad Ls e

(12)

From (12), the system stability criterion is defined as: † −1 Π = Ls A−1 d A A Ad Ls > 0, ∀t

(13)

Π > 0 is a sufficient condition to obtain global asymptotic stability (GAS). It can be seen from the closed-loop equation (12) that if Π is positive definite, then the control scheme will ensure an exponential convergence of the error e to 0. However, if it is negative then the error e will increase and the system may diverge from the goal. Indeed, (13) is only a sufficient condition, therefore the stability of the system is uncertain once the condition is not held. 3.1

Estimated Parameters

Given the closed loop equation (12), the following variables are estimated and can therefore affect the system stability: – sˆ – object pose in Fc is computed from image features, so it will not be the exact pose s – p T c – the pose of the camera in the MP frame Fp . We have an idealistic model, however due to manufacturing imprecisions, the actual camera pose will be a little bit different from the modeled pose. i – the Cartesian coordinates of cable anchor points to the MP, expressed – pB in Fp . Due to mechanical solution of the anchor points, the actual point is a point located on a sphere around the nominal point. i – the Cartesian coordinates of cable exit points, expressed in the base – bA frame Fb . We are using the simplified CDPR model, that does not include pulleys actually located at the exit points.

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pulley base cable moving platform camera

prilTags

(a)

(b)

Fig. 3. CDPR prototype: (a) ACROBOT; (b) V-REP model of ACROBOT

– b T p , b Tp is estimated by exponential mapping: (b Tp )t+Δt = (b Tc )t+Δt c Tp = (b Tc )t exp(

c

vc ,Δt) c

Tp

(14)

Since velocity c vc , necessary for exponential mapping, is computed from the object pose measurement sˆ, which we admit to be different from s, then b Tp = b Tp . Furthermore, initial b Tp is only coarsely known6 .

4

Case Study

The stability condition (13) is applicable to any CDPR with a PBVS control. However, this analysis includes model-related parameters, thus it should be done for each CPDR separately. Hence, in the following sections the chosen CDPR is presented and results of stability analysis are shown. 4.1

ACROBOT and Simulation in V-REP

ACROBOT For this paper, a CDPR prototype, named ACROBOT and shown in Fig. 3(a), is used. Its WS is a 1 m3 cube. The robot is assembled in a suspended configuration. A simple webcam AUTOPIX MT4018 is mounted on the MP in the eye-in-hand configuration facing the ground. To simplify the computer vision part, AprilTags [14] are used instead of real objects. They are especially convenient to use in combination with ViSP library [15], because the latter contains functions that allow to recognize the tags and retrieve their 3D pose. 6

For this CDPR, the initial pose was defined at the center of the WS, which itself was measured by hand with a measurement error of ±2 cm along X and Y axes, resp.

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Fig. 4. The representation of a cable and its pulley in V-REP

V-REP Model The V-REP simulation environment [16] was used in order to create a dynamic model of ACROBOT including the vision sensor. This gives us the capacity to use the same software to control the real and the virtual hardware. As a result we can speed up the development time, test and debug our algorithms in simulation (with a perfectly known ground truth), then use the real robot for final verifications. To create a dynamical simulation, the pulleys and cables are modeled as a sequence of joints and mass objects, as shown in Fig 4. The current model does not take into account the pulley diameter and the cable sag. The pulleys are represented as a vertical revolute passive joint followed by a small spherical mass and a horizontal revolute passive joint. The cables are modeled as a sequence of prismatic joint, cylindrical mass, prismatic joint, cylindrical mass and a final spherical joint attaches the cable to the MP. The first prismatic joint is used to change the cable length. The second prismatic joint is responsible for the cable behavior through a specific joint control callback script, which models the cable forces as either an elastic spring, when in tension, or an element transmitting zero force, when in compression. To have a stable simulation some model design rules need to be considered [17]. In our tests, the Vortex physical engine was used. 4.2

Numerical Analysis

It is not possible to express analytically the pseudo-inverse of the Jacobian A† , thus stability analysis is only possible in numerical form. In this paper, we will only study two parameters, namely p Tc and bAi . The Cartesian coordinates of bAi are known, we need to find out the range of perturbation that does not destabilize the system. In Fig. 5(a) the perturbation range is defined by the radius rAi of the sphere centered at point Ai . We want to find the maximum value rAi,max so that for any point within a sphere of radius rAi ≤ rAi,max the system is stable. Unlike cable exit points, camera pose in Fp could in fact be changed, if necessary. Therefore, here we first define the range, where the camera could

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i ri

Op

Rpc rpc

(a)

(b)

Fig. 5. Perturbation ranges for: (a) cable exit points Ai ; (b) camera pose in Fp

be positioned, which is defined in Fig. 5(b) by radius Rpc of the largest sphere around origin Op of Fp . Then for any camera position, the perturbation range is defined by the small sphere radius rpc . Our goal is to find rpc,max so that if rpc ≤ rpc,max and camera position is within the sphere defined by radius Rpc , then the system will be stable. Finally, we transform pRc into axis-angle representation θpc u. We define the perturbation as δθpc , that can be made about any unit vector uδ . Therefore, our goal is to find δθpc,max so that |δθpc | ≤ δθpc,max for any uδ , u and θpc . To find the range of perturbation, we define the system as stated in Table 1. Then we portray two distinct cases: (i) The system is assumed to be ideal, no perturbations other than the one we are studying; (ii) the system has small perturbations in all parameters. The perturbation values are chosen either based on the mechanical errors (such as pBi and bAi ) or as 5 to 10 percent of the actual parameter value. For the latter, we also distinguish the results depending on the desired range of motion of the MP in Fb expressed as Rbp . The perturbation limits that we have established based on condition (13) are shown in Table 2. Here are the observations: – None of the parameters are affected by the actual value of p tc or pRc . This is especially surprising for the perturbations on these parameters defined by rpc and δθpc . – To keep the full motion range Rbp and the full rotational range bRp , perturbations on bAi must not be larger than 0.01 m. – In the ideal case, the perturbation values for camera pose in Fp are large. In fact these perturbations do not appear separately. Indeed, all the considered perturbations are affecting the system at the same time, not independently. For this reason it was important to do the second part of this study while considering perturbations in all parameters. – When the perturbations, defined in Table 1, are added to the system, the latter remains stable for all tested motion range of MP. – However, once the MP motion range is reduced, it is possible to increase the perturbation levels without making the system unstable. – As soon as Rbp is reduced, rpc increases significantly, as well as δθpc . This allows to conclude that perturbations in the respective variables have little effect on the stability of the system.

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– There is a considerable increase in rAi as well. This means that the modeling accuracy of exit points is not important. Even at WS borders the tolerated perturbation easily covers the corresponding model inaccuracies. That is, even though pulleys that are present on the cable exit points are not modeled, the small changes in the actual exit point location on the pulley are covered by the perturbation rAi = 0.01 m and do not affect the system stability. Table 1. Variable and perturbation ranges variable family

variable value

perturbation value

Rst = 0.5 m

rst = 0.05 m

object pose in Fc

|θω | ≤ 50

◦

|δθω | ≤ 2.5◦

camera pose in Fp

  −0.032 −0.026 −0.01 180◦ 0◦ 180◦

cable anchor points pBi cable exit points bAi

In eight corners of MP of size 0.1 × 0.1 × 0.05 m Two exit points at each top corner of ACROBOT

MP pose in Fb

rpc = 0.01 m δθpc = 3◦ rBi = 0.008 m rAi = 0.01 m

Rbp is equal to 0.5 m, 0.3 m and 0.1 m, resp. rbp = 0.02 m Rotation about global axes: 45◦ about Z, 5◦ about Z axis, 3◦ 20◦ about Y, 20◦ about X about Y and X axes

Table 2. Perturbation change depending on MP motion range Condition

Camera pose in Fp

Ideal robot, no other perturbarpc = 0.5 m tion in the system, Rbp = 0.5 m Minimal perturbations from rpc = 0.03 m Table 1, Rbp = 0.5 m Minimal perturbations from rpc = 0.5 m Table 1, Rbp = 0.3 m Minimal perturbations from rpc = 0.8 m Table 1, Rbp = 0.1 m

4.3

Cable exit points bAi

|δθpc | ≤ 55◦

rAi = 0.01 m

|δθpc | ≤ 3◦

rAi = 0.01 m

|δθpc | ≤ 16◦

rAi = 0.09 m

|δθpc | ≤ 24◦

rAi = 0.18 m

Experimental Validation

Some experiments have been conducted to validate the theoretical results 7 . First, we assume that the rotational error of the camera in Fp and about Z axis is out 7

Please also see the accompanying video at https://youtu.be/tfiTDlp1ZIY

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0.6

400

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Fig. 6. CDPR behavior depending on added perturbations. (a) and (b): the AprilTag trajectory and error e over time in V-REP simulation with δθ = 85◦ ; system is not stable (only center-point trajectory shown). (c) and (d): the AprilTag trajectory and error e over time in V-REP with δθ = 55◦ ; system is stable; (e) and (f ): the AprilTag trajectory and error e over time on ACROBOT with δθ = 55◦ ; robot does not converge because AprilTag leaves the camera field of view. (g) and (h): the AprilTag trajectory and error e over time on ACROBOT with δθ = 16◦ ; system is stable.

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of bounds of stability, i.e. δθpc = 85◦ . As shown in Fig. 6(a) the robot diverges from its goal position and error oscillates and slowly increases (Fig. 6(b)), the system is not stable. If δθpc = 55◦ , the V-REP model will successfully reach the target as shown in Fig. 6(c), though the trajectory is far from optimal (a straight line). Some error oscillations can be observed in Fig. 6(d). The same rotational error on the actual robot is shown in Fig. 6(e). The initial behavior is similar, but AprilTag leaves the image and the task is failed. This is not surprising, given the additional uncertainties of the actual robot and the lower image quality. Finally, δθpc was set to its maximum for a noisy system with range of motion reduced to Rbp = 0.3 m, i.e. δθpc = 16◦ . Once implemented on ACROBOT, we see that though the trajectory is perturbed, the MP reaches its targeted pose (Fig. 6(g)). Indeed, error e converges to zero without oscillations (Fig. 6(h)). To summarize, for an ideal robot the range of perturbation on a single parameter is very large. On the contrary, as soon as we acknowledge that all the system parameters are noisy, then each individual perturbation has quite a limited range within the bounds of stability.

5

Conclusions

This paper proposed a method to analyze the stability of Pose-Based Visual Servoing (PBVS) control of Cable-Driven Parallel Robots (CDPRs). A general stability criterion was introduced. The stability of ACROBOT, a CDPR prototype located at IRT Jules Verne, was analyzed. Two CDPR model-related parameters were studied and their maximum perturbation range was found both for ideal robot and for a noisy one. A dynamic CDPR model in V-REP was presented. It was found that for an ideal system any one parameter could be highly perturbed without making the system unstable. This result was successfully validated in simulation. When adding a large perturbation (validated in simulation) to ACROBOT, the system does not converge. However, if the perturbation is kept within the corresponding (noisy system) range, the robot will be able to complete its task. For ACROBOT, the tolerated perturbation on cable exit points is rather large. This is beneficial, because it allows to avoid adding pulley kinematics to the model. Indeed, cable exit point variations on the pulley are smaller than the tolerated perturbation, which does not affect the stability of the system. The added perturbations affect the trajectory to the goal and the system’s ability to actually reach it. However, perturbations in the robot model do not affect the final accuracy. This is because the final pose is considered to be reached only when the error between the current and the desired object pose becomes smaller than the threshold defined in the visual servoing loop. Future work will deal with the implementation of PBVS control on a large semi-industrial CDPR prototype as well as its stability analysis. It is also of interest to consider real objects instead of AprilTags to validate this approach for more realistic use cases.

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Acknowledgment This work is supported by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing) in the framework of the PERFORM project.

References 1. L. Gagliardini, S. Caro, M. Gouttefarde, and A. Girin, “Discrete Reconfiguration Planning for Cable-Driven Parallel Robots”, in Mechanism and Machine Theory, vol. 100, pp. 313–337, 2016. 2. V. L. Schmidt, “Modeling Techniques and Reliable Real-Time Implementation of Kinematics for Cable-Driven Parallel Robots using Polymer Fiber Cables”, Ph.D. dissertation, Fraunhofer Verlag, Stuttgart, Germany, 2017. 3. E. Picard, S. Caro, F. Claveau, and F. Plestan, “Pulleys and Force Sensors Influence on Payload Estimation of Cable-Driven Parallel Robots”, in IROS, Madrid, Spain, October, 1–5 2018. 4. A. Fortin-Cˆ ot´e, P. Cardou, A. Campeau-Lecours, “Improving Cable-Driven Parallel Robot Accuracy Through Angular Position Sensors”, in IROS, pp. 4350–4355, 2016. 5. T. Dallej, M. Gouttefarde, N. Andreff, R.Dahmouche, and P. Martinet, “VisionBased Modeling and Control of Large-Dimension Cable-Driven Parallel Robots”, in IROS, pp. 1581–1586, 2012. 6. R. Chellal, L. Cuvillon, and E. Laroche, “A Kinematic Vision-Based Position Control of a 6-DoF Cable-Driven Parallel Robot, in Cable-Driven Parallel Robots, pp. 213–225, Springer, Cham, 2015. 7. R. Ramadour, F. Chaumette, and J.-P. Merlet, “Grasping Objects With a CableDriven Parallel Robot Designed for Transfer Operation by Visual Servoing”, in ICRA, pp. 4463–4468, IEEE, 2014. 8. F. Chaumette, S. Hutchinson, “Visual servo Control I. Basic Approaches”, in IEEE Robotcs & Automation Magazine, vol. 13, no. 4, pp. 82–90, 2006. 9. Z. Zake, F. Chaumette, N. Pedemonte, S. Caro, “Vision-based Control and Stability analysis of a Cable-Driven Parallel Robot”, submitted to ICRA 2019. 10. A. Pott, “Cable-Driven Parallel Robots: Theory and Application”, vol. 120., Springer, 2018, pp. 52–56. 11. W. Khalil, E. Dombre, “Modeling, Identification and Control of Robots”, Butterworth-Heinemann, 2004, pp. 13–29. 12. H. K. Khalil, Nonlinear systems 2nd ed., Macmillan publishing Co., New York 1996. 13. C.R. Johnson, “Positive definite matrices”, in The American Mathematical Monthly, vol. 77, no. 3, pp. 259–264, 1970. 14. E. Olson, “AprilTag: A robust and flexible visual fiducial system”, in ICRA, pp. 3400–3407, IEEE, 2011. ´ Marchand, F. Spindler, F. Chaumette, “ViSP for visual servoing: a generic 15. E. software platform with a wide class of robot control skills”, in IEEE Robotics & Automation Magazine, vol. 12, no. 4, pp. 40–52. 16. E. Rohmer, S. P. Singh, M. Freese, “V-REP: A versatile and scalable robot simulation framework”, in IROS, pp. 1321–1326, 2013. 17. V-REP, Designing Dynamical Simulations. http://www.coppeliarobotics.com/ helpFiles/en/designingDynamicSimulations.htm

Practical Stability of Under-Constrained CableSuspended Parallel Robots Dragoljub Surdilovic and Jelena Radojicic Fraunhofer Institute for Production Systems and Design Technology IPK-Berlin, Department Robotics and Automation, Pascalstr. 8-9, 10587 Berlin, Germany, E-mail: [email protected]

Abstract This paper, motivated by the development of a novel gait rehabilitation system, presents a mechanical approach for the dynamic modelling and analysis of equilibrium stability of under-constrained cable suspended parallel robots. These types of cable robots exhibit interesting characteristics of self-motion in the Jacobian null-space. Modelling and understanding of this motion is essential for their applications. It is demonstrated that both a wrench consistency test and proof of stability conditions, derived for real robots with a pulley mechanism, play a crucial role for practical equilibrium stability assessment. Thereby dynamic simulation of the null-space motion help to analyse robustness of the equilibrium against perturbations. Several examples with a 4-4 type robots illustrate the theoretical analysis. Key Words: cable-driven parallel robots, under-constrained cable suspended structures, equilibrium stability analysis, gait rehabilitation

1.

Introduction

A specific class of cable-driven parallel robots (CDPR) with all cables located above the common platform, referred to as force-supported or cable-suspended parallel robots (CSPR), has been recently addressed in numerous researches [1-2] focusing on their advantages for implementing simple, large span, fast moving, lightweight and heavy-duty active overhead mechanisms. A major problem with cable-suspended structures is ensuring tension forces for any admissible motion, as well as realizing a stable static equilibrium (rest poses) and a desired rest to rest motion. It is well recognized that gravitation force of a cable platform and payload plays an essential role in stabilizing CSPR, acting as an additional virtual wire that ensures cables tension. In contacts with environment, however, the external process/contact forces become also relevant and may jeopardize stability and cause undesired, even uncontrollable motion. The contact applications of CSPR have been not intensively investigated in the past, despite some attractive presentations such as grinding of large objects with NIST RoboCrane [3], or transportation and assembly tasks performed by cooperative aerial towing [4-5] (CSPR with cables attached to aerial robots). This paper deals with structurally under-constrained CSPRs, where the platform is suspended with less than 6 cables which are often applied in practice [1, 2]. It is worth mentioning that CSPR with 6 or more cables © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_8

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can also become under-constrained, when the Cartesian wrench-space spanned by taut cables become less than 6 [2]. The stability questions concerning mechanical equilibrium of underconstrained CSPRs have been well investigated in various systems using numerous approaches, such as analysis of net forces and torques, momentum, virtual displacements and velocities, or potential energy gradient. The stability of a pose/configuration have been mainly investigated in the context of solving direct and inverse kinematic problems. In order to determine a feasible configuration, without cable slack or overloaded cables (aerial robots), the kinematic analysis, typically for general CDPR, cannot be considered independently of the system dynamic, i.e. quasi-static analysis. Consequently these problems have been referred to as geometric(o)-static analysis [6]. In general case with fixed cable length, relative motion of an under-constrained CSPR platform (with less than 6 taut cables) around a possible stable equilibrium remains still possible, under action of external perturbation forces. Thereby the robot can switch to other equilibrium in an unpredictable way. This motion has been referred to as self-motion in singular configuration of parallel robots [7], however it is also characteristic for underconstrained CSPRs in which all Cartesian DOFs cannot be controlled. A typical example is so called 2-2 CSPR configuration, in which by analogy with the coupler of an overhead four-bar planar mechanism [4, 6], the platform can move in the working plane with fixed cables length. Moreover, the “cranks” of this CSPR are not rigid, the cables may become slack or the platform can move outside the plane, which is likely to occur in practice. Several researches [6, 8-9] have tried to analyse these effects in underconstrained CSPR by solving the direct kinematic problem for different standard structures with 2 to 5 cables, computing all solutions for given parameter sets and analysing their stability for a given payload. The background for stability analysis of under-constrained CSPRs has been established by Carricato and Merlet [6] based on static forces virtual works around an equilibrium (i.e. pose candidate). In [4] an energetic approach has been applied considering that an equilibrium pose corresponds to the minimum in the potential energy. The Hessian of payload potential energy expressed in terms of independent angular rotations around equilibrium has been used to derive complex analytical expressions and analyse stability (eigenvalues of Hessian matrix). Collard and Cardou [10] have pursued a similar approach trying to compute lowest equilibrium pose of CSPR by minimizing potential energy subject to complex algebraic constraints. In previous work the wire-robot motion effects occurring in commonly applied pulley elements (e.g. due to pulley rolling and wire coil) have been neglected [12]. In this paper we will derive dynamic equilibrium models of a general underactuated CSPR system, taking also into account the pulley motion effects, and apply them for stability assessment and analysis of robustness against perturbations. A novel algorithm has been proposed for efficient numerical finding of potential equilibrium configurations and assessing their stability. The stability of underconstrained CSPR has been considered in the domain of practical stability (notion introduced by LaSalle and Lefschetz [13]) of dynamical systems, providing a

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practical framework for assessing stability and robustness of an equilibrium against perturbations. This work is motivated by a further development of CSPR technology for gait rehabilitation by simplification an over-constrained system STRINGMAN [14] to a CSPR structure with cable implemented by pneumatic artificial muscles (Fig. 1), which provide an intrinsic safety to patient. Analysis and understanding of possible system motion and equilibrium stability require complete dynamic models, including human dynamics (MATMAN model developed in Matlab with 40 DOFs using bio-mechanical human data), that will be in the paper simplified by considering only possible trunk motion.

Fig. 1 4-4 CSPR with pneumatic artificial muscles as wires

Fig. 2 MATMAN model (left) and simplified configuration used in the paper

2.

Kinematic Analysis

A model of under-constrained CSPR with n-wires ( n  6 ) is given in (Fig. 3). The position of i-th wire platform attachment point B i is defined by

pi

a i  Li

p  bi

(1)

where constant vector a i defines “attachment” Ai of pulley (the bearing block of the pulley is mounted around fixed cable axis e i , in front of the pulley), b i is platform attachment Bi position vector wrt local platform frame, p is the position vector of the platform reference frame and Li Ai Ci  CiTi  l i , where l i is the effective wire-length vector, while Ci and Ti denote centre of the pulley and actual wire tangent points (instant center of rotation of cable l i , i.e. point Bi) (Fig. 3).

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Fig. 3 Cable-suspended robot structure In general the wire performs a complex composite motion that can be decomposed into transferred motion, representing rotation (rolling motion of the pulley) of the entire wire plane {AiBiCiTi} around the fixed pulley axis ei, and relative motion in the wire plane (Fig. 4). The relative motion consists of relative translation i.e. change of the relative length in the actual cable direction (due to cable control, i.e. via a winch that are not presented in Fig. 3-4), and a relative rotation of the wire around the pulley (i.e. instantaneous center of rotation Ti that changes its position during winding). Differentiating (1) twice with respect to time the expressions, absolute velocities and accelerations of wire end-point (Bi) are obtained

v i p i ω e i u Li  ω r i u l i  l i v p  ω p u b i (2) ai

 i p

ε e i u Li  ε r i u l i  ω e i u (ω e i u L i )  ω e i u (ω r i u l i )  ω r i u (ω r i u l i ) 



 2(ω e i  ω r i ) u l i  l i a p  ε p u b i  ω p u (ω p u b i )

where ω e i and ε e i

(3) denote pulley rotation velocity and acceleration around e i

(Fig. 4), ω r i and ε r i relative wire rotation velocity and acceleration around wire

plane normal n i , l i and l i are linear wire relative velocity and acceleration due to cable length changes, v p and ω p , a p and ε p are platform linear and angular velocities and acceleration vectors respectively. Relative velocity components and their directions are shown in (Fig. 4). The projections of the velocity and acceleration vectors (3, 4) into wire-length vector direction, defined by unit vector l i 0 l i li , i.e. scalar multiplication of these equations by l i 0 yields the magnitudes of wire linear relative velocity and acceleration respectively

li

[l i 0

T

 l i0 bi ]t p T

(4)

Practical Stability of Under-Constrained Cable-Suspended Parallel Robots

&

&



&

&

&

&

Ze 2 [( ei u L i ) ˜ n i ][( ei u n i ) ˜ li 0 ]  Zr 2 li  [l i 0 T

li

i

i

x p

where t p

[v p

T

T T  l i 0 b i ] t p  l i 0 ω p b i ω p

89

(5)

ω p ]T is the platform twist vector, where xp denotes platT

form pose (both position and orientation), while b i indicates the skew-symmetric 3x3 matrix formed from the elements of the vector b i in order to represent the vector product in the matrix form. The first two components of the right-hand side of (5) represent the projections of centrifugal accelerations components (corresponding to the pulley and relative wire rotations), while the remaining parts define projections of platform tangential and centrifugal accelerations into the wire directions. The expressions for vectors of angular pulley and wire relative rotations in terms of platform twist vector, are obtained by scalar multiplication of (2, 3) by vectors n l n i u l i 0 and n i respectively i

Ze

i

1 & & & (ei u L i ) ˜ n i

ªn T «¬ i

n i b i ºt p ; Zr »¼ T

i

1 T [l i 0 n i li

T

l i 0 n i b i ]t p

(6)

Fig. 4 Velocity vectors components (in wire-pulley plane) The relationship between relative wire velocity, defining the cable length variations, and platform twist vector is defined by the wire-robot Jacobian J  ƒnx 6

l

ª l10  l i 0  l n 0 º «b l » «¬ 1 10  b i l i 0  b n l n 0 »¼

JT

Jt p ;



(7)

where l [l 1 l i l n ]T . The time derivative of CSPR Jacobian is obtained by differentiating (7)

l

Jt p  J t p ;

J T

l l l º ª   i0 n0 10 « l  l  b l  l  b l » b l b b b    i i0 i i0 n n0 n n0 » 1 10 «¬ 1 10 ¼

(8)

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where the time derivatives of the constant intensity vectors l i 0 and b i (considering an ideal rigid platform and cables) are

l i0

b i

ω ei u l i 0  ω ri u l i 0 ;

ω p u bi

(9)

and substituting (6) and (9) yields T T T T T J t p … J t p … [J1 Ji J n ]T

(10)

where J is a n(6)x1(6) block matrix (numbers outside parenthesis define block matrix dimension, while within parenthesis the dimension of each block-matrix element has been given), … is the Kronecker’s product (each block-element of J T

is multiplied by t p ), and the block element Ji  ƒ6 x 6 has the form Ji

1 1 ª T T & n i l i0 e i «  l n i l i0 l i0 n i  & & ( u e L i i i ) ˜ ni « 1 T « 1 b n l l T n  bnl e & « l i i i 0 i 0 i (e& u L ) ˜ n& i i i 0 i i i i ¬ i

1 1 º T T n i l i0 l i0 n i b i  & & & n i l i0 e i b i » (11) li (e i u L i ) ˜ n i » 1 1 T T » b i n i l i0 l i0 n i b i  & & & b i n i l i0 e i b i  b i l i0 » li (e i u L i ) ˜ n i ¼

3. Stability of equilibrium The equilibrium between internal cable tension forces, grouped in the wire tension vector f [f1 f i f n ]T with elemental forces acting along l i 0 (i=1,…,n), and Cartesian wrench w [FT MT ]T , involving gravity and external net static force/torque acting on platform frame, is defined by the balance of virtual works of these forces on virtual displacements compatible with the constraints around an equilibrium

GL

G l T f  G xT w

>

(12)

0

@

T T

where G x G pT G o , Gp and Go are possible relative translational and rotational virtual displacements of platforms around an equilibrium that are compatible with the constraints, G l >G li @ . G L represents variation of a Lagrange function, where the first term characterizes the work of internal cable forces (representing Lagrange constraints multipliers), and the second one is variation of potential energy (virtual work of gravitation forces and torques). Substituting G l JG x p in (12), yields the known Jacobian mapping between internal cable forces and corresponding external wrench, defining a static equilibrium condition

w

JT f

(13)

The second order variation of L is given by

G 2 L G 2 lT f  G 2 xTp w;

G 2l

JG 2 x p  G JG x p

Where the variation of Jacobian based on (10) may be written in the form

(14)

Practical Stability of Under-Constrained Cable-Suspended Parallel Robots

GJ

G xTp … J G x p T … [J1T Ji T Jn T ]T

91

(15)

Substituting (13) and (15) in (14) yields

G x

G 2L

T p



JT … G x p f  G 2 xTp J T f  G 2 xTp w

G x

T p



JT … G x p f

G xTp JT $ f G x p

(16) where the product “ $ ” defines the multiplication between block-vectors and matrices applying standard rules on block-elements, i.e. the multiplication of raw block-vector JT and f yields n

JT $ f

T ¦ Ji f i

(17)

i 1

and J i is given in (11) for a general CSPR with pulleys. Finally

G 2 L GxTp HG x p

(18)

with 6x6 Hessian matrix H

JT $ f

n

T ¦ Ji f i

(19)

i 1

Orthogonal projection of (18) into null-space of robot Jacobian N vides symmetric reduced Hessian [6]

HN

n T NT HN NT §¨ ¦ J i f i ·¸N ©i 1 ¹

null J pro-

(20)

A sufficient condition for a stable equilibrium of CSPR consists in H N being positive definite, i.e. having all positive eigenvalues Oi (i=1,…,6-n).

4.

Practical tasks and stability

The stability assessment based on (20) requires determination of equilibrium configurations for known cable-lengths li and static external wrench w (e.g. given payload w G in the free space), satisfying equilibrium condition (13). However, this is not a trivial task. This problem, referred to as direct geometrico-static problem DGP, has been intensively investigated in [6, 8-9] etc., specifically for 4-4 CSPR robots in [15]. The inverse geometrico-static problem IGP assumes known platform poses [6]. Thereby mostly analytic approaches based on exact arithmetic elimination procedure, by analogy with overhead rigid parallel robot structures, also using the same SW packages, have been applied to find distinct real solutions of high-order polynomials (with degrees 12, 156, 216 and 140, respectively, for two, three, four and five taut cables) obtained from algebraic constraints equations. However, finding all solutions of direct DGP is computationally expensive and difficult for real-time applications (though assessing the stability may be compatible with RT requirements). Recent improvements [9, 11] have focused on

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reduction of computational burdens.The problem in real CSPR with pulleys become considerably more complex and cannot be solved in closed form. In the following, alternative numeric approaches with additional tasks and stability assessment methods, based on dynamic model and above analysis, are proposed. As common in dynamic systems we can obtain a static equilibrium condition in a considered pose from the dynamic model by nullifying the motion terms. Assuming for the case of simplicity that the platform-reference frame and center of mass coincide P=C, the dynamic model of a CSPR may be written as Mt p  t Tp … C t p M

ªm p E 0 º « 0 I p »¼ ¬

w G  J T f C

­ Cv ½ ® ¾ ¯CZ ¿

(21) Cv

06 x 6

CZ

ª0 0 º «0 I » p¼ ¬

E and 0 denote 3x3 unity and zero matrices respectively, mp and Ip are platform mass and inertia, C is a 2(6)x1(6) block vector of centrifugal forces and torques, T

ª v T  ωT º is a 6x3 matrix composed of skew-symmetric velocities. p» «¬ p ¼ Nullifying twist vectors t p t p 0 in (25) yields and t p

wG

JT f

(22)

This condition (22) in an arbitrary pose x p and for a given static wrench w G (IGP), different from articulated robots, or fully- and over-constrained CDPRs, in general cannot be fulfilled. Therefore x p will be referred to as quasi-equilibrium. For the considered under-constrained CSPRs, the above mapping defines an overdetermined system of linear equations where J T  ƒ6 xn and n0 0  294.3 0 0 0@ is consistent to (13) and falls into column span of J T (22), providing the tension solution fi 95.98 (N) in each cable. The equilibrium is stable with eigenvalues of null-space Hessian H N λ >26.48 55.4@ . However, if the relative attachments b i are selected bellow the center of mass (Fig. 5 right), the equilibrium becomes obviously unstable λ > 31.5  5.3@ (the consistency is further ensured). Cable a b e r 1 [-0.65 -2.11 2.28] [0.1 -0.1 0.1] [0 -1 0] 0.015 2 [-1.8 -2.11 2.28] [-0.1 -0.1 0.1] [0 -1 0] 0.015 3 [-0.65 -0.29 2.28] [0.1 0.1 0.1] [0 1 0] 0.015 4 [-1.8 -0.29 2.28] [-0.1 0.1 0.1] [0 1 0] 0.015 Table 1: Geometric cables parameters of the robot with platform mass and inertia matrix m=30 (kg) and Ic=diag[1.25 2.67 1.42] (kgm2), respectively, r is the pulley radius (m)

In an asymmetric configuration (Fig. 6 left) wrench w G is inconsistent to the mapping (16), a solution for f doesn’t exist. In this configuration we can solve stabilization task by modifying w G into a virtual consistent wrench w con involving the same gravitation force components from w G and adding two additional torques, around x and y axes respectively, computed from (26). Basically an external torque w conx 10.73 (Nm) around lateral trunk x-axis is only needed to virtually stabilize the equilibrium.

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Fig. 5 Consistent stable (left) and unstable (right) configurations Without this stabilization torque, the current w G , i.e. its projection on the nullspace (mainly torque component around lateral axis x) causes rotation in the nullspace in the sagittal plane until a new stable and for w G consistent configuration has been reached (Fig. 6 right). This new really stable equilibrium configuration is determined by simulation using dynamic model (29), and has been defined by a relative rotation ox 0.195 (rad) around lateral x-axis wrt. the initial pose. A completely asymmetric real-equilibrium pose consistent with w G obtained also by simulation model (29) starting from vertical initial posture is presented in (Fig. 7). For the considered robot all consistent stable poses in a workspace of interest that can be reached by the trunk rotations and pelvis (body) inclination are presented in (Fig. 8). However, different to stable symmetric configuration (Fig. 6), the practical stabilities bounds ( G and Q) of these poses are relatively small, and the platform is prone to switch to other neighborhood equilibrium poses. The estimation of these practical stability measures is the topic of the current work.

6. Conclusion This paper has presented the detailed modeling and analysis of practical stability of general under-constrained CSPRs with pulley elements. The practical stability analysis includes both consistency tests of the applied wrenches (gravity and external platform forces and torques) and assessment of eigenvalues of Hessian projected on the null-space of robot Jacobian. For quasi-stable configurations a stabilization task has been proposed, which can find attractive applications in future CSPRs systems with additional drives embedded in the platform. Despite these conditions have been fulfilled, a dynamic motion-oscillations in the nullspace due to perturbation effects can destabilize robot and bring it in another steady-state stable pose. The practical stability framework has been proposed in conjunction with the null-space dynamic model to assess robustness of equilibrium against perturbations. For the gait rehabilitation robot under development this self-motion may be very useful to support patient efforts to stabilize trunk posture and to make training of balance more efficient based on “assist as needed, less as possible” para-

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digm. Thereby patient fall is prevented by fixed-controlled cable lengths. However in other under-constrained CSPRs the self-motion can be quite undesirable. An external damping is needed to dynamically stabilize the null-space motion using additional actuators, since the cable drives act in the Jacobian column space. The assessment of practical dynamic stability of an equilibrium in rehabilitation and transportation applications is a subject of the on-going work.

Fig. 6 Initial (inconsistent, left) and final stable (right) configurations

Fig. 7 Asymmetric gravity wrench consistent configuration

Fig. 8 Stable configurations in the trunk-pelvis workspace

7. References [1] B. Zi and S. Qian, Design, Analysis and Control of Cable-Suspended Parallel Robots and Its Application, Springer-Verlag, 2017.

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[2] J-P. Merlet, On the robustness of cable configurations of suspended cable-driven parallel robots, Proc. 14th IFTOMM World-Congress, Taipei, October 2015. [3] J. Albus, R. Bostelman and N. Dagalakis, The NIST Robocrane, Journal of Robotic Systems, Vol. 10, No. 5, 1993, pp.709-724. [4] N. Michael, S. Kim, J. Fink and V. Kumar, Kinematics and Statics of Cooperative Multi-Robot Aerial Manipulation with Cables, Proceedings of ASME 2009 IDETC/CIE Conference, San Diego, pp. 83-91. [5] J. Fink, N. Michael, S. Kim and V. Kumar, Planning and control for cooperative manipulation and transportation with aerial robots, The Int. Journal of Robotic Research, 30(3), 2015, pp.324-334. [6] M. Carricato and J-P. Merlet, Stability Analysis of Underconstrained Cable-Driven Parallel Robots, IEEE Transaction on Robotics, Vol. 29, No.1, February 2013, pp. 288-296. [7] Karger, A. and Husty, M.L. (1998). Classification of all self-motions of the original Stewart-Gough platform. Computer-Aided Design 30(3), 1998, pp. 205–215. [8] G. Abbasnejad and M. Carricato, Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots with n-Cables, IEEE Transaction on Robotics, Vol. 31, No. 2, April 2015, pp.468-478. [9] A. Berti, J-P. Merlet and M. Carricato, Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval-analysis, The Int. Journal of Robotic Research, Vol. 36, 2017, pp. 723-729. [10] J-F. Collard and P. Cordou, Computing the lowest Equilibrium Pose of a CableSuspended Rigid Body, Optim. Eng. 14, 2013, pp. 457–476. [11] G. Liwen, X. Huayang and L. Zhihua, Kinematic analysis of cable-driven parallel mechanisms based on minimum potential energy principle, Advances in Mechanical Engineering, Vol. 7(12), 2015, pp. 1-11. [12] A. Pott, Influence of pulley kinematics on cable-driven parallel robots, In Latest Advances in Computation Kinematics (Ed. J. Lenarcic), Springer, 2012, pp. 197-204. [13] J. LaSalle and S. Lefshetz, Stability by Lyapunov’s Direct Method, Academic Press, New York, 1961. [14] D. Surdilovic. R. Bernhardt, T. Schmidt and J. Zhang, STRING-MAN: A Novel Wire Robot for Gait Rehabilitation, in Advances in Rehabilitation Robotics, Lecture Notes in Control and Information Sciences, Vol. 306, 2004, pp. 413-424. [15] M. Carricato and G. Abasnejad, Direct Geometrico-Static Analysis of UnderConstrained Cable-Driven Parallel Robots with 4 Cables, Cable-Driven Parallel Robots (Eds. T. Bruckmann and A. Pott), Springer, 2013, pp. 269-285. [16] G. Stepan, A. Toth, L. Kovacs, G. Bolmsjo, G. Nikoleris, D. Surdilovic, A. Conrad, A. Gasteratos, , N. KyriakoulisR, J. Canou, T. Smith, W. Harwin, R. Loureiro, R. Lopez and R. Moreno, ACROBOTER: a ceiling based crawling, hoisting and swinging service robot platform, Proceedings of beyond gray droids: domestic robot design for the 21st century workshop at HCI 2009. [17] O.A. Bauchau and A. Laulusa, Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems, Journal of Computational and Nonlinear Dynamics, 3(1), 2008, pp. 1-8. [18] X-S. Yang, Practical stability in dynamical systems, Chaos, Solitons and Fractals, Pergamon, 11 (2000), pp. 1087-1092.

Singularity Characteristics of a Class of Spatial Redundantly actuated Cable-suspended Parallel Robots and Completely actuated ones Lewei Tang1, Xiaoyu Wu1, Xiaoqiang Tang2, and Li Wu3 1

College of Mechanical Vehicle Engineering, Hunan University, Changsha, 410082, China 2 Department of Mechanical Engineering, Tsinghua University, Beijing, 100084, China 3 Department of Project Engineering &Technology Center, GAC FIAT CHRYSLER Automobiles Co., Ltd., Changsha, 410100, China [email protected]

Abstract. This paper aims to study singularity characteristics of a class of spatial redundantly actuated Cable-Suspended Parallel Robots(CSPRs) and completely actuated ones with pairwise cables. This study focuses on the CSPRs with purely three translational degrees of freedom using redundant actuators or complete actuators. One class of CSPRs is able to perform the translational movement with pairwise cables as parallelograms. There are two types of singularity to be discussed, which result from dynamic equations of CSPRs and parallelogram pairwise cables configurations. To assure three-translational dofs without the rotation of the end-effector, the matrix formed by normals of pairwise cables should maintain in full rank. In the case study, one type of CSPRs with a planar end-effector is discussed to clarify and conclude the singularity features. The results show that in some configurations of CSPR there exists singularity of completely actuated CSPR but redundantly actuated ones are able to fulfill the three-translational-dof movement. Keywords: Cable-suspended Parallel Manipulator, Singularity analysis, Redundant Actuator.

1

Introduction

Cable mechanism has been used in a large range of engineering field due to its flexible feature and suitable strength with lightweight structure such as construction [1], transmission unit of anthropomorphic manipulator [2], leg exoskeleton [3], material manipulation [4] and astronomical observation [5]. Compared with rigid link parallel robots, the cable-driven parallel robot employs flexible cables as force transmission and position constraint parts to manipulate the end-effector in workspace. Since cables can be wound around winches, cable robot features large-scale workspace and ease of reconfiguration. As shown in Fig. 1, the workspace below line p2p3 is divided into four areas A1, A2, A3 and A4, where the area above line p2p3 is not available because it is not fully controllable. A1 is formed as a polygon with the pul-

© Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_9

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leys as vertices, which is referred to as static workspace or force-closure workspace [6]. A2 is the area limited by lines p1p2, p3p4 and p1p4, where only the gravity can exert force downward so that in this area the mechanism with this configuration is also named as cable-suspended parallel robot. The other areas as A3 and A4 is defined as the dynamical workspace with considering inertial effect from acceleration of end-effector. Group of Gosselin has presented several literature regarding dynamical workspace of CSPRs [7-9]. p3

p2

p(xo,yŽ) A1

p1

A3

G

p4

A2

A4

Fig. 1. Workspace of Cable-driven Parallel Robots.

A gripper or grasping mechanism is fixed on the end-effector to perform pick-andplace movement in general. It is difficult to assemble grippers or other tools on the end-effector as a mass point. To solve this issue, several new designs of CSPRs are presented by Behzadipour in [10], where the parallelogram structure with pairwise cables is adopted to constrain rotational DOFs and perform only translational DOFs such as BetaBot and planar cable-based manipulators. Pairwise cables in parallelogram are able to constrain the rotational DOF around the normal of plane formed by these cables. Considering the gravity of the end-effector, a class of CSPRs similar to BetaBot is proposed by Vu [11] in place of the rigid link spine to extend the workspace with only translational DOFs. Due to this parallelogram design, there is no need to limit the end-effector as a mass point, which is beneficial for designing end-effector structures with different tools. Gosselin[12] first studied point-to-point dynamic trajectory planning method of two and three DOFs cable-suspended robots. Additionally, some papers focus on redundantly-actuated cable-suspended robots other than completely-actuated ones. In the paper [8], a planar two-DOF redundantly actuated cable-suspended parallel robot with the end-effector as a mass point was studied on dynamic trajectory planning and its characteristics had been discussed in details. In this paper, a class of new cable-suspended robot with actuator redundancy is studied by considering structure of the end-effector with parallelogram pairwise cables attached. Compared to the structure mentioned in [11], redundant actuation is introduced in cable-suspended robots to alleviate the issues of both system vi-

Singularity Characteristics of a Class of Spatial Redundantly…

101

bration and abrupt tensions. Moreover, the redundant cables are able to exert additional tensions with various combinations and better performance can be obtained[13]. Singularity is another issue to work on for new mechanisms before trajectory planning since singular positions should be avoided as static state during motion. In the range of achievable workspace, there may exist some singular positions where the end-effector can not be controllable. Therefore, before discussing the trajectory planning of this class of CSPRs, singularity analysis should be finished. Many researchers had studied the singularity issue of rigid parallel mechanisms. Zlatanov defined constraint singularity of parallel mechanisms, and noted that constraint singularity only occurs in constrained parallel mechanisms[14]. Screw theory is also used in singularity analysis of 3-dof planar parallel mechanisms[15], where the singular loci of PPMs with parallelograms have been discussed. Liu proposed a geometric treatment to identify singularities of parallel manipulators[16]. Since CSPRs are driven by cables, tension should be considered as a constrain for these mechanisms. In this paper, a class of CSPRs with parallelogram structure is studied with redundant actuators on singularity analysis and the results are compared with CSPRs using complete actuators. The rest of this paper is organized as follows. Section 2 introduces a spatial redundantly-actuated cable-suspended parallel robot with parallelogram pairwise cables as the research object. In what follows, Section 3 provides the kinematic and dynamic modeling of the cable-suspended robots with redundant actuators. Singularity analysis is implemented in Section 4. Next, one type of CSPRs is discussed in Section 5. Finally, the conclusion is drawn in Section 6.

2

Spatial Redundantly-actuated CSPR Description with Parallelogram Pairwise Cables

For a spatial redundantly-actuated CSPR, one end of pairwise cables as parallelogram (as shown in Fig. 2) is attached to pulleys and the other end of cables is anchored at the end-effector. Each pair of cables is employed as a parallelogram during motion because one servo motor is able to be applied to control the extension and retraction of pairwise cables in synchronization. The global coordinate system is defined with z-axis along the opposite direction of gravity acceleration in this paper. Due to the parallelograms[11], rotational degrees of freedom around the normal of each plane of parallelogram pairwise cables can be forbidden so that rotational DOFs of the end-effector can be constrained in space at any non-singularity position.

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Pulleys

Parallelogram

...

Cables

Attachment points

End-effector

Fig. 2. Parallelogram Pairwise Cables of CSPR.

3

Kinetic Modelling

Kinematic model of the CSPR is illustrated in Fig. 3, where the relative coordiQDWHV\VWHP2¶-[¶\¶]¶LVGHILQHGZLWKWKHRULJLQDWWKHFHQWHURIPDVVRIWKHHQGeffector. A12 is denoted as the middle point of line A1A2 (attached points of pulleys at the parallelogram) and a12 is the middle point of line a1a2 (anchors of parallelogram on the end-effector ). The position vector of two origins is defined as p, where A and a are position vectors in global and relative coordinate systems respectively. At any non-singular position, this robot is able to realize three translational-dof motion using the parallelogram design. Thus, the length of cable 1&2 is denoted as

A 1&2

A p a .

(1)

As shown in Fig. 4, the wrench (force and moment) is exerted on the end-effector by the pairwise cables such as cable 1&2. Tensions of cable 1&2 are referred to as t1 and t2 with the directions along each cable. Since A1A2a2a1 is constrained as a parallelogram during motion, the directions of tensions of pairwise cables are identical. Forces exerted on the end-effector by tensions of cable 1&2 are (t1+t2), where the moments are (a1xt1+a2xt2) with respect to the center of mass of the end-effector. The force and moment from tensions on the end-effector are denoted as

F12 M12

t1  t 2

a1 u t1  a2 u t 2

where l1 is defined as the unit vector of t1.

(t1  t2 )l1 , (t1a1  t2a2 ) u l1 ,

(2) (3)

Singularity Characteristics of a Class of Spatial Redundantly…

Fig. 3. Kinematic Modeling.

103

Fig. 4. Dynamical Modelling.

Assume that the tensions of pairwise cables have the same value as t1=t2. Considering the acceleration of the end-effector, the dynamic modeling of this CSPR is obtained n

2¦ t(2i 1)l i

  g , mp

(4)

i 1

n

¦ t

( 2 i 1)

a ( 2i 1)(2i ) u l i 0 ,

(5)

i 1

where m is the mass of the end-effector, ‫݌‬ሷ ൌሾ‫ݔ‬ሷ ǡ ‫ݕ‬ሷ ǡ ‫ݖ‬ሷ ሿ் ,܏ ൌ ሾͲǡͲǡ െ‰ሿ் (g is the gravitational acceleration) and n is the number of pairwise cables. This CSPR is referred to as the completely actuated mechanism with three-translational-DOF as n is equal to three, while the redundantly actuated CSPR is denoted as n is larger than three.

4

Singularity Analysis

For this class of redundantly-actuated CSPR, singular positions may exist over the workspace. When it happens, the three-translational-dof of the end-effector can not be maintained. In this section, singularity analysis is proceeded to classify the feasible positions to fulfill the requirement of translational motion. Accordingly, at least three pairs of cables are required to keep taut to control motion of this CSPR. Each pair of cable with parallelogram structure restrains the rotational degree of freedom around normal to the parallelogram. In order to constrain all rotational dofs, the first singularity of this robot is deduced as Rank([(a2-a1) xl1,..., (a2i-a(2i-1))xli,..., (a2n-a(2n-1))xln ]) x y z I T \ @T  ƒ6 represents the pose vector of the end-effector, M ( q) ƒ6u6 represents a symmetric positive definite inertia matrix, C ( q, q )q ƒ6

represents the vector of centripetal and Coriolis forces, G ( q) ƒ6 represents the gravitational force vector, Fd ƒ6u6 denotes the constant viscous friction coefficient matrix, Fs ƒ6 represents the Coulomb friction, and Td (t ) ƒ6 is a time-varying disturbance term. In addition, the matrix J  ƒ6un is the Jacobian of the robot which is defined as in [12], and W ƒn represents the vector of tension forces. Furthermore, rp

represents the radius of pulley, W m  ƒ n represents the motor torque vector,

W g ƒn represents the viscous and Coulomb friction torques of gearboxes which

assumed to be differentiable and bounded, Z ƒn denotes the vector of shaft speed, Fdg ƒnun is a diagonal constant matrix, and Fsg (Z ) ƒn denotes the Coulomb fric-

tion. Dynamic Equation (1) has the following properties, which are useful in the stability analysis [12]. Property 1: The inertia matrix M ( q ) is symmetric, positive definite, and both M ( q ) and M 1(q) are uniformly bounded as a function of q  ƒ6 . Property 2: The matrix M ( q )  2C ( q , q ) is skew symmetric, which satisfies y T  M ( q )  2C ( q, q ) y

3

0,

 y  ƒ6 .

Control design

In this section, a back-stepping-like control scheme will be used for the actuated CDPRs. First, the desired tension W d ƒn is designed to ensure the convergence of q to the desired trajectory qd . Next, the control input W m is constructed to ensure the convergence of tension tracking error, eW W  W d ƒn , around zero. Before going to the details of controller derivation, we present the following assumption. Assumption 1: The desired trajectory of the end-effector, and its first two time derivatives are continuous and uniformly bounded. 3.1

Outer-loop controller design

Let us define an error vector as

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S q  /q

(3)

where q q  qd ƒ6 , and / diag( /1 ,..., / 6 ) ƒ6u6 with / i ! 0 for all i 1,..., 6 . Adding and subtracting the same term W d to the right hand side of Equation (1), and introducing Q

qd  /q ƒn , we have

M (q)S  C(q, q )S  M ( q)Q  C(q, q )Q  G( q)  Fd q  Fs ( q )  Td  JeW  JW d

(4)

The objective of the controller design is to regulate the dynamics in Equation (4) by designing an appropriate tension vector W d , keeping in mind that the cables should always be in tension. To satisfy the later, the following general solution is adopted, which follows the internal forces concept [3]: Wd

Jˆ †W 0  N n

(5)

where Jˆ  ƒ6un denotes an estimate of Jacobian matrix, N ƒ is an arbitrary value chosen in a way to yield a positive tension for the cables, n ƒn is the null space of ˆ Jˆ satisfying Jn

0 , and Jˆ †W 0 is the particular solution for the tension, in which

Jˆ †  ƒnu6 denotes the pseudo-inverse of Jˆ , and W 0  ƒ6 is the applied wrench on the

end-effector, which is proposed as W0

Mˆ (q)v  Cˆ ( q, q )v  Gˆ (q)  Hˆ  Kd S

(6)

In the last equation, Mˆ ( q ) , Cˆ ( q, q ) , Gˆ ( q ) , and Hˆ denote the estimates of M ( q ) , C ( q, q ) , G ( q ) , and H

N ( Jˆ  J )n  ( I  JJˆ † )W 0  Fd q  Fs ( q )  Td ƒ6 , respectively;

in which I  ƒ6u6 represents the identity matrix, and K d ƒ6u6 is a constant diagonal gain matrix. Now, plugging (5) and (6) into (4), and introducing ( < ) ( 0.7 0.7 0.05@ , which is bounded with finite energy. Furthermore, we assume %20 uncertainty in the Jacobian matrix. Comparisons between the proposed controller and the robust trajectory control presented in [12] have also been done. Assume that a circle with the radius of 0.25m is defined as the desired trajectory for the manipulator end-effector. The orientation was also commanded to stay constant ( I 0 ) throughout the circular path. The initial value of the end-effector posi-

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tion is set at q(0)  [0.1 0.55 0]T , that is far from the desired one. The controller parameters were selected as Kd diag (100,100,100) , Kc diag (20,20,20,20) , and / diag (10,10,10) . Assume that the first term of FS are used for approximation of 9u3 3u3 3u3 Wˆ M , WˆG , Wˆ H , and WˆW , respectively. Therefore, WˆM ƒ , WˆG ƒ , WˆH ƒ

, and WˆW ƒ4u4 . The initial conditions for the Fourier series coefficients have been set to zero and the convergence rate in the adaptation rule (19) are defined as QM 1000 u I 9 , QG QH 0.033 u I3 , and QW 0.033 u I 4 , where I ( x ) indicates identity



1RUPRISRVLWLRQHUURU

P@

matrix. Under the aforementioned settings, Fig. 2a shows the actual and desired motions in the x-y plane tracked by the proposed adaptive controller, and the robust control scheme [12]. As it can be seen, the proposed approach has better performance compared with the control system presented in [12]. Also, Fig. 2b represents norm of position error. Finally, Fig. 3 shows the applied positive tensions to all cables. According to this figure, the control signals are smooth during the robot manoeuvres. The large initial value of the control signal is due to difference between the initial positions of the end-effector and desired one.

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6

Conclusion

A FAT-based robust adaptive controller using the FS has been developed for CDPRs. The controller has been designed so that the information of the system parameters is not required. The system nonlinearities including inertia matrix, Coriolis/centrifugal matrix, and gravitational torques vector, have been estimated using the FS. Satisfactory performance in the transient state has also been investigated. Simulation results on CDPRs with three DOF verify the efficiency of the proposed controller.

References 1. Abbasnejad, G., Yoon, J., Lee, H.: Optimum kinematic design of a planar cable-driven parallel robot with wrench-closure gait trajectory. Mech. Mach. Theory 99, 1–18 (2016) 2. Oh, S.-R., Ryu, J.-C., Agrawal, S.K.: Dynamics and control of a helicopter carrying a payload using a cable-suspended robot. J. Mech. Des. 128(5), 1113–1121 (2006) 3. Kawamura, S., Kino, H., Won, C.: High-speed manipulation by using parallel wire-driven robots. Robotica 18(01), 13–21 (2000) 4. You, X., Chen, W., Yu, S., Wu, X.: Dynamic control of a 3-DOF cable-driven robot based on backstepping technique. 6th IEEE Conference on Industrial Electronics and Applications, pp. 1302–1307 (2011)

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5. Fang, S., Franitza, D., Torlo, M., Bekes, F., Hiller, M.: Motion control of a tendon-based parallel manipulator using optimal tension distribution. IEEE/ASME Trans. Mechatron. 9(3), 561–568 (2004) 6. Oh, S.-R., Agrawal, S.K (2005) Cable suspended planar robots with redundant cables: controllers with positive tensions. IEEE Trans. Robot. 21(3), 457–465. 7. Babaghasabha, R., Khosravi, M.A., Taghirad, H.D (2015) Adaptive robust control of fullyconstrained cable driven parallel robots. Mechatronics 25, 27–36 8. R. J. Caverly, J. R. Forbes, and D. Mohammad shahi.: (2015) Dynamic Modeling and Passivity-Based Control of a Single Degree of Freedom CableActuated System. IEEE Transactions on Control Systems Technology, Vol: 23, pp. 898 909 9. Tiechao Wang, Shaocheng Tong, Jianqiang Yi, and Hongyi Li.: 2015. Adaptive Inverse Control of CableDriven Parallel System Based on Type2 Fuzzy Logic System s. IEEE Transactions on Fuzzy Systems. Vol: 23. Pp. 1803 – 1816. 10. Oh, S.-R., Albus, J.S., Mankala, K., Agrawal, S.K (2005) A dual-stage planar cable robot: dynamic modeling and design of a robust controller with positive inputs. J. Mech. Des. 127, 612 11. Jabbari Asl, H, and F. Janabi-Sharifi (2017) Adaptive neural network control of cabledriven parallel robots with input saturation. Engineering applications of artificial intelligence. Vol: 65, pp. 252-260. 12. Jabbari Asl, H, and J. Yoon (2017) Robust trajectory tracking control of cable-driven parallel robots. Nonlinear Dynamics, Vol. 89, pp. 2769-2784. 13. A. Izadbakhsh, and P. Kheirkhahan, Adaptive Fractional-Order Control of electrical Flexible-Joint Robots: Theory and Experiment, Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering, DOI: 10.1177/0959651818815384. 14. A. Izadbakhsh, and P. Kheirkhahan (2019) An alternative stability proof for "Adaptive Type-2 fuzzy estimation of uncertainties in the control of electrically flexible-joint robots", Journal of Vibration and Control, Vol: 25. pp. 977–983. 15. B. Chen, C. Lin, X. Liu, and K. Liu (2016) Observer-based adaptive fuzzy control for a class of nonlinear delayed systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, Vol: 46. pp. 27-36. 16. Q. Zhou, H. Li, C. Wu, L. Wang, and C. K. Ahn (2017) Adaptive fuzzy control of nonlinear systems with unmodeled dynamics and input saturation using small-gain approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems. 17. A. Izadbakhsh, S. Khorashadizadeh, and P. Kheirkhahan (2018) Tracking control of electrically driven robots using a model free observer, Robotica, DOI:10.1017/S0263574718001303. 18. A. Izadbakhsh (2018) Robust adaptive control of voltage saturated flexible joint robots with experimental evaluations,AUT Journal of Modeling, and simulation, Vol: 50. pp. 3138. 19. A. Izadbakhsh, and S. M. R. Rafiei (2009) Endpoint perfect tracking control of robots – A robust non inversion-based approach, International Journal of Control, Automation, and systems, Springer, vol.7, no.6, pp. 888-898. 20. A. Izadbakhsh (2017) FAT-based robust adaptive control of electrically driven robots without velocity measurements, Nonlinear Dynamics, vol. 89, pp. 289-304.

Model Predictive Control of Large-Dimension Cable-Driven Parallel Robots Jo˜ ao Cavalcanti Santos, Ahmed Chemori, and Marc Gouttefarde LIRMM, University of Montpellier, CNRS, Montpellier, France {joao.cavalcanti-santos, ahmed.chemori, marc.gouttefarde}@lirmm.fr

Abstract. A Model Predictive Control (MPC) strategy is proposed in this paper for large-dimension cable-driven parallel robots working at low speeds. The latter characteristic reduces the non-linearity of the system within the MPC prediction horizon. Therefore, linear MPC is applied and compared with two commonly used strategies: Sliding mode control and PID+ control. The simulations aim at comparing disturbance rejection performances and the results indicate a superior performance of the proposed controller. Indeed, MPC takes into account control limits (cable tension limits) directly in the control design which allows the controller to better exploit the robot capabilities. In addition, actuation redundancy is resolved as an integral part of the control strategy, instead of calculating the desired wrench and then applying a tension distribution method. Keywords: Cable-driven parallel robots, model predictive control, disturbance rejection

1

Introduction

A rather typical control strategy for Cable-Driven Parallel Robots (CDPRs) is the combination of a linear feedback control with computed torque applied as a feedforward term [1]. Indeed, the computed torque (also known as feedback exact linearization) enables the application of usual linear control methods [2, 3]. Besides, Sliding Mode Control (SMC) is an advanced nonlinear feedback control that has been implemented successfully in CDPRs [4–7]. The main advantages of SMC are the possibility to attain finite time convergence, simple implementation and robustness to uncertainties. However, recent and advanced SMC methods still present chattering issues in experimental setups [6] even if some previous studies presented methods in order to reduce it [7]. Since a fully-constrained CDPR has more cables than degrees of freedom (DOF), there are infinitely many possible combinations of cable tensions generating a desired wrench. The choice of one of these combinations is an actuation redundancy resolution problem where cable tension lower and upper limits should be taken into account. The lower limit is a positive tension to avoid cable slackness. The upper limit is set in order to account for the mechanical © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_19

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limitations of the cables, motors, etc. Several previous works deal with CDPR actuation redundancy resolution, e.g. [8, 9]. Usually, the control strategy and the resolution of actuation redundancy are addressed separately. This paper proposes the use of Model Predictive Control (MPC) as the control strategy, which has the advantage of solving the tension distribution problem as an integral part of the control strategy. MPC is a feedback control design which, at each decision instant, computes the sequence of future controls that optimizes a cost function satisfying the system constraints. The cost function is formed by a weighted sum of individual costs (tracking errors, control input and other performance measures). MPC is considered as one of the most general way of posing a control problem in the time domain [10, 11]. It defines an optimized admissible control sequence if the considered model is sufficiently close to reality. Moreover, control limits can be directly handled, which is an important advantage since the optimized performance is often obtained with active constraints. If the MPC is applied to a linear optimization problem without constraints, the solution is analytic, e.g. in [12], where Generalized Predictive Control (GPC) is presented. Nevertheless, constraints [13] and nonlinear systems [14] have been successfully addressed as well. This paper presents the design, implementation and simulation of an MPC scheme for motion control of large-dimension CDPRs. To the best of our knowledge, MPC has never been used to control CDPRs. Several applications of CDPRs involve a large workspace and relatively low velocities of the mobile platform [15–18]. Hence, within a reasonable prediction horizon, the CDPR dynamics may be approximated as being a linear time invariant system. Based on this assumption, linear MPC is applied in this paper. Simulation results compare the performances obtained with the proposed MPC and two other control strategies, namely SMC and PID+ Controller [19]. The performance evaluation focuses mainly on external disturbance rejection. The paper is organized as follows. The modeling of a spatial CDPR is introduced in Section 2. Two state-of-the-art control schemes, SMC and PID+ Control, and the proposed MPC strategy are introduced in Section 3. These three methods are compared through numerical simulations in Section 4. Conclusions and future works are drawn in Section 5.

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Dynamic Modeling of CDPRs

The dynamic modeling of a spatial CDPR is presented in this section. The control schemes introduced later in Section 3 are based on this model. Figure 1 illustrates a CDPR with the notations used in its kinematic modeling. The robot consists of a n-DOF mobile platform driven by m cables. The spatial case where n = 6 is considered and the m cables (m ≥ 6) are considered massless and inextensible. Each cable has one end attached to the platform and the other end wound on a winch drum. The cables are responsible for transmitting to the platform the efforts applied by the winches. Cable tensions τ applied on

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the platform generates the wrench f . These variables are related linearly by the wrench matrix W, so that f = W τ [9]. The length li is defined as the distance between the drawing point Ai and and the attachment point Bi . Point Ai is the drawing point defined by the pulley attached to the base frame. This point is considered as being fixed. Point Bi is the attachment point of cable i on the platform. The cable length vector is l = [l1 , ... , lm ]T .

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˙ x ¨ be the platform pose, velocity and acceleration, respectively1 . The Let x, x, pose (position and orientation) of the platform is then defined by x = [pT , ψ T ]T . p is the position vector of the reference point of the platform. Vector ψ defines the orientation of the platform. Typically, it is composed of three Euler angles. Using Newton-Euler formalism, the dynamics of the platform can be written as [20] ¨ + C(x, x) ˙ x˙ = g(x) + f M(x) x (1) where matrices M and C are given by     ˆ ˆω ˆc mp ω mp I3 −mp c ˙ x˙ = , C(x, x) M(x) = ˆ Hω ˆ mp c ω H

(2)

Scalar mp is the platform mass and I3 is the identity matrix of dimension 3. The present model considers that the platform geometric center and its center T  of mass may not be coincident and, accordingly, c = cx cy cz is the vector ˆ and going from the platform geometric center to its center of mass. Matrices ω ˆ are the skew-symmetric matrices associated to ω and c, respectively, with c ω the angular velocity of the platform. The matrix H is defined as H = I + ˆc ˆT where I is the platform inertia matrix relative to its center of mass. mp c 1

We highlight that x˙ = dx , since angular velocity is not equal to the derivative of the dt 2 ¨ = ddtx . vector of Euler angles. Similarly, x

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 T The vector of gravitational forces is g(x) = mp g 0 0 −1 −cy cx 0 . The wrench applied by the cables on the platform is f = W τ .

3

Proposed Control Schemes

The goal of this paper is to compare MPC performance to those obtained with strategies commonly used for CDPRs. Namely, MPC is compared to: (i) A linear PID+ controller [19], based on the controller proposed in [20], and (ii) SMC. These two strategies were recently validated experimentally, demonstrating applicability and good performances. A brief description of these control methods is presented in the following section. 3.1

Background on the State of the Art Controllers

For a given reference trajectory in time ti  t  tf , the desired poses, velocities ¨ d (t), respectively. At a given and accelerations are denoted as xd (t), x˙ d (t) and x instant t, the error in the Cartesian space is expressed as ex (t) = xd (t) − x(t). Similarly, the error in joint space is ej (t) = ld (t) − l(t), where ld (t) is the vector of desired cable lengths obtained with the inverse kinematics. PID+: The PID+ control strategy applies the following wrench    t ˙ x˙ d − g(x) + W Kp ej + Ki ej (τ ) dτ + Kd e˙ j (3) f = M(x)¨ xd + C(x, x) ti

where Kp , Ki and Kd are diagonal matrices containing the linear feedback PID gains. Sliding Mode Control: The SMC strategy defines a sliding surface s = ex + Ce e˙ x ,

with Ce = diag(c1 , ..., cn ). The wrench to be applied on the platform is

˙ + K sat(s) + Q s + C(x, x) ˙ x˙ d − g(x) ¨ d + Cd (x˙ d − x) f =M x

(4)

where K, Q and Cd are diagonal gain matrices. The function sat(s) is a continuous approximation of the sign function. Each component of this vector-valued function is calculated as follows ⎧ ⎨ 1, if si > Δ si , if |si |  Δ sat(si ) = (5) ⎩ Δ −1, if si < Δ

The resulting function presents the same output than the sign(s) function except for the interval −Δ  s  Δ in which a linear interpolation eliminates the discontinuity.

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Redundancy Resolution: The two above control strategies define wrench f . The final control output is the vector of cable tensions τ (or motor torques). For fully-constrained CDPRs, m > n and actuation redundancy shall be resolved to determine τ being given f , i.e., the equation system W τ = f is underdetermined and an appropriate vector of cable tensions τ shall be determined among the infinitely many possible ones (assuming a non-singular pose). In this work, the following common optimization problem is used to resolve actuation redundancy minτ 2

(6)

s.t. W τ = f τmin  τ  τmax

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τ

Using (6), the 2-norm of τ is minimized. As constraints, the tension distribution shall generate the desired wrench (7) and the cable tensions shall be in an admissible interval (8). The constraint (8) is necessary mainly for two reasons: avoiding cable slackness (0  τmin  τ ) and not violating mechanical limitations of the cables, motors, etc (τ  τmax ). In some cases, the wrench demanded by the controller is not feasible. In the space of cable tensions, the intersection of the subspaces defined by the constraints (7) and (8) is empty. Another strategy should then be defined and the following optimization problem is proposed minW τ − f P τ

s.t. τmin  τ  τmax

(9)

where the subscript P indicates that the 2-norm is calculated with a weighting positive definite diagonal matrix P, which is necessary since f has components with inconsistent units (forces and moments). 3.2

Proposed Model Predictive Controller (MPC)

A model predictive control strategy is proposed in the present section. The continuous model (1) may be approximated as a discrete-time system   x(t + Δt) y(t + Δt) = = A(t) y(t) + B(t) τ (t) + v(t). (10) ˙ + Δt) x(t where the vector v and the matrices A and B are given by   0nx1 v= Δt M−1 g   Δt In In A= 0nxn In − Δt M−1 C   0nxm , B= Δt M−1 W

(11) (12) (13)

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where time t has been omitted, being given that v, A and B are calculated at time t. MPC predicts the states over a given horizon t + N Δt. Equation (10) is applied N times in order to obtain y(t + i Δt), with i = 1, ..., N . As mentioned earlier, the proposed MPC strategy is applied to large-dimension CDPRs moving at low velocities. Therefore, for a small horizon N Δt, the linear time varying system (10) may be approximated by a linear time invariant system. Matrix A is considered as A(t + i Δt) = A(t), i = 1, ..., N . Similarly, B and v are considered constant over the MPC optimization horizon. With this approximation, linear MPC can be applied. A vector Y(t) containing the states over the prediction horizon is calculated as follows

⎡ ⎤ B A ⎢ AB y(t + Δt) ⎢ 2⎥ ⎢ ⎢ ⎥ ⎢A ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎢ .. Y(t) = ⎢ ⎥ = ⎢ . ⎥ y(t) + ⎢ A B . ⎢ ⎣ ⎦ ⎢ . ⎥ .. . ⎢ ⎦ ⎣ ⎣ y(t + N Δt) . AN AN −1 B    D ⎡

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Similarly, the reference trajectory yd = [xTd , x˙ Td ]T should be defined over the prediction horizon ⎤ ⎡ yd (t + Δt) ⎢ yd (t + 2 Δt) ⎥ ⎥ ⎢ (14) w(t) = ⎢ ⎥ .. ⎦ ⎣ . yd (t + N Δt)

For a given current state y(t), the control is responsible for finding a trade-off between minimizing the predicted error w(t) − Y(t) and minimizing the control effort U(t). To this end, the following cost function can be considered

J y, U = (w − Y)T KY (w − Y) + UT KU U

(15)

where Ky and KU are diagonal weight matrices. The minimization of J for a given current state is equivalent to   min UT (ET KY E + KU ) U + 2 (D y + F − w)T KY E U U       Hc

dT

(16)

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This minimization is a case of quadratic programming (QP). Constraints on cable tension limits and rates of change can be easily added  τmin  τ  τmax (17) ⇒ Aineq U  bineq |τ˙i |  τ˙max Accordingly, at each time step, a QP problem is solved in order to determine the optimal control output τ (t). This problem is stated as 1 min UT Hc U + dT U U 2 s.t. Aineq U  b

(18)

The QP problem (18) can be efficiently solved with the Interior Point method implemented in the MATLAB function quadprog. The optimal argument U(t) = [τ T (t), τ T (t + Δt), ..., τ T (t + (N − 1)Δt)]T contains all vectors of future control outputs over the control prediction horizon. The controller applies the first sample of the obtained sequence and maintains this action until the next time step. After that, the algorithm is repeated. The new state y(t+Δt) is measured and the states Y(t+Δt) = [y(t+2 Δt)T ... y t+

T T (N + 1) Δt ] are estimated and optimized taking U(t + Δt) as argument. The procedure is repeated until t  tf .

4

Simulation Results

This section presents simulation results that compare the performances obtained with the three control strategies presented in Section 3. The context of these simulations is the project Hephaestus [21] where a large-dimension CDPR is intended to automatize several tasks in the construction and maintenance of building facades. The main task of the CDPR is the installation of curtain wall modules. The robot workspace is a rectangular region in front of the building facade. Thereby, the CDPR mobile platform can get curtain wall modules on the ground and position them where needed on the building facade. Since the CDPR will operate in an outdoor environment, it will be subjected to external disturbances. One of the main concerns is the incidence of wind gusts. For this reason, the simulations presented in this section focus on external disturbance rejection performances of the proposed control strategies. An impulsive disturbance is applied and the response of the CDPR is analyzed. Note that the simulated trajectory is relatively short. However, the simulation of a longer trajectory would not affect the results which highlight the disturbance rejection capabilities of each control strategy. The initial and final positions are depicted in Fig. 2. The path between these two positions is a straight line segment. The trajectory is the fastest possible respecting upper bounds on linear velocities, accelerations and jerks. These bounds are 0.3m/s, 0.3m/s2 and 1.0m/s3 , respectively. The resulting trajectory has continuous derivatives up to the acceleration level. The desired orientation of the

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platform is constant along the trajectory. An impulsive disturbance fd is applied  T at the instant t = 2 s. This impulsive wrench is fd = 55 55 550 0 0 0 (N) and it is applied at the reference point of the platform. The CDPR configuration (cable drawing points, cable-platform attachments, and cable arrangement) can also be seen in Fig. 2. Moreover, the parameters of the CDPR dynamic model are the following: τmin = 100N, τmax = 14kN, mp = 1000kg, [c]Fp = [0 5 0]T m, I = diag([400 100 400])kg.m2 .

Fig. 2. Initial and final positions of the simulated trajectory

In the following, the obtained results with the three proposed motion control strategies of Section 3 are presented and discussed. Control parameters used for the simulations are the following: Kp = 71400 I8 , Ki = 71400 I8 , Kd = 71400 I8 , Ce = 2 × 10−3 I6 , Cd = 36 I6 , Q = 40 I6 , K = 0.2 I6 , N = 20, Δt = 6 × 10−4 , Δr = 5 × 10−1 , KY p = 6 × 109 , KY v = 1 × 10−2 , KU = 2 × 10−5 I8N . Scalar KY p are the components of KY related to pose errors, whereas KY v is related to velocity errors. Value of Δt is used in (5) as Δ for translational inputs of sat(s), and Δr is used for rotational inputs. Figure 3 shows the evolution versus time of the norms of the translational and rotational errors. All the control strategies are able to cancel the tracking error caused by the applied external disturbance but PID+ presents an oscillatory behavior and increased settling time. SMC responds faster and without oscillations. MPC presents the fastest response, resulting in the smallest tracking errors along the whole trajectory. The histograms of Fig. 4 present a performance comparison on different aspects. Let et be the 2-norm of the translational error. Histogram (a) quantifies the maximum value et over the trajectory. Histogram (b) compares the RMS

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value of et along the trajectory. Taking these two performance measures, MPC leads to the smallest error.

Fig. 4. Comparative results: (a) maximum error, (b) error RMS, (c) maximum cable tensions, (d) maximum cable tension derivatives, (e) RMS of cable tension derivatives, and (f) consumed energy.

Regarding cable tension values, as shown in Fig. 4-(c), MPC demands the maximum allowed value τmax = 14 kN. This is also shown in Fig. 5, which depicts the cable tensions near the instant of application of the impulsive disturbance. Indeed, as discussed earlier, the main advantage of MPC is that the controller takes into account the constraints of the system and optimizes the control actions

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in order to reduce the tracking errors. Here, the maximum allowed cable tension is an active constraint just after the impulsive disturbance is applied. In the case of SMC, the maximum tension value is 13.7 kN along the trajectory which indicates that this controller response is close to the largest admissible value τmax = 14 kN. If higher gains were used, the wrench f would then be unfeasible. In order to not exceed τmax , the strategy described in Eq. (9) would be necessary and the resulting wrench would not be equal to f . However, the MPC strategy does not lead to this risk. The PID+ controller has |τmax | = 9.1 kN. This relatively low tension is a consequence of the use of small gains Kp . Indeed, small gains were used because larger gains lead to high frequency oscillations of cable tensions. For instance, increasing the gains of less than 1% with this strategy, RMS(τ˙ ) is equal to 86 kN (more details about this measure are discussed in the following paragraph). The time derivative of the cable tensions is a measure of the aggressiveness of the control action. Large values of this variable may excite high frequency dynamics which are difficult to control. Figure 4-(d) presents the maximum derivative of cable tensions over the trajectory considering

all cables. Mathematically, values in Fig. 4-(d) are equal to maxt maxi |τ˙i (t)| . The smallest maximum cable tension derivative is obtained for the PID+, which is an advantage of this method. SMC is the most aggressive controller considering this performance measure. The proposed MPC strategy constraints this variable (Eq. (17)). Therefore, any value can be imposed independently of the rest of the controller parameters. In simulations, the value used is 800 kN/s and Fig. 4-(d) shows that this value is reached. Fig. 4-(e) presents the values of the maximum RMS value of all cables i = 1, ...8. More precisely, Fig. 4 (e) presents values τ˙i considering of maxi RMS(τ˙i ) . MPC presents the largest RMS of the cable tension derivatives. Note that the MPC optimization cannot take as constraint variables that depends on the system states beyond the prediction horizon. The prediction horizon covers only a small part of the whole trajectory over which the RMS values are calculated.

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The consumed energy over the trajectory (Fig.4-(f)) is roughly the same for all the control schemes. Nevertheless, MPC leads to the highest consumption. t ˙ T τ (t)|dt. The consumed energy is calculated as tif |l(t)

5

Conclusions and Future Work

An MPC strategy was proposed for large-dimension CDPRs. Considering that the robot works at low velocities, a linear MPC approach was selected. A linear time invariant system is considered along the MPC optimization horizon. MPC allows to minimize the error between the desired trajectory and the predicted positions while taking into account constraints on cable tensions. Since cable tensions are the argument of this optimization, actuation redundancy resolution is integrated within the MPC optimization. Simulation results validate these advantages by comparing the disturbance rejection performances of the proposed MPC strategy with two other commonly used controllers (PID+ and SMC). MPC yields the smallest errors when compared to these two other controllers. In future works, the modeling errors due to the assumption of a linear time invariant system should be quantified. Indeed, the non-linear dynamics of the CDPR will differ from the proposed linear time invariant system according to the mobile platform dynamics and the MPC optimization horizon. These modeling errors may be predicted and the relevance of the proposed linear MPC may be evaluated in a general case. Additionally, experimental tests should be conducted to validate the simulations presented in this paper. Experimental validation may also clarify the feasibility regarding computational burden, which is a critical aspect of using MPC for real-time control of CDPRs.

ACKNOWLEDGMENT The research leading to these results has received funding from the European Union’s H2020 Programme (H2020/2014-2020) under grant agreement No. 732513 (Hephaestus project).

References 1. R. L. Williams, P. Gallina, and J. Vadia, “Planar translational cable-direct-driven robots,” Journal of Robotic Systems, vol. 20, no. 3, pp. 107–120, 2003. 2. M. H. Korayem, H. Tourajizadeh, and M. Bamdad, “Dynamic load carrying capacity of flexible cable suspended robot: Robust feedback linearization control approach,” Journal of Intelligent and Robotic Systems: Theory and Applications, vol. 60, no. 3-4, pp. 341–363, 2010. 3. M. A. Khosravi and H. D. Taghirad, “Robust PID control of fully-constrained cable driven parallel robots,” Mechatronics, vol. 24, no. 2, pp. 87–97, 2014. 4. A. Alikhani and M. Vali, “Sliding Mode Control of a Cable-driven Robot via Double-Integrator Sliding Surface,” in International Conference on Control, Robotics, and Cybernetics, 2012.

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Linearised Feedforward Control of a Four-Chain Crane Manipulator Michael Stoltmann1 , Pascal Froitzheim2 , Normen Fuchs2 , Wilko Fl¨ugge2 , and Christoph Woernle1

2

1 University of Rostock, 18059 Rostock, Germany, {michael.stoltmann,woernle}@uni-rostock.de, WWW home page: http://www.ltmd.uni-rostock.de Fraunhofer Research Institution for Large Structures in Production Engineering IGP, 18059 Rostock, Germany WWW home page: https://www.igp.fraunhofer.de

Abstract. For a crane manipulator suspending a flexible metal plate by four chains feedforward control is derived that moves the payload along desired spatial trajectories. Due to the statically indeterminate suspension, the stiffness of the payload has to be taken into account. The actuator coordinates can be algebraically calculated from the desired trajectory of the plate at the position, velocity and acceleration levels exploiting the flatness property of the system. As the system is kinematically redundant, a technically meaningful solution like minimal inclination angles of the chains are determined by optimisation. As the iterative calculation of the nonlinear constrained optimisation problem is computationally expensive and not suitable for implementation on the crane controller, a linearised inverse dynamics model is derived that describes small sway motions of the flexible payload around a static equilibrium state. Linearised and nonlinear feedforward control are compared in a numerical simulation of the crane system. Keywords: crane manipulator, elastic payload, underactuated system, linearisation, inverse dynamics, quadratic programming, feedforward control

1

Introduction

Metal plates can be formed by means of a so-called ship building press in a cold plastic forming process. Hereby the workpiece is positioned above the forming tool by four chain hoists that are mounted on trolleys of an overhead crane with two bridges each with two trolleys as schematically shown in Fig. 1. Bridges, trolleys and chain hoists are individually controllable to achieve the desired workpiece position. In addition to the flexibilty of the workpiece itself the four chain attachment points at the metal plate are compliant in order to isolate the crane gear from process forces. This crane manipulator can be considered as a 4-4 cable-driven parallel manipulator (CDPM) with movable winch positions that is underconstrained (incompletely restrained) or underactuated [12, 13]. Compared to completely constrained and overconstrained CDPMs the pose of the payload is not fully defined by the actuator positions, here the bridge and trolley positions and the chain lengths, but by the static equilibrium © Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_20

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u1

y0 z0

x0 u12

u11

u2

u22

u21

w22

w11

w12

a η

b

w21 y1

z1

x1

Fig. 1. Cable manipulator with four load chains suspending a flexible rectangular metal plate [15].

of the system under gravity. Accordingly the definitions of direct and inverse kinematics of completely constrained manipulators are extended to direct and inverse statics, also designated as geometrico-static analysis in [6]. The task of direct statics of a CDPM is to find stable equilibrium poses of the payload for given actuator coordinates, while inverse statics comprises the determination of the actuator positions for a desired payload pose. Solution of direct statics with finding all possible stable equilibrium poses of CDPMs was solved with different methods e. g. in [1, 2, 9]. Inverse statics for a fourcable CDPM was considered in [5]. The payload trajectory of an underactuated CDPM cannot be kinematically controlled but requires dynamic control in order to avoid undesired residual sway motions. Feedforward and feedback approaches for anti-sway control of cranes are mostly based on linearised equations of motion that enable linear controller design, see e.g. [14, 11], or the development of input shaping strategies [10]. Dynamic models of cranes and CRPMs with idealised massless cables exhibit the property of flatness [7] that can be exploited to derive a nonlinear feedforward control model [8]. The actuator coordinates and actuator forces are algebraically calculated from the desired trajectory of the payload and their time derivatives up to the second and fourth order, respectively. The twice time differentiation of the equation of motion here required can be avoided by applying an implicit integration procedure [4, 18]. For the crane manipulator shown in Fig. 1 direct and inverse statics and flatnessbased feedforward trajectory control were formulated in [15] using the nonlinear equations of motion. However, this requires the iterative solution of a nonlinear constrained optimisation problem that is computationally expensive and not directly suitable for implementation into a plant controller. Therefore a simplified feedforward control scheme is studied in the present contribution. It is derived from the linearised equations of mo-

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tion under the assumption that the motion takes place in the neighbourhood of a static equilibrium position of the system. The contribution is organised as follows. In section 2 the nonlinear dynamic model of the crane manipulator formulated in [15] is summarised. The linearised equations of motion are analytically derived by Taylor series expansion in section 3. Feedforward control based on the linearised equations of motion is described in section 4. A simulation example comparing linear and nonlinear feedforward control is shown in section 5.

2

Nonlinear Dynamics of the Crane Manipulator

The equations of motion of the crane manipulator are formulated in terms of the absolute pose coordinates of the flexible plate. The masses of the chains are neglected. The bridge and trolley displacements and the chain lengths are treated as kinematic inputs of the dynamic model under the assumption of individual drive controllers. 2.1

Coordinates

According to Fig. 1 the independently controllable actuator coordinates are, with i = 1, 2 and j = 1, 2, the positions of the two bridges ui , the positions of the four trolleys ui j and the lenghts of the four chains wi j including the lengths of the unloaded attachment springs. They are summarised in the vectors u = [ u1 u2 u11 u12 u21 u22 ]T ,

w = [ w11 w12 w21 w22 ]T .

(1)

The spatial pose of the flexible plate relative to K0 is described by the six position coordinates of the body frame K1 , comprising the three Cartesian coordinates r1 = [ r1x r1y r1z ]T of O1 , three Cardan angles ϕ = [ ϕ1 ϕ2 ϕ3 ]T , defined in the rotation sequence around the follower axes z1 –y1 –x1 , and a coordinate to describe the elastic deformation of the plate η defined in the following subsection. The pose coordinates are summarised in the 7-vector  T rˆ = rT1 ϕ T η . (2) To describe the spatial velocity of the flexible plate relative to K0 , the velocity v1 = r˙ 1 of O1 , the angular velocity vector ω of K1 and the time derivative vη of η are used, summarised in the 7-vector T  (3) vˆ = vT1 ω T vη . The relation between rˆ and vˆ is given by the kinematic differential equation [17] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ I 0 0 v1 r˙ 1 ⎣ ϕ˙ ⎦ = ⎣ 0 Hϕ (ϕ ϕ) 0 ⎦ ⎣ ω ⎦ η˙ 0 0 1 vη r˙ˆ

=

H(ˆr)

(4)

vˆ

ϕ ) relating the time derivatives with the (3, 3) identity matrix I and the (3, 3) matrix Hϕ (ϕ of the Cardan angles ϕ to the angular velocity ω .

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Nonlinear Equations of Motion

The workpiece is modelled by a rectangular Kirchhoff plate (edge lengths a, b, thickness h and density ρ being constant over the area A, body coordinate system K1 located in the mass center O1 ) that is suspended at its corners (Fig. 2). The displacement of the plate in z1 -direction is only modelled by the shape function φ (x, y) and the elastic coordinate η(t) describing the torsion [3], W (x, y,t) = φ (x, y) η(t)

φ (x, y) = 4

with

x y . ab

(5)

Compared to other flexible modes of the workpiece, the torsion most strongly influences the force distribution among the four load chains. The vectors from O1 to the corners Qi j then are (6) di j = di j,0 + φi j ez1 η with vectors di j,0 of the planar plate and φi j = 1 for i = j and φi j = −1 for i = j. The nonlinear equations of motion of the crane manipulator are [15] M v˙ˆ = kc (ˆv) + ke (ˆr, u, w) + kel (ˆr)

(7)

with the mass matrix M and the generalised centrifugal and Coriolis forces kc according to ⎡ ⎤ ⎡ ⎤ mI 0 0 0 ⎢ ⎥ ⎢ ⎥ M = ⎣ 0 Θ 1 0 ⎦, kc = ⎣ −ω˜ Θ 1 ω ⎦ . (8) 0 0 0 91 m

Neglecting quadratic terms in the thickness the flexible coordinateη, the inertia

1h and 1 1 mb2 , 12 ma2 , 12 m(a2 + b2 ) , represented tensor of the undeformed plate Θ 1 = diag 12 in K1 , can be used. The vector of generalised external forces on the plate ke in (7) is governed by the gravity force fG = m g ez0 at the mass center O1 and the four suspension forces fi j at the

−f11

−f12

η

−f21

Q12

η

d11 fG

Q11

d12

Q21

η

d 21

z1 e z1

a 2

y1

a 2

O1

d22 η Q22

b 2

−f22

Fig. 2. Model of the flexible metal plate [15].

b 2

x1

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O0 y0

x0 z0

ui

Pij trolley ij

uij

eij

bridge i

r1

chain ij

O1 wij lij

dij

c Qij

z1

x1

y1

Fig. 3. Geometric relations for calculation of the suspension forces [15].

attachment points Qi j for i = 1, 2 and j = 1, 2, ⎡ e⎤ ⎡ ⎤ kf fG − f11 − f12 − f21 − f22 ⎢ ⎥ ⎢ ⎥ ke = ⎣ keτ ⎦ = ⎣ −ττ 11 −ττ 12 −ττ 21 −ττ 22 ⎦ e T kη −ez1 (−f11 + f12 + f21 − f22 )

with τ i j = d˜ i j fi j .

(9)

Here kef is the resulting force, keτ the resulting moment of the suspension forces with respect to O1 , and kηe is the generalised force on the elastic coordinate η due to the suspension forces. With the position vectors rPi j = ey0 ui + ex0 ui j of points Pi j and rQi j = r1 + di j of points Qi j according to Fig. 3 and under consideration of linear attachment springs (stiffness c) the chain forces are fi j = c (li j − wi j ) ei j

with ei j =

li j , li j

li j = rQi j − rPi j ,

li j = |li j | .

(10)

The generalised elastic force of the plate kel in (7) is related to the bending stiffness of the plate K under consideration of Young’s modulus E and Poisson’s ratio ν [3], ⎡ ⎤ 0 32 K (1 − ν) E h3 , K= . (11) kel = ⎣ 0 ⎦ with kηel = −cf η , cf = ab 12 (1 − ν 2 ) kel η

2.3

Direct and Inverse Dynamics and Statics

Direct dynamics of the crane manipulator yields the spatial motion of the plate rˆ (t) for given time trajectories of the actuator coordinates u(t), w(t) by means of numeri-

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cal integration of the equations of motion (7) together with the kinematic differential equations (4). Inverse dynamics enables feedforward control by calculating the actuator positions u(t), w(t) for a desired spatial trajectory of the plate given in terms of position rˆ d (t), velocity vˆ d (t) and acceleration v˙ˆ d (t). This calculation exploits the flatness property of the system [7]. The equations of motion (7) represent a system of seven nonlinear equations for the ten actuator positions u, w. Thus the problem is kinematically triple redundant. Appropriate solutions can be found by optimisation [15]. A technically meaningful objective criterion are minimal chain inclinations. Direct and inverse statics is based on the static equilibrium conditions ke (ˆr, u, w) + kel (η) = 0

(12)

included in (7). Direct statics comprises calculation of the static equilibrium pose of the plate T  (13) r¯ˆ = r¯ T1 ϕ¯ T η¯

¯ w. ¯ Here the and of the forces of the chains for given stationary actuator coordinates u, seven equilibrium equations (12) have to be iteratively solved with respect to the seven unknowns r¯ˆ under the condition of positive chain forces. Inverse statics yields the ten ¯ w ¯ to achieve a desired equilibrium pose of the plate r¯ˆ . As for actuator coordinates u, inverse dynamics redundancy resolution by optimisation is necessary.

3

Linearised Dynamics of the Crane Manipulator

The linearised equations of motion are formulated to describe small oscillations of the platform around a static equilibrium pose r¯ˆ according to (13). The small deflections of the platform from its equlibrium posen are described by T  ψ T Δη Δz = ΔrT1 Δψ (14)

ψ with ω = ψ˙ describing comprising the deflection Δr1 of point O1 , the rotation vector Δψ small rotations from the equilibrium orientation, and the increment of the elastic coordiψ nate Δη. According to the kinematic differential equation (4) the rotation increment Δψ ψ. ϕ by Δϕ ϕ = Hϕ (ϕ¯ ) Δψ is linked to the corresponding increments of the Cardan angles Δϕ With the increments of the actuator coordinates Δu and Δw from the respective station¯ a Taylor series expansion of the nonlinear equations of motion (7) ary values u¯ and w up to first-orders in the Δ-variables yields the linearised equations of motion in the form M Δ¨z + K Δz = Bu Δu + Bw Δw

(15)

the stiffness matrix K ∈ and the input matrices with the mass matrix M ∈ Bu ∈ R7,6 , Bw ∈ R7,4 . The linearised equations of motion (15) are obtained by successively linearising all calculations during formulation of the nonlinear equations of motion (7). The increments of the chain forces fi j from (10) become (magnitudes related to the equilibrium values denoted by an upper bar) R7,7 ,

R7,7

Δfi j = c (Δli j − Δwi j ) e¯ i j + c (l¯i j − w¯ i j ) Δei j .

(16)

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239

The increments of the suspension lengths Δli j and of the chain unit vectors Δei j are obtained by calculating firstly from (10) the increments of the suspension vectors Δli j = ΔrQi j − ΔrPi j

with ΔrPi j = ey0 Δui + ex0 Δui j ,

ΔrQi j = Δr1 + Δdi j . (17) Using the tensor representation of a vector product a × b ≡ a˜ b ≡ −b˜ a with the skewsymmetric tensor a˜ = −˜aT , the vector increments Δdi j are expressed by the rotation ψ according to increment Δψ ψ d¯ + φ e Δη = −d¯˜ Δψ ψ + φ e Δη , (18) Δd = Δψ ij

ij

i j z1

ij

i j z1

again with φi j = 1 for i = j and φi j = −1 for i = j. The increments of the suspension vectors then are ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Δu1 Δl11 0 0 ⎢ Δu2 ⎥ I |−d˜¯ 11 | ez1 ⎡ ⎤ −ey0 0 | −ex0 0 ⎥ ⎢Δl ⎥ ⎢ ⎥ Δr1 ⎢−e ⎥⎢ ˜ ⎥ ¯ 0 | 0 −e 0 0 x0 ⎢ 12 ⎥ ⎢ I |−d12 |−ez1 ⎥⎣ ⎦ ⎢ y0 ⎥⎢ ⎢Δu11 ⎥. (19) ψ Δψ + ⎢ ⎥ =⎢ ⎥ ⎢ ⎥ ⎢ ˜ ¯ ⎣Δl21 ⎦ ⎣ I |−d21 |−ez1 ⎦ ⎣ 0 −ey0 | 0 0 −ex0 0 ⎦⎢Δu12 ⎥ ⎥ Δη ⎦ ⎣ ˜ I |−d¯ 22 | ez1   Δl22 0 0 −ex0 Δu21 0 −ey0 | 0

  Δz

  Δu22

  Jlz Jlu Δu The increments of the suspension lengths Δli j are obtained by Δli j = e¯ Ti j Δli j = e¯ Ti j Jlz,i j Δz + e¯ Ti j Jlu,i j Δu

(20)

whereby Jlz,i j and Jlu,i j are the row entries of Jlz and Jlu , respectively, in (19). The increments of the chain unit vectors Δei j are obtained from the taylor series expansion of Δli j = li j ei j yielding Δli j e¯ i j Δei j = ¯ − ¯ Δli j (21) li j li j and with (19) and (20) Δei j =

I − e¯ i j e¯ Ti j I − e¯ i j e¯ Ti j J Jlu,i j Δu . Δz + lz,i j l¯i j l¯i j

 

  Jez,i j Jeu,i j

Inserting (20) and (21) into (16) yields the increments of the chain forces ⎡ ⎤ ⎡ ⎡ ⎤ ⎤ c (s¯11 Jez,11 + e¯ 11 e¯ T11 Jlz,11 ) c (s¯11 Jeu,11 + e¯ 11 e¯ T11 Jlu,11 ) Δf11 ⎢Δf12 ⎥ ⎢c (s¯12 Jez,12 + e¯ 12 e¯ T12 Jlz,12 )⎥ ⎢c (s¯12 Jeu,12 + e¯ 12 e¯ T12 Jlu,12 )⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎣Δf21 ⎦ = ⎣c (s¯21 Jez,21 + e¯ 21 e¯ T Jlz,21 )⎦ Δz + ⎣c (s¯21 Jeu,21 + e¯ 21 e¯ T Jlu,21 )⎦ Δu 21 21 Δf22 c (s¯22 Jez,22 + e¯ 22 e¯ T22 Jlz,22 ) c (s¯22 Jeu,22 + e¯ 22 e¯ T22 Jlu,22 )    

Jfz Jlu ⎤⎡ ⎡ ⎤ 0 0 −c e¯ 11 0 Δw11 ⎢ 0 −c e¯ 12 0 ⎢ ⎥ 0 ⎥ ⎥ ⎢Δw12 ⎥ . +⎢ ⎦ ⎣ 0 ⎣ 0 −c e¯ 21 0 Δw21 ⎦ 0 0 0 −c e¯ 22 Δw22

    Jfw Δw

(22)

(23)

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The increments of the generalised forces kef in (9) become with the row entries J f z,i j , Jlu,i j and J f w,i j of J f z , Jlu and J f w , respectively, 2

Δkef = − ∑

2

2

i=1 j=1

2

2

2

∑ J f z,i j Δz − ∑ ∑ J f u,i j Δu − ∑ ∑ J f w,i j Δw .

 Jk f ,z

i=1 j=1



 Jk f ,u

(24)

i=1 j=1



 Jk f ,w



The increments of the moments τ i j = d˜ i j fi j of the chain forces fi j with respect to O1 are (25) Δττ i j = Δd˜ i j fi j + d˜ i j Δfi j = −˜fi j Δdi j + d˜ i j Δfi j and after inserting (18) and (23)    Δττ i j = 0 | ˜¯fi j d˜¯ i j | φi j ez1 + d˜¯ i j JFz,i j Δz + d˜¯ i j J f u,i j Δu + d˜¯ i j J f w,i j Δw .

 

 

  Jτu,i j Jτw,i j Jτz,i j

(26)

In analogy to (24) the increment of the resulting torque keτ in (9) reads 2

Δkeτ = − ∑

2

2

i=1 j=1

2

2

2

∑ Jτz,i j Δz − ∑ ∑ Jτu,i j Δu − ∑ ∑ Jτw,i j Δw .

 Jkτ,z

i=1 j=1



 Jkτ,u

(27)

i=1 j=1



The increment of the generalised force kηe in (9) becomes

 Jkτ,w



Δkηe = − (−¯f11 + ¯f12 + ¯f21 − ¯f22 )T Δez1 − e¯ Tz1 (−Δf11 + Δf12 + Δf21 − Δf22 )

  ¯fTη

(28)

with   ψ e¯ z1 = −e˜¯ z1 Δψ ψ = 0 | −e˜¯ z1 | 0 Δz Δez1 = Δψ

  Jψ z

(29)

and together with Δfi j from (23)

 Δkηe = − ¯fTη Jψ z + e¯ Tz1 (−JFz,11 + JFz,12 + JFz,21 − JFz,22 ) Δz

  Jkη,z − e¯ Tz1 (−JFu,11 + JFu,12 + JFu,21 − JFu,22 ) Δu

  Jkη,u

(30)

− e¯ Tz1 (−JFw,11 + JFw,12 + JFw,21 − JFw,22 ) Δw .

  Jkη,w

The increment of the elastic force kηel from (11) is directly obtained by   Δkηel = −cη Δη = − 0 | 0 | cη Δz .

  Jηz

(31)

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With (24), (27), (30) and (31) the linearised equations of motion (15) become ⎤⎡ ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ mI 0 0 Δ¨r Jk f ,z Δr −Jk f ,u −Jk f ,w ⎣ 0 Θ 1 0 ⎦⎣Δψ¨ ⎦ + ⎣ Jkτ,z ⎦⎣Δψ ψ ⎦ = ⎣ −Jkτ,u ⎦Δu + ⎣ −Jkτ,w ⎦Δw Jkη,z + Jηz Δη −Jkη,u −Jkη,w 0 0 91 m Δη¨ M

4

Δ¨z +

K

Δz

=

Bu

Δu +

Bw

(32)

Δw .

Feedforward trajectory control

The objective of feedforward control is to move the workpiece along a desired trajectory between two rest positions with pose rˆ d (t), velocity vˆ d (t) and acceleration v˙ˆ d (t). The desired trajectory is discretised in N intervals with a time increment Δt leading to pose increments between subsequent time steps tk and tk+1 = tk + Δt, k = 0, 1, . . . , N, according to Δˆrdk = rˆ d (tk+1 ) − rˆ d (tk ) and Δzdk = H−1 (ˆrdk ) Δˆrdk

(33)

with the kinematic matrix H from (4). The desired acceleration at time tk is Δ¨zd = v˙ˆ dk . The linearised equations of motion (32) represent conditions for increments of the actuator positions Δuk and Δwk that are consistent with the desired trajectory,     Δuk Bu Bw (34) = M Δ¨zdk + K Δzdk . Δwk This is an underdetermined system of seven linear equations for the increments of the ten actuator coordinates Δuk and Δwk , corresponding to the triple kinematic redundancy. Solutions can be found by quadratic programming using an objective function being quadratic in terms of the actuator coordinate increments Δuk and Δwk and (34) as linear constraint equations. The actuator coordinates are then updated according to uk+1 = uk + Δuk ,

wk+1 = wk + Δwk ,

k = 0, 1, . . . , N,

(35)

with the actuator coordinates of the initial rest position u0 , w0 . The position-dependent system matrices in (34) are calaculated with the values of the initial rest position and kept constant during execution of the trajectory. The actuator coordinates and the increments are used as inputs for individual axis controllers that compensate disturbing effects like inertia, friction and varying loads. Within the following numerical example it is assumed that the axis controllers make the actuators track the desired trajectories ideally. In a next step simplified dynamic transfer behaviours of the controlled drives will be incorporated into the control scheme [11].

5

Numerical Example

As an example a rectangular plate with a = 2 m, b = 3 m, h = 0.01 m, ρ = 7800 kg m−3 , E = 2.1 · 1011 N m−2 , ν = 0.3 and c = 1.5 · 104 N m−1 is considered. The numerical calculations were done in Matlab. According to Fig. 4a the plate is moved from the initial

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1.5

r1dy

O0

r1dz

0.5

u1

u2

1

[m]

[m]

1

y0

bridge positions

interpolated plate position

1.5

0.5 0

x0

z0

r1dx 0

5

0

10

ϕ d3

[m]

ϕ d2

u22 u21

-0.6

5

0

10

5

ηd

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w21 w12

w22

1

0.015

10

chain lengths

interpolated plate torsion

0.02

[m]

final position t N = 10s

u11

u12

-0.4

0

0

initial position t0 = 0s

10

-0.2

2

-2

O1

5

0

ϕ 1d

[m]

[deg]

4

0

trolley positions

interpolated plate Cardan angles 6

w11

0.5

0.005 0

b

a

0

5 time [s]

0

10

c

0

5 time [s]

10

Fig. 4. Motion between prescribed rest positions of the plate. a Initial and final rest positions. b Time trajectories of the interpolated plate position r˘ d1 (t) = rd1 (t) − rd1 (t0 ), plate Cardan angles ϕ˘ (t) = ϕ d (t) − ϕ d (t0 ), and plate torsion η d (t). c Time trajectories of the bridge coordinates u˘i (t) = ui (t) − ui (t0 ), trolley coordinates u˘i j (t) = ui j (t) − ui j (t0 ), and chain lengths w˘ i j (t) = wi j (t) − wi j (t0 ).

rest position at t0 = 0 s into the final rest position at tN = 10 s with the pose coordinates rd1 (t0 ) = [ 2.0 3.0 3.0 ]T m , rd1 (tN )

T

= [ 1.6 4.5 4.0 ] m ,

ϕ d (t0 ) = [ 0 0 0 ]T deg , T

η d (t0 ) = 0.02 m ,

ϕ (tN ) = [ 5 − 3 6 ] deg , d

η (tN ) = 0 m . d

(36) (37)

The actuator coordinates belonging to the initial plate position are calculated by the nonlinear optimisation procedure with minimised chain lengths described in [15] that leads here to almost vertical chains. The time trajectories of the seven position variables of the plate in Fig. 4b are interpolated by functions being steady up to the fourth-order time derivatives. The time trajectories of the ten actuator coordinates in Fig. 4c are obtained as described in section 4 by calculating the minimum norm solution of the actuator coordinates in (34). For feedforward control simulation the trajectories of the actuator coordinates from Fig. 4c were applied as rheonomic constraints to a Simpack multibody model of the crane manipulator. The time history of point O1 in the horizontal x,y-plane is shown in Fig. 5. The comparison with the results from interpolations with simple trapezoidal velocity profiles in Fig. 5a shows the effect of the feedforward control. There are only small differences between the nonlinear and linear inverse dynamic models that are seen in detail in Fig. 5b. The residual sway motions can be traced back to the stiff plate representation used in the Simpack forward dynamics model. Thus the linearised inverse dynamics model well approximates the nonlinear one and can be used for further investigations.

Linearised Feedforward Control of a Four-Chain Crane Manipulator

1.6002

1.9

1.6

trapezoidal velocity profile [m]

[m]

residual sway motions r1x(t)

plate position r1x(t)

2

243

1.8 nonlinear/linear feedforward control

1.7

nonlinear feedforward control

1.5998 linear feedforward control

1.5996 1.5994

1.6 0

5

10

15

20

10

plate position r1y(t)

15

20

residual sway motions r1y(t)

4.5 4.5004

[m]

[m]

4

4.5 4.4998

3.5

trapezoidal velocity profile

linear feedforward control

4.4996 4.4994

3 0

a

nonlinear feedforward control

4.5002

nonlinear/linear feedforward control

5

10 time [s]

15

20

b

10

15 time [s]

20

Fig. 5. Feedforward control trajectories applied to a Simpack forward dynamic model of the crane manipulator. a Time trajectories r1x (t), r1y (t) of point O1 between the prescribed initial and final rest positions from (36), (37) with interpolated actuator coordinates by linear feedforward control according to Fig. 4c, nonlinear feedforward control calculated from (12) according to [15] (not distinguishible from linear feedforward results in this representation) and, for comparison, with trapezoidal velocity profiles. b Zooms on residual sway motions of r1x (t), r1y (t) under linear and nonlinear feedforward control.

6

Conclusion and Outlook

The nonlinear equations of motion of a CDPM suspending a flexible workpiece by four load chains are analytically linearised around a static equilibrium state of the system. From the linearised equations of motion a flatness-based linear feedforward control scheme is derived that calculates the actuator coordinates for given trajectories of the payload between rest positions. The linearised model is parametrised by the positions of the actuator coordinates and the equilibrium position of the workpiece that are to be measured during operation of the system. A numerical simulation of the crane system shows negligeable differences between nonlinear and linearised feedforward control. In a next step simplified transfer behaviours of the controlled axes will be incorporated into the control scheme. Model incertainties and external disturbances in the industrial environment will have to be compensated by an additional feedback controller that actively dampens sway motions of the payload. For this purpose a control design model will be used that is reduced to the three dominant coordinates of the sway motion Δzsway = [ Δrx Δry Δψz ]T that have to be estimated from measurements [16]. The control design model is obtained by static (Guyan) condensation of the linearised equations of motion (32). A prototype industrial crane manipulator is being prepared for experimental validations.

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Acknowledgement This research and development project was supported by the European Regional Development Fund (EFRE). Support was also provided by the lead partner Technologie-Beratungsinstitut (TBI) according to the directive for support, development and innovation of the Ministry of Economics, Construction and Tourism of Mecklenburg-Vorpommern.

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© Springer Nature Switzerland AG 2019 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 74, https://doi.org/10.1007/978-3-030-20751-9_21

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