ROMANSY 24 - Robot Design, Dynamics and Control: Proceedings of the 24th CISM IFToMM Symposium (CISM International Centre for Mechanical Sciences, 606) [1st ed. 2022] 9783031064081, 9783031064098, 3031064089

Table of contents :
Preface
Contents
Part I: Keynote Lectures
On the Stability of Macro-Mini Robotic Systems for Physical Human-Robot Interaction
1 Introduction
2 Dynamic Model of the Macro-Mini System
2.1 Dynamic Model
2.2 State-Space Formulation
2.3 Controllability, Observability and Transfer Function
3 Closed-Loop System
3.1 Feedback Control
3.2 Stability of the Closed-Loop System
4 Application to a Multi-dof Macro-Mini Robotic System for pHRI
4.1 Architecture of the Mini Robot
4.2 Control of the Macro-Mini Robot
5 Conclusion
References
A Method for the Automatic Design of Mechanisms on the Example of a Four-Bar Linkage
1 Motivation
2 Definition of Motion Tasks for Planar and Spherical Four-Bar Linkage
3 Formal Description of Four-Bar Linkages and Their Motion
4 Description of the Automated Pose Synthesis
4.1 Calculation of the Pivot Point Grid and Rack Point Grid
4.2 Pose Synthesis Instead of Coupled Curve Syntheses
4.3 Testing a Four-Bar Linkage Solution Option for Pose Synthesis
5 Constraints During Synthesis
6 Sorting Solutions After Synthesis
7 Contouring of Effector, Crank and Swing Arm
8 Layering and Collision-Free Movement of Four-Bar Linkages
9 Final Construction of the Four-Bar Linkage with Standardized Assembly Solutions
10 Fits and Assembly with Standard Parts
10.1 Printer Specific Fits
10.2 Tables for DIN and ISO
10.3 Relative Orientation of Bodies and Coordinate Systems
10.4 Subtraction Solids
10.5 Variants for Mounting and Axial and Radial Bearing Arrangements
10.6 Montage Views and Motion Videos
10.7 Slicing and Manufacturing with the 3D Printer
References
Bio-inspired Structural Intelligence for Miniature Robots in Minimal-Invasive Surgery
1 The Potential of Bio-inspired Designs for Minimal-Invasive Robots (In Surgery)
References
Part II: Wearable, Humanoid and Surgery Robotics
Development of Wearable Assist Robot for Twist and Leaning Motion
1 Introduction
2 Motion
3 Robot
3.1 Body Assist Robot
3.2 Leg Assist Robot
3.3 Wearable Assist Robot
4 Experiment
5 Conclusion
References
Preliminary Mechanical Design of a Wearable Parallel-Serial Hybrid Robot for Wrist and Forearm Rehabilitation with Consideration of Joint Misalignment Compensation
1 Introduction
2 Design and Implementation
2.1 Requirement to the Wearable Robot
2.2 Details of Mechanical Design
2.3 Design of Joint/Axis Compensation Mechanisms
3 Evaluation of the Robot Design
4 Conclusion
References
Simple Method for Humanoid Robot Posture Design
1 Introduction – State of the Art
2 Research Problem
3 Measuring Equipment and the Human Body Model
4 The Method
5 The Tests
6 Discussion and Conclusions
References
Wearable Exoskeletons for Hand Assistance: Concept and Design of a Thumb Module with Hybrid Architecture
1 Introduction
2 Requirements and Design Choices
3 Design of the Thumb Module
3.1 Rigid Architecture
3.2 Soft Architecture
3.3 Preliminary Evaluation of the New Thumb Module
4 Conclusions
References
Dynamic Modeling of a Human-Inspired Robot Based on a Newton-Euler Approach
1 Introduction
2 Description of the Multibody Model
3 Equations of Motion
4 Inverse Dynamics for Closed and Overconstrained Loops
5 Results and Discussion
References
New Hand Exoskeletons for Four Long Finger Rehabilitation
1 Introduction
2 One Finger Exoskeleton
2.1 Exoskeleton for Finger Extension
2.2 Exoskeleton for Finger Extension and Finger Flexion
3 Exoskeletons for the Four Long Fingers
3.1 First Hand-Exoskeleton for Synchronous Finger Motion
3.2 Second Hand-Exoskeleton for Simple Finger Movement
4 Conclusions
References
Model-Based Control for Arm Support Exoskeleton
1 Introduction
2 Mechanical Model
3 Control
4 Discussion
References
Design and Characterization of Modular Soft Components for an Exoskeleton Glove with Improved Wearability
1 Introduction
2 Methods
2.1 Finger Modules and Soft Exoskeleton Design
2.2 Finite Element Model
2.3 Experimental Tests on the Open Rings and Soft Exoskeleton Glove Preliminary Test
3 Results
4 Conclusions
References
Kinematic Modelling of a Parallel Robot Used in Single Incision Laparoscopic Surgery
1 Introduction
2 Kinematic Description of the Parallel Robot for SILS
3 The Geometric Modelling of the Parallel Robot for SILS
4 The Kinematic Modelling of the Parallel Robot for SILS
5 Conclusion
References
Part III: Sensorics
An Optical Curvature Sensor for Soft Robots
1 Introduction
2 Sensor Design and Fabrication
2.1 Measurement Principle
2.2 Measurement Circuit
2.3 Fabrication
3 Experimental Evaluation
3.1 Constant Curvature
3.2 Non-constant Curvature
4 Application to a Simple Soft Robot Segment
5 Conclusion
References
Methods Comparison and Proposition of New Quaternion-Based Approach for Extraction of In-plane Rotations Out of 3D Rotations
1 Introduction
2 Planar Rotation Extraction Methods
2.1 Euler-Angle Decomposition
2.2 Axis Projection Method [Jakob et al. (ch14Jakob)]
2.3 New Approach Using Quaternion Decomposition
3 Examples
4 Conclusion
References
Low-Cost Gyroscope and Accelerometer Calibration with Free Fall Pendulum Motion: Results and Sensitivity
1 Introduction
2 Sensors Error Model
3 Pendulum Model
4 Proposed Calibration Procedure
5 Results
5.1 Pendulum Parameters Model
5.2 Inertial Sensor Calibration
6 Conclusions
References
A COBOT-IMU Hand-Guiding System with Online Collision Avoidance in Null Space
1 Introduction
2 Experimental Setup
3 MoCap Calibration and Algorithm
4 Robot Control Strategy
5 Results
6 Conclusions
References
Part IV: Path Planning and Control of Robots
Finding Formations for the Non-prehensile Object Transportation with Differentially-Driven Mobile Robots
1 Introduction and Transportation Approach
2 Assumptions
3 Formation Synthesis
4 Simulation Results
5 Outlook
References
On-line Velocity Scaling Algorithm for Task Space Trajectories
1 Introduction
2 Trajectory Scaling
3 Proposed Algorithm
3.1 Underlying Ideas
3.2 Essential Concepts
3.3 Detailed Algorithm
3.4 Remarks—Algorithm Enhancements
4 Numerical Example
5 Final Comments and Conclusions
References
Path Quality Improvement of Sampling-Based Planners: An Efficient Optimization-Based Approach Using Analytical Gradients
1 Introduction
2 Problem Formulation
2.1 Path Length Post-Processor
2.2 Path Smoothness Post-Processor
3 Results
3.1 Analytical Gradient Formulation Efficiency
3.2 Optimization-Based Path Post-Processing
3.3 Task Board Assembly
4 Conclusion
References
Hexapod Posture Control for Navigation Across Complex Environments
1 Introduction
2 Model Description
3 Control Strategy
4 Results and Discussion
5 Conclusions and Future Work
References
Control Performance Improvement in Dynamically Decoupled Manipulators
1 Introduction
2 Decoupled Dynamics of the 2R Spatial Serial Manipulator and Its Motion Generation via Bang-Bang Profile
3 Design of Linear Controller
3.1 Performance Indices
3.2 Pole Placement
3.3 Lead Compensation
4 Conclusion
References
Reliable Time-Optimal Point to Point Handling of Unmounted Objects with Industrial Robots
1 Introduction
2 Inverse Dynamics Model
3 Task Constraints
4 Optimal Control Problem and Numerical Solution
5 Experimental Results
6 Conclusion
References
Design of a Weed-Control Mobile Robot Based on Electric Shock
1 Introduction
2 Selection of the Weeding Method
3 Electrode Design
3.1 Electrocution Strategy
3.2 Electrode Shape
4 Electrocution Equipment Design
5 Mobile Robot Design
6 Conclusions
References
Part V: Modeling and Synthesis of Robotic Devices
Development of Joint Mechanism that can Achieve Both Active Drive and Variable Joint Quasi-stiffness by Utilizing 4-Bar and 5-Bar Composite Linkage
1 Introduction
2 Design and Mechanism
3 Verification
4 Conclusion
References
Kinematic Synthesis of a Tendon-Driven 4R Planar Robotic Arm
1 Introduction
2 Tendon-Driven 4R Planar Robotic Arm
3 Simulations and Validation
4 Conclusions
References
Inverse and Forward Kinematics of a Reconfigurable Spherical Parallel Mechanism with a Circular Rail
1 Introduction
2 Mechanism Design
3 Inverse Kinematics
4 Forward Kinematics
5 Conclusions
References
Novel Gripper Design for Transporting of Biosample Tubes
1 Introduction
2 Calculation Model of the Gripping Device
3 Simulation Results
4 Conclusion
References
Techniques for Synthesis of Inherently Force Balanced Mechanisms
1 Introduction
2 Principal Vector Linkages
3 Techniques to Modify Principal Vector Linkages
3.1 Constraining Link Motions
3.2 Modifying Link Positions
3.3 Substituting Links
4 Example of a 2-DoF Inherently Force Balanced Mechanism
5 Discussion and Conclusion
References
A Preliminary Study of Factors Influencing the Stiffness of Aerial Cable Towed Systems
1 Introduction
2 System Modeling
3 Stiffness Modeling
4 Effect of the Number of Cables and Their Arrangement on the ACTS Stiffness
5 Conclusion and Future Works
References
Design Optimization of a 6-DOF Cable-Driven Parallel Robot for Complex Pick-and-Place Tasks
1 Introduction
2 Design, Kinematics and Statics
3 Design Optimization
4 Results
5 Conclusions and Future Work
References
Improved Resolution of a Forward Kinematic Model of Parallel Manipulators Using a Serial Approach
1 Introduction
2 Kinematic Study of TPMs
2.1 3-UPU Robot
2.2 RAF Robot
3 Comparison Between the Serial and the Classical Approach for Solving the FKM
4 Conclusion
References
Artificial Neural Network Modelling of Cable Robots
1 Introduction
2 Platform Deflections
3 ANN Modelling and Prediction
4 Discussion and Conclusions
References
Part VI: Technological Applications and Teaching Aspects
Design of a Robot for the Automatic Charging of an Electric Car
1 Introduction
2 Design Requirements of a Robot for the Automatic Charging of an Electric Car
3 Mechanism Understudy
3.1 The Mechanism: P-Pi-R-P Planar Parallel Robot
3.2 The Mechanism: Serial Robot
3.3 The Mechanism
4 Workspace Analysis
5 Stiffness Analysis
6 Hardware and Software
7 Conclusions
References
A New Parameter Selection Method for Power Skiving Tools
1 Introduction
2 Kinematics of Helical Gears with Skew Axes
2.1 The Infinitesimal Teeth
2.2 The Case of Power Skiving
3 Tool Parameters Selection Method
3.1 Requirement of Good Surface Finishing
3.2 Requirement of Low Cutting Forces
3.3 Constraint for Collision Avoidance
4 Working Example
5 Conclusions
References
Low Cost Delta Robot for the Experimental Validation of Frame Vibration Reduction Methods
1 Introduction
2 Vibration Reduction and Its Application to Delta Robots
3 The Suisui Bot
4 Experimental Results
5 Conclusion
References
Influence of the Receiving Timing of an Online Lecture in Actual Practical Mechanical Engineering Training
1 Introduction
2 Practical Training
3 Influence of Receiving Timing
4 Teaching Material to Aid in Understanding
5 Conclusions
References
Author Index

Citation preview

CISM International Centre for Mechanical Sciences 606 Courses and Lectures

Andrés Kecskeméthy Vincenzo Parenti-Castelli Editors

ROMANSY 24 Robot Design, Dynamics and Control Proceedings of the 24th CISM IFToMM Symposium International Centre for Mechanical Sciences

CISM International Centre for Mechanical Sciences Courses and Lectures Volume 606

Managing Editor Paolo Serafini, CISM—International Centre for Mechanical Sciences, Udine, Italy Series Editors Elisabeth Guazzelli, Laboratoire Matière et Systèmes Complexes, Université Paris Diderot, Paris, France Alfredo Soldati, Institute of Fluid Mechanics and Heat Transfer, Technische Universität Wien, Vienna, Austria Wolfgang A. Wall, Institute for Computational Mechanics, Technische Universität München, Munich, Germany Antonio De Simone, BioRobotics Institute, Sant'Anna School of Advanced Studies, Pisa, Italy

For more than 40 years the book series edited by CISM, “International Centre for Mechanical Sciences: Courses and Lectures”, has presented groundbreaking developments in mechanics and computational engineering methods. It covers such fields as solid and fluid mechanics, mechanics of materials, micro- and nanomechanics, biomechanics, and mechatronics. The papers are written by international authorities in the field. The books are at graduate level but may include some introductory material.

More information about this series at https://link.springer.com/bookseries/76

Andrés Kecskeméthy Vincenzo Parenti-Castelli •

Editors

ROMANSY 24 - Robot Design, Dynamics and Control Proceedings of the 24th CISM IFToMM Symposium

123

Editors Andrés Kecskeméthy Department of Mechanical Engineering University of Duisburg-Essen Duisburg, Germany

Vincenzo Parenti-Castelli Department of Industrial Engineering University of Bologna Bologna, Italy

ISSN 0254-1971 ISSN 2309-3706 (electronic) CISM International Centre for Mechanical Sciences ISBN 978-3-031-06408-1 ISBN 978-3-031-06409-8 (eBook) https://doi.org/10.1007/978-3-031-06409-8 © CISM International Centre for Mechanical Sciences 2022 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

Preface

ROMANSY 2022, the 24th CISM-IFToMM Symposium on the Theory and Practice of Robots and Manipulators, returns to Udine celebrating its 50 year edition of the series that started in 1973 under the joint patronage of International Centre for Mechanical Science (CISM) and International Federation for the Promotion of Mechanism and Machine Science (IFToMM). The first symposium was held on 5–8 September 1973 in the main hall of the Palazzo del Torso, Udine. It was the first research-based conference dealing with robot design, mechanics and control that brought together researchers from Eastern Europe and the West. Since its birth, the acronym ROMANSY stands for emphasizing the spirit of the series by combining the first letters of the words: “Robots”, “Manipulators” and “Symposium”. It was thanks to world-famous scientists in the area of Robotics such as Academician I.I. Artobolevskii (USSR), Prof. Bianchi (Italy), Prof. A. P. Bessonov (URSS), Prof. Kato (Japan), Prof. A. E. Kobrinsii (USSR), Prof. M. S. Konstantinov (Bulgaria), Prof. A. Morecki (Poland), Prof. A. Romiti (Italy), Prof. B. Roth (USA), L. Sobrero (Italy), Prof. M. W. Thring (UK), Prof. M. Vucobratovic (Jugoslavia), Prof. H. J. Warnecke (Germany) and Prof. Withney (USA) that this symposium was brought into live, and it is in the spirit of this origination that these series continue to contribute to the development of robotic technology. The CISM-IFToMM ROMANSY Symposia have played from the outset a significant role in the development of the theory and practice of Robotics, opening and maintaining a strong multinational scientific exchange across the world. Recalling the series of ROMANSY gives an impression of the strong impact and actuality these series have had and has over the many decades since its beginning, with its basic biennial nature missed in the presence only in 2020 due to the then raging historic worldwide COVID-19 pandemic: 1973: 1976: 1978: 1981:

ROMANSY ROMANSY ROMANSY ROMANSY

1 2 3 4

in in in in

Udine, Italy, with chairmanship of A. E. Kobrinskii Jadwisin, Poland, with chairmanship of B. Roth Udine, Italy, with chairmanship of L. Sobrero Zaborow, Poland, with chairmanship of A. Morecki

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1984: ROMANSY 5 in Udine, Italy, with chairmanship of G. Bianchi 1986: ROMANSY 6 in Cracow, Poland, with chairmanship of A. Morecki 1988: ROMANSY 7 in Udine, Italy, with chairmanship of G. Bianchi and A. Morecki 1990: ROMANSY 8 in Cracow, Poland, with chairmanship of A. Morecki and G. Bianchi 1992: ROMANSY 9 in Udine, Italy, with chairmanship of G. Bianchi and A. Morecki 1994: ROMANSY 10 in Gdansk, Poland, with chairmanship of A. Morecki and G. Bianchi 1996: ROMANSY 11 in Udine, Italy, with chairmanship of G. Bianchi and A. Morecki 1998: ROMANSY 12 in Paris, France, with chairmanship of A. Morecki, G. Bianchi and J. C. Guinot 2000: ROMANSY 13 in Zakopane, Poland, with chairmanship of A. Morecki and G. Bianchi 2002: ROMANSY 14 in Udine, Italy, with chairmanship of G. Bianchi and J. C. Guinot 2004: ROMANSY 15 in Montreal, Canada, with chairmanship of J. Angeles and J. C. Piedboeuf 2006: ROMANSY 16 in Warsaw, Poland, with chairmanship of T. Zielinska 2008: ROMANSY 17 in Tokyo, Japan, with chairmanship of A. Takanishi and Y. Nakamura 2010: ROMANSY 18 in Udine, Italy, with chairmanship of W. Schiehlen and V. Parenti-Castelli 2012: ROMANSY 19 in Paris, France, with chairmanship of P. Bidaud and O. Khatib 2014: ROMANSY 20 in Moscow, Russia, with chairmanship of M. Ceccarelli and V. Glazunov 2016: ROMANSY 21 in Udine, Italy, with chairmanship of V. Parenti-Castelli and W. Schiehlen 2018: ROMANSY 22 in Rennes, France, with chairmanship of V. Arakelyan and P. Wenger 2020: ROMANSY 23 in Tokyo, Japan (online), with the chairmanship of G. Venture, J. Solis, Y. Takeda and A. Konno Currently, there is a wide choice of conferences on Robotics. However, despite continued growth of scientific conferences, after 49 years, the ROMANSY Symposium remains an important event for the Robotics community that brings together researchers from all around the world. The present proceedings volume contains 36 papers that have been selected after review. These papers cover several areas of the wide field of Robotics concerning wearable, humanoid and surgery robotics, sensorics, path planning and control of robots, modelling and synthesis of robotic devices, as well as technological applications and teaching aspects.

Preface

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We are grateful to all reviewers for the time and efforts they spent in evaluating the papers. Their valuable comments were helpful for the improvement of papers. A very special thanks to the Advisory Board and International Scientific Committee members for their active participation in the review process. The Advisory Board of the 24th ROMANSY Symposium is composed of the following members: Vincenzo Parenti-Castelli (Italy) CISM Representative Philippe Bidaud (France) Marco Ceccarelli (Italy) I-Ming Chen (Singapore) Victor Glazunov (Russia) Oussama Khatib (USA) Werner Schiehlen (Germany) Atsuo Takanishi (Japan) Yukio Takeda (Japan) Teresa Zielinska (Poland) and the International Scientific Committee is composed of the members: Giuseppe Carbone (Italy) IFToMM Representative (TC Chair) Vigen Arakelyan (France) Greg Chirikjian (Singapore) Paulo Flores (Portugal) Manfred Husty (Austria) Qiang Huang (China) Yan Jin (UK) Andres Kecskemethy (Germany) IFToMM President Igor Orlov (Russia) Gentiane Venture (Japan) Marek Wojtyra (Poland) The Programme Committee Co-chairs Werner Schiehlen (Germany) Honorary Chair Andres Kecskemethy (Germany) Co-Chair Vincenzo Parenti-Castelli (Italy) Co-Chair are thankful for the support of International Federation for the Promotion of Mechanism and Machine Science (IFToMM) and International Centre for Mechanical Science (CISM). We would also like to thank the members of the Local Organizing Committee, in particular Dr Marika Minisini and Prof. Paolo Serafini, for all their efforts in planning this event in Udine. In closing, we would like to remember the important message conveyed by the then President of IFToMM, Prof. I.I. Artobolevskii, in his preface to the first ROMANSY conference:

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“Objects under our study - robots and manipulators - are just such objects. And for all of us - specialists working in adjoining fields of science - they are of “ burning” interest. We listened to and discussed with great interest and advantage many reports devoted to the structure, kinematics and dynamics of robots and manipulators, as well as to biomechanics and to problems of such systems control and to the creation of man-made (artificial) intellect, that may be used in these systems. That is why it is possible to draw the conclusion that this Symposium was very important not only for the development of problems connected with the theory and principles of robot and manipulator design, but was also very useful and fruitful for all participants from the point of view of interchange of experience, as well as interchange of new results and ideas.” We believe that this message has lost none of its freshness and timeliness, not only from the viewpoint of the farsightedness of the topics addressed in this first Symposium, but also from its wise vision of fostering academic liaison over borders, which is now more important than ever. We hope and look forward that this will continue not only now but also for the years to come, for the good of science and humankind. April 2022

Andrés Kecskeméthy Vincenzo Parenti-Castelli

Contents

Part I: Keynote Lectures On the Stability of Macro-Mini Robotic Systems for Physical Human-Robot Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clément Gosselin

3

A Method for the Automatic Design of Mechanisms on the Example of a Four-Bar Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tim Lueth and Franz Irlinger

12

Bio-inspired Structural Intelligence for Miniature Robots in Minimal-Invasive Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Georg Rauter, Lorin Fasel, Manuela Eugster, and Nicolas Gerig

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Part II: Wearable, Humanoid and Surgery Robotics Development of Wearable Assist Robot for Twist and Leaning Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koji Makino, Yuta Dobashi, Masato Karaki, Xiao Sun, and Hidetsugu Terada Preliminary Mechanical Design of a Wearable Parallel-Serial Hybrid Robot for Wrist and Forearm Rehabilitation with Consideration of Joint Misalignment Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . Ying-Chi Liu, Andrea Botta, Giuseppe Quaglia, and Yukio Takeda Simple Method for Humanoid Robot Posture Design . . . . . . . . . . . . . . . Teresa Zielinska and Oguz Kahraman Wearable Exoskeletons for Hand Assistance: Concept and Design of a Thumb Module with Hybrid Architecture . . . . . . . . . . . . . . . . . . . Chiara Brogi, Nicola Secciani, Lorenzo Bartalucci, Marco Pagliai, Benedetto Allotta, Alessandro Ridolfi, and Andrea Rindi

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Dynamic Modeling of a Human-Inspired Robot Based on a Newton-Euler Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fernando Gonçalves, Tiago Ribeiro, António Fernando Ribeiro, Gil Lopes, and Paulo Flores

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New Hand Exoskeletons for Four Long Finger Rehabilitation . . . . . . . . Antonello Vuocolo, Lei Cao, Andrea Petinari, Nicola Sancisi, Yukio Takeda, and Vincenzo Parenti-Castelli

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Model-Based Control for Arm Support Exoskeleton . . . . . . . . . . . . . . . Mikhail Chumichev and Igor Orlov

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Design and Characterization of Modular Soft Components for an Exoskeleton Glove with Improved Wearability . . . . . . . . . . . . . . 106 Tommaso Bagneschi, Daniele Leonardis, Domenico Chiaradia, and Antonio Frisoli Kinematic Modelling of a Parallel Robot Used in Single Incision Laparoscopic Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Paul Tucan, Bogdan Gherman, Iulia Andras, Calin Vaida, and Doina Pisla Part III: Sensorics An Optical Curvature Sensor for Soft Robots . . . . . . . . . . . . . . . . . . . . 125 Malte Grube and Robert Seifried Methods Comparison and Proposition of New Quaternion-Based Approach for Extraction of In-plane Rotations Out of 3D Rotations . . . 133 Mehdi Ghiassi, Joey Maibaum, and Andrés Kecskeméthy Low-Cost Gyroscope and Accelerometer Calibration with Free Fall Pendulum Motion: Results and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . 142 Giulia Avallone, Lorenzo Agostini, Michele Conconi, Vincenzo Parenti-Castelli, Rocco Vertechy, and Nicola Sancisi A COBOT-IMU Hand-Guiding System with Online Collision Avoidance in Null Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Giulia Avallone, Eugenio Monari, Yi Chen, Lorenzo Agostini, Nicola Sancisi, and Rocco Vertechy Part IV: Path Planning and Control of Robots Finding Formations for the Non-prehensile Object Transportation with Differentially-Driven Mobile Robots . . . . . . . . . . . . . . . . . . . . . . . . 163 Henrik Ebel, Daniel Niklas Fahse, Mario Rosenfelder, and Peter Eberhard On-line Velocity Scaling Algorithm for Task Space Trajectories . . . . . . 171 Marek Wojtyra and Łukasz Woliński

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Path Quality Improvement of Sampling-Based Planners: An Efficient Optimization-Based Approach Using Analytical Gradients . . . . . . . . . . 182 Jonas Wittmann, Corina Klein, and Daniel Rixen Hexapod Posture Control for Navigation Across Complex Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Joana Coelho, Bruno Dias, Gil Lopes, Fernando Ribeiro, and Paulo Flores Control Performance Improvement in Dynamically Decoupled Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Yaodong Lu, Yannick Aoustin, and Vigen Arakelian Reliable Time-Optimal Point to Point Handling of Unmounted Objects with Industrial Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Hubert Gattringer and Andreas Müller Design of a Weed-Control Mobile Robot Based on Electric Shock . . . . . 220 Giuliano Morara, Alberto Baldassarri, Jens Diepenbruck, Tobias Bruckmann, and Marco Carricato Part V: Modeling and Synthesis of Robotic Devices Development of Joint Mechanism that can Achieve Both Active Drive and Variable Joint Quasi-stiffness by Utilizing 4-Bar and 5-Bar Composite Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Hiroki Mineshita, Takuya Otani, Yuji Kuroiwa, Masanori Sakaguchi, Yasuo Kawakami, Hun-ok Lim, and Atsuo Takanishi Kinematic Synthesis of a Tendon-Driven 4R Planar Robotic Arm . . . . . 238 Chiara Lanni, Giorgio Figliolini, and Michele D’Angelo Inverse and Forward Kinematics of a Reconfigurable Spherical Parallel Mechanism with a Circular Rail . . . . . . . . . . . . . . . . . . . . . . . . 246 Pavel Laryushkin, Anton Antonov, Alexey Fomin, and Victor Glazunov Novel Gripper Design for Transporting of Biosample Tubes . . . . . . . . . 255 Artem A. Voloshkin, Larisa A. Rybak, Giuseppe Carbone, and Vladislav V. Cherkasov Techniques for Synthesis of Inherently Force Balanced Mechanisms . . . 263 Lorenzo Girgenti and Volkert van der Wijk A Preliminary Study of Factors Influencing the Stiffness of Aerial Cable Towed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Vincenzo Di Paola, Edoardo Idà, Matteo Zoppi, and Stéphane Caro Design Optimization of a 6-DOF Cable-Driven Parallel Robot for Complex Pick-and-Place Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Luca Guagliumi, Alessandro Berti, Eros Monti, and Marco Carricato

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Contents

Improved Resolution of a Forward Kinematic Model of Parallel Manipulators Using a Serial Approach . . . . . . . . . . . . . . . . . . . . . . . . . 292 Majdi Meskini, Houssem Saafi, Abdelfattah Mlika, Marc Arsicault, Saïd Zeghloul, and Med Amine Laribi Artificial Neural Network Modelling of Cable Robots . . . . . . . . . . . . . . 301 Leila Notash Part VI: Technological Applications and Teaching Aspects Design of a Robot for the Automatic Charging of an Electric Car . . . . . 311 Damien Chablat, Riccardo Mattacchione, and Erika Ottaviano A New Parameter Selection Method for Power Skiving Tools . . . . . . . . 322 Enea Olivoni, Rocco Vertechy, and Vincenzo Parenti-Castelli Low Cost Delta Robot for the Experimental Validation of Frame Vibration Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Christian Mirz, Mathias Hüsing, Yukio Takeda, and Burkhard Corves Influence of the Receiving Timing of an Online Lecture in Actual Practical Mechanical Engineering Training . . . . . . . . . . . . . . . . . . . . . . 340 Koji Makino, Yasutake Haramiishi, Chaofeng Zhang, and Hiroshi Hashimoto Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Part I: Keynote Lectures

On the Stability of Macro-Mini Robotic Systems for Physical Human-Robot Interaction Cl´ement Gosselin(B) Laboratoire de Robotique, D´epartement de G´enie M´ecanique, Universit´e Laval, Qu´ebec, QC, Canada [email protected]

Abstract. This paper addresses the stability of macro-mini robotic systems in a context of physical human-robot interaction (pHRI). A simple model including a one-degree-of-freedom (one-dof) macro robot and a one-dof mini robot is investigated using a state-space representation in order to assess the controllability and stability of the system. The conclusions of the analysis can be used in the design of multi-dof macro-mini systems. An example of a 12-dof macro-mini system is briefly described to illustrate potential applications.

1

Introduction

The complete automation of some industrial tasks remains difficult to implement due to the complexity and variability of the interactions. Physical human-robot interaction (pHRI) is therefore viewed as a means of combining the adaptation capabilities of humans with the effectiveness of robots (Kr¨ uger et al. 2009). However, the interaction with large robots can be uncomfortable for human operators. In this context, the concept of macro-mini robotic system (Sharon et al. 1988) emerges as a possible avenue to allow human operators to interact with a low impedance mini-robot mounted on a larger high-impendance robot (Labrecque et al. 2016). In (Labrecque et al. 2016, 2017; Badeau et al. 2018), active macro/passive mini systems are shown to yield very intuitive interaction and effective implementation for industrial tasks. However, for large payloads, a passive mini robot is not ideal because the human user must handle the inertia of the payload. In such a context, an active macro/active mini system is preferable so that the active mini robot can assist the human user in handling the inertia of the payload. This active macro/active mini paradigm raises the issue of stability due to the interaction between the actuators of the macro, the actuators of the mini and the human input. This paper studies the dynamics and stability of active macro/active mini robotic systems. First, the dynamic model of a simple one-dof macro/ one-dof mini system is established. The model obtained is formulated using the statespace approach in order to facilitate the analysis. Controllability and observability of the model are then verified and the transfer function is obtained. Feedback c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 3–11, 2022. https://doi.org/10.1007/978-3-031-06409-8_1

4

C. Gosselin

control is then included in the model to simulate the effect of the controllers. Assuming a certain behaviour of the human input, stability is then studied. It is shown that the macro-mini system is generally stable and that its behaviour can be adjusted using the control gains. Finally, the application of this approach to a multi-dof system is briefly presented.

2 2.1

Dynamic Model of the Macro-Mini System Dynamic Model

For purposes of analysis, a simple translational macro-mini robotic system is used here. Consider the macro-mini robotic system represented schematically in Fig. 1. A macro robot positions a first body of mass m1 along the direction of motion. A mini robot is mounted on the macro robot and positions a second body of mass m2 with respect to the first body. A human user is interacting with the end-effector of the mini robot and displaces it along the direction of motion. The mini robot’s task is to accommodate the human user by following the imparted motion while rendering a prescribed low impedance to the user. In a real setting, the motion range of the mini robot with respect to its central position is limited (here the limits are ±L) while the motion range of the macro robot can be large. The task ascribed to the macro robot is to ensure that the mini robot remains within its range limits so that the proper prescribed impedance is rendered to the user. The position of the macro robot with respect to the fixed frame is noted s1 while the position of the mini robot with respect to the macro robot is noted s2 . The position of the mini robot with respect to the fixed frame is noted s3 , where s3 = s1 + s2 . The macro robot, of mass m1 , is subjected to an actuating force noted f1 while the mini robot, of mass m2 , is subjected to an actuating force noted f2 . The force applied by the human user on the mini robot is noted fh . Referring to the free-body diagram of Fig. 1 and applying Newton’s second law, the following dynamic model is readily obtained

L

s3 s1

s2

f1 − f2 = m1 s¨1

(1)

fh + f2 = m2 s¨3 .

(2)

L

m2 m1

fh

m2

f2 f2 m1 f1

Fig. 1. Schematic representation of the macro-mini robotic system and free-body diagram.

On the Stability of Macro-Mini Robotic Systems for pHRI

2.2

5

State-Space Formulation

Globally, the macro-mini robotic system described above has two degrees of freedom and can therefore be cast in the state-space formulation using four state variables. One has x˙ = Ax + Bu (3) where x = [s1 , s˙ 1 , s3 , s˙ 3 ]T is the state vector, u = [f1 , f2 , fh ]T is the input vector and matrices A and B are written as ⎤ ⎡ ⎤ ⎡ 0 0 0 0100 1 ⎥ ⎢0 0 0 0⎥ ⎢ 1 ⎥ , B = ⎢ m1 − m1 0 ⎥ . (4) A=⎢ ⎣0 0 0 1⎦ ⎣ 0 0 0 ⎦ 1 1 0000 0 m2 m2 Also, considering the output vector of the system as the position of the macro and the position of the mini, namely y = [s1 , s3 ]T , one has y = Cx with

 1000 C= 0010

(5)

(6)

which completes the state-space formulation. 2.3

Controllability, Observability and Transfer Function

The controllability of the system can be established by determining the rank of the controllability matrix, written as

 rank B, AB, A2 B, A3 B . (7) Knowing that A2 = A3 = 0, it is apparent that the rank of the controllability matrix is dependent on the first two blocks only (matrices B and AB), where ⎡ 1 ⎤ 1 m1 − m1 0 ⎢ 0 0 0 ⎥ ⎥ (8) AB = ⎢ 1 1 ⎦. ⎣ 0 m2 m2 0 0 0 By inspection of Eqs. (4) and (8), it is clear that the rank of the controllability matrix is equal to four (full rank), which means that the system is controllable. Similarly, the observability of the system can be established by determining the rank of the observability matrix, written as ⎛⎡ ⎤⎞ C ⎜⎢ CA ⎥⎟ ⎢ ⎥⎟ rank ⎜ (9) ⎝⎣ CA2 ⎦⎠ . CA3

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C. Gosselin

Since A2 = A3 = 0, only the first two blocks are significant. Also, one has  0100 CA = . (10) 0001 By inspection of Eqs. (6) and (10), it is clear that the rank of the observability matrix is equal to four (full rank), which means that the system is observable. The transfer function matrix G between the input and the output can also be obtained based on the above model. One has y = Gu,

with

G(s) = C(s1 − A)−1 B

(11)

where s is the Laplace variable and 1 stands for the identity matrix. One then obtains  1 1 0 2 − m s2 1 G(s) = m1 s . (12) 1 1 0 m2 s2 m2 s2

3

Closed-Loop System

3.1

Feedback Control

As mentioned above, the task assigned to the macro robot is to ensure that the mini robot remains as close as possible to its reference position (centre of its range or motion). Also, the task assigned to the mini robot is to follow the user’s input while rendering to the user a prescribed mass, noted mr , and a prescribed damping, noted cr . Therefore, the feedback controllers are chosen as follows f1 = kp (s3 − s1 ) + kd (s˙ 3 − s˙ 1 )

(13)

f2 = (m2 − mr )¨ s3 − cr s˙ 3

(14)

where kp and kd are the gains of the PD controller used for the macro robot. Substituting Eqs. (13) and (14) into Eqs. (1) and (2) then yields s3 + m1 s¨1 = kp (s3 − s1 ) + kd (s˙ 3 − s˙ 1 ) + cr s˙ 3 (m2 − mr )¨ mr s¨3 = fh − cr s˙ 3 .

(15) (16)

Considering the input of this closed-loop system as fh , namely setting u = fh , the state-space formulation of the closed-loop system is obtained by solving Eq. (16) for s¨3 and substituting into Eq. (15) to obtain an expression for s¨1 , leading to x˙ = A x + b u (17) where

⎡

0

1

0

⎢ kp k − kd p ⎢− A = ⎢ m1 m1 m1 ⎣ 0 0 0 0 0 0

1 m1



0



⎤

r) ⎥ cr + kd + cr (mm2 −m ⎥ r ⎥ ⎦ 1 cr − mr

(18)

On the Stability of Macro-Mini Robotic Systems for pHRI

and

 b= 0

(mr −m2 ) m1 mr

0

1 mr

T

.

7

(19)

Considering the output vector defined in Eq. (5) and the matrix defined in Eq. (9) constructed with matrix A , it can be readily shown that the system is observable. Also, considering the matrix defined in Eq. (7) and vector b , it can be shown that the system is controllable. In other words, the system can be controlled from the user input given the controllers defined in Eqs. (13) and (14). The eigenvalues of matrix A are readily obtained, namely  cr kd 1 kd2 4kp , λ3,4 = − ± − . (20) λ1 = 0, λ2 = − 2 mr 2m1 2 m1 m1 It is readily observed that all eigenvalues are located in the left half-plane of the complex plane. 3.2

Stability of the Closed-Loop System

In order to investigate the stability of the system, it is now assumed that, starting from arbitrary initial conditions, the human user tries to bring the end-effector back to the reference position, s3 = 0, by behaving like a proportional controller. The control equations then become f1 = kp (s3 − s1 ) + kd (s˙ 3 − s˙ 1 ) f2 = (m2 − mr )¨ s3 − cr s˙ 3

(21) (22)

fh = −kh s3 .

(23)

Substituting Eqs. (21), (22) and (23) into Eqs. (1) and (2) then leads to s3 + m1 s¨1 = kp (s3 − s1 ) + kd (s˙ 3 − s˙ 1 ) + cr s˙ 3 (m2 − mr )¨ mr s¨3 = −cr s˙ 3 − kh s3 .

(24) (25)

Solving Eq. (25) for s¨3 and substituting into Eq. (24) to obtain an expression for s¨1 , the state-space formulation of the closed-loop system can then be written as x˙ = A x with

⎡

0

1

⎢− kp − kd ⎢ A = ⎢ m1 m1 ⎣ 0 0 0 0

1 m1

0   kh (m2 −mr ) kp + mr 0 kh −m r

1 m1

(26)



⎤  r) ⎥ cr + kd + cr (mm2 −m ⎥ r ⎥. ⎦ 1 cr − mr 0

(27)

The stability of the system of Eq. (26) is determined by the eigenvalues of matrix A , which are readily obtained as   kd 1 kd2 4kp cr 1 c2r 4kh λ1,2 = − ± − , λ3,4 = − ± − . (28) 2 2 2m1 2 m1 m1 2mr 2 mr mr

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C. Gosselin

By inspection of Eq. (28), it is clear that the four eigenvalues always have a negative real part. Indeed, if the term under the square root is positive, it is clearly smaller than the square of the first negative term. Therefore, it can be concluded that the system is stable. Obviously, simplifying assumptions have been made regarding the behaviour of the human user, which may not be exact. Nevertheless, the results provide confidence in the stability of the real system. Moreover, the inspection of Eq. (28) shows that the behaviour of the closed-loop system can be adjusted by selecting the gains and the prescribed rendered inertia and damping. It is observed that the rendered mass mr cannot be set to zero but that it can be set to a value smaller than the real mass of the mini robot m2 , thereby making the interaction easy and friendly for the human user. Selecting the rendered mass and damping should be done through experimental validation for better results.

4

Application to a Multi-dof Macro-Mini Robotic System for pHRI

Figure 2 shows a macro-mini robotic system that consists of a 3-dof translational gantry robot (the macro) on which a (6+3)-dof hybrid parallel robot (the mini) is mounted. The architecture of the mini robot is described in detail in (Wen et al. 2020) and is briefly recalled here for quick reference.

Fig. 2. Macro-mini robotic system for pHRI and close-up view of the mini robot.

On the Stability of Macro-Mini Robotic Systems for pHRI

4.1

9

Architecture of the Mini Robot

The architecture of the mini robot consists of a moving platform connected to the base by three identical kinematically redundant R(RR-RRR)SR legs. One of the legs is shown in Fig. 3. Here, R stands for an actuated revolute joint, R stands for a passive revolute joint and S stands for a passive spherical joint. In each of the legs, a first actuated revolute joint is mounted at the base. Then, two actuated revolute joints with collinear axes are mounted on the first moving link and are used to actuate two bars of a planar 5-bar linkage (see Fig. 3). The combination of these three actuated joints is used to position the spherical joint of the ith leg (point Si ) in space, as illustrated in Fig. 3. The spherical joint at point Si is then connected to a link that is in turn connected to the moving platform using a revolute joint. Globally, the mini robot has 9 degrees of freedom. Six degrees of freedom correspond to the positioning and orientation of the end platform and the additional three degrees of freedom correspond to the redundant coordinates, which are used to avoid singularities and operate the gripper. The velocity equations of the mini robot can be written as Jt = Kθ˙

(29)

where J is a 6×6 Jacobian matrix, t is the six-dimensional Cartesian twist vector of the platform, K is a 6 × 9 Jacobian matrix and θ˙ is the nine-dimensional actuated joint velocity vector. In (Wen et al. 2020), it is shown that by using the redundant degrees of freedom, it is possible to alleviate all singularities. A mechanism is also presented to use the redundant degrees of freedom to operate a gripper so that no actuator is needed at the gripper. The mini robot has a direct-drive architecture, which makes it completely backdrivable. The mini robot can therefore render a very low impedance to the human user, as demonstrated in videos (see for instance: https://www.youtube. com/watch?v=29pGFE3Xy2M). Moreover, the mini robot is kinematically redundant and provides a very large range of rotational motion by alleviating singularities. 4.2

Control of the Macro-Mini Robot

The macro robot provides a large workspace while the mini robot provides a low-impedance interaction to the human user. Since the macro robot is a purely translational robot, the macro-mini control approach described above applies only to the translational dofs while the orientational dofs are provided solely by the mini robot. The controller of the macro-mini robot takes advantage of the formulation given in this paper. With proper gain adjustments, the macro robot handles the large amplitude low-frequency displacements imparted by the human user while the mini robot handles the small amplitude high-frequency motions. This approach is akin to the concept of mid-ranging (Badeau et al. 2015) and is currently being demonstrated experimentally. The general architecture of the controller is illustrated in Fig. 4. The human user handles the end-effector of

10

C. Gosselin n Si si3

si7

li5 si5

si2

li3

z

li4 βi si4 di

θi2

y O

ri

x

Q p

θi3 θi1 si6 ei1

ei2 ei3

z

y

si1 α

O

x

Fig. 3. Geometric description of one leg of the mini robot. Prescribed redundant coordinates

Direct Kinematics User input

Robot

Redundancy mini orientation mid-ranging position

macro-mini IK

Fig. 4. Architecture of the controller of the macro-mini robotic system for pHRI.

the mini robot. The motion imparted to the robot is determined based on the actuated joint encoders (9 joints for the mini robot and 3 joints for the macro robot). The controller then applies feedback to control the redundant degrees of freedom and the orientation of the mini robot. For the position, the macro-mini approach described in this paper is used so that the macro robot can mid-range the mini robot. As a result, the mini robot reacts to the high-frequency motion while the macro robot takes care of the large amplitude low-frequency motion.

5

Conclusion

The stability of macro-mini robotic systems for pHRI was addressed in this paper. Using a simple model, it was shown that an active mini robot can be used to assist the human user in the manipulation of a payload while ensuring stability, which is highly relevant for large payloads. An example of a multi-dof system was described to illustrate the applicability of the concept.

On the Stability of Macro-Mini Robotic Systems for pHRI

11

Acknowledgements. The financial support of the Natural Sciences and Engineering Research Council of Canada and of the Fonds de Recherche du Qu´ebec, Nature et Technologie is gratefully acknowledged.

References Badeau, N., Gosselin, C., Foucault, S., Lalibert´e, T., Abdallah, M.E.: Intuitive physical human-robot interaction: using a passive parallel mechanism. IEEE Robot. Autom. Mag. 25(2), 28–38 (2018) Kr¨ uger, J., Lien, T.K., Verl, A.: Cooperation of human and machines in assembly lines. CIRP Ann. Manuf. Technol. 58(2), 628–646 (2009). ISSN 0007-8506. https:// doi.org/10.1016/j.cirp.2009.09.009 Labrecque, P.D., Hach´e, J.M., Abdallah, M.E., Gosselin, C.: Low-impedance physical human-robot interaction using an active-passive dynamics decoupling. IEEE Robot. Autom. Lett. 1(2), 938–945 (2016). ISSN 2377-3766. https://doi.org/10.1109/LRA. 2016.2531124 Labrecque, P.D., Lalibert´e, T., Foucault, S., Abdallah, M.E., Gosselin, C.: uMan: a low-impedance manipulator for human-robot cooperation based on underactuated redundancy. IEEE/ASME Trans. Mechatron. 22(3), 1401–1411 (2017) Ma, Z., Hong, G.-S., Ang, M.H., Poo, A.-N.: Mid-ranging control of a macro/mini manipulator. In: 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM), pp. 755–760. IEEE (2015) Sharon, A., Hogan, N., Hardt, D.E.: High bandwidth force regulation and inertia reduction using a macro/micro manipulator system. In: IEEE International Conference on Robotics and Automation (ICRA), vol. 1, pp. 126–132 (April 1988). https://doi. org/10.1109/ROBOT.1988.12036 Wen, K., Nguyen, T.-S., Harton, D., Lalibert´e, T., Gosselin, C.: A backdrivable kinematically redundant (6+3)-degree-of-freedom hybrid parallel robot for intuitive sensorless physical human-robot interaction. IEEE Trans. Rob. 37(4), 1222–1238 (2020)

A Method for the Automatic Design of Mechanisms on the Example of a Four-Bar Linkage Tim Lueth(B) and Franz Irlinger Chair of Micro Technology and Medical Device Technology, Technical University of Munich, Munich, Germany [email protected]

1

Motivation

Mechanical motion automata such as four-bar linkages will regain much of their importance in the 21st century due to the now very inexpensive and widely available 3D printing technology. With appropriate design methodology, such motion automata can be designed and manufactured in the same time it would take to program a robot for this motion task. However, motion automata are smaller, stiffer, more durable and, due to the reduced number of drives, significantly less expensive than today’s robots. Today’s robot technology with motors in the joints also cannot be easily scaled in size for many reasons. In the future, we will increasingly see robots in which the simple mechanical joints are replaced by four-bar joints and the mechanical degrees of freedom will have many more different joint and drive variants. Motion automata are also the basis for complex mechanical algorithms or self-reproducing machines, which in the future will be sought more in the world of microtechnology and nanotechnology than in the time of Konrad Zuse’s mechanical computers. A prerequisite for this development, however, is that the design of motion automata has become understood and automatable. This article is intended to contribute to this. For 30 years, we have designed, developed and transferred robots for medical applications into clinical use. For 20 years, we have been striving to develop medical robots that can be used as needed and to make the size of these devices as small as possible. If a minimum size specified by the application must be achieved, a classic 6DoF robot is usually no longer an option for a movement. Instead, a task-specific mechanism must be designed. This must be able to be manufactured in a few hours and validated under real conditions (Fig. 1).

c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 12–36, 2022. https://doi.org/10.1007/978-3-031-06409-8_2

A Method for the Automatic Design of Mechanisms

13

Fig. 1. Required sequence of steps to achieve an automated design process.

In the following, we will restrict ourselves exclusively to the design of simple planar or spherical four-bar joints. While in the past the design of four-bar joints was characterized by link lengths, pole curves, coupling curves, center curves, nowadays, due to the computer-aided design methodology, the two most important questions are: • Which body should be moved with the four-bar linkage and in which form? • What are the dimensions of the smallest possible linkage joint?

2

Definition of Motion Tasks for Planar and Spherical Four-Bar Linkage

In contrast to other representations, in this article we start from the task of moving not a point but a body in space with a four-bar linkage. We call this moving body an effector. The area of the effector is described by a circumferential closed contour (Closed Polygon List: CPL) CPLE, which is defined relative to an origin coordinate system of the unit matrix eye (3). We use in 2D instead of complex numbers, the 3 × 3 homogeneous transformation matrices to describe coordinate systems and poses (translations and rotations). For further understanding of the motion task, it is useful to think of a revolving 4bar linkage, such as a walking mechanism. In this text, we mainly use two

14

T. Lueth and F. Irlinger

examples: A revolving walking mechanism and an angle-constrained Lemans table. A third example is a box lid. A motion task is now defined by at least one given pose (one-pose synthesis) where the effector CPLE must be located. This path, defined as positions and orientations, can basically be traversed in two directions. If two poses are specified (two-pose synthesis), at which the effector must be able to be located, an additional boundary condition results from a later angular limitation of the crank revolution movement, which part of the rotational movement between the two poses should be selected. If three poses (three-pose synthesis) are specified, which the effector must reach, the sequence in which the three poses are to be run through results as an additional boundary condition in the case of a later angle limitation of the crank revolution movement. In the case of non-rotating four-bar linkages, on the other hand, the sequence of the three poses is already mechanically predetermined. I do not consider the fourpose synthesis here, because in my experience the solution space is too small for practical tasks, and therefore there is little potential for optimization with limited rack space. An implementation can easily be added. The design of pure paths can be done efficiently by search, starting from the one-pose synthesis. To describe poses, we use only the origin point and any second point on the x-axis. Since in R2 the y-axis orthogonal to the x-axis can in principle be formed from the x-axis ex = (xex yex ) and ey = (−y ex xex ) exactly two points A and B with which a pose can be specified are sufficient to describe a pose. The 3 × 3 matrix can then be easily determined by the translation vector to A, and the length-normalized ex-axis direction ex = (xex yex ) and the ey vector orthogonal to it       B−A xA −yex ex ey tA and ey = , with tA = , ex = (1) 0 0 0 yA +xex |B − A| In the left image of Fig. 2 (Walking Mechanism) one sees a walking machine and in the right image (LeMans-Table Mechanism) one sees a LeMans table as a kitchen cabinet solution next to a stove top and a wash basin. Even though the three poses are numbered, it must be clear that neither the order nor the limitation of the movement to the distance between the poses is implicit. All examples presented in this text can be reproduced using SG-Lib 5.1 (www.sg-lib.org) for Matlab 2021b. The library can be downloaded from Github or Matlab Central. In the right image of Fig. 2, in addition to the green surface (CPLE) of the effector, there is also a red surface (CPLW) to describe restricted areas, a blue surface (CPLM) for potential rack points, and a black contour (CPLB). All areas are described by circumferential contours in our calculations. The black contour CPLB is only used to illustrate the design task and is not used in calculations.

A Method for the Automatic Design of Mechanisms

15

Fig. 2. By shifting the effector contour CPLE into the coordinate systems of the poses, the required motion of the body can be defined or understood.

The consideration of the area CPLE of the effector is absolutely necessary, because the anchor points A1 and B1 can only be attached to this area. The consideration of the area CPLM of the effector is absolutely necessary, since the base points A0 and B0 can be attached only in this area. A certain exception is the three-position synthesis, as we will see later. The consideration of the area CPLW is necessary, because it restricts the solution space for the movement paths of the linkage points of crank and swingarm, restricts the shape of the effector and forces the shape of the two linkages, crank A0A1 and swingarm B0B1. Likewise, it is absolutely necessary to know in advance the minimum possible radius R12 of the links, which is enforced by the real technical design of the joints, since the surfaces described above must be reduced or enlarged by this value R12 in many design steps, so that movement is possible without collision or an appropriate link contour can be found automatically. In contrast to these geometrically excluding boundary conditions, which are therefore mandatory for the task description, there are many other constraints, such as the minimum distance of the frame points, path courses, etc., which will first come into play later when selecting a solution. These will be discussed later.

3

Formal Description of Four-Bar Linkages and Their Motion

Four-bar linkages can basically be described in two ways: • by the lengths of the four links (4 values) of the four-bar link and • by the coordinates of the four pivot points (8 values) of the four-bar linkage!

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T. Lueth and F. Irlinger

The second variant describes basically a realizable four-bar linkage and has considerable advantages over the first variant. All properties, which are determined by the lengths alone, can be determined immediately also for the second variant, if one calculates the distances between the joint points. Therefore, we always use the frame points A0 (crank) and B0 (rocker) as well as two real articulation points A1 (crank) and B1 (rocker) for the description of four-bar joints. For the motion calculation we consider the four-bar link as a 2R chain. It consists of the rocker with length L1 = |B1 − B0| and coupler with length L2 = |A1 − B1|, attached to the frame point B0. This 2R chain is forced into a circular path by the crank with length R = |A1 − A0|, attached to the frame point A0 and to the coupler link point A1. Using the inverse kinematics of the 2R chain, the joint angles of the four-bar linkage can be determined. As is known, there are two solution configurations here with elbow-up and elbowdown. However, since the coordinates of the articulation points A1 and B1 were already a default, the elbow configuration is uniquely determined for this fourbar linkage definition. Exceptions would be singularities like branching positions and plugging positions, which one would like to recognize and exclude earlier in reality. The position of point B1 uniquely defines the elbow configuration while A0, B0, A1 are identical for both elbow configurations.

Fig. 3. The definition of four joint point coordinates clearly defines the elbow configuration of the four-bar linkage.

Another important point is that with four-bar joints not only 2 elbow configurations are possible, but also potentially two separate motion segments exist for each elbow configuration, which can overlap, but can only be switched between by disassembling and reassembling the four-bar joint. Exceptions would again be singularities such as branching positions and plugging positions, which, as mentioned, one would like to recognize and exclude earlier in reality (Figs. 3 and 4).

A Method for the Automatic Design of Mechanisms

17

Fig. 4. Of the two elbow configurations “up”/green (a) and “down”/red (b) of the four-bar linkage, only one fulfills the necessary condition that the linkage runs from A1 to B1. The two motion segments of the crank (blue) that cannot be switched between are drawn in yellow and cyan.

To allow a four-bar linkage to be a valid solution, all the poses (1–3) to be held must be in the-same motion segment of the same elbow configuration and must be approached one after the other in the desired order when moving the crank. For revolving four-bar joints, any order is possible. For non-revolving four-bar linkages, an incorrect sequence is a criterion for exclusion. Critical are always poses which are taken in stretching positions or branching positions. The calculation is done in our implementation in Matlab using numerical methods of analytical geometry.

4

Description of the Automated Pose Synthesis

The procedure of the pose synthesis presented here is to construct a virtual coupler (A1B1) for all possible combinations of frame points and hinge points within the given areas CPLM or CPLE, which includes both hinge points and the respective point A of the pose specification. Except for special cases, the coupler is always a triangular body with which the effector can assume the desired poses. In one-pose synthesis, the surfaces CPLM and CPLE are rasterized. In two-position synthesis, the frame points must be located on the perpendiculars between the hinge points and therefore the entire surface of CPLM cannot be rasterized, but the perpendiculars on the lines that intersect CPLM are rasterized. In the case of three-pointer synthesis, the rack points must lie on the intersections of the center perpendiculars between the link points and thus can only be selected if they lie within CPLM. It is therefore reasonable to automatically extend the domain CPLM to a limited extent in order to detect at least the spatially closest rack points to CPLM. Since the solution space for one-pose synthesis is the most extensive, we describe the procedure for one-pose synthesis below. The other two methods are actually faster in computation because they reduce the solution space early. In our considerations we assume a maximum of three poses, i.e. three points with directions. Three starting points A and three end points B are defined. Figure 7 shows an example of opening a box lid, where

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Fig. 5. Attachment point GPLA = [−10 +10] and attachment point GPLB = [+10 −10], both relative to CPLE: a) without rotation, b) with rotation of the pose with respect to the originating coordinate system.

Fig. 6. In the picture you can see the possible frame points A0 and pivot points A1 for the crank in blue and the possible frame points B0 and pivot points B1 for the swing arm in cyan. Only the points for which solutions exist are shown. a) 1-pose synthesis b) Two-pose synthesis, c) Three-pose synthesis

three arrows between the start and end points indicate the three poses by means of start and end points. The green surface CPLE is the effector surface, which in any case must include the pivot points for the crank and swing arm. The blue area CPLM describes the mounting area in which the frame points for crank and swing arm may be located. The red area CPLW describes the restriction area in which the pivot points may never be moved during a movement. At a later stage, the contours of the effector, crank and swing arm can be formed by collision considerations with this area CPLW. The black contours are only intended to illustrate the task and are not considered in the calculations (Figs. 5 and 6).

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Fig. 7. Three poses of a crate lid during opening of a crate. You can also see: effector surface CPLE (green), rack point surface CPLM (blue), blocking surface CPLW (red).

4.1

Calculation of the Pivot Point Grid and Rack Point Grid

First the effector is gridded. A grid point list GPL of potential crank and rocker points is calculated. In my experience, the grid is not a major limitation of the procedure. The list of potential rack points is achieved in one-pose synthesis by a uniform gridding of the mounting surfaces CPLM. The exact number of desired points cannot always be achieved. There is usually a threshold above which the number of points increases abruptly. Then the small number is used, because the number of the combination increases fast and the computation time should remain estimable (Fig. 8).

Fig. 8. a) End effector end point grid CPLE, b) End effector grid in the frame surface CPLM

The number of four-bar linkages now possible is the product of the square of the number of rack points and the square of the number of linkage points minus the nonsensical use of identical points for the two rack points or linkage points, since these lead to rigid/immobile linkage chains or prevent the achievement of orientation specifications.

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Pose Synthesis Instead of Coupled Curve Syntheses

It must be stated once again quite clearly that the pose is fixed relative to the effector and the orientation of the effector is fixed. Pose synthesis always assumes a real motion task where effector is to be moved relative to its environment. Therefore, the shape of the effector CPLE is also known and defined relative to a pose and not to a point. The surface contour CPLE is chosen so that the motion task is technically possible and the position of the articulation points within CPLE is described as flexibly as possible. The position of the effector contour CPLE to the pose plays a decisive role. The two representations in Fig. 9 are completely different synthesis tasks. This becomes immediately clear when you move the effector contour to the defined poses. The two representations in Fig. 9 are completely different synthesis tasks. This becomes immediately clear when you move the effector contour to the defined poses. Thus, effector form and item definitions are always firmly connected with each other and jointly define a real task. The idea of a free coupling plane is basically obstructive. Only the position of the end effector relative to its environment is important (Figs. 10 and 11).

Fig. 9. Two identical effector shapes and identical articulation point grids describe different motion tasks due to different position definitions in a) and b).

Fig. 10. An identical real design task with three poses represented on the left and right by two different pose triples. Without the shape and position of the end effector, it is not possible to tell that this is an identical task.

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Fig. 11. In this illustration for a walking machine, it becomes completely clear that a pose specification without an effector shape cannot give any conclusions about the task.

4.3

Testing a Four-Bar Linkage Solution Option for Pose Synthesis

In fact, we have here a very large number of possible four-bar linkages due to the grid of the surfaces CPLE and CPLM. They are defined by two arbitrarily selected frame points A0 and B0 from the frame point grid ABL as well as two articulation points A1 and B1 which result from the back-calculated position of two arbitrarily selected articulation points from the articulation point grid under consideration of the pose. We must now check whether all poses can be reached with this four-bar linkage! If there is only one pose specification (onepose synthesis), the hinge points can always be reached, since they are calculated from exactly this pose specification. From the distances of the four points, the lengths of the swing arm L1, the coupler L2 as well as the length of the crank R and the distance of the two frame points D can be determined. All further analysis is done with the calculation of the inverse kinematics of a robot at position B0 with lower arm L1 and upper arm L2, which can reach a point on a circular path with radius R and center at position A0 in two joint configurations (Fig. 12).

Fig. 12. In this example, the pose target is placed between the pivot points and slightly twisted.

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First Subcondition: A four-bar link can therefore potentially be operated in two different elbow configurations. Since the forearm of the rocker-coupler chain corresponds to the copula, only one of the two solutions can fulfill the necessary pose specification. When the crank is at A1, the coupler must run along the A1 to B1 specification. This is only given with one configuration (Fig. 13).

Fig. 13. Of the two elbow configurations “up”/green (a) and “down”/red (b) of the four-bar linkage, only one fulfills the necessary condition that the linkage runs from A1 to B1. The two motion segments of the crank (blue) that cannot be switched between are drawn in yellow and cyan.

Second Subcondition: A four-bar joint can potentially contain two motion segments between which it is not possible to switch without disassembling the four-bar joint. While the change of elbow configurations with one motion can be achieved by a possible stretch position (singularity), this is not possible for a segment change. Therefore, we have to check if the one desired pose is in the segment for a one-pose synthesis or if all poses are in the same segment for a two- or three-pose synthesis. With a one-pose synthesis, we have calculated the points A0, B0, A1, B1, from the pose and the linkage grid points and, after calculating the elbow configuration, we only have to determine the segment in which the linkage point A1 of the crank is located. With both subconditions, all four-joints can be found that pass through all given poses in the correct order.

5

Constraints During Synthesis

The solutions are stored in a uniform format. For each four-bar linkage, the four points A0, B0, A1 and B1 are stored as well as the coordinates GPLA and GPLB, which describe the position of the linkage points A1 and B1 relative to the effector origin coordinate system that was also used to define the poses. In addition, the maximum angle range of the crank for the motion segment covering all poses is stored. In special cases, this can also be two motion segments.

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While all solutions found in the previous section allow the effector CPLE to reach all three poses in the correct order and all rack points lie in the predicted area CPLM, there are some constraints that reject a solution early on. For example, in the case of the walking motion, Posesample(12), only circumferential four-bar linkages are useful, where the coordinate of the pose does not touch the area CPLW, while in the case of the LeMans table, when designing all solutions, the motions should first be restricted to the area between pose 1 and pose 3.

Fig. 14. The restriction to revolving four-bar linkages alone is not sufficient. Only the linkage curve of four-bar (c) does not enter the restricted area CPLW.

When constraints are introduced, the solution can now be limited to only revolving four-bar joints. This is done with a small constraint language, with which collisions and angle constraints can be prescribed. Sensible solutions must not touch the surface of CPLW in the running machine additionally with the coupling curve CPL0. Likewise, the end point of CPLE (CPL2) should not touch the surface of CPLW (Fig. 14).

These conditions can be taken into account either subsequently or immediately at design time. In the case of three-pose synthesis, it must be remembered to automatically extend the CPLM area, since inexperienced four-joint designers usually do not succeed in defining the poses to fit CPLM without computer assistance. The LeMans table also has concurrent constraints. The pose sequence and the angular stops are also given (Figs. 15 and 16).

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Fig. 15. The frame area CPLM is automatically extended to allow at least a minimum number of frame points of the crank (blue) and the swing arm (cyan).

Fig. 16. a) Not all possible rack points for crank (blue) and rocker (cyan) are located in the area CPLM (blue). Only the right side of the effector CPLE comes into question for the attachment of the pivot point. b) The few rack points within CPLM.

6

Sorting Solutions After Synthesis

As mentioned before, after taking into account the concurrent secondary constraints, the four-bar solutions are available as a list of coordinates: A0, B0, A1, B1, GPLA, GPLB, and the angle limits wlim. Likewise, angles are already limited to collision-free or colliding solutions are excluded. Further criteria can now be used as sorting criteria: Distances between frame points, distances between articulation points, lengths of crank or swing arm, the length of the motion path and much more. In practice, SG-Lib is constantly being extended to increase the number of these criteria. In the case of the LeMans table, for example, the solution with the maximum effector area that can be moved without collision is interesting (Fig. 17).

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Fig. 17. Two positions of the Lemans table with the largest possible area.

7

Contouring of Effector, Crank and Swing Arm

When sorting solutions in the previous section, it has already been mentioned that collision-free movement, taking into account workspace restrictions, influences the shape of the effector. In the case of a Lemans shelf, one naturally wants the shelf area to be as large as possible, and then the rack points can be selected for this large area. This leads to three tasks: a) Trimming a maximum collision-free effector shape. b) design of a crank shape for collision free motion c) design of a swing arm for collision free motion In fact, the circular segment path of points A1 and B1 must always remain collision-free, but neither the crank nor the rocker must be designed as straight links. Linear links are special shapes. In order to form the effector and the crank or the swing arm, the motion paths of the pose and the two pivot points A1 and B1 must be calculated. Then the contour of the moved paths must be calculated, here subtracted. The crank and swing arm are designed similarly to the effector, but the aim here is to achieve a minimum size. In a first step, the collision-free area is therefore determined for the crank based on a large circular handlebar contour. In a second step, a path planning algorithm is used to find the shortest distance from A0 to A1 within the collision-free area and this path is extended with the radius required for the assembly. When designing the swing arm, the moving contour CPLC of the crank on the path of the crank must then be subtracted from the circular initial contour in addition to subtracting the

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Fig. 18. a) Maximization of the effector area in the Lemans table, b) Occurrence of collisions of the effector, crank trajectory and oscillator shape.

Fig. 19. Subtraction of the static surface CPLW from the moving surface CPLE to form the effector, a) trajectories of motion, b) cutting via motion and subtraction, c) motion of the body.

static contour CPLW. The contours of the path planning can be smoothed with Bezier splines, but should be checked again for collisions afterwards. If there is no restriction of the frame points to a surface CPLM, then the automatic design of the forms can create extreme crank and swing contours that look strange and still solve the task (Figs. 18, 19, 20, 21 and 22).

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Fig. 20. Path planning of the shortest distance (b) from A0 to A1 in the collision-free maximum crank contour (a). Path planning of the shortest distance (d) from B0 to B1 in the collision-free maximum crank contour (c).

Fig. 21. Collision-free area maximization and trajectory planning for tracker size minimization: a) A* trajectory planning b) Bezier smoothing, c) final tracker shape.

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Fig. 22. Two extremely shaped wings (cyan-colored), which also do not touch the restricted area CPLW, magenta-colored, during the collision-free movement of the effector CPLE.

8

Layering and Collision-Free Movement of Four-Bar Linkages

The problem of collision-free motion always depends on which bodies are in the same plane of motion. The next problem to be solved, in automated design, is therefore the control of the layering of crank, swing arm, effector, frame and the collision-free movement of crank, swing arm and effector. For explanation, the example of the running machine instead of the LeMans table is useful here, since frame points and linkage points are closer together, making the layering easier to see. In addition, the shape of the crank and swing arm do not change, since the movements can be carried out collision-free between the crank and swing arm in the selected example, as long as no spacers are used. The following illustration shows the six of eight layering solutions that are possible without cutting the effector if the frame is always in plane 0 and the effector is always in plane 2 and the crank and rocker are allowed to be in planes −1, 2 and 3. It is important to understand that a four-bar linkage consists of eight moving bodies and not only four moving bodies. The four bodies usually omitted are spacers, but they also run on paths of motion and or can be intersected by motions of other bodies. In principle, these spacers could be bent between planes to avoid collisions. This has not yet been implemented. The following figure shows the two of eight solutions where the spacers are not intersected if the above solutions are implemented with spacers of equal length. A four-bar linkage consisting of only four members is a special form reduced in complexity. The general four-bar linkage consists of eight bodies of which two spacers can be regarded as stationary at positions A0 and B0 and two spacers move and rotate along with positions A1 and B1! The co-moving spacers can be extended to the shape of the effector (Fig. 23, 24 and 25).

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Fig. 23. Transition from 2D contour (a) to 2.5D layering (b) of links using (c) spacers, drawn in red.

Accordingly, it is also true that in general the frame plane of the swing arm does not have to be identical with the frame plane of the crank. In this article, the rack points are always attached to the same yellow rack. This does not have to be the case.

Fig. 24. Six of eight principle layering solutions when using a maximum of 5 layers. In the two missing representations, the crank has intersected the effector. In solutions a) and c), the crank has intersected the rack.

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Fig. 25. Two solutions in which you can see all eight bodies of the four-bar linkage.

9

Final Construction of the Four-Bar Linkage with Standardized Assembly Solutions

The shaping of the effector based on the movement near the limiting contours CPLW or CPLM. The shaping of the crank and rocker due to the collision-free movement depending on the respective selected layering. Now the bodies still need to be extruded from the contours and a screw connection or bearing of the joints. The implementation of the design is done by using a library of standard parts, of standardized assembly and bearing variants, taking into account fits, which will be explained in the next section. As an example, the four-bar linkage has been implemented here once with a cylinder head screw-nut connection and once with a countersunk screw connection with self-tapping threads (Fig. 26).

Fig. 26. a) Automatic design for DIN912 cylinder head M2.5 × 6 mm with DIN 985 nut b) Automatic design for DIN965/7991 with self-separating thread M1.6 × 6 mm. c) Finished four-joints after printout (40 min) and assembly.

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Also clearly visible in the figure and in the photo are the automatically designed stops of the motion automaton, which are attached to the frame and can be designed not only as a stop, but also as a cage or housing.

10

Fits and Assembly with Standard Parts

The SG-Lib design functions described above for the automatic design of fourbar joints for immediate printing on a 3D printer use even more methods that can only be touched upon here. 10.1

Printer Specific Fits

If swivel joints are implemented by screw connections or ball bearing solutions, fits between standard parts and the 3D printed part play a decisive role. These are clearance fits (assembly and non-contact), transition fits (ball bearing seat) or interference fits (shafts). 3D printers currently have an accuracy in the 1/10 mm range and do not achieve 1–10 µm like the fits for steel parts such as m6/h7 range. Nevertheless, fits can also be realized very well with 3D printers. When designing with SG-Lib, an Excel spreadsheet is stored with which printing process which oversizes and undersizes occur and through experiments the successful fit dimensions were stored. A hole with a radius of 10 mm receives the offset required for a fit as an addition (Fig. 27):

Fig. 27. Three designed transition fits for the 3D printers: “Formlab FORM2, Anycubic Predator, EOS Formiga 100”.

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10.2

Tables for DIN and ISO

In order to be able to offer a wide range of assembly solutions, both the DIN tables for design tasks considering fits and constructions are accessible in SG-Lib: For standard parts such as cylinder head screws, countersunk screws, washers, etc., both the DIN tables and functions for designing the screws are stored. The standard parts are of course not printed, but purchased and used for assembly (Fig. 28).

Fig. 28. Three designed transition fits for the 3D printers: “Formlab FORM2, Anycubic Predator, EOS Formiga 100”.

10.3

Relative Orientation of Bodies and Coordinate Systems

Basically, for all tasks of relative orientation of bodies either spatial relations or for precise tasks always frames (coordinate systems) are used, which point out of a body surface (Fig. 29).

Fig. 29. Arrangement of solids using spatial relations and frames: a) cuboid with frames like ‘X+’, b) cylinder with frames like ‘F’, c) arrange cuboid on cylinder by centering, aligning on top edge and connecting on right side. d) aligning 2 frames, d) overlapping 2 frames.

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Subtraction Solids

The arrangement over frames is especially important when the space for the creation of joint structures must be created for the automatic design. For this purpose, so-called subtraction bodies are subtracted from a body with Boolean operations that include geometry, movement and fit for creating a degree of freedom. In the simplest case, for a bolted joint with a DIN912 cylinder head bolt with a DIN 985 self-locking nut, which is to be inserted from the side, a subtraction body must be subtracted from one body to be joined, for the bolt and its insertion path, and in the other body to be joined, a subtraction body, for the nut and the insertion path (Fig. 30).

Fig. 30. a) Subtraction bodies for the head side (red) and for the nut side (green). b) Cuboid and cylinder after subtraction of the subtraction bodies and representation of the standard parts.

10.5

Variants for Mounting and Axial and Radial Bearing Arrangements

Based on the subtraction bodies, the SG-Lib now offers different variants for mounting or bearing bodies using standard parts such as screws, nuts, radial, and thrust bearings. For loaded connections, cylinder head bolts with self-locking nuts are useful for ball bearing solutions from M2.5; for smaller dimensions, nuts and ball bearings become very expensive. For M2, screw connections using drive-in sleeves are still an option. For very small links without bearings, M1.6 countersunk screws can be used. Since the fits described above are used here on a large scale, the interpretation of the graphical representation always requires increased attention (Fig. 31).

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Fig. 31. Subtraction bodies for a) screw-nut connections, b) self-tapping threads, c) drive-in sleeves, d) radial bearings, e) axial bearings, f) axial and radial bearings

10.6

Montage Views and Motion Videos

Four-bar linkages have the problem that they consist of several parts, where the crank and swing arm often look very similar and these can also be easily assembled incorrectly, for example mirrored. For this reason, SG-Lib generates an STL file in the assembled state, 4 views for assembly, in addition to the STL files for the individual parts when preparing for 3D printing. Likewise, a video is generated showing how the mounted four-bar linkage must move (Figs. 32 and 33).

Fig. 32. Automatically generated views as assembly aid for the four-bar linkage with standard parts.

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Fig. 33. Automatically generated video as assembly control for the four-bar movement.

10.7

Slicing and Manufacturing with the 3D Printer

The validation of the methods took place on the Form2+, Form3, Eos Formiga 100, and Anycubic Predator printers. Two different resin grades were used for the SLA method and eight different PLA grades for the FDM method. For the Anycubic Predator, the Ultimaker CURA slicer was used. For the Formiga, the material used was PA2200. By far the cheapest, fastest and most reliable process for single four-joints is the Anycubic Predator (330 euros/Oct. 2020) compared to the Form3 (4000 euros/Aug. 2020) and the Formiga 100 (300,000 euros/Mar 2013). All printers are in permanent use at the chair. The four-bar linkages can be printed on a Predator in about 40 min and assembled within 5 min. Electricity costs are 5 euros per kilo and filament is about 20 euros per kilo. A four-bar link with 20% infill and 1.2 mm wall thickness weighs about 20 g. Acknowledgement. Tim Lueth and Franz Irlinger thank for the many discussions. For example Alexandra Mercader for the use of the functions for running machines, Christoph Parhofer for hints on small connections, Simon Laudahn for hints on the bearings and Yannick Krieger for the considerations to achieve a smooth motion with non-circular gears.

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References Craig, J.J.: Introduction to Robotics: Mechanics and Control. Pearson Education (1989) Denavit, J., Hartenberg, R.S.: A kinematic notation for lower-pair mechanisms based on matrices. J. Appl. Mech. 77(2), 215–221 (1955) Shen, Q., Russel, K., Sodhi, R.: Mechanism Design. CRC Press (2014) Lueth, T.: Automated planning of robot workcell layouts. In: Proceedings of the 1992 IEEE International Conference on Robotics and Automation, pp. 1103–1108 (1992) Lueth, T., Irlinger, F.: Berechnete erzeugung von dreidimensionalen oberfl¨ achenmodellen im stl-format aus der beschreibung planarer mechanismen f¨ ur die generative fertigung durch selektives-lasersintern [computational 3d surface generation of planar mechanismus using stl file format for generative manufacturing by selective laser sintering. In: Konferenzband 10. Kolloquium Getriebetechnik, TU Ilmenau, pp. 267–284 (2013) Reuleaux, F.: Die Praktischen Beziehungen der Kinematik zu Geometrie und Mechanik. Vieweg und Sohn (1900) Sun, Y., Lueth, T.: SGCL: a b-rep-based geometry modeling language in MATLAB for designing 3D-printable medical robots. In: IEEE 17th International Conference on Automation Science and Engineering (CASE), pp. 1388–1393 (2021)

Bio-inspired Structural Intelligence for Miniature Robots in Minimal-Invasive Surgery Georg Rauter(B) , Lorin Fasel, Manuela Eugster, and Nicolas Gerig BIROMED-Lab, Department of Biomedical Engineering (DBE), University of Basel, Basel, Switzerland [email protected]

Abstract. Being able to perform a surgical intervention in a minimalinvasive way depends on the availability of miniature surgical devices. However, miniaturization per se bears many challenges. In particular, when functionality, versatility, as well as properties like accuracy, precision, and high forces are key requirements for the miniaturized tools. At the BIROMED-Lab, we are convinced that nature is the best inspiration for tackling the challenge of miniaturization while increasing functionality and improving device performance. Nature has spent millions of years evolving different solutions for the daily challenges of life forms in different orders of size. This short paper intends to highlight a few possibilities, where we can profit from nature’s evolution for accelerating the development of miniature surgical devices for the benefit of patients.

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The Potential of Bio-inspired Designs for Minimal-Invasive Robots (In Surgery)

Minimal-invasive surgery (MIS) has been shown to be less invasive than open surgery, and therefore, results in faster patient recovery (Wei et al. 2011). Expanding MIS by using robotic tools allows to further enhance task performance. RAMIS, i.e. robot-assisted minimal-invasive surgery, is typically realized by replacing the direct manual guidance of instruments with telemanipulated or robot-guided instruments. Hand tremor can be filtered (Veluvolu and Ang, 2010), motions can be scaled (Prasad et al. 2004), and the fulcrum effect of MIS procedures (i.e., the tool handle motion has to be performed in the opposite direction compared to the tool tip motion inside the patient due to motion inversion at the incision point) can be compensated (Vitiello et al. 2013).

c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 37–40, 2022. https://doi.org/10.1007/978-3-031-06409-8_3

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Fig. 1. Inspiration on structural intelligence from a comparison of large serial robotic structures with humans performing the same task. Alternative robotic structures fixed in close vicinity of the point of intervention could provide important advantages in accuracy and disturbance rejection.

(a) Miniature parallel robot

(b) Series Elastic Actuation (SEA) endoscope

Fig. 2. Parallel robotic concept with relative fixation for stabilization and accuracy increase (a). SEA endoscope for automated inspection of structures inside the human using safety limits for interaction forces (b). “M” indicates motors/actuation, “A” are anchor points, “S” are springs, and “T” indicates tendons.

Up to today, the concepts for robots in surgery were mostly built on serial robots that were composed by rigid links with dedicated joints in between. Rigid serial robots are well-suited for repetitive tasks in a large workspace, but do not perform well in terms of end-effector precision, since errors accumulate along their length. In MIS, however, high accuracy is required in a rather small workspace in the vicinity of the surgical site. We believe that small structures that fix close to the target tissue at the surgical site (e.g., inside the patient) allow to reduce the robot’s structure and footprint in the OR. As an example, we have developed a small parallel robotic device that fitted inside the knee joint (Eugster et al. 2020), which could fix itself close to the target tissue to provide stability for laser-based bone ablation with an accuracy of 0.176 mm

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for continuous cuts (Eugster et al. 2022). The robot could “walk” in order to expand its workspace (Eugster et al. 2020). Our miniature robot guides the laser with a parallel structure similar to how humans guide a pen with several fingers. Structural intelligence realized through local fixation and parallel actuation increases accuracy, stability, and device robustness. Errors of one drive train are compensated by others and higher interaction forces from the environment can be counter-acted despite the robot’s miniature structure (see Fig. 1 and Fig. 2a). As a second example, we have drawn inspiration from the human finger for surgical robot design. Fingers can be positioned precisely because of closedloop position feedback, they can safely interact with the environment thanks to their compliance, and provide force feedback. These advantages can be transferred to surgical robots by replicating this structurally intelligent actuation. We designed an articulated robotic endoscope with joints driven by antagonistic tendons based on series elastic actuation (SEA). This configuration is suitable for closed-loop position and force control, while also allowing compliance (Fasel et al. 2022). Due to force control, directly limiting interaction forces with the environment is possible. Applications of such smart SEA-endoscopes are interesting in particular for automated tracking of structures (visual servoing (Fasel et al. 2020)), while ensuring safety for the surrounding tissues (Fasel et al. 2021) (Fig. 2b). Acknowledgements. This work was supported as a part of NCCR Robotics, a National Centre of Competence in Research, funded by the Swiss National Science Foundation (grant number 51NF40 185543) and the MIRACLE project, funded by the Werner Siemens Foundation.

References Eugster, M., et al.: Quantitative evaluation of the thickness of the available manipulation volume inside the knee joint capsule for minimally invasive robotic unicondylar knee arthroplasty. IEEE Trans. Biomed. Eng. 68, 2412–2422 (2020). https://doi. org/10.1109/TBME.2020.3041512 Eugster, M., Oliveira Barros, M., Cattin, P.C., Rauter, G.: Design evaluation of a stabilized, walking endoscope tip. In: Rauter, G., Cattin, P.C., Zam, A., Riener, R., Carbone, G., Pisla, D. (eds.) MESROB 2020. MMS, vol. 93, pp. 127–135. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-58104-6 15 Eugster, M., Merlet, J.-P., Gerig, N., Cattin, P.C., Rauter, G.: Miniature parallel robot with submillimeter positioning accuracy for minimally invasive laser osteotomy. Robotica 40(4), 1070–1097 (2022). https://doi.org/10.1017/S0263574721000990 Fasel, L., et al.: Visual servoing for tracking cartilage with a robotic endoscope. In: Proceedings on Automation in Medical Engineering, AUTOMED 2020 (2020). https:// doi.org/10.18416/AUTOMED.2020 Fasel, L., Gerig, N., Cattin, P.C., Rauter, G.: The SEA-scope: torque-limited endoscopic joint control for telemanipulation or visual servoing through tendon force control with series elastic actuation. In: 2021 International Symposium on Medical Robotics (ISMR), November 2021. IEEE (2021). https://doi.org/10.1109/ismr48346. 2021.9661497

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Fasel, L., Gerig, N., Cattin, P.C., Rauter, G.: Control evaluation of antagonistic series elastic actuation for a robotic endoscope joint. J. Bionic Eng. (2022). https://doi. org/10.1007/s42235-022-00180-6 Prasad, S.M., Prasad, S.M., Maniar, H.S., Chu, C., Schuessler, R.B., Damiano, R.J.: Surgical robotics: impact of motion scaling on task performance. J. Am. Coll. Surg. 199(6), 863–868 (2004). https://doi.org/10.1016/j.jamcollsurg.2004.08.027 Veluvolu, K.C., Ang, W.T.: Estimation and filtering of physiological tremor for realtime compensation in surgical robotics applications. Int. J. Med. Robot. Comput. Assist. Surg. 6(3), 334–342 (2010). https://doi.org/10.1002/rcs.340 Vitiello, V., Lee, S.-L., Cundy, T.P., Yang, G.-Z.: Emerging robotic platforms for minimally invasive surgery. IEEE Rev. Biomed. Eng. 6, 111–126 (2013). https://doi.org/ 10.1109/rbme.2012.2236311 Wei, B., et al.: Laparoscopic versus open appendectomy for acute appendicitis: a metaanalysis. Surg. Endosc. 25(4), 1199–1208 (2011). ISSN 0930-2794

Part II: Wearable, Humanoid and Surgery Robotics

Development of Wearable Assist Robot for Twist and Leaning Motion Koji Makino1(B) , Yuta Dobashi1 , Masato Karaki2 , Xiao Sun1 , and Hidetsugu Terada1 1

2

University of Yamanashi, Kofu, Yamanashi, Japan [email protected] TOYO KOKEN K.K., Minami-Alps, Yamanashi, Japan

Abstract. It is important to reduce the human workload by developing technologies such as a robot. However when developing a robot, one of main issues is that it is difficult to automate the loading task due to different shapes of object. The reasons are that it is difficult to detect the shape, to grab it, and to stack them efficiently. Therefore, it is desirable to use the flexibility of a human recognition and judgement by using a wearable assist robot. Many kinds of wearable assist robots are studied and developed. In certain working environment such as luggage stacked into a cargo in the airport, a twist motion is important for the laborer. In this paper, a wearable assist robot for a twist motion and a leaning motion is developed. It utilizes a motor unit with a lack spur gear for the safety and power efficiency is developed. The effectiveness of the robot is verified by the experiments. Keywords: Wearable assist robot Loading task

1

· Twist and leaning motion ·

Introduction

Loading tasks as when luggage and other objects stacked in the cargos, shelves or cages are performed not by a robot but a human in general. One of the reasons is that it is difficult for the robot to perform this task automatically, since the shapes of the loads are different, and the position of the stacking is not decided in advance. It is important to reduce the human workload, since loading tasks are hard and pose some risks [1]. There are many studies for the reduction of the workload of the loading task, which are classified into two methods. One is a method using an automatic robot. In the automation of the loading task, it is difficult to detect the shape of the load, take hold of it, and stack it efficiently. Recent studies, the target object can be detected with high accuracy by the deep learning [2], and it can be grabbed by the robot hand using the deep reinforcement learning [3]. However, the flexibility of the robot is not sufficient in the real world. And there is not an exact solution but an approximate solution, since the stacking task is an NP-hard c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 43–52, 2022. https://doi.org/10.1007/978-3-031-06409-8_4

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problem. The time interval for obtaining the approximate solution is too long even if a high performance computer is used. Therefore many problems are still remaining for the automation of the loading task. The other is a method using wearable assist robots. A human can stack the target objects suitably. Therefore, various wearable assist robots are proposed and developed to utilize human ability [4]. The general-purpose wearable assist robots are useful for various tasks [5]. However, some their functions are not necessary if target tasks are limited to the loading. Therefore, various robots specialized in loading task are studied and developed. For example, the robots for the lumbar support are studied [6], the robots for the arm are proposed [7] and the robots for the leg are developed [8]. Some of them are manufactured and used in the real field. Some kinds of actuators to assist the lumber are used; air force type [9] (for example, McKibben artificial muscle), electric motor [10], and passive type [11] (spring etc.). Especially, the lumbar support by the robots are important for the loading task, since the low back pain often occurs [12,13]. Most of them assist only in a pull-up motion or a leaning motion. It is sufficient for many tasks, however other motions are necessary in some cases. For example, a laborer is stacking a luggage into a cargo in the airport as shown in Fig. 1. In this case, it is important to assist not only in a pull-up motion but also a twist motion. We studied a wearable assist robot for this kinds of motions using an air force [14]. However, there are some problems such as force is not enough and working time is too short by the air pressure force using a portable air tank.

4. R eturn the bod y position Labor

3. Tw ist it to th e co ntrary and put it on the other luggage w ith lea ning forw ard m otion 2. Lift up the luggage and tw ist it

Luggage

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Fig. 1. The concept of the new switchable robot.

This paper describes the development of the wearable assist robot for a leaning motion and a twist motion using the electric motor. And the lack spur gear that has some advantage points for the assist motion is used newly. The Sect. 2 describes the target motion. Next, the new wearable assist robot is shown in Sect. 3. And the result of experiment using the new wearable robot is shown in Sect. 4. Finally, Sect. 5 is the conclusion.

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45

Motion

It is necessary to analyze the motion again, since the actuator for assist the motion and sensor for detect the motion are changed from previous model. The actuator is changed from the air pressure unit (McKibben artificial muscle) to the electric rotational motor, since the working time is too short if the portable air tank is used. The difference of the sensor is as follows; the sensor of the previous model is rotation sensor, the sensor of the proposed model is load cell. Because, we confirm that the force is more important than angle by the following motion analysis. Therefore, in this section, the loading motion of stacking a luggage in the cargo is described and analyzed in order to adapt the proposed model to the motion. In the loading motion, it is necessary to lift up the luggage and to put it on the top of other ones as shown in Fig. 1. The motions of the laborer who works in the back yard of the airport are recorded by a movie camera, and are analyzed. We confirm that there are the common motions as shown in Fig. 2, though the motions of each laborer are slightly different. The laborer lifts up the luggage, and twists it at the same time, first. The twist motion is performed as a preliminary motion for stacking the other luggage, since the position that the luggage is put on is inside of the cargo. Next, the reverse twist motion and the stacking motion are performed at the same time. And the laborer hold the leaning forward motion to stack it. Finally, the leaning backward motion is performed to return to the initial position. Initial posture

G rasp

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Fig. 2. Procedure of the target loading task including the twist motion and leaning motion.

As a result of the analysis of the motion, it is clear that the twist motion and the leaning forward motion while holding an object are important. It is necessary to decide the timing for the assistance in the case of usage of the load cells, newly. We investigated the motion using pre-developed robot that was equipped

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with two sensors (load cells). The data were measured again using new robot described in the next section, however the position of the sensor is almost same. The result of three motions (right and left twist motion, and leaning motion) are shown in Fig. 3(a). The horizontal axis means the time, and vertical axis means the force obtained by the load cell. It is clear the difference of twist motion and the leaning motion.

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Next, the sequential loading motions shown in Fig. 2 are performed, and the forces in the motion are shown in Fig. 3(b). The horizontal axis means the time, and the vertical axis means force obtained by the load cell. The force becomes plus value if the load cell is twisted to clock wise by the top view as shown in Fig. 4, and the sign of the both value of the force is same. Similarly, the force becomes minus value in the case of twist to counter clock wise. And the sign of the each value of the force becomes different in the leaning motion.

Load cell values Initial posture

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Fig. 4. Relation between the motion and the sign of the load cells.

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In the loading motion, the luggage (H 500 × W 350 × D 200 × mm, 10 kg) is used. First, the leaning forward motion to grab the handle of the luggage is performed. In this motion, one on the sensor value is plus and the other is minus. Next the leaning back motion and the twist motion for the lift-up of the luggage occur at the same time. Therefore, one of the sensor value becomes minus rapidly, and the both sensor values becomes same sign. The motion is named as “Lift-up motion”. And the reverse twist motion and the leaning motion are executed. The each value becomes different for the leaning motion. The motion is named as “stacking motion”. Finally, a leaning back motion is performed for the returning to the initial position. The both values return to zero in this case. The assist motion is identified by the motion shown in Fig. 3(b). The timing to start assistance is when the handle is grabbed. The time interval of the lifting motion is approximate 1.4 s, and the time is almost same even if the some other motions are tried. Therefore, the assistance for the stacking motion is started at 1.4 s later after the handle is grabbed. And, the assist for the leaning back motion is also necessary. Finally, the assistance ends, when the initial position is returned and the subject does not grab it.

3

Robot

The wearable assist robot consisting of the upper body and the leg assisting instrument for the twist motion and the leaning motion is proposed and developed, which consists of body assist and leg assist instrument. The main objective is development of the upper body assist in this paper. 3.1

Body Assist Robot

The developed wearable robot for the upper body to perform the twist motion and the leaning motion is shown in Fig. 5. The robot consists of the back shoulder plate, the waist belt with the motor unit for reel up the wire, and the spring plate made by the stainless steel for the parts’ connection. The two wires connected to the motor units that can be controlled independently are attached on the shoulder plate. Therefore, the twist motion is performed if either motor is driven, the leaning motion is performed if both motors are driven. And the reverse twist motion for returning back to the initial posture occurs naturally by the torsion force of the spring plate. The development of the motor unit is the key issue of this study. It is described below. The motor unit consists of a motor, pulley for reel-up of the wire, a one-way clutch, and two spur gears as shown in Fig. 6. One of the gear connected with the pulley is a normal spur gear. Some teeth of the other gear connected with the motor are lacked. The role of the pulley is to wind up the wire connected to the shoulder part. The shoulder is pulled if the wire is reel up. The twist motion occurs in the case of reeling up either wire by the pulley. On the other hand, the leaning motion occurs if both wires are reeled.

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B ack sholder plate

Tw ist m otion (right) F ront posture R ight w ire is pulled.

R ev. Tw ist m otion (left) Left tw ist posture B oth w ires are pulled.

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B oth w ires are pulled.

F ront posture M otor unit for reel up the w ire

B oth w ires are pulled.

Fig. 5. The developed wearable assist robot, and the relation between the force by the wire and motion of twist and leaning.

Next, the procedure of rotation of the gear is described. In the initial state, the gears are meshed (Fig. 7 left). And the assist force is applied by the reel-up of the wire for the rotation of the pulley, when the motor is driven (Fig. 7 center). After the gears are rotated, they are not meshed because of lack of the teeth (Fig. 7 right). The assist force applied to the shoulder is released. And the gears are meshed again, if the motor is driven. There are three merits of this unit. First, the control is simplified, since the motor rotates in only one direction. Second is the safety. The release of the reel-up always occurs, even if the motor is driven continuously because of the breakdown of the controller. Therefore, the magnitude of the assist force is limited to a certain value. This feature is important to use the robot in the real field. Third, the angle of the pulley is held without motor driven by using the one-way clutch. In the case of the rotation of the motor, a one-way clutch can be rotated freely. On the other hand, the one-way clutch is locked, if the pulley is tried to be rotated inversely by pulling of the wire. For this reason, the angle can be held without the electric brake. 3.2

Leg Assist Robot

In our wearable assist robot, there is also the leg assist part that consists of hip joint and knee joint parts as shown in Fig. 8. The extension of the hip joint and knee joint can be assisted by the leg robot. These parts are driven by the coil spring, and there are not necessary for an electrical actuator. Users can walk freely, since the upper body and lower robot are driven independently.

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P ulley for reel-up N orm al spur gear Lack spur gear M otor

O ne-w a y clutch (in the gear box)

Fig. 6. Structure of the motor unit. H ip jo in t sup po rt u n it

and reel-up of the wire

Release of the wire

Fig. 7. State of the mesh of the gear. K n ee jo int su p p o rt un it

C oil spring C o il sp ring

A syn chronized gear for a roll back m otion of the knee

Fig. 8. The lower body assist robot that is driven by the spring force.

3.3

Wearable Assist Robot

Finally, Fig. 9 (left) shows the wearable assist robot. The micro controller unit and battery is attached at either side of the waist belt, respectively. And the motor unit is set at the back of the waist belt. The block diagram of the robot is illustrated in Fig. 9 (left). The input of a touch sensor to detect the grasp of the handle of the luggage, load cells to measure the condition of the motion of the twist and leaning, and rotation sensors to detect the angle of the lack spur gear are operated by the microcontroller (Arduino nano). And the motor (maxon, EC-i 30 Φ30 mm) is driven by the motor controller (maxon, ESCON Module 50/5) through the command from the microcontroller. The motor state to assist the three kinds of motions are decided. The twist motion is the motion that either motor is rotated to pull one shoulder. The reverse twist motion is the motion that both two motors pull the corresponding shoulders in the twist state. The holding leaning forward motion is that both motors are stopped to pull both shoulders up.

4

Experiment

The effectiveness of the robot is confirmed by the experiments. The set-up of the experiment is described as follows. The luggage is the same as in Sect. 2. The height of the table is 700 mm, and the surface of the table is large enough, so

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Fig. 9. Each part of the developed wearable assist robot, and its block diagram.

the luggage does not fall from it. The goal is set to put the target object 850 mm from the edge near the operator to imitate the motion of the laborer who works in the back yard in the airport to put the far position from the edge of the cargo to put many luggage. The participant of the experiment is one, who is not used this developed assist robot many times, and 167 cm height and 67 kg weight. The sequential motions are shown in Fig. 10. It was confirmed that the same motion as shown in Fig. 2 is traced.

Initial posture

G rasp of a handle

Lift-up m otion

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Leaning back

Fig. 10. Sequential motions that the subject equipped with the assist robot puts the luggage on the table.

In the experiments, the participants performed the sequential motions twelve times with and without the proposed robot, respectively. The myoelectricity was measured to investigate the effectiveness. The position of the measurement is erector spinae maximus, since the position is worked strongly. The typical results are shown in Fig. 11. The horizontal axis means time, and the vertical axis means %MVC. Three peaks depict the maximum % MVC on the lift-up motion, stacking motion, and leaning back motion, respectively. The maximum peaks among the motions becomes small if the robot is equipped. In this result, the average values of maximum % MVC is decreased from 39.8% to 35.0%. Moreover, the average values of the peaks of each motion repeated for twelve times and the p-values of them were compared. The results are shown in Fig. 12. The p-values are addressed as shown in the figure. In the lift-up motion and

Development of Wearable Assist Robot for Twist and Leaning Motion

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the leaning back motion, there is significant difference. However, there is not significant difference in the stacking motion. One of reasons is that the timing is not suitable, since the timing is determined by only interval time. Therefore, the timing adjustment by the sensor value of the load cell is the goal for the future. On the other hand, all average values becomes smaller if the robot is used. As a result, the effectiveness of the robot is verified by the experiments. 50

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5

* p 1). Integer c is used to count the number of these “not-slowed” points within the buffer. Next, from among all stored λi and the current λ (that corresponds to the new point appended to the buffer), the largest one is selected and denoted as λM for further use. This scale factor is used to speed up the motion (using a factor less than λM would violate the limits). In the case of speeding up, the cumulative scale factor σj is updated; however, it cannot drop down below 1 since the motion should not be accelerated above the initially planned velocity. The formula for the cumulative scale factor reads: σj = max(1, σj−1 λM ). To calculate scaled time samples between the current time and the prediction horizon Eq. (4) is used. For the new values of scaled time, the points stored in the s buffer are recalculated. Finally, as soon as calculations for speeding up (or slowing down) are finished, the robot controller utilizes the first value stored in the memory buffer to execute the motion. Then, the values kept in both buffers (s and λ) are shifted to the left, and new values—corresponding to the prediction horizon—are placed at the ends of the buffers. Additionally, values stored in the scale factors buffer are updated (i.e., divided by λM ) to account for the most recent scaling. Afterward, calculations are repeated for the next control step. 3.3

Detailed Algorithm

For the proposed trajectory scaling algorithm, it is crucial to check the limits for joint velocity and acceleration and calculate the scale factor. An auxiliary

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procedure Limits is used to find the scale factor λ for the given motion advancement measure si (λ = Limits(si )). The following calculations are performed: – Generate a point of the trajectory in the Cartesian space: pi = T rajGen(si ). – Solve the inverse kinematics problem: qi = InvKin(pi ). T  ¨i – Calculate joint velocities and accelerations: q˙ i q = V elAcc(qi , qi−1 , qi−2 ). ¨ i into Eqs. (1) and (2), – Calculate the scale factor λ—substitute q˙ i and q respectively, then use Eq. (3). Algorithm 1. Trajectory scaling algorithm [Initialization] , σ = 1, τcur = 0, τhor = N Δτ , λM = 1 Δτ = dt T for i = 1, . . . , N do sBuf (i) = s (iΔτ ), λBuf (i) = 1 end for [Main loop] while τhor < 1 do  sN = s τhor + σ1 Δτ , λ = Limits(sN ), λM = λ (slow down) if λ > 1 then σ = σλ, c = 0 for i = 1, . . . , N do τi = τcur + i σ1 Δτ , sBuf (i) = s(τi ) end for end if (speed up) if λ < 1 and c > N − 1 and σ > 1 then λM = max(λBuf (1), . . . , λBuf (N ), λ) σ = max(1, σλM ) for i = 1, . . . , N do τi = τcur + i σ1 Δτ , sBuf (i) = s(τi ) end for end if (execute motion) send sBuf (1) to the robot position controller (shif t buf f ers) for i = 2, . . . , N do sBuf (i − 1) = sBuf (i), λBuf (i − 1) = λBuf (i)/λM end for   sBuf (N ) = s τhor + σ1 Δτ , λBuf (N ) = λ/λM (update) c = c + 1, τhor = τcur + N σ1 Δτ , τcur = τcur + σ1 Δτ end while [Finalization] for i = 2, . . . , N do send sBuf (i) to the robot position controller (execute motion) end for For the sake of simplicity, in the presented algorithm it was assumed that scaling of the trajectory is not needed during the first N − 1 steps.

On-line Velocity Scaling Algorithm for Task Space Trajectories

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In the algorithm proposed here, the consecutive values of motion advancement parameters si , for all time steps in the prediction horizon, are stored in the memory buffer sBuf . The scale factors λi calculated for points stored in sBuf are kept in another buffer, λBuf . Note that in procedure Limits, to estimate velocities and accelerations, preceding values of q are used (backward finite difference formulae are utilized). Prior values of s (si−1 , si−2 , etc.) retrieved from sBuf can be used to calculate qi−1 and qi−2 . Since the values of s stored in the buffer change during the scaling process, there is no point in storing q. 3.4

Remarks—Algorithm Enhancements

In the classic method, the whole trajectory is scaled down uniformly (and only once), which guarantees that the joint limits for velocities and accelerations are not violated. In the proposed method, however, the trajectory is scaled down partially. Moreover, the same point may be scaled several times within the prediction horizon (within the memory buffer). As a result—when the cumulative scale factor σ changes—at the transition point between the previously scaled (or not yet scaled) trajectory and the newly scaled trajectory, the required motion deceleration may be large. Consequently, the joint limits for velocities or accelerations may be violated. In an enhanced version of the discussed algorithm, such problems are detected and eliminated at the expense of additional calculations. As some details of calculations are still under testing and development, only an outlook of these enhancements is presented here. Note that—at the given control step—the transition point between the not scaled and scaled part of the trajectory is located at the beginning of sBuf . In order to avoid excessive accelerations at the transition point, the following procedure is applied: – Scale factor λend is calculated for the newly added point at the end of sBuf . – Maximum feasible scale factor λstart is computed for the first point in sBuf . – To scale  within sBuf , smaller of the two lambdas is utilized  the points (min λstart , λend ). – In the case of λstart < λend , the last point in the buffer is insufficiently scaled down and should be scaled again in consecutive steps. A pointer is introduced to indicate the position of the not sufficiently scaled down point within the buffer. – When the buffer is shifted to the left, the pointer is shifted as well. Consequently, after several steps, the pointer indicates the leftmost point in sBuf that is not sufficiently scaled down. – At the beginning of each step, scale factors should be calculated for points between the pointer and the buffer end. If for a certain point the scale factor is less than 1 (i.e., no further scaling is needed), then the pointer should be shifted one position to the right. Shifting the pointer must be ceased when the first point with a scale factor greater than 1 is encountered.

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– The largest of scale factors (calculated for all points between the pointer and the buffer end) is selected as λend for further use. again and all points within – Maximum feasible scale factor   λstart is computed the buffer are scaled by min λstart , λend factor. – If λstart < λend , then the pointer should be shifted one position to the left; otherwise, the pointer should be shifted to the buffer end. – Obviously, the entire procedure is repeatedly executed. An important remark should be made. If the initially planned trajectory is too demanding, the pointer is shifted to the very beginning of the buffer. In such a case, the continuation of the scaling process is impossible. The occurrence of this situation indicates that the prediction horizon is too short, and the robot cannot slow down its motion fast enough. The procedure described above is executed in the slowing down phase of the trajectory scaling process. An analogous procedure, with necessary amendments, was also devised for the speeding up phase.

4

Numerical Example

The proposed algorithm was tested on a model of a 2-DOF planar manipulator. The lengths of the links were chosen as l1 = 1 m and l2 = 0.5 m. The kinematic  T  T 1, 2 rad limits of the actuators were vmax = 1, 1.5 rad s and amax = s2 . The desired end effector trajectory was generated as motion along a straight line from  T  T PA = −1, 0.505 m to PB = 1, 0.505 m with trapezoidal velocity profile with nominal time T = 5 s and acceleration and deceleration periods both equal to 0.2 T . The number of prediction steps was N = 10 with dt = 0.01 s. The manipulator in its initial and final configurations is shown in Fig. 1, along with the linear path and the workspace limits (represented by circular arcs).

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The desired trajectory is too demanding and violates the velocity and acceleration limits. Application of the classic method, briefly summarized in Sect. 2,

On-line Velocity Scaling Algorithm for Task Space Trajectories 2

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slows down the motion and extends its duration to 11 s. Meanwhile, the proposed algorithm from Sect. 3 results in the motion duration of 5.59 s. In Fig. 1 (right) time evolution of the path velocity s(t) ˙ is shown. The desired trapezoidal velocity profile is tracked up to 2.23 s when the algorithm slows down the motion to avoid exceeding the joint limits. It is worth noting that the velocity is gradually slowed down to the level of the classic method. After a while, the velocity is speeded up to resume the initially planned value at time 4.14 s. Then, the initially planned velocity is realized again. Joint velocity q˙1 (t) and acceleration q¨1 (t), seen in Fig. 2, for the proposed algorithm and the classic method are safely within their limits (shown as red lines). Note that, with the proposed algorithm, joint velocity saturates the limit between 3.25 s and 3.69 s. The acceleration limit is saturated from 2.67 s to 3.23 s.

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The method was tested with different numbers of prediction steps. For N < 9, the prediction horizon was too short, and no feasible trajectory was found. For larger values of N , the method was able to slow down and then accelerate motion. Figure 3 presents the comparison of path velocities and joint accelerations for different N . As N grows, slowing down commences a bit earlier whereas speeding up slightly later. Moreover, a larger N requires more computations.

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Final Comments and Conclusions

A new algorithm for trajectory scaling was proposed and tested on a numerical example illustrating its advantages over the classic method. There are two crucial features of the algorithm: (1) the possibility to traverse the originally planned path with velocity only reduced when it is necessary, and (2) the realtime capabilities. A natural extension of the algorithm will be devising a method for choosing the best number of the prediction steps N , representing the horizon’s length. It is worth noting that if the initially planned velocity is large, the algorithm will tend to provide time sub-optimal solutions. Another potential direction of development builds upon the observation that the initially planned path and velocity profile (being on-line adjusted by the described method) must be known in advance only within the prediction horizon. Outside this horizon, the trajectory may be replanned if needed. Some amendments to the proposed algorithm can be made in order to transform the proposed method into a typical receding horizon strategy.

References 1. Biagiotti, L., Melchiorri, C.: Trajectory Planning for Automatic Machines and Robots. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85629-0 2. Bobrow, J., Dubowsky, S., Gibson, J.: Time-optimal control of robotic manipulators along specified paths. Int. J. Robot. Res. 4(3), 3–17 (1985) 3. Broquere, X., Sidobre, D., Herrera-Aguilar, I.: Soft motion trajectory planner for service manipulator robot. In: IROS 2008, pp. 2808–2813 (2008) 4. Dahl, O., Nielsen, L.: Torque-limited path following by online trajectory time scaling. IEEE Trans. Robot. Autom. 6(5), 554–561 (1990) 5. Faroni, M., Beschi, M., Visioli, A.: Predictive inverse kinematics for redundant manipulators: evaluation in re-planning scenarios. IFAC-PapersOnLine 55(22), 238–243 (2018) 6. Gerelli, O., Guarino Lo Bianco, C.: Nonlinear variable structure filter for the online trajectory scaling. IEEE Trans. Ind. Electron. 56(10), 3921–3930 (2009) 7. Guarino Lo Bianco, C., Wahl, F.M.: A novel second order filter for the real-time trajectory scaling. In: ICRA 2011, pp. 5813–5818 (2011) 8. Haschke, R., Weitnauer, E., Ritter, H.: On-line planning of time-optimal, jerklimited trajectories. In: IROS 2008, pp. 3248–3253 (2008) 9. Hollerbach, J.M.: Dynamic scaling of manipulator trajectories. In: 1983 American Control Conference, pp. 752–756 (1983) 10. Kaserer, D., Gattringer, H., M¨ uller, A.: Nearly optimal path following with jerk and torque rate limits using dynamic programming. IEEE Trans. Robot. 35(2), 521–528 (2019) 11. Kr¨ oger, T., Tomiczek, A., Wahl, F.M.: Towards on-line trajectory computation. In: IROS 2006, pp. 736–741 (2006) 12. Lange, F., Albu-Sch¨ affer, A.: Path-accurate online trajectory generation for jerklimited industrial robots. IEEE Robot. Autom. Lett. 1(1), 82–89 (2016) 13. Lange, F., Albu-Sch¨ affer, A.: Iterative path-accurate trajectory generation for fast sensor-based motion of robot arms. Adv. Robot. 30(21), 1380–1394 (2016)

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14. Shin, K., McKay, N.: A dynamic programming approach to trajectory planning of robotic manipulators. IEEE Trans. Autom. Control 31(6), 491–500 (1986) 15. Wojtyra, M., Woli´ nski, L  .: Proposition of on-line velocity scaling algorithm for task space trajectories. In: Venture, G., Solis, J., Takeda, Y., Konno, A. (eds.) ROMANSY 2020. CICMS, vol. 601, pp. 423–431. Springer, Cham (2021). https:// doi.org/10.1007/978-3-030-58380-4 51

Path Quality Improvement of Sampling-Based Planners: An Efficient Optimization-Based Approach Using Analytical Gradients Jonas Wittmann(B) , Corina Klein, and Daniel Rixen Chair of Applied Mechanics, Munich Institute of Robotics and Machine Intelligence, Department of Mechanical Engineering, TUM School of Engineering and Design, Technical University of Munich, Garching, Germany [email protected]

Abstract. Modern collaborative robotic applications require feasible robot motions that are predictable for human coworkers. We propose two path post-processing modules based on nonlinear optimization to improve path quality: the Path Length Post-Processor (PLPP) minimizes the path length; the Path Smoothness Post-Processor (PSPP) improves the path smoothness. Both keep a clearance to obstacles. We derive the analytical gradients of the optimization problem for computational efficiency and we validate our approaches in simulation.

1

Introduction

State-of-the-art sampling-based path planners like PRM and RRT efficiently find collision-free motions in cluttered environments. However, the obtained path is generally of low quality and includes jerky motions. Approaches to optimize corresponding path metrics are either incorporated in a dedicated post-processing module or included in existing modules: path planner; (path post-processor;) trajectory generator and motion controller. RRT* and PRM* are quality-aware extensions of their geometric versions (Karaman and Frazzoli 2011). Instead of the path length in configuration space (C-space), they implement user-defined quality measures in the state propagation and converge to the globally optimal path as a function of time. Kim et al. (2003) define a higher cost for roadmap edges close to obstacles and thus maximize the clearance. However, the solution path is restricted to the randomly sampled nodes and therefore the achievable path quality is limited. Optimal control approaches directly compute a trajectory and thereby incorporate path quality. CHOMP (Zucker et al. 2013) optimizes path smoothness and collision clearance using covariant Hamiltonian gradient descent. STOMP (Kalakrishnan et al. 2011) uses several stochastically generated initial guess paths to avoid local minima. Quality-aware planners and optimization-based planners may be combined either bottom-up or top-down (Willey 2018). The former approach uses optimization-based approaches within c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 182–190, 2022. https://doi.org/10.1007/978-3-031-06409-8_19

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the state propagation of sampling-based planners. The latter uses them to postprocess paths that were generated by sampling-based planners which is also the focus of our work. Common path post-processing approaches for manipulators are geometric simplifiers that reduce unnecessary motion segments. Path pruning subsequently checks whether a waypoint can be removed while maintaining the path feasibility. The shortcut approach checks path waypoints randomly. Extensions like partial shortcutting (Geraerts and Overmars 2007) remove waypoints only of specific degrees of freedom (DoFs). Hybridization Graphs combine best segments of multiple solution paths to increase the path quality (Raveh et al. 2011). Geometric simplifiers fail to incorporate further criteria like safety distances in the path quality optimization. Optimal control approaches often come along with custom solvers and avoid the exploitation of state-of-the-art optimization solvers. We propose two path post-processing modules: PLPP focuses on path length reduction only; PSPP incorporates path smoothness and integrates postprocessing and trajectory generation in one module. As a second contribution, we derive the analytical gradients to enable an efficient implementation. The remainder of this paper is organized as follows. In Sect. 2, we present our problem formulation for PLPP and PSPP as well as the derivation of the analytical gradients for the underlying optimization problems. Section 3 evaluates the proposed methods in simulations on a planar robot model with four DoF and a FRANKA EMIKA robot. Conclusions follow in Sect. 4.

2

Problem Formulation

PLPP and PSPP are both part of a general motion planning pipeline as described in Sect. 1. In specific, we assume that there is a path planner that outputs a feasible solution path according to Definition 2.1. Definition 2.1. A path Q in C-space is defined by n waypoints qi that represent the joint configurations: Q = {qi } ∀i ∈ [1, n]. Further, we require a collision detector implementing a proximity method that, given a configuration q, computes the distance between the simplified robot geometry and the environment model, and the corresponding nearest points of the robot model and the environment model (see Fig. 1): prox : q → d, rrob , renv

(1)

In Eqs. (3) and (4) we derive the distance gradient ∇q d which is required to efficiently solve the optimization problems in Sects. 2.1 and 2.2. Thereby, the kinematic robot model provides the translational and rotational Jacobians of the colliding link j as well as the coordinates of its joint j.

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Fig. 1. The collision detector computes the closest distance d between the robot and the environment and the closest points rrob and renv .

d(q) = ∇q d =

 T (rrob (q) − renv ) (rrob (q) − renv )

(2)

1 T ∂rrob (rrob (q) − renv ) d ∂q

(3)

∂rrob = Jtrans,j − (rrob − rj ) × Jrot,j ∂q

2.1

(4)

Path Length Post-Processor

Equations (5–9) formulate the optimization problem to minimize the geometric path length defined as the Euclidean distance in the robot’s task space. Thereby, wi = w(qi ) computes the forward kinematics for the end effector. Joint limits are expressed by Eq. (6) and Eq. (9) ensures the minimum clearance dobs to obstacles according to Eq. (2). We define the additional constraint (8) to restrain the distance between two waypoints in C-space below a maximum limit δmax . That way, we maintain a somewhat uniform waypoint distribution during the optimization and thus avoid collisions resulting from waypoints that are too far apart. min f (Q) = Q

n−1  i=1

1 T (wi+1 − wi ) (wi+1 − wi ) 2

|qi | ≤ qmax q1 = qstart ;

i ∈ [1, n]

(6) (7)

∀i ∈ [1, n − 1] ∀i ∈ [1, n]

(8) (9)

qn = qgoal

2 ≥ (qi+1 − qi )T (qi+1 − qi ) δmax d(qi ) ≤ dobs

(5)

Note that we assume symmetric joint limits to simplify notations. The optimized path so obtained is then forwarded to a trajectory generator.

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We derive the analytical formulation of the derivatives of the optimization problem w.r.t. Q to ensure a computationally efficient solution. Equations (6) and (7) directly depend on the waypoints qi and their Jacobians are composed of identity matrices. The Jacobian of the collision constraint (9) was already derived in Eq. (3). To derive the objective function’s gradient ∇Q f , we as apply the chain rule to Eq. (5) and use the end effector’s Jacobian J = ∂w(q) ∂q obtained from the forward kinematics: ∇Q f = [(w2 − w1 )T J, . . . , (2wi − wi−1 − wi+1 )T J, . . . , (wn − wn−1 )T J]T

(10)

A vector δ ∈ R(n−1)×1 stores the squared distances between the waypoints on the right-hand side of Eq. (8) for all i ∈ [1, n−1]. Its Jacobian is given as follows: ⎤ ⎡ 0 A1 B 1 ∂δ ⎥ ⎢ .. .. =⎣ (11) ⎦ . . ∂Q 0 An−1 Bn−1 with 2.2

Ai = −diag(qi+1 − qi ) and Bi = diag(qi+1 − qi )

(12)

Path Smoothness Post-Processor

Equations (13–19) formulate the optimization problem to maximize the path smoothness in C-space by incorporating squared accelerations into the cost function. In contrast to Sect. 2.1 the optimization variable incorporates not only the path Q but the whole trajectory T according to Definition 2.2, and the trajectory’s total time duration D. We use the weighting factor α to combine the two independent components time duration and smoothness. Zucker et al. (2013) use the same approach to combine smoothness and obstacle avoidance and propose weight scheduling, i.e. a modification of the weighting factor during the optimization for a more robust solution. We use a constant α that we choose based on a parameter study for a given motion planning problem such that it provides a good trade-off between smooth trajectories and short execution times. Using T as the optimization variable implies that we use a direct collocation to solve the underlying optimal control problem. The trajectory computation is already part of the post-processing module and no subsequent trajectory planner is required (unlike for the PLPP ). Definition 2.2. A trajectory T in C-space is defined by a path Q and the joint velocities and accelerations at the waypoints: ¨ i } ∀i ∈ [1, n]. T = {qi , q˙ i , q

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min f (T , D) = D + α T ,D

n 

¨ Ti q ¨i q

(13)

i=1

|qi | ≤ qmax ;

|q˙ i | ≤ q˙ max ;

¨ max |¨ qi | ≤ q

q1 = qstart ; qn = qgoal q˙ 1 = 0; q˙ n = 0 q˙ i + q˙ i+1 D 0 = qi+1 − qi − 2 n−1 ¨i + q ¨ i+1 D q 0 = q˙ i+1 − q˙ i − 2 n−1 d(qi ) ≤ dobs

∀i ∈ [1, n]

(14) (15) (16)

∀i ∈ [1, n − 1]

(17)

∀i ∈ [1, n − 1]

(18)

∀i ∈ [1, n]

(19)

Equation (14) defines the actuator limits and Eq. (16) ensures zero velocities at the start and goal waypoint. They directly depend on the optimization variables, i.e. T , and therefore their Jacobians are composed of identity matrices. Equation (19) again ensures the minimum clearance dobs to obstacles with its gradient derived in Sect. 2.1. The gradient of the PSPP objective function in a k-dimensional C-space is derived as:

T qT1 , . . . , 2α¨ qTn , 1 ∇T ,D f = 01×2nk , 2α¨

(20)

As we include the time parameterization in the optimization procedure, an additional constraint as Eq. (8) is not required. Instead, two additional collocation constraints (17) and (18) are used that define the trapezoidal integration. We stack the two collocation constraints in a vector λ ∈ R2k(n−1)×1 that composes the right-hand sides of Eqs. (17) and (18) for all i ∈ [1, n − 1]. Its Jacobian w.r.t. the trajectory and time duration is derived as follows: ⎤ ⎡ −I I C C 0 0 D1 ⎢ 0 0 −I I CC E1 ⎥ ⎥ ⎢ ∂λ ⎢ .. ⎥ .. .. .. =⎢ . . . . ⎥ ⎥ ∂T , D ⎢ ⎣ (21) −I I C C 0 0 Dn−1 ⎦ 0 0 −I I C C En−1 ¨ i+1 ) D (q˙ i + q˙ i+1 ) (¨ qi + q with C = − I, Di = − , Ei = − . 2(n − 1) 2(n − 1) 2(n − 1)

3

Results

We validate the PLPP and PSPP in a MATLAB simulation on a planar robot model with four DoF as shown in Fig. 2. We implemented the nonlinear optimization expressed by Eqs. (5–9) and Eqs. (13–19) using MATLAB ’s sequential quadratic programming (SQP) solver. We used an equal weighting of trajectory time and smoothness for the PSPP, i.e. α = 1. Note that, although the computed

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path of the PRM planner consists of only four waypoints, we further discretize it by linear interpolation using steps of the order of the collision objects’ sizes before passing it to the PLPP or PSPP. The post-processors satisfy collisions only at the waypoints, not in between. Thus, the modification of the waypoints could result in collisions when the initial path waypoints are too far apart.

Fig. 2. Computed paths for the evaluation scenario with two rectangular collision objects. The discretized input path consists of 41 waypoints and was computed with a standard PRM planner. The orange and green line show the post-processed paths for our two proposed approaches. We implemented the standard short cutting (SC ) described in Sect. 1 as a benchmark.

3.1

Analytical Gradient Formulation Efficiency

To demonstrate the computational efficiency of our analytical gradient-based optimization approach, we ran the scenario in Fig. 2 with the same solver once with the analytical gradient formulation and once using finite differences instead. Table 1 compares the analytical gradient formulation with numerical approximations using finite forward differences and finite central differences for the PLPP. The simulation runs on a computer with 16 GB RAM with a quad-core Intel Xeon E3-1240 v5 CPU running at 3.5 GHz. The computation time is averaged over ten runs and the mean value and the standard deviation are given. For the numerical approximation of the gradient information, finite differences with a step size of 1e−6 s have been used. In a comparison of analytical and numerically approximated gradients, we validated the correct formulation of the former. In each iteration, the numerical approximation of gradients requires a much higher number of evaluations of the optimization and constraint functions compared to the analytical gradient formulation. Therefore, the computation times increase significantly and restrict the formulation with numerical approximated gradients to offline applications. Even with the analytical gradient formulation, the optimization time of 66.4 s is still too long for an online application. The cause for the long computation time lies in the implementation of the collision detector within MATLAB : the Gilbert-Johnson-Keerthi (GJK) distance algorithm is

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used for distance computations which make up 90.4 % of the computation time. In robotics, however, environmental models are often generated using simplified geometries like swept sphere volumes which allow more efficient distance computations than GJK. Table 1. Computation time for path post-processing with PLPP with and without analytical gradients. Analytical Forward diff.

Central diff.

Computation time [s] 66.4 ± 0.8 4487.6 ± 22.8 8570.1 ± 96.7 49/8102 47/15481 Iterations/Function eval. 56/119

3.2

Optimization-Based Path Post-Processing

The post-processed paths in Fig. 2 demonstrate the post-processing capabilities of PLPP and PSPP. To evaluate the computational efficiency of the SQP solver, it is crucial to identify appropriate termination conditions and optimization parameters. Further, a benchmark of different problem formulations, i.e. of PLPP and PSPP, is not straightforward as the different cost functions, i.e. Eqs. (5) and (13), have different convergence characteristics. For the PLPP, a parameter study showed that a relative tolerance of 1e−3 for the first-order optimality provides a good trade-off between computation time and optimization. Thereby, PLPP decreases the task space length of the input path from 1.81 m to 0.83 m, i.e. by 54.1 %, within 66.4 s computation time for 56 iterations. With the same number of iterations as termination criterion, PSPP decreases the path length to 0.85 m, i.e. by 53.0 %, within 246.7 s. However, PSPP shows a faster convergence rate and provides a similar result already after 21 iterations and 54.0 s. With respect to computation time, PLPP and PSPP cannot compete with the SC approach that minimizes the path length to 0.75 m within 2.12 s. However, in contrast to PLPP and PSPP, SC does not maintain a defined safety distance and generates a much jerkier path. The smoothness optimization within PSPP is validated in Fig. 3. Due to the collocation constraints in Eqs. (17) and (18), PSPP computes a cubic position profile which results in a smoother motion than the linear profile of PLPP and the input path. The joint limit constraints are respected at every waypoint. 3.3

Task Board Assembly

We implemented PLPP in our C++ control framework for a FRANKA EMIKA robot using the NLopt library. In the tested application scenario, the robot solves assembly operations on a task board using a geometric RRT planner. We use the Flexible Collision Library for the distance computations and the 3D simulation shows that this increases the computational speed compared to the

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GJK algorithm that we use in the MATLAB simulation. The result is shown at: https://youtu.be/Vu9me5HIFr4. Note that we did not implement analytical gradients of the optimization problem in C++ yet. Therefore, the post-processing times for the optimization in between the robot motions is still considerably long.

Fig. 3. Position profile of the path waypoints for the second and third joints. We implemented symmetric joint limits.

4

Conclusion

Our proposed PLPP is able to post-process the path computed by state-of-theart sampling-based path planners. Thereby, it minimizes the path length and keeps a defined clearance to obstacles. The incorporation of distance computation within the post-processing step requires the derivation of the analytical gradients for the underlying nonlinear optimization problem. Numerically approximated gradients result in computation times that are several orders higher and that prevent an application even in offline path planning, e.g. with PRM. Our proposed PSPP incorporates a smoothness metric and thus generates a high quality path while keeping a defined clearance to obstacles. Still, PLPP and PSPP are time-consuming and restrict online applications in which motion planning times are supposed to be of the same magnitude as motion execution times. In future work, we plan to implement more efficient distance computation algorithms that make up a large proportion of the current computation time.

References Geraerts, R., Overmars, M.H.: Creating high-quality paths for motion planning. Int. J. Robot. Res. 26(8), 845–863 (2007). https://doi.org/10.1177/0278364907079280 Kalakrishnan, M., Chitta, S., Theodorou, E., Pastor, P., Schaal, S.: Stomp: stochastic trajectory optimization for motion planning. In: 2011 IEEE International Conference on Robotics and Automation, pp. 4569–4574 (2011). https://doi.org/10.1109/ICRA. 2011.5980280

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Karaman, S., Frazzoli, E.: Sampling-based algorithms for optimal motion planning. Int. J. Robot. Res. 30(7), 846–894 (2011). https://doi.org/10.1177/0278364911406761 Kim, J., Pearce, R.A., Amato, N.M.: Extracting optimal paths from roadmaps for motion planning. In: 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422), vol. 2, pp. 2424–2429 (2003). https://doi.org/10.1109/ ROBOT.2003.1241956 Raveh, B., Enosh, A., Halperin, D.: A little more, a lot better: improving path quality by a path-merging algorithm. IEEE Trans. Robot. 27(2), 365–371 (2011). https:// doi.org/10.1109/TRO.2010.2098622 Willey, B.S.: Combining sampling and optimizing for robotic path planning. Master’s thesis, Rice University (2018) Zucker, M., Ratliff, N., Dragan, A.D., et al.: Chomp: covariant hamiltonian optimization for motion planning. Int. J. Robot. Res. 32(9–10), 1164–1193 (2013). https:// doi.org/10.1177/0278364913488805

Hexapod Posture Control for Navigation Across Complex Environments Joana Coelho1(B) , Bruno Dias2 , Gil Lopes3 , Fernando Ribeiro4 , and Paulo Flores1 1

2

CMEMS UMinho, Department of Mechanical Engineering, University of Minho, Guimaraes, Portugal [email protected] Center Algoritmi, Informatics Department, University of Minho, Braga, Portugal 3 University of Maia, Maia, Portugal 4 Center Algoritmi, Industrial Electronics Department, University of Minho, Guimaraes, Portugal

Abstract. Hexapod locomotion in unstructured environments relies on an efficient posture adjustment with the terrain topology. This paper presents a strategy to adapt the hexapod torso orientation through ground plane estimation. With an Inertial Measurement Unit (IMU) and the robot kinematic model, the current supporting feet coordinates are calculated, and the relative inclination between the ground and the torso angular position can be obtained. This information is used to adjust the novel foothold positions, in order to ensure the hexapod posture remains stable. The torso height is also controlled to avoid collisions with the ground asperities and decrease its deviation during motion. The proposed method is evaluated in a complex terrain made of 0.1×0.1 m blocks with variable height, causing different slopes across the field. Through result analysis, a significant behavior improvement is observed, due to the reduction of the torso posture oscillation and the increase of its locomotion efficiency.

1

Introduction

The design of hexapods has an increasing interest for the execution of tasks in uncontrolled and dangerous environments (Rubio et al. 2019). Nonetheless, the generation of adaptive locomotion remains an important research question in the control of these robots (Coelho et al. 2021). In an unstructured ground, a hexapod deals with an oscillation of its posture, due to the variable feet forces The first author received funding through a doctoral scholarship from the Portuguese Foundation for Science and Technology (FCT) (Grant No. SFRH/BD/145818/2019), with funds from the Portuguese Ministry of Science, Technology and Higher Education and the European Social Fund through the Programa Operacional Regional Norte. This work has been supported by FCT within the R&D Units Project Scope: UIDB/00319/2020, UIDB/04436/2020 and UIDP/04436/2020. c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 191–198, 2022. https://doi.org/10.1007/978-3-031-06409-8_20

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distribution, which compromises the system stability (Irawan and Nonami 2011). Furthermore, the field slopes cause the torso Center of Mass (CM) to translate to the robot rear side, having also a negative influence in the hexapod locomotion (Molnar et al. 2017). Thereby, the body orientation control is important for the adaptive locomotion efficiency. Amongst the existent solutions, Molnar et al. (2017) proposes a body levelling method for climbing ramps which translates the ˇ ıˇzek (2019) hexapod CM to ensure its quasi-static stability. Likewise, Faigl and C´ also increased a hexapod stability in unstructured environments by centering the CM position with the legs support polygon. Besides that, the presented method also ensures that the body remains at a certain relative height through the ground plane estimation based on the feet coordinates. Nonetheless, the robot resorts to inverse dynamics calculations to detect the limbs interaction with the ground, which increases the system computational complexity. The main disadvantage of these researches is not assuming the torso pose adjustment with the slope variation. This is important not only when the hexapod must transport hardware or cargo in this type of environment but also to improve the system efficiency. In a different approach, Bjelonic et al. (2016) proposes a redundant limb kinematic design with five Degrees of Freedom (DOF) to adjust the body posture and feet orientation by forcing the legs tarsus to be aligned with the gravity vector for efficiency improvement. However, the increase of the hexapod DOF results in more complex kinematic calculations for the foot trajectory generation, requiring a higher computational efficiency. This article aims at improving the hexapod pose for navigation across unstructured terrain. The proposed method avoids increasing the system kinematic design complexity by resorting to an optimal simplification of the insects appendages, i.e. three DOF per leg (Buschmann and Trimmer 2017). The torso pose is adjusted through estimation of the plane formed by the footholds in the stance phase, using the robot kinematic model. Moreover, this method also introduces the usage of an infrared sensor to control the body height. In sum, the advantage of this strategy is to ensure a stable body pose despite the slope variation. To study the impact of this approach in the hexapod locomotion, the proposed method is added to a control framework which applies a reactive behavior to the limbs, to improve the system adaptability. This architecture is tested through computational simulations in CoppeliaSim, which provides a diverse library of sensors and is commonly used for mobile robots to test navigation planning in real-time (Ivaldi et al. 2014; Collins et al., 2021). Besides that, a Robot Operating System (ROS) framework communicates with the remote control API, which facilitates the transition of the simulation to the real world once the algorithm is ready to be implemented. Since the goal of this paper is to study the proposed method efficiency, the controller is implemented in a hexapod model provided by CoppeliaSim. The results obtained confirm an improvement of the hexapod performance in a slope variable irregular ground. The structure of this article is as follows. Section 2 describes the hexapod kinematic design and Sect. 3 presents the proposed control framework. Section 4 discusses the obtained results from the computational simulation. Finally, Sect. 5 presents the concluding remarks and future developments.

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(a)

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(b)

Fig. 1. Model description: (a) hexapod design and (b) limb kinematic design with the relative references for the forward kinematics calculation.

2

Model Description

The model description includes the following aspects: presentation of the hexapod kinematic design and discussion of the limbs forward kinematics. Figure 1(a) presents the hexapod used in this study, which is provided by the software CoppeliaSim and has three DOF per limb. Regarding the model dimensions, the torso radius (l0 ) is 0.0799 m, and the coxa (l1 ), femur (l2 ) and tibia (l3 ) lengths are, respectively, 0.0502, 0.0723 and 0.1159 m. The revolute joints nomenclature is as follows: Thorax-Coxa (TC), Coxa-Trochanterofemur (CTr) and Femur-Tibia (FTi). For this work, the feet relative position in respect to the torso coordinates is important not only to plan the hexapod locomotion but also to adjust the foothold positions based on the field slope. Using the Denavit-Hartenberg convention, the foot relative position of the ith leg is expressed as, ⎤ ⎡ cos (θ0r + θ1r ) (l3 cos (θ2r + θ3r ) + l2 cos (θ2r ) + l1 ) + l0 cos (θ0r ) pTi = ⎣ sin (θ0r + θ1r ) (l3 cos (θ2r + θ3r ) + l2 sin (θ2r ) + l1 ) + l0 sin (θ0r ) ⎦ l3 sin (θ2r + θ3r ) + l2 sin θ2r

(1)

where θ0r is the relative angle between the references of the torso and the TC joint, and is expressed as θ0r = π3 i, i = 1, ..., 6.

3

Control Strategy

Figure 2 portrays the system control framework. The higher layer plans the robot motion and synchronizes the limbs actuation based on the desired footholds and the gait phase. For the swing phase, the leg executes a B´ezier curve, while for the stance phase the motion is similar to the one proposed by Liu et al. (2020). Along with sending the joints desired angular positions to the actuators, the low level control also gathers the data from the hexapod sensors for the adaptive behavior. The force sensors placed on the feet measure the normal contact forces, and the legs change from the swing to the stance phase when these sensors

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Fig. 2. Hierarchical control architecture.

detect an unexpected foot-ground interaction. Besides this reactive behavior, the impedance controller adapts the foot coordinates during contact detection, to adjust the leg stiffness and absorb some impact energy. The posture controller is responsible for adjusting the torso angular position with the terrain topology. The goal is to ensure the minor fluctuation of the torso angular position when walking across complex environments. Thereby, the approach is as follows. The current hexapod orientation [αT , βT , γT ]T is obtained using an IMU placed on the hexapod CM. With this information, the foot coordinates are expressed as, T pW i = Rotz (γT )Roty (βT )Rotx (αT )pi , i = 1, ..., 6

(2)

where Rot is the rotation matrix and pTi is obtained using the forward kinematics. From Fig. 3, the supporting feet coordinates form a relative plane ap x + bp y + cp z = d, and its respective roll (αp ) and pitch (βp ) are obtained through the norm vector. Thus, the angular position adjustment can be obtained through the following translation for each TC joint, T ransT C = Roty (βp − βT )Rotx (αp − αT )T CiW − T CiW , i = 1, ..., 6

(3)

The obtained translation matrix is applied to the foot desired coordinates, and the limb trajectory is re-adjusted. Besides the posture control, since the hexapod relies only on proprioceptive information, it is important to maintain a body height high enough to avoid obstacle collisions. Thus, this framework controls the torso relative height through an infrared sensor placed on the body CM, which measures its distance to the floor, despite the slope and terrain

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Fig. 3. Representation of the relative ground plane formed by the limbs 2, 4 and 6.

topology. By setting up a threshold value, the foot final Z-coordinate of each gait phase is adjusted to ensure the hexapod height remains equal or higher than the pre-defined goal.

4

Results and Discussion

This section describes the implementation of the proposed method through computational simulations. The proposed framework was implemented using the simulator CoppeliaSim, with the Bullet physic engine version 2.78 and a 50 ms simulation time-step. A ROS framework with a rate 20 Hz establishes the communication between the simulator and the designed control, which is a remote Python API. It is important to mention that during the transition between the simulation and the real word, the framework rate needs to be re-adjusted to synchronize with the real-time sensors frequency. Figure 4 presents the tested scenario. It consists of a 0.1 × 0.1 m blocks test bed with a height value within 0.01 and the 0.09 m (Fig. 6b). The hexapod adopts a tripod gait, with a phase period of 1 s, step height of 0.2 m and a stroke of 0.015 m. The proposed control strategy has a force threshold of 1 N for the contact detection, and the impedance controller damping ratio, characteristic frequency and virtual mass are, respectively, 0.8, 280 s−1 and 0.4491 kg. For the body leveler, the height threshold is 0.1241 m. For result analysis, the designed approach behavior is compared to the one of a non-adaptive controller. Besides the posture evaluation, the Cost of Transport (CoT) is also analysed, which is expressed as (Bjelonic et al. 2016), CoT =

P mgv

(4)

where P is the overall power consumption, m is the system mass and v is its velocity. This parameter gives an insight into the locomotion efficiency, based on

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Fig. 4. Scenario where the blocks with variable height are visible.

Fig. 5. Results obtained for the torso orientation: (a) roll angle and (b) pitch angle.

the energy required to actuate the limbs. The value of P was calculated using the torque and angular velocity of each joint. Figures 5(a) and 5(b) portray the torso angular position throughout the computational simulations respectively for the non-adaptive control and the designed architecture. As it can be observed, the difference between the obtained results is significant. Due to the field height variation, the feet height is not constant, which has a strong influence on the torso posture. Since the proposed method estimates the ground relative plane at each time-step, the hexapod footholds are constantly adjusted to the terrain slope, leading to a lower posture variance. The hexapod reactive behavior and the feet impedance controllers also provide a better response to the field irregularities, which aids with the robot stability during its locomotion. The behavior difference is also observed in Fig. 6. In opposition to Fig. 6(a), the adaptive control forces the torso to remain horizontal while traversing the test bed (Fig. 6(b)). This adjustment also influences the locomotion performance. The results presented in Table 1 highlight the higher performance of the designed method when walking across this environment. The average value of CoT is smaller for the posture control, thus, the hexapod joints require less power to actuate its limbs in the same conditions. The smaller CoT standard deviation observed in the adaptive controller also shows evidences of a higher locomotion stability.

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(a)

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(b)

Fig. 6. Hexapod posture during the simulation: (a) non-adaptive control and (b) with posture control. Table 1. CoT results. Test

Average CoT

Non-adaptive control 7.9873 ± 36.6401 With posture control 6.7998 ± 12.8485

5

Conclusions and Future Work

This article proposes a posture controller based on the estimation of the plane formed by the supporting feet. The designed system also levels the torso height to increase the locomotion stability and avoid its collision with the ground. This module is implemented in a proprioceptive adaptive control with reactive behavior, to detect unexpected foot-ground interactions, and feet impedance controllers, to adapt both the limbs trajectory and stiffness to the terrain topology. By comparing the results with the ones obtained through simulation of a non-adaptive control system, it is concluded that the proposed model reduces significantly the torso posture fluctuation, ensuring that it remains horizontal during most of the study. Since the purpose of this research is to verify the proposed controller behavior, the computational simulations were conducted in CoppeliaSim, which allows to use ROS for the communication establishment between the software and the remote control API. The advantage of this methodology is simplifying the implementation of the controller in the real world prototype. The values obtained for the CoT corroborate the system improvements, since the hexapod requires less actuator power to walk across the same scenario. Since this paper is inserted in an ongoing research, the further stage of this work aims at studying the control architecture in different terrain topologies to evaluate both the consistency of the posture adapter and the gait efficiency. Furthermore, the controller will be implemented in a real world prototype and the obtained results will be compared with the ones obtained through computational simulations.

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References Bjelonic, M., Kottege, N., Beckerle, P.: Proprioceptive control of an over-actuated hexapod robot in unstructured terrain. In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 2042–2049 (2016). https://doi.org/ 10.1109/IROS.2016.7759321 Buschmann, T., Trimmer, B.: Bio-inspired robot locomotion. In: Hooper, S.L., B¨ uschges, A. (eds.) Neurobiology of Motor Control, pp. 443–472. Wiley, Hoboken (2017). https://doi.org/10.1002/9781118873397. ISBN:9781118873397 Coelho, J., Ribeiro, F., Dias, B., Lopes, G., Flores, P.: Trends in the control of hexapod robots: a survey. Robotics 10(3) (2021). https://doi.org/10.3390/robotics10030100. ISSN:2218–6581 Collins, J., Chand, S., Vanderkop, A., Howard, D.: A review of physics simulators for robotic applications. IEEE Access 9, 51416–51431 (2021). https://doi.org/10.1109/ ACCESS.2021.3068769 ˇ ıZek, ˇ Faigl, J., C´ P.: Adaptive locomotion control of hexapod walking robot for traversing rough terrains with position feedback only. Robot. Autonom. Syst. 116, 136–147 (2019). https://doi.org/10.1016/j.robot.2019.03.008. ISSN:0921–8890 Irawan, A., Nonami, K.: Optimal impedance control based on body inertia for a hydraulically driven hexapod robot walking on uneven and extremely soft terrain. J. Field Robot. 28(5), 690–713 (2011). https://doi.org/10.1002/rob.20404 Ivaldi, S., Peters, J., Padois, V., Nori, F.: Tools for simulating humanoid robot dynamics: a survey based on user feedback. In: 2014 IEEE-RAS International Conference on Humanoid Robots, pp. 842–849 (2014). https://doi.org/10.1109/HUMANOIDS. 2014.7041462 Liu, Y., Fan, X., Ding, L., Wang, J., Liu, T., Gao, H.: Fault-tolerant tripod gait planning and verification of a hexapod robot. Appl. Sci. 10(8) (2020). https://doi.org/ 10.3390/app10082959. ISSN:2076–3417 Molnar, T., Steindl, R., Kottege, N., Talbot, F., Elfes, A.: Steep terrain ascension controller for hexapod robots. In: Australasian Conference on Robotics and Automation, ACRA, December 2017, pp. 1–7 (2017). ISBN:9781510860117 Rubio, F., Valero, F., Llopis-Albert, C.: A review of mobile robots: concepts, methods, theoretical framework, and applications. Int. J. Adv. Robot. Syst. 16(2), 1729881419839596 (2019). https://doi.org/10.1177/1729881419839596

Control Performance Improvement in Dynamically Decoupled Manipulators Yaodong Lu1,2(B) , Yannick Aoustin1 , and Vigen Arakelian1,2 1

LS2N-ECN UMR 6004, 1 rue de la No¨e, BP 92101, 44321 Nantes, France 2 Mecaproce/INSA-Rennes, 20 av. des Buttes de Coesmes, CS 70839, 35708 Rennes, France [email protected]

Abstract. The control of industrial robot manipulators presents a difficult problem for control engineers due to the complexity of robot dynamics models. Nonlinear controls based on feedback linearization are developed to meet control requirements. Model-based nonlinear control is highly sensitive to parameter errors and leads to problems of robustness for tracking trajectories at high speeds, and there is the additional problem of a heavy computational burden to consider in the design of nonlinear controllers. In this paper, a mechatronics design approach is proposed, which aims to facilitate controller design by redesigning the mechanical structure. The problem is approached in two steps: first, the dynamic decoupling conditions of manipulators are described and discussed, involving redistribution of the moving mass, which leads to the decoupling of motion equations. A classical linear control technique is then used to track the desired efficient bang-bang profile trajectory. An analysis of the results from a simulation of this approach demonstrates its effectiveness in controller design. The proposed improvement in control performance is illustrated via a two-revolute joint spatial manipulator.

1

Introduction

In recent years, industrial robots have been widely used for high-precision applications [1]. Different types of industrial robots are developed to carry out various tasks such as painting, welding, manufacturing, assembling and so on. The requirements for industrial robot performance are accuracy, speed and versatility of manipulation. To meet this demand, robot control is a key element for robot manufacturers, and a great deal of development work is carried out to increase the performance of robots, reduce their cost and introduce new functionalities. In terms of controller design, robustness, tracking accuracy, energy consumption and computational burden must be taken into account [2,3]. However, robot manipulator dynamics are known to be highly nonlinear and coupled, which results in poorer control performance at high speeds [4,5]. The complicated dynamics are the result of varying inertia and interactions between the different joints, and the nonlinear coupled dynamics of the robot manipulator c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 199–209, 2022. https://doi.org/10.1007/978-3-031-06409-8_21

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also increase the energy consumption and computational burden of the controller [4]. Various nonlinear control strategies have been developed to resolve this problem, such as feedback linearization [6]. Feedback linearization is a tool widely used for controlling non-linear systems, and operates by canceling the nonlinearities in robot dynamics [7,8]. With this method, the closed-loop system becomes nominally linear and classical linear system control techniques can then be applied in the controller design, with relative efficiency. However, feedback linearization requires an accurate model and high sampling rate for successful implementation [9]. In addition, the computational burden caused by nonlinear and coupled terms cannot be avoided with complex robot structures in feedback control [10,11]. For example, computed torque (CT) control is a well-known control law that consists of a propotional-derivative (PD) term and the feedback dynamic compensation term calculated from the actual velocity and desired acceleration signals [12]. CT control offers a wide range of advantages, such as high tracking accuracy, low energy consumption, and more compliant control. However, its major drawback is the precise analytical dynamic models required for torque generation, which cannot always be obtained due to modeling errors and unknown disturbances [13]. Inaccurate models will lead to low tracking performance in practice, and a complex robot structure also requires an enormous calculations in the CT controller. The nonlinear model can be said to increase the difficulty in designing suitable controllers for robot manipulators. Some researchers suggest canceling the nonlinearities by removing the complex inertia terms in dynamic models, as these are negligible at low speeds [14,15]. However, at high motion speeds, the errors in the model resulting from this approach become significant and would probably reduce stability, leading to loss of effectiveness in tracking performance. As described above, it is challenging to find a simple and effective strategy for controlling robot manipulators due to the presence of model nonlinearities and errors. To resolve this problem, a mechatronic design methodology known as Design For Control (DFC) was proposed by [16]. The method points out that the performance of the mechatronic system relies not only on the design of the controller but also on the design of its mechanical structure. Controller design is usually considered subsequent to mechanical design, but an appropriate mechanical structure design will lead to a simple dynamic model which makes design of the controller easier. In this paper, an effective design approach is proposed to improve the control performance of a 2R robot manipulator, based on development of the mechanical structure design. With dynamic decoupling, a simplified system is obtained and effective conventional control techniques, rather than complex nonlinear control techniques, are applied. There are three ways to create dynamically decoupled manipulators through mechanical transformation: i) via mass redistribution; ii) via actuator relocation; iii) via the addition of auxiliary links.

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To eliminate the coupling and nonlinear torques via mass redistribution, the inertia matrix must be diagonalized and made invariant for all arm configurations [17–20]. The linearization and dynamic decoupling of 2-DOF manipulators via mass redistribution has been considered previously [20]. In this study, all arm constructions yielding decoupled inertia matrices were identified. The proposed approach was applied to serial manipulators in which the axes of joints were not parallel; with parallel axes this approach allows linearization of the dynamic equations but not their dynamic decoupling [21] and cannot therefore, be used with planar serial manipulators. As a result, the inertia matrix cannot be decoupled unless the joint axes are orthogonal to each other in serial manipulator arms with an open kinematic chain structure. As regards the decoupling of dynamic equations via actuator relocation, a review [18] has shown that the remote-actuation design concept is not optimal from the point of view of precise reproduction of the end-effector tasks, because it accumulates all errors due to intermediate transmissions. It is evidently much better to connect actuators directly to the links than to use transmission mechanisms. The clearance, flexibility, manufacturing and assembly errors of the added transmission mechanisms have a negative impact on the robot’s precision. The linearization of dynamic equations and their decoupling by adding auxiliary links to redesign the manipulator has also been developed [22–24]: dynamic decoupling of the manipulator by connecting a two-link group to the initial structure, forming a Scott-Russell mechanism, was proposed by [25]. Noted that dynamic decoupling via redesign of the manipulator by adding auxiliary links is a promising new approach in robotics. As discussed above, taking into account the structural features of the 2R spatial serial manipulator, i.e., orthogonality of the joint axes, it can be confidently asserted that dynamic decoupling via mass redistribution is the most expedient approach. This paper is organized as follows: Sect. 2 contains a review of the dynamic decoupling conditions of 2R spatial serial manipulators based on mass redistribution, and the generation of motion by “bang-bang” profile, enabling reduction of the maximal input torque values. In Sect. 3, different conventional linear controllers are used to stabilize the decoupled system and track the desired trajectory. Numerical simulations are also carried out. Section 4 presents conclusions and perspectives.

2

Decoupled Dynamics of the 2R Spatial Serial Manipulator and Its Motion Generation via Bang-Bang Profile

Before addressing the problem of control performance improvement, the dynamic decoupling conditions of the 2R spatial serial manipulator (Fig. 1) are reviewed below.

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Fig. 1. 2R spatial serial manipulator

The manipulator consists of two orthogonal links, 1 and 2, with rotating angles θ1 and θ2 . We will distinguish the relative angular velocities of vectors ¨r1 , θ ¨r2 , and the vectors θ˙1 and the absolute angular velocities θ˙1r and θ˙2r with θ θ˙2 with θ˙1 = θ˙1r and θ˙2 = θ˙1r + θ˙2r . In the study [17], it was reported that the 2R spatial serial manipulator can be dynamically decoupled completely if the following conditions are satisfied: – the potential energy of the manipulator is constant (or canceled), i.e. the manipulator is statically balanced; – Ix2 = Iy2 = I ∗ , where Ix2 and Iy2 are the axial moments of inertia of link 2 relative to the corresponding coordinate axes of the syst- em associated with link 2; As a consequence of such a redistribution of moving masses, the motion equations become linear and decoupled: τ1 = (I1 + I ∗ )θ¨1r

(1)

τ2 = Iz2 θ¨2r

(2)

The non-linear dynamic system is thus transformed into a double integrator model and the state space equation of each link can be defined: x˙ = Ax + Bu with

    01 0 A= ,B = 00 1/I

(3)

(4)

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where, I is calculated with inertia moment of each link in Eqs. 8 and 9. I = (I1 + I ∗ ) for the first link and I = Iz2 for the second link. As discussed in [17], to generate motion in the dynamically decoupled 2R spatial serial manipualtor, it is preferable to apply the “bang-bang” profile (Fig. 2), which enables reduction of the maximal input torque values.

Fig. 2. “Bang-bang” profile used for generation of motion in the dynamically decoupled 2R spatial serial manipulator.

Bang-Bang motion profile:  θ(t ) + 2( ttf )2 θf , t0 < t < tf /2   0 θ(t) = t t 2 θ(t0 ) + −1 + 4( tf ) − 2( tf ) θf , tf /2 < t < tf

3 3.1

(5)

Design of Linear Controller Performance Indices

For the controller design, performance indices are introduced to quantify and evaluate system performance. Two performance indices were considered in our

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case: the integral of the square of the error (ISE) and the maximum input torque. The form of the ISE is defined as:  tf e2 dt (6) ISE = t0

The ISE discriminates between excessively over-damped and excessively underdamped systems. Tracking accuracy can be evaluated using this criterion. The other criterion introduced was the maximum input torque, which represents input energy. The actuator capacity requirement also depends on the maximum input torque. 3.2

Pole Placement

Since the double integrator model system is unstable, the control technique should increase the stability of the system. Therefore, the speed of response and tracking error requirement should be improved in order to track the bang-bang profile. Pole placement is a well-established design method for linear control systems. We assumed that all state variables were measurable and also available for feedback. It will be shown that if the system considered is completely state controllable, then pole feedback is through an appropriate state feedback gain matrix. The system must be completely state-controllable for arbitrary pole placement. The controllability matrix is given by:    0 1/I 2 n−1 B = (7) C = B AB A B ... A 1/I 0 Since the control matrix C of the state space equation obtained has full row rank 2, this decoupled dynamic system is controllable. The system input can therefore be defined as u = −Kx, where K is the state feedback gain matrix. The state space equation is obtained as follows: x˙ = (A − BK)x

(8)

To ensure stability and speed of response, the desired closed-loop eigenvalues (poles) of A − BK must be negative. Two eigenvalues, λ1 and λ2 , were set at −120. The state feedback gain matrix K can then be determined by direct substitution. The characteristic polynomial for the desired system is:

     

λ0

k1 k2 01 0  k1 k2

= λ2 + λ + − + (9) |λI − A + BK| =

0λ 00 1/I I I This characteristic polynomial must be equal to: (λ + 120)2 = λ2 + 240λ + 14400

(10)

By equating the coefficients of the terms of the like powers of λ, we obtain:   K = k1 k2 = 240I 14400I (11)

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The simulation was carried out in Matlab software with I = 0.4 kgm2 and the time-step set at 0.00001. The initial and final values of the rotating angles were as follows: θi = 0◦ and θf = 30◦ . Tracking performance and input torque are obtained as follows (Figs. 3 and 4):

Fig. 3. Trajectory tracking of pole placement

Fig. 4. Torque generation of pole placement

From the simulation results, it was observed that the decoupled dynamic system with the pole placement method results in excellent tracking performance by the bang-bang profile. The ISE is 0.0182 and the maximum torque 3.1933 Nm. However, this is a full state feedback control method; when the system is not measurable for velocity, other linear control techniques are considered. 3.3

Lead Compensation

The primary function of the lead compensator is to reshape the frequencyresponse curve to provide a phase lead angle sufficient to offset the excessive lag phase associated with the components of the fixed system. Since our new linear system, obtained by dynamic decoupling, was a double integrator model, a lead compensator could be used to stabilize the system by increasing the phase margin. A lead compensator in the following form will be used: Gc (s) = Kc

Ts + 1 αT s + 1

(12)

where Kc , T and α are the coefficients determined with the maximum phase lead angle. Note however that the system’s dynamic characteristics need to be modified by increasing the cut-off frequency, which increases the dynamic response speed to track the bang-bang profile. The modified dynamic system expression is then: I (13) G(s) = k 2 s

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where the value of k is determined to increase the system cut-off frequency wc . The lead compensator is then applied to help the modified system track the desired trajectory by increasing a certain phase margin. In this case, the selected value of k is 20,000. The gain crossover frequency is therefore increased to 89.44 rad/s. This implies an increase in the speed of response. We assumed that the necessary maximum phase lead angle φm is 70◦ , and therefore the coefficients in the lead compensator, could be determined. The Bode plot of the compensated system is as follows:

Fig. 5. Frequency domain response with and without a lead compensator

As observed in the Bode plot, the maximum phase lead margin was increased to 70◦ at the gain crossover frequency, which improved the stability. To see the tracking performance and input torque, the simulation was carried out in Matlab. The value of the time-step was set at 0.00001 s.

Fig. 6. Trajectory tracking of lead compensator

Fig. 7. Torque generation of lead compensator

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It is evident that the system with the modified gain crossover frequency and lead compensator with 70◦ phase is capable of tracking the desired bang-bang profile trajectory. The integral square of errors is 0.0399 and the maximum torque 3.6289. In addition, a table with a variety of phase lead angles was created to investigate the influence of different phase lead angles on two criteria (Table 1). Table 1. Results of different phase lead angles Tracking Performance ISE (rad2 s) Maximum of Torque (Nm) 6◦ Phase Lead Angle

2.10 × 10−3

7.18

10 Phase Lead Angle 2.30 × 10−3

6.89

20◦ Phase Lead Angle 3.20 × 10−3

6.32

◦

◦

30 Phase Lead Angle 4.50 × 10

−3

40◦ Phase Lead Angle 6.80 × 10−3 ◦

50 Phase Lead Angle 1.08 × 10

−2

60◦ Phase Lead Angle 1.90 × 10−2 ◦

70 Phase Lead Angle 3.99 × 10

−2

80◦ Phase Lead Angle 1.18 × 10−1

5.78 5.27 4.76 4.23 3.63 2.87

The table shows that the ISE is diminished and the maximum torque increased when the phase lead angle is increased. In application, a large phase lead angle is required to lower the actuator capacity and energy consumption. However, for a system requiring high tracking accuracy, a small phase lead angle is needed. Obviously, there is a trade-off between the two criteria, which also means that the required performance can be achieved by adjusting the phase lead angle, which facilitates the design of the robot manipulator controller.

4

Conclusion

The control of robot manipulators for high-performance and high-speed tasks has always been a challenge for control engineers. Nonlinear control has been developed but it can encounter difficulties such as tracking inaccuracy at high speed and a heavy computational burden. It can be difficult to obtain satisfactory control performance. To resolve this problem, a new mechatronics approach is proposed to meet the demand for control performance. Rather than concentrating on the design of the control algorithm, this approach focuses on redesigning the mechanical structure to obtain a linear and decoupled dynamic system. This offers greater convenience for controller design. The arrangement of centers of mass and inertia redistribution for the links were described to obtain the decoupled and linear dynamic equations for the manipulator. It was demonstrated that the input torques in the dynamically decoupled manipulator we obtained are directly proportional to the input angular accelerations. The classical linear control techniques of pole placement and lead compensation were therefore

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adopted to track the desired bang-bang profile trajectory. With these classical linear control strategies, the decoupled system can be stabilized rapidly and the desired control performance obtained. The results of the simulations demonstrate that the proposed method reduces the computational burden and provides an improved control performance in tracking accuracy and maximum input torque.

References 1. Brog˚ ardh, T.: Present and future robot control development-an industrial perspective. Annu. Rev. Control 31(1), 69–79 (2007) 2. Luh, J., Fisher, W., Paul, R.: Joint torque control by a direct feedback for industrial robots. IEEE Trans. Autom. Control 28(2), 153–161 (1983) 3. Luh, J.: Conventional controller design for industrial robots-a tutorial. IEEE Trans. Syst. Man Cybern. 3, 298–316 (1983) 4. Kuo, C.-Y., Wang, S.-P.T.: Nonlinear robust industrial robot control (1989) 5. Freund, E.: Fast nonlinear control with arbitrary pole-placement for industrial robots and manipulators. Int. J. Robot. Res. 1(1), 65–78 (1982) 6. Poignet, P., Gautier, M.: Nonlinear model predictive control of a robot manipulator. In: Proceedings of the 6th International Workshop on Advanced Motion Control (Cat. No. 00TH8494), pp. 401–406. IEEE (2000) 7. Wang, D., Vidyasagar, M.: Control of a class of manipulators with a single flexible link: part i-feedback linearization (1991) 8. Lewis, F., Abdallah, C., Dawson, D.: Control of robot. In: Manipulators, pp. 25–36. Editorial Maxwell McMillan, Canada (1993) 9. Spong, M., Vidyasagar, M.: Robust linear compensator design for nonlinear robotic control. IEEE J. Robot. Autom. 3(4), 345–351 (1987) 10. Slotine, J.-J.E.: The robust control of robot manipulators. Int. J. Robot. Res. 4(2), 49–64 (1985) 11. Tarn, T.-J., Bejczy, A. K., Isidori, A., Chen, Y.: Nonlinear feedback in robot arm control. In: The 23rd IEEE Conference on Decision and Control, pp. 736–751. IEEE (1984) 12. Shang, W., Cong, S.: Nonlinear computed torque control for a high-speed planar parallel manipulator. Mechatronics 19(6), 987–992 (2009) 13. Nguyen-Tuong, D., Seeger, M., Peters, J.: Computed torque control with nonparametric regression models. In: American Control Conference IEEE 2008, pp. 212–217 (2008) 14. Sage, H., De Mathelin, M., Ostertag, E.: Robust control of robot manipulators: a survey. Int. J. Control 72(16), 1498–1522 (1999) 15. Barjuei, E.S., Boscariol, P., Gasparetto, A., Giovagnoni, M., Vidoni, R.: Control design for 3d flexible link mechanisms using linearized models (2014) 16. Zhang, W., Li, Q., Guo, L.: Integrated design of mechanical structure and control algorithm for a programmable four-bar linkage. IEEE/ASME Trans. Mechatron. 4(4), 354–362 (1999) 17. Arakelian, V., Geng, J., Lu, Y.: Torque minimization of dynamically decoupled RR spatial serial manipulators via optimal motion control. Mechanism Design for Robotics: MEDER 2021, vol. 103, p. 20 (2021) 18. Arakelian, V., Xu, J., Le Baron, J.-P.: Dynamic decoupling of robot manipulators: a review with new examples. Dynamic Decoupling of Robot Manipulators, pp. 1–23 (2018)

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19. Youcef-Toumi, K., Asada, H.H.: The design of open-loop manipulator arms with decoupled and configuration-invariant inertia tensors. In: Proceedings. 1986 IEEE International Conference on Robotics and Automation, vol. 3, pp. 2018–2026 (1986) 20. Youcef-Toumi, K., Asada, H.: The design of open-loop manipulator arms with decoupled and configuration-invariant inertia tensors (1987) 21. Gompertz, R.S., Yang, D.C.: Performance evaluation of dynamically linearized and kinematically redundant planar manipulators. Robot. Comput. Integrat. Manufact. 5(4), 321–331 (1989) 22. Coelho, T.A., Yong, L., Alves, V.F.: Decoupling of dynamic equations by means of adaptive balancing of 2-dof open-loop mechanisms. Mechan. Mach. Theory 39(8), 871–881 (2004) 23. Arakelian, V., Sargsyan, S.: On the design of serial manipulators with decoupled dynamics. Mechatronics 22(6), 904–909 (2012) 24. Arakelian, V., Xu, J., Le Baron, J.-P.: Mechatronic design of adjustable serial manipulators with decoupled dynamics taking into account the changing payload. J. Eng. Design 27(11), 768–784 (2016) 25. Xu, J., Arakelian, V., Le Baron, J.-P.: The design of planar serial manipulators with decoupled dynamics taking into account the changing payload. J. Robot. Mech. Eng. Resr. 1(4), 38–45 (2016)

Reliable Time-Optimal Point to Point Handling of Unmounted Objects with Industrial Robots Hubert Gattringer and Andreas M¨ uller(B) Institute of Robotics, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria {hubert.gattringer,a.mueller}@jku.at

Abstract. Handling of parts that are loosely placed at the end-effector (EE) of a robot is a task with high practical relevance, which is synonymously known as the waiter motion problem, and the time optimal motion planning is an important task for factory automation. In the past, this problem was often relaxed to a time-optimal path following problem by assuming that the geometric path of the EE is prescribed. This limits the flexibility and overall task efficiency, and requires manual teaching of the path. In this paper, the general point to point (PtP) time-optimal motion planning problem (only initial and terminal position is prescribed) is formulated and solved with a multiple shooting method. The formulation includes all relevant dynamic effects (joint friction, contact conditions, motor limits, etc.). The trajectory is C 2 continuous, which avoids a bang-bang behavior of the motor torques. The reported experimental results confirm applicability of the trajectory to real robotic setup. The CasADi framework and the Ipopt solver are used for numerical computations.

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Introduction

Manipulation of objects with industrial robots is a central task in factory automation. Ideally, the handling procedure involves the handling of rigidly grasped rigid objects. Clearly, a key element is the stable grasp of the part to be manipulated. Recent research addressed the problem of grasping flexible objects. A problem with higher practical relevance is the handling of rigid objects that are not rigidly fixed to the robot. The parts are for instance ‘affixed’ to the robot by means of suction grippers or loosely placed in a container or at a hold. The hold is thus achieved via force closure. When using suction grippers, for instance, the part is pushed toward the EE by the suction force that in turn produces a tangential force due to friction, while for loosely placed parts the net normal force is only due to gravitation and EE acceleration. Figure 1 shows a situation, where several cups are placed on an EE-fixed tray. To ensure efficiency, the handling task must be performed within the shortest time possible. In case of rigidly grasped parts, the minimum time is dictated by the robotic c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 210–219, 2022. https://doi.org/10.1007/978-3-031-06409-8_22

Reliable Time-Optimal Point to Point Handling q3

q2

fz q5 C2C3

q4

C1 Iz

q1

211

C4

q6

fN fT

rE rEO,4 Iy Ix

Fig. 1. Waiter motion; industrial robot and 4 cups; friction cone in detail

manipulator. However, if the objects are only attached to the robot via force closure, limits on the contact force must be taken into account for the timeoptimal motion planning. This problem is known as the ‘waiter motion problem’ [9]. This terminology stems from the observation that manipulating unmounted parts resembles the task of a waiter: transporting glasses, dishes, etc. placed at a tray from the restaurant’s kitchen to the customer’s table as fast as possible. In a number of publications, the waiter motion problem has been formulated as a path-following problem, i.e. the geometric path of the Cartesian position of object or EE is prescribed, and the task is to follow this path in minimum time while optimally aligning the EE. In [10], the numerical integration method was used, where the optimal switching points are determined to switch from maximal acceleration to deceleration. The path following problem was solved in [14] with a dynamic programming method. In this publication experimental result of a real system operating at the physical limits were reported. A critical aspect of these methods is that it leads to a bang-bang control, i.e. the drive torques alternate between their maximal and minimal values. A convex formulation of the problem was used in [4] where the position and the orientation of the object/EE are prescribed, and constraints for avoiding tip-over are included. A formulation as convex second-order cone program was introduced in [2] to solve the waiter motion problem using direct transcription method. These methods rely on a convex formulation obtained by omitting joint friction. A direct multiple shooting method was applied to the problem including joint friction, and experimental results were reported in [7,8]. Including joint friction is decisive in order to obtain realistic trajectories, which can actually be executed on the real robot. In most applications, the path is not prescribed by the handling task, rather the start and terminal poses are defined. Therefore, and to fully exploit the capability and to ensure flexibility of the robot, the realistic waiter motion problem is a point-to-point (PtP) optimal control problem. This PtP problem was addressed in a few publications. In [17], a two step optimization procedure was proposed where intermediate points were introduced between which the optimal

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path following problem is solved. Time-optimal transportation of goods using suction cups is addressed in [15], which is a continuation of [16] where the tasks of manipulating loosely placed objects on a tray underneath an obstacle was discussed. The unilateral contact and the friction cone condition are the peculiarities of the waiter motion problem. In context of general trajectory optimization, they have been included in [3,12]. In [3], the non-sliding constraints are incorporated into a second order cone problem which is solved using a sequential solution algorithm. To simplify the non-linear time-optimal path following problem, the friction cone constraints were linearized in [10]. In this paper, a numerical approach for the solution of the time-optimal PtP waiter motion problem and experimental results are presented. The optimal control problem includes all task constraints (normal and friction forces of objects), in order to prevent the objects to twist and tip over, and all limitations of the robot (max. speed, acceleration, jerk, and drive torque). The jerk of the joint coordinates is used as the control input to the optimization problem. The timeoptimal trajectory is thus C 2 continuous (continuous acceleration), and does not cause jumps of the drive torques. The problem formulation includes viscous friction, and thus yields non-convex optimization problem. The latter is solved numerically with a direct multiple shooting method, which is implemented in the CasADi framework [1], and the optimization problem is solved with Ipopt [18]. Feasibility of the time-optimal PtP motion is verified experimentally, which confirms that the robot can in fact operate almost at the physical limits.

2

Inverse Dynamics Model

Basis for the optimal control problem is the inverse dynamics model of the controlled robot, which serves to deliver the vector of motor torques, denoted with QM (t) for a given motion of the robot, represented by the vector of joint variables q (t). Moreover, since this inverse dynamics model is used to evaluate the constraints on the motor torques, the identification of the corresponding parameters is crucial. Not all dynamic parameters are independently identifiable, only a set of so-called base parameters can be identified [8,11,13]. This is sufficient to obtain a validated inverse dynamics model. The latter can be written in the form ˙ q ¨ )p (1) QM = Θ(q, q, where Θ is the regressor matrix as function of state q, q˙ and its derivatives, and the vector p comprises the identifiable base parameters. Since (1) is linear in the ˙ q ¨) base parameters p, it provides the basis for identification. To this end, Θ(q, q, and QM are measured along optimized identification trajectories q (t), and the corresponding regression problem is solved. An important aspect for the efficient generation of EOM and the corresponding inverse dynamics model is to follow a modular modeling approach. This is particularly relevant in order to admit easy substitution of the equations accounting for the dynamics of the parts to be handled. The latter admits representing

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complicated contact geometries but also internal dynamics as in case of containers filled with liquid [5,6]. Depending on the environmental conditions, the subsystem representing the object to be handled could also take into account aerodynamic forces. Denote with NObj the number of objects to be handled. The dynamics model of the ith manipulated object, which is loosely placed at the EE, has the general form ¨ i + Gi y˙ i − Qi , i = 1, . . . , NObj Qzi = Mi y

(2)

where Mi is the generalized mass matrix, Gi y˙ i is the vector of generalized Coriolis and centrifugal forces, and Qi represents generalized forces due to gravity and further external loads. Further, y˙ i is the vector of generalized velocities of the object, and Qzi is the vector of generalized constraint forces. The generalized velocities y˙ i are independent, and the system (2) serves as EOM of the separated object. In general, the DOF of this subsystem depends on the object and the dynamic effects represented [8]. In the following the ith handled object is a single T  rigid body. Then y˙ i = (viT ω Ti )T is the rigid body twist, and Qzi = fizT MzT i is the contact wrench, with constraint force fiz and torque Mzi , respectively. More precisely, vi is the velocity of the contact point of object i. If (1) represents the robot dynamics, then the overall motor torques necessary for executing a task with trajectory qd (t) are determined by the inverse dynamics model   ∂ y˙ i T ¨ i + Gi y˙ i − Qi ) (3) ¨ d ) = Θ(qd , q˙ d , q ¨ d )p + (Mi y QM,ff (qd , q˙ d , q ∂ q˙ i=1 NObj

which is obtained assuming perfect contact (no slipping) of the object and EE, so that the object velocities are determined by y˙ i = ∂∂y˙q˙i q˙ (see Sect. 3). The motor torques QM,ff are used as feed-forward command in the model-based controller.

3

Task Constraints

The overall motion must respect several constraints so to maintain contact via force closure. These are briefly summarized here, their derivations can be found in [8]. The object and the surface it is placed at are assumed to be flat. The EEframe is introduced so that its z-axis is normal to the surface and pointing away from the surface (see Fig. 2). Denote with fT = (fx , fy , 0)T and fN = (0, 0, fz )T the contact force tangential and normal to the contact plane (omitting index i), i.e. the relevant components of fiz in Qzi . Non-lifting condition: Denote with fz the normal component of the contact force. The condition for preventing that the object lifts off the surface is fz ≤ 0. Non-slipping condition: Denote with μ0 the static friction coefficient of the object-surface contact. This must be determined experimentally, and is possibly different for the different objects. It can be identified via tan ρ = μ0 , where ρ is the angle where sliding starts. The condition for non-slipping is fT  ≤ −μ0 fz .

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Non-tip-over condition: When the EE is accelerated, a torque is generated acting at the object. The torque depends on the contact geometry. The condition for that the object does not tip over is that this torque is less than the torque due to the normal force. For a circular footprintwith radius r0 (assumed to be the contact area) this condition is expressed as

My2 + Mx2 ≤ −ro fz .

object My

Ez

tray 0

Ex

fx fz

ro

A

Fig. 2. Contact forces and torques between object and tray (E x-E z view).

Non-twisting condition: In general, also a contact torque about the surface normal is generated, which may cause the object to twist about the z-axis of the EE-frame. The condition to avoid twisting is that |Mz | ≤ Mz,max , where Mz,max is the maximal magnitude of the torque up to which the object sticks. For a circular footprint with radius r0 , this maximal torque is Mz,max = 23 μ0 fz ro3 .

4

Optimal Control Problem and Numerical Solution

T T T ¨T A dynamic control system ) ... is formulated in terms of the state x = (q q˙ q and control input u = q , in the form of a simple integrator chain ⎡ ⎤ ⎛ ⎞ 0I0 0 x˙ = ⎣ 0 0 I ⎦ x + ⎝ 0 ⎠ u =: f (x, u). (4) 000 I

Using the jerk as input ensures smooth torques. Let q0 and qe be the initial and terminal configuration of the robot, respectively. The optimal PtP control problem for a rest-to-rest motion is then  te   1 + kuT u dt (5) min te ,u

0

subject to

˙ ≤ q˙ max |q| u ≤ umax ˙ q ¨ )| ≤ QM,max |QM,ff (q, q,

 2 + f2 + μ f fx,i 0 z,i ≤ 0, i = 1, . . . , NObj y,i fz,i ≤ 0, i = 1, . . . , NObj  2 + M2 + r f Mx,i o z,i ≤ 0, i = 1, . . . , NObj y,i

(6) (7) (8) (9) (10) (11)

Reliable Time-Optimal Point to Point Handling

|Mz,i | − Mz,max ≤ 0, i = 1, . . . , NObj x˙ = f (x, u) ˙ ¨ (0) = 0 q(0) = q0 , q(0) = 0, q q(te ) = qe ,

˙ e ) = 0, q(t

¨ (te ) = 0 q

215

(12) (13) (14) (15)

where QM,ff is the inverse dynamics solution (3), and QM,max is the maximal magnitude of motor torques. The objective function (5) accounts execution time as well as smoothness of the trajectory. The priority of either goal is encoded in the weighting factor k (k = 0 implies time-optimal motion). There are various methods for solving optimal control problems. Here it was solved with a multiple shooting method using N = 150 shooting intervals. The ODE (13) and the objective function (5) are integrated numerically with a Runge-Kutta scheme. The multiple shooting method is implemented in the CasADi optimization framework [1]. The interior point method Ipopt [18] is used to solve the optimization problem. The weight was set to k = 10−6 to get a nearly time-optimal solution. The time for solving the optimization problem is about 13 s on a standard PC.

5

Experimental Results

Experiments are preformed with a St¨ aubli RX130L robot controlled with a B&R controller system. Detailed hardware description can be found in [8]. The dynamic robot parameters were identified. A tray is mounted at EE, as shown in Fig. 4, and NObj = 2 cups are placed at the tray. The mass of each cup is mObj = 0.12 kg. The cups have a height of h = 0.06 m, and a footprint radius position of of ro = 0.025 m, and are filled with  water. The  initial and terminal  the EE in task space is rTE,0 = 0.01 1.2 0.2 m and rTE,e = 0.01 − 1.2 0.2 m, respectively. In both configurations, the tray is horizontally aligned. The initial and terminal orientation of  the tray  (i.e. the EE) is described by  the vector of Cardan angles ϕTE,0 = 0 0 1.56 rad and ϕTE,e = 0 0 − 1.56 rad, respectively. The corresponding joint coordinates are determined by solving the   1.56 0.57 − 0.20 0 − 0.36 0 rad and inverse kinematics, which yields qT0 =  qTe = −1.56 0.57 − 0.20 0 − 0.36 0 rad. The positions of the cups relative to the EE-frame are E rTEO1 = (0.14 − 0.065 0.02) m, E rTEO2 = (0.14 0.065 0.02) m, where E rEOi is the position vector of cup i . Figure 3 shows the joint trajectory corresponding to the computed time-optimal motion with optimized task duration of te = 1.23 s. Compared with the time-optimal solution following a prescribed path, which was computed in [8] as te = 1.32 s, the PtP solution is about 10% faster. The jerks are shown in Fig. 4. The computed feed-forward torque commands and the measured motor torques are shown in Fig. 5, which

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match well. The importance of a model-based controller is clearly visible, which is also apparent from the small joint tracking error even for this highly dynamic motion. The remaining model uncertainties cause slight deviations at t = 1 s. This is attributed to elastic effects that are not included in dynamics model. The speed limiting constraints are the friction forces at the cups, as apparent from Fig. 6. The limits of the tipping and twisting torques are not active, see Fig. 7. A video of the experiment can by found at https://youtu.be/C6XpyTvGCAQ.

q1 q2 q3 q4 q5 q6

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Fig. 3. Joint coordinates q (l) and normalized velocities q˙ (r)

u1 u2 u3 u4 u5 u6

u normalized

0.6 0.4 0.2 0

-0.2 -0.4 -0.6 -0.8 0

0.2

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... Fig. 4. Jerk u = q (l) and a snapshot of an extreme position (r).

1.2

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10-3

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Fig. 5. Measured and computed feed-forward motor torques (l), tracking error (r)

-0.5 -1

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Conclusion

The PtP waiter motion problem was formulated as optimal control problem, and solved with a multiple shooting method. The formulation accounts for all relevant dynamic phenomena, such as joint friction, limits on the joint rates, accelerations, and jerks, as well as the motor torque. Particularly crucial is the modeling of unilateral contact conditions and the corresponding limits. This naturally leads to a non-linear non-convex optimization problem. Experimental results show the feasibility of the computational approach and the practical applicability of the obtained time-optimal trajectories, which can only be ensured by using an identified model. The approach is applicable to general robotic systems including parallel manipulators. Future research will address the improvements of the robot model, in particular so to account for joint/gear compliance. The modular subsystem modeling admits seamless replacement of the EOM of the manipulated objects, and thus to represent the internal dynamics of the objects. This is particularly important when handling liquid-filled containers [5,6]. A further extension will be incorporating collision avoidance conditions. Acknowledgments. This work has been supported by the “LCM-K2 Center for Symbiotic Mechatronics” within the Austrian COMET-K2 program.

References 1. Andersson, J.A.E., Gillis, J., Horn, G., Rawlings, J.B., Diehl, M.: CasADi - a software framework for nonlinear optimization and optimal control. Math. Prog. Comput. 11(1), 1–36 (2019) 2. Csorv´ asi, G., Nagy, A., Vajk, I.: Near time-optimal path tracking method for waiter motion problem. IFAC-PapersOnLine 50(1), 4929–4934 (2017) 3. Csorv´ asi, G., Vajk, I.: Sequential time-optimal algorithm for extended path tracking problem. J. Dyn. Sys. Meas. Control 142, 081005 (2020) 4. Debrouwere, F., Loock, W.V., Pipeleers, G., Diehl, M., Swevers, J., Schutter, J.D.: Convex time-optimal robot path following with cartesian acceleration and inertial force and torque constraints. J. Syst. Control Eng. 227, 724–732 (2013) 5. Di Leva, R., Carricato, M., Gattringer, H., M¨ uller, A.: Sloshing dynamics estimation for liquid-filled containers under 2-dimensional excitation. In: ECCOMAS Thematic Conference on Multibody Dynamics, pp. 80–89 (2021) 6. Di Leva, R., Carricato, M., Gattringer, H., M¨ uller, A.: Time-optimal trajectory planning for anti-sloshing 2-dimensional motions of an industrial robot. In: 2021 20th International Conference on Advanced Robotics (ICAR), pp. 32–37. IEEE (2021) 7. Gattringer, H., M¨ uller, A., Oberherber, M., Kaserer, D.: Time-optimal path following for robotic manipulation of loosely placed objects: modeling and experiment. 21th IFAC World Congress 53(2), 8450–8455 (2020) 8. Gattringer, H., M¨ uller, A., Oberherber, M., Kaserer, D.: Time-optimal robotic manipulation on a predefined path of loosely placed objects: modeling and experiment. Mechatronics (2021)

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9. Geu Flores, F., Kecskem´ethy, A.: Time-optimal path planning for the general waiter motion problem. In: Advances in Mechanisms, Robotics and Design Education and Research, pp. 189–203. Springer, Heidelberg (2013). https://doi.org/10. 1007/978-3-319-00398-6 14 10. Geu Flores, F., Kecskem´ethy, A., P¨ ottker, A.: Time-optimal motion planning along specified paths for multibody systems including dry friction and power constraints. In: 13th World Congress in Mechanism Machine Science (2011) 11. Khalil, W., Dombre, E.: Modeling, Identification and Control of Robots. Hermes Penton, London (2002) 12. Luo, J., Hauser, K.: Robust trajectory optimization under frictional contact with iterative learning. In: Robotics, Science and Systems (RSS) (2015) 13. Neubauer, M., Gattringer, H., Bremer, H.: A persistent method for parameter identification of a seven-axes manipulator. Robotica 33, 1099–1112 (2014) 14. Oberherber, M., Gattringer, H., Springer, K.: A time optimal solution for the waiter motion problem with an industrial robot. In: Proceedings of the Austrian Robotics Workshop (ARW) 2013, pp. 55–60 (2013) 15. Pham, H., Pham, Q.-C.: Critically fast pick-and-place with suction cups. In: 2019 International Conference on Robotics and Automation (ICRA), pp. 3045–3051. IEEE (2019) 16. Pham, Q.-C., Caron, S., Nakamura, Y.: Kinodynamic planning in the configuration space via admissible velocity propagation. In: Proceedings of Robotics: Science and Systems (2013) 17. Van Duijkeren, N., Debrouwere, F., Pipeleers, G., Swevers, J.: Cartesian constrained time-optimal point-to-point motion planning for robots: the waiter problem. In: Benelux Meeting on Systems and Control, Belgium (2015) 18. W¨ achter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Prog. 106, 25–57 (2006)

Design of a Weed-Control Mobile Robot Based on Electric Shock Giuliano Morara1 , Alberto Baldassarri1(B) , Jens Diepenbruck2 , Tobias Bruckmann3 , and Marco Carricato1 1

DIN, University of Bologna, Bologna, Italy [email protected], {alberto.baldassarri,marco.carricato}@unibo.it 2 Mercatronics GmbH, Bocholt, Germany [email protected] 3 University of Duisburg-Essen, Duisburg, Germany [email protected] Abstract. Weed control is a long-standing problem and plants, if left unchecked, can cause serious damage to crops or compromise the functionality of a working environment. Nowadays, mobile robotics is offering solutions in the agricultural sector to relieve the operator from this tedious and repetitive task, but at the same time, there are no solutions specifically designed for the industrial field. In this paper, the study and design of a remote controlled mobile robot based on electric shock for weed control for industrial sites is presented. The robotic tool is composed of two electrodes whose shape and arrangement was chosen based on experiments performed on clover plants, in order to identify the most effective solution.

1

Introduction

A weed is fundamentally any plant that develops in a spot where it is not required. Examples come from unwanted plants in human-controlled environments, such as farm fields, gardens, lawns, and parks. Weeds, in general, are undesirable plants which meddle and contend with the crop for use of land, supplements, daylight, and water assets [1]. Many automated weed-control solutions for the agricultural sector are already on the market, with some robotic systems performing various farming tasks autonomously, such as seeding, irrigating, and eradicating weeds. In this field of application, AgroBot [2] is a robotic system composed of a mobile base and an anthropomorphic robotic arm equipped with a herbicide spray tool; thanks to vision sensors weeds can be detected and subsequently treated. A different approach is presented in [3], where rotating blades are installed beneath a mobile robot. A similar solution is presented in [4], where weeds are eliminated by a delta-arm equipped with a blade on its end-effector. In both solutions, weeds are detected via vision sensors and algorithms. Weeding is a problem not only in agriculture, but also in industrial sites that require to be kept free of vegetation for different reasons, such as aesthetics, protection of the hardware, access to sidewalks and parking lots. Substantial differences c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 220–227, 2022. https://doi.org/10.1007/978-3-031-06409-8_23

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emerge between agricultural and industrial fields of application, such as the soil composition, consisting of concrete or asphalt pavement in industrial sites. This means that limited space is available for weeds to grow, for instance in the gap between two paving stones. These differences also affect the characteristics and peculiarities of a mobile robot meant to work in such an environment. Although some agricultural technologies can be applied in the industrial field, nowadays there are no robotic solutions on the market specifically designed for industrial sites. The aim of this paper is therefore to design a robotic system for weedcontrol able to operate remotely controlled in industrial areas with hard and flat pavements.

2

Selection of the Weeding Method

Firstly, it is necessary to define which weed control method is the most suitable for the proposed application. Currently, weed control methods mainly include chemical, mechanical, thermal, and electrical weeding. The long-term overuse of the former by means of herbicides, not only increases the resistance of weeds, but also causes serious ecological problems [5]. For these reasons this method will not be taken into account here. Mechanical weeding is a common practice and has been performed for very long time, with techniques like hoeing and tilling. This method is based on the physical removal of the whole plant or part of it, e.g., breaking stems using rotary blades or steel bristle brushes. It allows difficult areas to be reached, such as walls or corners, and can be easily automated. On the other hand, brushes or blades suffer wear and, normally, only the upper part of the plant is removed, letting plants regrow easily from roots. Thermal weeding is based on the principle of thermal energy exchange at high temperatures, which disturbs the functioning of the plant above the ground. Even if this is a highly effective method, some drawbacks are evident. The employment of high temperature sources might be harmful for a human operator and heat generation requires high power. In case of gas- or steambased method, a tank transportation is also needed. Finally, electrical weeding is based on the principle of using electrical power for treating weeds. When a high-voltage electrode touches a weed, electric current passes through the plant. Due to the electrical resistance of the plant, the energy is converted into heat resulting in plant death [6]. This method can be easily automated and simple precautions can be taken to guarantee safety. Moreover, mechanical parts are not affected from wear. Considering pros and cons of the various techniques for the proposed application, electrical weeding was chosen as the best method. Its ease of implementation on a mobile device and its low risk for the human operator make it the most suitable choice.

3

Electrode Design

An electrical discharge through a body is generated by the contact of a pair of electrodes subject to an electrical potential difference. Two types of current

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generator are available, DC (direct current) or AC (alternating current). According to industry standards, the safe limit value for continuous operation is 10mA for DC applications, and 5mA for AC devices [7]. Higher currents would be harmful to a human operator and would require grounding systems. Since a smaller current determines a lower chance of electrocuting the weeds, the DC solution is preferred. 3.1

Electrocution Strategy

Once the type of current generator has been determined, the next step is to choose the most suitable electrocution strategy (Fig. 1). Contact points between plants and electrodes can be multiple and it is essential to identify the most effective option. In the first electrocution strategy (Fig. 1(a)) one electrode touches the plant leaves while the other is in contact with the ground. The current will flow through leaves, the stem, and the soil. The second strategy (Fig. 1(b)) is based on the principle of touching two different plants, letting the current stream through leaves, roots, the stem, and the ground. These two options have a disadvantage. Generally, the electrical conductivity of the ground is extremely low, especially for industrial pavements such as concrete or asphalt. The magnitude of pavement electrical resistance can be in the order of kΩ or even M Ω [8]. As a result, a small increase in the distance the current has to travel causes a large increment in the voltage required to ensure an effective current flow. In the third strategy (Fig. 1(c)) the first electrode is placed on the leaves while the second touches the plant roots. Although this is a highly effective strategy, as the current flows through the whole plant, reaching the roots can be difficult, especially in soils with hard surfaces. The last solution (Fig. 1(d)) involves touching both electrodes with plant leaves, so that current will flow through leaves and upper stem only. Although the latter strategy causes reduced damages to plants, it was chosen as the best option, due to its ease of application makes it the most implementable solution on a mobile robot. Its reduced effectiveness can be countered by increasing the frequency of treatment cycles.

Fig. 1. Different electrocution strategies: (a) leaf-ground (b) two plants (c) leaf-roots (d) leaf-leaf.

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Electrode Shape

This Subsection discusses some possible solutions with regard to electrode shape. Each design phase is followed by plant-electrocution test to verify the effectiveness of the proposed solution. Experiments were performed using clover plants as target weeds. A 12KV generator designed for pasture fences was employed with electrical energy being delivered in short pulses. • Point electrodes: a pair of point electrodes, made of two conductors separated by a rigid support, were initially tested (Fig. 2(a)). Electrocuted plants are marked in yellow while healthy plants are marked in light blue (Fig. 2(b)). It turns out that leaves wilt in the following days but the effects are localised, and reaching all specimens might be arduous. • Rectangular bars: two rectangular aluminum bars were adopted to improve the previous solution (Fig. 2(c)). Electrocuted plants (yellow in Fig. 2(d)) show more damage and, thanks to a wide conductor, the treatment is more uniform. However, it was noticed that the lower electrode tends to bend the base of the plant, affecting the upper electrode contact with leaves and thus compromising the effectiveness of the electrical discharge. • Thin lower bar : the lower electrode was replaced with a thin cylindrical brass bar in order to reduce the bending effect (Fig. 2(e)). Although the lethal effects on the electrocuted clovers (yellow in Fig. 2(f)) remain satisfactory, disadvantages emerged. The difference in thickness of the two electrodes can cause a drop in the effectiveness of the electrical discharge when approaching a flat surface, like a wall, as the electrode simultaneous contact can be lost. • Cylindrical bars: to overcome such limitations, the upper electrode was replaced with a thin cylindrical brass bar (2(g)). Experiments results are shown in Fig. 2(h), where we can note that all plants in the picture are electrocuted (healthy clover plants highlighted in blue in the previous figures can be used as reference). Damages are still consistent and, in addition, the distance between electrodes was reduced allowing the system to electrocute smaller plants. Considering the different shapes analysed, the cylindrical bars configuration is considered to be the best as it combines excellent weeds damage with reduced issues in reaching plants.

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Fig. 2. Different electrodes shapes and experiments: (a) point electrodes (b) point electrodes tests (c) rectangular bars (d) rectangular bars tests (e) thin lower bar (f) thin lower bar tests (g) cylindrical bars (h) cylindrical-bar tests.

4

Electrocution Equipment Design

After having identified the most suitable electrode shape, the next step is to study the electrocution equipment, i.e. how the electrodes, orange in Fig. 3, are integrated onto the mobile robot. Different mounting solutions are analysed in order to select the best one. • Double rotating electrodes: electrodes are installed on rotating supports driven by actuators (Fig. 3(a)). After the robot reaches the targets the movement of the electrodes allows to electrocute weeds treating multiple plants with one single robot positioning. Despite this advantage, an actuated motion requires additional motors and sensors. Moreover, this solution does not reach corners,

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where weeds are likely to grow, and rotating tools might be harmful to human operator s. • Translating electrodes: the conductors are placed on a support that may translate along two horizontal directions (Fig. 3(b)). The operation is similar to the previous option, with a first phase of robot positioning followed by an electrocution phase through the movement of the electrodes. Although weed targets are always shocked, regardless of their location, this solution requires a sophisticated sensor apparatus to guide the electrodes into the right position in order to perform effective electrocution. • Fixed front bars: electrodes are fitted on a fixed support and plants are touched while the robot is going forward (Fig. 3(c)). A passive solution allows avoid sensors and actuators, benefiting simplicity of construction. In addition, it can be considered human-operator-safe, as there are no bulging parts, and multiple plants can be shocked simultaneously. • Fixed V-bars: for the second passive electrode arrangement, a V structure is considered (Fig. 3(d)). This configuration ensures a larger contact surface between plants and, with no actuations involved, it is simple to design. On the other hand, it shows reduced reachability near walls and edges and, moreover, the prominent equipment could harm an operator.

Fig. 3. Different electrocution equipment design: (a) double rotating bars (b) translating equipment (c) fixed front bars (d) fixed V-bars.

Taking into account pros and cons of each electrocuting system, the fixed front bars were deemed the most effective solution, due to its high reachability, combined with a reduced human operator risk. In addition, the absence of actuators ensures ease of design and implementation.

5

Mobile Robot Design

Once the electrocution system has been defined, it is possible to proceed with the design of the mobile robot (Fig. 4). A 900 × 500 × 8 mm acrylonitrile butadiene styrene (ABS) panel acts as the support base for all other components. Electrodes are installed both on the front and the rear side of the robot, in order to shock plants also in reverse motion. Their support consists of a 3D printed L-shape component hinged on a pin. The rotation makes it possible to overcome obstacles on the robot path (Fig. 5). If the encountered hurdle is too tall,

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a switch limit sensor is triggered: in this case, the robot will go backward to avoid any possible damage to components. The electrode support is divided into three parts to allow the robot to keep on electrocuting plants even if one section is rotated due to an obstacle. Rotation is counterbalanced by a pair of springs constrained to the end of the support, that bring back and hold electrodes in the rest position. The elastic constant is 0.15 N/mm for each spring. The relative distance between the electrodes is 14 mm, while the lower electrode is 6 mm far from the ground. As a result, plants below 20 mm in height cannot be electrocuted, as simultaneous contact between the two electrodes does not occur. Regarding the traction system, two motorized wheels are placed on the robot front side, while one caster wheel is installed on the rear side centreline. Two brushless motors are installed and the power is delivered to wheels through belt transmissions. The robot can reach a top speed of 7 km/h and it is able to climb a 15◦ slope. Antennas and a remote receiver allow the user to remotely control the robot. Finally, whole system power is supplied by a 48V battery pack.

Fig. 4. Proposed weeding mobile robot.

Fig. 5. Electrode rotational support.

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Conclusions

In this paper, the conceptual design of a weed-electrocuting mobile robot was presented. Starting from the analysis of the best electrocution strategy, different tests were performed on clover plants in order to identify which shape of the electrodes was most effective. Then, the best arrangement of the electrodes on board was analysed and chosen. Finally, a preliminary design phase of the mobile robot was carried out. Although the initial tests showed promising results, it is necessary to perform additional experiments using the mobile robot to consolidate the results and, if possible, to test different electrocution strategies. Acknowledgments. This work was supported by MERCATRONICS GMBH (www. mercatronics.de) in collaboration with the University of Bologna and the University of Duisburg-Essen.

References 1. Singh, G., Moses, S., Dahate, H.: Study of low land rice weeder and development of fing cutting attachments. Int. J. Agricul. Sci. Res. 5(4), 315–322 (2015) 2. Sarri, D., Lombardo, S., Lisci, R., De Pascale, V., Vieri, M.: AgroBot smash a robotic platform for the sustainable precision agriculture. In: Coppola, A., Di Renzo, G.C., Altieri, G., D’Antonio, P. (eds.) MID-TERM AIIA 2019. LNCE, vol. 67, pp. 793– 801. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39299-4 85 3. Quan, L., Jiang, W., Li, H., Li, H., Wang, Q., Chen, L.: Intelligent intra-row robotic weeding system combining deep learning technology with a targeted weeding mode. Biosys. Eng. 216, 13–31 (2022) 4. Sethia, G., Guragol, H.K.S., Sandhya, S., Shruthi, J., Rashmi., N.: Automated computer vision based weed removal bot. In: 2020 IEEE International Conference on Electronics, Computing and Communication Technologies, CONECCT, pp. 1–6 (2020) 5. Hajjar, A., Mourtada, A.: Central receiver power plant: a feasibility study based on the central receiver technology. In: 2012 International Conference on Renewable Energies for Developing Countries, REDEC, pp. 1–7 (2012) 6. Vigneault, C., Benoˆıt, D.L.: Electrical Weed Control: Theory and Applications, pp. 174–188. Springer Berlin Heidelberg (2001). https://doi.org/10.1007/978-3-66204584-8 12 7. Cadick, J., Capelli-Schellpfeffer, M., Neitzel, D.K., et al.: Electrical Safety Handbook. McGraw-Hill Education (2012) 8. Samou¨elian, A., Cousin, I., Tabbagh, A., Bruand, A., Richard, G.: Electrical resistivity survey in soil science: a review. Soil Tillage Res. 83(2), 173–193 (2005)

Part V: Modeling and Synthesis of Robotic Devices

Development of Joint Mechanism that can Achieve Both Active Drive and Variable Joint Quasi-stiffness by Utilizing 4-Bar and 5-Bar Composite Linkage Hiroki Mineshita1(B) , Takuya Otani2,3 , Yuji Kuroiwa1 , Masanori Sakaguchi4 , Yasuo Kawakami5 , Hun-ok Lim3,6 , and Atsuo Takanishi3,7 1 Graduate School of Creative Science and Engineering, Waseda University, Tokyo, Japan

[email protected]

2 Research Institute for Science and Engineering, Waseda University, Tokyo, Japan 3 Humanoid Robotics Institute, Waseda University, Tokyo, Japan

[email protected]

4 Institute of Sports Science, ASICS Corporation, Kobe, Hyogo, Japan 5 Faculty of Sports Science, Waseda University, Saitama, Japan 6 Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, Japan 7 Department of Modern Mechanical Engineering, Waseda University, Tokyo, Japan

Abstract. A large joint output is required for the robot to perform dynamic motions. Most previous robots output power only with actuators, with a lot of energy. On the other hand, humans perform with high efficiency by storing and releasing energy by tendons. When trying to simulate it with robots, the actuator and elastic element are arranged in series. This mechanism becomes complicated, large and heavy. Therefore, we propose a novel joint mechanism reducing the size and weight by designing using a 5-bar linkage. To make it easy to use for the joint mechanism, we propose to combine the 5-bar linkage and the 4-bar linkage. We developed an ankle joint of the robot with this mechanism. While demonstrating the same performance as the previous mechanism, we succeeded in reducing the size by 20% and the weight by 0.7 kg. Keywords: Joint stiffness mechanism · Humanoid · Dynamic motion robot

1 Introduction In recent years, humanoid robots that perform various dynamic motions have been developed. For example, Honda’s ASIMO (2011) is running and jumping, and Boston Dynamics’ Atlas (2018) has realized running, parkour and so on. However, these robots move Research was conducted with the support of Research Institute for Science and Engineering, Waseda University; Humanoid Robotics Institute, Waseda University; Human Performance Laboratory, Waseda University; Future Robotics Organization, Waseda University. It was also financially supported in part by NSK Foundation for the Advancement of Mechatronics and the JSPS KAKENHI Grant No. 21H05055. Further, 3DCAD software SolidWorks was provided by SolidWorks Japan K.K.; the force sensors received cooperation from Tec Gihan Co., Ltd. © CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 231–237, 2022. https://doi.org/10.1007/978-3-031-06409-8_24

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only with actuators and require a large amount of energy to move. On the other hand, in addition to the output from the muscles, humans store and release energy in the tendons to achieve highly efficient and dynamic motions (Sugisaki et al. 2004). Especially when running, the knee and ankle joint behave like torsion springs (McMahon and Cheng 1990), and their quasi-stiffness are appropriately changed by speed (Arampatzis et al. 1999). Therefore, we have developed joint mechanisms that enables efficient and dynamic motion by mimicking a human muscle-tendon unit by arranging actuators and elastic elements in series (Otani et al. 2014). In addition, by making the joints quasistiffness variable as human joints, it is possible to store and release maximum energy according to the strength of the recoil (Otani et al. 2015). However, since the actuators and elastic elements are mounted in series, the mechanism becomes large and heavy, and the number of moving parts increases. These cause a decrease in controllability. On the other hand, the linkage is often used as a transmission element of the driving force to the joints. The linkage can transmit a large force. Therefore, power transmission after deceleration is possible, and it can be made joints smaller and lighter. In addition, the linkage can change its reduction ratio by setting the length of the link appropriately. This characteristic is best suited for joints such as humans whose torque and angular velocity change with angle (Mineshita et al. 2019). Of the linkage, the 5-bar linkage requires less two-axis control in order to fix the posture of the link. This can be regarded as controlling one axis with two inputs. These two inputs can be driven independently, both receive torque, and both speeds are combined. As a result, it is possible to enhance the joint output by using recoil, as in the case where the actuator and elasticity are mounted in series. Therefore, in this research, we develop a joint mechanism that achieves both active drive and variable joint quasi-stiffness by utilizing 2 inputs and 1 output with a 5-bar linkage and adapt it to the ankle joint mechanism corresponding to the lower leg and foot of a humanoid robot.

2 Design and Mechanism 4-Bar and 5-Bar Composite Linkage Design. When considering a 5-bar linkage, the axes at both ends of the fixed link are often used as inputs (see Fig. 1a). However, in this mechanism, the output axes always move from the fixed link. Therefore, both ends of the fixed link are used as one input and output, and other axis is used as another input (see Fig. 1b). If either input element does not require a large amount of space, it is possible to mount the element on an axis whose position is variable. However, when the joint quasi-stiffness is made variable, the mechanism tends to be large. In addition, since the required torque is large, a large deceleration is required, and the mechanism becomes large. Therefore, it is difficult to mount elastic and active elements on this axis. Therefore, by moving this axis to the same position as the input axis at the fixed link by the 4-bar linkage, the two inputs can be combined in one place while keeping the joint part simple. This allows both two inputs to be entered at angle from the fixed link. This mechanism is called the 4-bar and 5-bar composite linkage in this paper (see Fig. 1c). In the case of a mechanism in which active and elastic elements are connected in series, either element moves with respect to the link and cannot be fixed, which requires strength and is large and heavy. Since both elements can be fixed to the link in this

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Input

Output

Input Inputs (a) normal 5-bar linkage

Output

Inputs

(b) proposed 5-bar linkage

Output (c) proposed composite linkage

Fig. 1. 5-bar linkage and 4-bar and 5-bar composite linkage.

mechanism, it is easy to secure the strength, which leads to the reduction in size and weight. Furthermore, since the mass that moves in the link during motion is reduced, the changes in physical parameters such as the inertia and center of gravity (COG) of the link become smaller. This is expected to improve controllability. In addition, since the actuators of both elements do not move from the link in this mechanism, it is easy to change the actuator position, and it is also easy to move the actuator to another link. This makes it possible to adjust the imbalance of the link mass due to the differences in the required elements. In this mechanism, the mass balance is adjusted by moving the motor from the lower leg link to the thigh link. The designed mechanism mounts joints and active elements at both ends of the fixed link and mounts elastic elements over the 4-bar linkage (see Fig. 2a). This 4-bar and 5-bar composite linkage can be calculated as follows. First, the length of each link is defined as shown in Fig. 2b. Also, the angles of the linkage are defined as shown in Fig. 2c and Fig. 2d. The links 4C and 5B in Fig. 2a are combined, and the angle θ45 between them is fixed. In addition, the input angle θ4a by the elastic element and the input angle θ5a by the active element can be measured. The joint angle θ5e is calculated from these angles and the lengths. Using the cosine formula, h4 , h5a , and h5b in Fig. 2c and Fig. 2d are calculated in the middle, and by using them, the calculation can be performed as follows:  h4 = l4a 2 + l4d 2 − 2l4a l4d cos(π − θ4a − θ5a ). (1)

Z X

(a) link name

(b) link length (c) 4-bar linkage angle (d) 5-bar linkage angle

Fig. 2. Definition of 4-bar and 5-bar composite linkage.

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 θ4d = cos

−1

h5a 

θ5a

⎧  2 2 2 ⎨ θ5a − cos−1 h5a +l4d −l5b if θ45 ≥ θ4d , 2h l  2 5a 24d 2 = ⎩ θ5a + cos−1 h5a +l4d −l5b if θ45 < θ4d . 2h5a l4d   h5b = h5a 2 + l5a 2 − 2h5a l5a cos θ5a . 

θ5e = cos

   2 2 2 h4 2 + l4d 2 − l4a 2 −1 h4 + l4c − l4b + cos . 2h4 l4d 2h4 l4c  = l4d 2 + l5b 2 − 2l4d l5b cos(θ45 − θ4d ).

−1

h5b 2 + l5d 2 − l5c 2 2h5b l5d



 + cos

−1

 h5b 2 + l5a 2 − h5a 2 . 2h5b l5a

(2) (3)

(4)

(5)

(6)

This makes it possible to calculate the joint angle from the angles of the elastic element and the active element. The reduction ratio R due to the link of the active element can be calculated from the balance of forces of links 5B and 5C, as follows:  2 

2 2 5d −h5b l5d sin cos−1 l5c +l 2l5c l5d . (7) R= l4d sin(θ4d − θ45 − θ5c ) when



θ5c 

θ5c

   2 2 2 h5a 2 + h5b 2 − l5a 2 −1 h5b + l5c − l5d = θ5c + cos + cos , 2h5a h5b 2h5b l5c ⎧  2 2 2 ⎨ cos−1 l5b +h5a −l4d if θ45 ≥ θ4d , 2l5b h5a = 2 2 2 ⎩ 2π − cos−1 l5b +h5a −l4d if θ45 < θ4d . 2l5b h5a 

−1

(8)

Design of Active Element. As a requirement for active output, the maximum torque is set to about 100 Nm, which is equivalent to the torque that humans can kick (Sugisaki et al. 2004). A worm gear is used as a reducer because of its torque resistance and low reverse transmission efficiency. The worm wheel is made lighter by cutting unnecessary parts according to the movable angle and by topology optimization. Design of Elastic Element. The elastic element makes the joint quasi-stiffness variable, and its minimum stiffness is 400 Nm/rad, which is the stiffness exhibited in the previous ankle joint mechanism other than the foot mechanism (Mineshita et al. 2019). In the previous research, it is possible to realize 250 Nm/rad of the ankle joint quasistiffness during running of a human with foot mechanism. A carbon fiber reinforced plastic (CFRP) laminated leaf springs that can store a lot of energy are used as elastic elements, and a trapezoidal shape that can be shortened while maintaining high strength and low stiffness is adopted (Mineshita et al. 2020). For the variable stiffness mechanism, the method of changing the load point was adopted.

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Adjustment of Design Parameters. Based on these designs, each parameter was adjusted to meet the required specifications. For the sake of simplicity, the 4-bar linkage of the mechanism is designed as a parallel link and the angle of the leaf spring when there is no load is set to 0 . First, the 5-bar linkage and gear reduction ratio were adjusted to meet the required specifications, and then the leaf spring was designed so that the stiffness range could be satisfied under the link conditions. As a result, the design parameters are as Table 1. θ4a0 , θ5a0 , and θ5e0 are the linkage angles when the joint angles are 0 and no load is applied to the joint at θ4a , θ5a , and θ5e , respectively. Table 1. Link length and link angle Length mm

Angle deg

l4a , l4c

l4b , l4d

l5a

l5b

l5c

l5d

θ45

θ4a0

θ5a0

θ5e0

45

90

250

50

245

35

60

0

70

90

CAD diagram of the developed mechanism is shown in Fig. 3a. Although the output of this mechanism is the same as that of the previous mechanism, we succeed in reducing the mass by 0.7 kg. In addition, the width of the lower leg link was reduced by 20% compared to the previous mechanism. Furthermore, the movement of the COG of the lower leg link when the joint rotates by 20 due to the additional torque is reduced to about 1/4 compared to the previous mechanism. The relationship between θ4a , θ5a , and joint angle is shown in Fig. 3b. The torque reduction ratio of the 5-bar linkage with respect to the joint angle of this mechanism when θ4a is 0, −10, and −20 is shown in Fig. 3b.

(a) CAD diagram

(b) angle relationship

(c) torque reduction

Fig. 3. Developed mechanism and the linkage angle relationship and torque reduction.

3 Verification Verification of Whether Active and Elastic Elements can Move at the Same Time. We confirm the motion of the developed mechanism (see Fig. 4a) in active and elastic

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elements. The developed mechanism is mounted on a table that moves only up and down, and a force plate is installed on the toes (see Fig. 4b). A weight is placed from the top of the table, and the joint quasi-stiffness is calculated from the joint angle and the force. Furthermore, in that state, an active kicking is performed to confirm the active element. The torque is calculated from the force and joint angle.

(a)) developed mechanism

(b) experimental apppearance

Fig. 4. Developed mechanism and experimental appearance.

The results are shown in Table 2. Although 100 Nm cannot be achieved by active kicking, we succeed in doing at 87.6 Nm. This is due to insufficient output of the motor driver, and it is considered that the required torque can be output by changing to a highpower motor driver in the future. In elastic element, the minimum quasi-stiffness of 400 Nm is achieved. Moreover, it is possible to further extend the load position to reduce the stiffness. Therefore, it is considered that this mechanism can meet the required ankle joint quasi-stiffness. Table 2. Experiment data: joint quasi-stiffness and kicking torque. Joint quasi-stiffness Nm/rad Minimum

Maximum

384

729

Kicking torque Nm 87.6

From this experiment, it can be confirmed that this mechanism can perform active and elastic motions independently and at the same time. It is thought that this makes it possible to strengthen joint output using recoil, like humans.

4 Conclusion In this study, we developed a joint mechanism that can perform active and elastic motions independently and at the same time by using a 4-bar and 5-bar composite linkage. First, we proposed that active and elastic motions can be achieved at the same time by adapting the 5-bar linkage to the joint mechanism. Furthermore, by combining with a 4-bar linkage, it is improved to be easier to use for the joint mechanism. We applied this

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mechanism to the ankle joint of a humanoid robot and confirmed that this can operate both active and elastic motions independently at the same time. This mechanism is 20% smaller and 0.7 kg lighter than the previous mechanism. By using this mechanism, it is expected that the weight can be reduced, and the controllability can be improved by reducing the change of physical parameters. In addition, since the layout of the elements is improved by the link design, it can be possible to mount it on small joints such as wrists, where it is difficult to simulate a muscle-tendon unit like humans. We will reduce the weight of the mechanism, further increase the output of active motions, and expand the range of joint quasi-stiffness so that it can be mounted on humanoid robots that perform dynamic movements in the future.

References Boston Dynamics: AtlasTM . https://www.bostondynamics.com/atlas. Accessed 15 Jan 2022 Honda Motor Co., Ltd. ASIMO. https://global.honda/innovation/robotics/ASIMO.html. Accessed 15 Jan 2022 Sugisaki, N., Okada, J., Kanehisa, H., Fukunaga, T.: Effect of elastic energy on the mechanical work and power enhancement in counter movement exercise of ankle joint. Jpn. J. Ergon. 40(2), 82–89 (2004) McMahon, T.A., Cheng, G.C.: The mechanics of running: how does stiffness couple with speed? J. Biomech. 23, 65–78 (1990) Arampatzis, A., Brüggemann, G.-P., Metzler, V.: The effect of speed on leg stiffness and joint kinetics in human running. J. Biomech. 32(12), 1349–1353 (1999) Otani, T., et al.: Hopping robot using pelvic movement and leg elasticity. In: Ceccarelli, M., Glazunov, V.A. (eds.) Advances on Theory and Practice of Robots and Manipulators. MMS, vol. 22, pp. 235–243. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07058-2_27 Otani, T., et al.: Knee joint mechanism that mimics elastic characteristics and bending in human running. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5156–5161 (2015) Mineshita, H., et al.: Robotic ankle mechanism capable of kicking while jumping and running and adaptable to change in running speed. In: 2019 IEEE-RAS 19th International Conference on Humanoid Robots (Humanoids), pp. 505–510 (2019) Mineshita, H., et al.: Development of a trapezoidal leaf spring for a small and light variable joint stiffness mechanism. In: ROMANSY 23 - Robot Design, Dynamics and Control, pp. 355–363 (2020)

Kinematic Synthesis of a Tendon-Driven 4R Planar Robotic Arm Chiara Lanni(B) , Giorgio Figliolini, and Michele D’Angelo Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, Cassino, Italy [email protected]

Abstract. This paper deals with the kinematic synthesis of a tendon-driven 4R planar robotic arm with 4-d.o.f.s and thus, two times redundant, which behavior is that of a deployable mechanism with wide workspace and small volume, when fully closed. The kinematic synthesis of the proposed 4R planar robotic arm has been developed by superposing three tendon-driven kinematic chains, in order to operate by means of a remote actuation, the last three links, since the first link is actuated directly. Moreover, a suitable algorithm has been formulated to operate the proposed 4R robotic arm in order to reach a given target point within the working area, as well as, to move inside a narrow environment in order to get a given configuration of the robotic arm. Examples are reported to simulate and test the working of the proposed robotic arm in different operating conditions.

1 Introduction Tendon-driven redundant robots are used in many fields for accessing harsh and confined environments that can be inaccessible or dangerous for the human beings. As stated by Tsai (1995, 1999), along with Tsai and Lee (1989), these serial robots are characterized by the following main advantages: 1) All actuators can be installed on the fixed base, resulting in a compact and lightweight design; 2) A well-designed tendon transmission system has little backlash. When these serial robots are also redundant, they can be considered deployable mechanisms, since characterized by wide workspace and small volume, when fully closed. Articulated tendon-driven robotic mechanisms were also investigated by Figliolini and Ceccarelli (1998), Krovi et al. (2002) and Uyguroglu and Demirel (2006). Ozawa et al. (2014) suggested a classification and design of tendon driven mechanisms, while Londi et al. (2004) and Okur et al. (2015) developed a control system. A tendon-driven robotic arm to remove skins by wine fermentation tanks was designed and tested by Figliolini et al. (2020), while a general equation matrix for different mechanisms is in Figliolini et al. (2021). Instead, a small workspace and high rigidity are typical of parallel robots, as in Figliolini et al. (2017). In this paper, the kinematic synthesis of a tendon-driven 4R planar robotic arm has been developed, in order to reach a given target point within the working area, as well as, to move inside a narrow environment in order to get a given configuration of the robotic arm, which is chosen inside this environment properly. © CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 238–245, 2022. https://doi.org/10.1007/978-3-031-06409-8_25

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2 Tendon-Driven 4R Planar Robotic Arm A tendon-driven 4R planar robotic arm can be considered an open kinematic chain with four links that are connected each other in series by revolute joints. This kinematic chain has 4 d.o.f.s, which can be operated by means of 4 independent actuators. Each actuator is usually installed far away from the actuated joints. Nevertheless, a suitable transmission system is required to transmit the motion from the actuators to the corresponding joints. This transmission system can be designed by using wires, chains, belts and timing belts, epicyclic gear trains and sprocket chains. From a kinematic point of view these mechanical devices can be considered equivalent among them according to a suitable design. Figure 1 shows an example of tendon-driven mechanism with 4-d.o.f.s, where both elements of parallel and cross types are connected to form a serial kinematic chain. These elements are equivalent by a kinematic point of view to a pair of internal gears (parallel-type) or external gears (cross-type) and they are composed by a rigid link and two pulleys that are connected by the tendon. Thus, one has R j θj,k = ±R j+1 θj+1,k

(1)

where θj,k and θj+1,k are the relative rotations of the pulleys j and j+1 of radii Rj and Rj+1 with respect to the link k. The sign of Eq. (1) is determined by the rotation of the tendon, which is positive for the parallel-type and negative for the cross-type. Moreover, the oriented angles of the relative rotations can be expressed as θj,j+1 = θj,k − θj+1,k

(2)

In particular, links 1, 2 and 3 are connected in series to shape an open-loop kinematic chain, where 0 is the fixed frame and pulley 4 ≡ j+3 is the end-effector. The rotation of the base pulley j can be related to that of pulley j+3 by means of the oriented angles

Fig. 1. Example of tendon-driven mechanism with 4-d.o.f.s.

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θ 1,0 , θ 2,1 , θ 3,2 and θ 4,3 by which the position equation can be obtained by considering the three kinematic elements of cross and parallel-type, (j, j+1, 1), (j+1, j+2, 2) and (j+2, j+3, 4) respectively, the first two terms denoting the pulleys and third one, their connecting link. Thus, one has R j θj,1 = −R j+1 θj+1,1 θj,0 = θ1,0 + θj,1

(3)

R j+1 θj+1,2 = −R j+2 θj+2,2 θj+1,1 = θj+1,2 + θ2,1

(4)

R j+2 θj+2,3 = R j+3 θj+3,3 θj+2,2 = θj+2,3 + θ3,2

(5)

which give the remote actuation angle θ j,0 of the base pulley j in the form θj,0 = θ1,0 −

R j+1 R j+2 R j+3 θ2,1 + θ3,2 + θ4,3 . Rj Rj Rj

(6)

The example of Fig. 1 can be generalized to any combination of parallel and crosstype elements by θj,0 = θ1,0 ±

R j+1 R j+2 R j+(m−1) θ2,1 ± θ3,2 ± . . . ± θm,m−1 Rj Rj Rj

(7)

where m denotes the number of links of the open-loop kinematic chain. Therefore, applying Eq. (7) to the planar tendon-driven 4R robotic arm of Fig. 2, one has, in matrix form ⎤⎡ ⎤ ⎡ ⎤ ⎡ 1 0 0 0 θ1,0 θ1,0 R ⎢ ⎥ ⎢ θ5,0 ⎥ ⎢ 1 R8 0 0 ⎥⎢ θ2,1 ⎥ 7 ⎥ ⎥ ⎢ (8) ⎥⎢ R4 R4 R10 ⎣ θ6,0 ⎦ = ⎢ 0 ⎦⎣ θ3,2 ⎦ ⎣ 1 R6 R 6 R6 θ7,0 θ4,3 1 RR2 RR 2RR9 RR2 RR9 RR3 7

7 7

7 7 7

which gives a lower triangular matrix of coefficients, when all pulleys have the same radius. Figure 2a shows the kinematic chain and Fig. 2b is related to a kinematic sketch to formulate the inverse kinematics for the remote actuation. The kinematic synthesis of the proposed 4R planar robotic arm has been developed by superposing three tendon-driven kinematic chains, in order to operate by a remote actuation, the last three links 2, 3, and 4, since the link 1 is actuated directly, where L 1 , L 2 , L 3 and L 4 are the link lengths, while θ 1,0 , θ 2,1 , θ 3,2 and θ 4,3 are the variable joint angles. Thus, for any assigned target point P with respect to the fixed frame OX 0 Y 0 , the absolute oriented angles θ 1 , θ 2 , θ 3 and θ 4 of links 1, 2, 3 and 4 of the 4R planar robotic arm, are computed by applying the inverse kinematics. The corresponding equations are non-linear and can be obtained by the direct kinematics, which is expressed through the Cartesian coordinates of point P, as xP = L1 cos θ1 + L2 cos θ2 + L3 cos θ3 + L4 cos θ4

(9)

yP = L1 sin θ1 + L2 sin θ2 + L3 sin θ3 + L4 sin θ4

(10)

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Moreover, the proposed 4R planar robotic arm is redundant and thus, provided of 4 d.o.f.s, which means to have four independent parameters, i.e. θ 1 , θ 2 , θ 3 and θ 4 . Consequently, in order to reduce the number of unknowns, the sketch of Fig. 2b has been assumed, where all links have the same length, the first link is supposed to be oriented toward the target point P and the remaining four-bar linkage is supposed to take the shape of an isosceles trapezoid. Thus, one has  −1 yP (11) θ3 = θ1 = tan xP while the angles θ 2 and θ 4 are obtained by applying the well-known loop-closure equation to the isosceles-trapezoidal four-bar linkage of Fig. 2b by giving

 √ 2 + C 2 − E2 D −D + σ θ2 = tan−1 (12) E−C  −1 B − L2 sin θ2 (13) θ4 = tan A − L2 cos θ2 where σ is equal to ±1 according to the particular assembly mode of the four-bar linkage and the coefficients A, B, C, D and E are expressed as follows A = xP − L1 cos θ1 − L3 cos θ3 , B = yP − L1 sin θ1 − L3 sin θ3

(14)

C = −2AL2 , D = −2BL2 , E = A2 + B2 + L22 − L24

(15)

3 Simulations and Validation The kinematic performances of the designed tendon-driven 4R planar robotic arm have been validated by means of different significant simulations, which have been carried out by implementing in Matlab the proposed inverse kinematic algorithm, along with the matrix Eq. (8), which expresses the remote actuation angles, as function of the joint angles. In particular, two different conditions have been considered for validation purposes, which are devoted to reach different target points that are chosen freely in the working area of the robotic arm and to move inside a narrow environment in order to reach a given configuration by avoiding any boundary collision. Thus, supposing all pulleys of Fig. 2a having the same radius and the links length L 1 = L 2 = L 3 = L 4 equal to 10u with u the unit length, when the Cartesian coordinates of the target point P (x P , yP ) are given, the remote actuation angles can be calculated as function of the absolute joint angles. In particular, referring to Fig. 2a, link 4 that is attached to pulley 3 is remotely driven by pulley 5 because of three kinematic elements of parallel-type, where pulleys 2 and 9 are idle on their axes. Link 4 rotates of the same angle of pulley 5. Similarly, link 3 is remotely driven by pulley 6, because of two kinematic elements of parallel-type, where pulley 4 is idle and pulley 8 is attached to link 2. Thus, link 3

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Fig. 2. Tendon-Driven 4R planar robotic arm: a) Kinematic chain; b) Kinematic sketch

rotates of the same angle of pulley 6 and again, the link 2 is attached to pulley 8 and remotely driven by pulley 7, because of a kinematic element of parallel-type. Finally, link 1 is driven directly around its fixed axis. The movements of this tendon-driven 4R planar robotic arm can be coupled, because the rotation of any link generates the translation of the successive part of the robotic arm and this characterize the proposed design. For example, the rotation of link 1 gives the translation of links 2, 3 and 4, which are moved as a unique rigid body, while the rotation of link 2 gives the translation of links 3 and 4. Thus, the last link 4 is pure rotated, because there is not any successive link. The simulations have been developed by considering each link i as actuated by a single motor Mi for i = 1, 2, 3, 4, which are installed on each driving pulley. In order to validate the proposed algorithm, along with the capability and functionality of the designed tendon-driven 4R planar robotic arm, two particular operating conditions have been considered, such as, to reach different target points that are chosen freely in the working area and to move inside a narrow environment in order to reach a given configuration by avoiding any boundary collision. Referring to Fig. 3, the first operating condition has been simulated to reach different target points that are indicated by Pi for i = 1, 2, 3, 4, which corresponding configurations of the tendon-driven 4R planar robotic arm allowed the validation of the proposed algorithm, according to the sketch of Fig. 2b. Moreover, the rotations of motors Mi for i = 1, 2, 3, 4 are also shown in each figure.

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Fig. 3. Simulation for given target points (where u is the unit length).

Referring to Fig. 4, the second operating condition has been simulated to move the robotic arm inside a narrow environment in order to reach a given configuration by avoiding any boundary collision. In particular, points Pi for i = 1, 2, 3, 4 have been chosen inside the given narrow environment, in order to reach the final configuration of the robotic arm, according to a specific path, which lets to calculate the absolute angles of each segment. Thus, the path is discretized and the intermediate absolute angles are calculated to determine the successive positions of each link of the robotic arm during its motion. In particular, the discretization between two consecutive points has been calculated by assuming a step equal to 1, while the rotation angles of the corresponding driving pulleys have been obtained by using Eq. (8).

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Fig. 4. Second simulation by given a set of points (where u is the unit length).

4 Conclusions In this paper, the kinematic synthesis of a tendon-driven 4R planar robotic arm with 4 d.o.f.s and thus, two times redundant, has been developed by superposing three tendondriven kinematic chains, in order to operate each link by means of a suitable remote actuation. Moreover, a specific algorithm has been formulated to reach a given target point within the working area and to move inside a narrow environment. Significant examples have allowed the validation of the proposed design of the tendon-driven 4R planar robotic arm and the related algorithm for two different operating conditions.

References Figliolini, G., Ceccarelli, M.: A motion analysis for one D.O.F. anthropomorphic finger mechanism. In: CD Proceedings of the 25th ASME Biennial Mechanisms Conference (DETC’98), Atlanta, USA, DETC98/MECH-5985 (1998) Figliolini, G., et al.: Mechatronic design of a robotic arm to remove skins by wine fermentation tanks. In: Carcaterra, A., Paolone, A., Graziani, G. (eds.) AIMETA 2019. LNME, pp. 271–277. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-41057-5_22

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Figliolini, G., Lanni, C., Di Donato, L., Melloni, R., Bacchetta, A.P.: Kinematic synthesis of a tendon-driven robotic arm. In: Niola, V., Gasparetto, A. (eds.) IFToMM ITALY 2020. MMS, vol. 91, pp. 386–393. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-55807-9_44 Figliolini, G., Lanni, C., Rea, P., Gallinelli, T.: Mechatronic design and control of a 3-RPS parallel manipulator. In: Ferraresi, C., Quaglia, G. (eds.) RAAD 2017. MMS, vol. 49, pp. 74–81. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-61276-8_9 Krovi, V., Ananthasuresh, G.K., Kumar, V.: Kinematic and kinetostatic synthesis of planar coupled serial chain mechanisms. ASME J. Mech. Des. 124(2), 301–312 (2002) Londi, F., Pennestrì, E., Valentini, P.P., Vita, L.: Control and virtual reality simulation of tendon driven mechanisms. Multibody Sys.Dyn. 12(2), 133–145 (2004) Okur, B., Zergeroglu, E., Tatlicioglu, E.: Nonlinear control of tendon driven robot manipulators: elimination of actuator side position measurements. In: 54th IEEE Annual Conference on Decision and Control (CDC), 15–18 December 2015, Osaka (Japan), pp. 1491–1496 (2015) Ozawa, R., Kobayashi, H., Hashirii, K.: Analysis, classification, and design of tendon-driven mechanisms. IEEE Trans. Rob. 30(2), 396–410 (2014) Tsai, L.-W.: Robot Analysis and Design: The Mechanics of Serial and Parallel Manipulators, 1st edn. Wiley, New York (1999) Tsai, L.-W., Lee, J.-J.: Kinematic analysis of tendon-driven robotic mechanisms using graph theory. J. Mech. Trans. Autom. Des. 111(1), 59–65 (1989) Tsai, L.W.: Design of tendon-driven manipulators. ASME J. Mech. Des. 117(B), 80–86 (1995) Uyguroglu, M., Demirel, H.: Kinematic analysis of tendon-driven robotic mechanisms using oriented graphs. Acta Mech. 182, 265–277 (2006)

Inverse and Forward Kinematics of a Reconfigurable Spherical Parallel Mechanism with a Circular Rail Pavel Laryushkin1 , Anton Antonov2 , Alexey Fomin2(B) , and Victor Glazunov2 1 Bauman Moscow State Technical University (BMSTU), Moscow 105005, Russia 2 Mechanical Engineering Research Institute of the Russian Academy of Sciences, Moscow

101000, Russia [email protected]

Abstract. The paper discusses inverse and forward kinematics of a 4-DOF parallel mechanism with a circular rail. The mechanism can perform an unlimited rotation around one of the axes and use one of its kinematic chains for reconfiguration. The inverse kinematics has a closed-form solution, while the forward kinematics implies solving coupled polynomial equations. The paper proves that there can be sixteen different assembly modes of the mechanism and verifies the result by a homotopy continuation approach. Keywords: Parallel mechanism · Spherical mechanism · Circular rail · Inverse and forward kinematics · MATLAB simulation · Homotopy continuation

1 Introduction Spherical mechanisms represent mechanical systems in which the axes of revolute joints intersect at a common point; there are many examples of such mechanisms (Furlong et al. 1999; Wampler 2004). The remote center of motion causes the extensive use of these systems, e.g., in medical applications (Baumann et al. 1997; Li and Payandeh 2002; Westwood 2005). Spherical mechanisms can also be composed of kinematic chains that do not constrain the output link motion. One example is a 3R1T mechanism with four degrees of freedom (4-DOF) designed for laparoscopic operations (Saafi et al. 2020). This mechanism has an asymmetric structure with one URU kinematic chain1 . Two remaining spherical RRR chains provide the spherical motion, while the URU one allows bypassing or avoiding singularities, which is important in many operations, including the surgery. Work (Wu et al. 2015) presents another asymmetric spherical mechanism. The mechanism has two actuated spherical RRR chains, one actuated RU chain, and one passive RRS chain, used to constrain the rotation of some links around the vertical axis. Such structure improves mechanism accuracy, provides partial kinematic decoupling, 1 R, P, U, and S designate revolute, prismatic, universal, and spherical joints, respectively.

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and allows the output link to have an unlimited rotation around the vertical axis. The paper also compares the presented mechanism with two other spherical systems: Agile Wrist (Staicu 2009) and CoSPM (Asada and Granito 1985). Among the spherical mechanisms, there are architectures with a circular rail (Xu et al. 2018; Zhao et al. 2014; Fang et al. 2013; Li et al. 2006). The rail provides the output link with an unlimited rotation around one of the axes and expands the applications of spherical mechanisms. Another technical solution that gives additional advantages is the reconfigurable design. Reconfiguration consists in changing workspace dimensions or modifying motion pattern (Borras et al. 2009). This property also allows the mechanisms to avoid singularities during their motion or find a way out of them (Kong et al. 2007). The authors of the current work, however, have found only one spherical mechanism with a circular rail and reconfigurability: a 3-RRR spherical mechanism with four-bar linkages (Wu and Bai 2019). The present work considers a novel reconfigurable spherical parallel mechanism with a circular rail proposed recently by the authors in (Laryushkin et al. 2022). In this work, we aim at its inverse and forward kinematic analysis. The paper is organized as follows. Section 2 discusses mechanism design and briefly analyzes its mobility considering constraints imposed by the mechanism kinematic chains. Section 3 shows solution of the inverse kinematics, in which we determine coordinates of the driven joints given the output link configuration. Section 4 studies the forward kinematics problem, where we compute output link orientation and position given the coordinates of the driven joints. This section also presents mechanism assembly modes based on the solutions of the forward kinematics problem. Section 5 summarizes the results.

2 Mechanism Design Figure 1a shows the mechanism under study. The mechanism has three identical RRRRR kinematic chains and an additional central SPS chain (the underline signifies an actuated joint). Each actuated revolute joint represents a carriage moving along a circular rail. The theoretical axes of these joints are collinear with a circular rail axis, which passes through rail center O orthogonally to the rail plane. Without loss of generality, we assume the rail axis is vertical. Axes of joints Ai , Bi , and C i (Fig. 1a), i = 1…3, are parallel to each other and the horizontal plane. Axes of joints Di intersect at point F on the rail axis. Note that point F does not necessarily lie on the mechanism moving plate (plane D1 D2 D3 ). To analyze mechanism mobility, let us ignore the SPS chain for a moment. Three remaining RRRRR chains provide a controlled 3-DOF spherical motion of the moving plate, with point F being the center of this motion. There, however, remains an uncontrolled translation along the vertical direction since the RRRRR chains prevent the moving plate translations only in horizontal directions. The fourth SPS chain, connected to the base and the moving plate at points M and N (Fig. 1a), allows us to deal with this uncontrolled motion. This chain constrains translation along the MN line when its actuated joint is locked. As long as this line has a non-zero projection on the vertical axis, the vertical translation of the moving plate can be controlled by the SPS chain. Moreover, if points F and N coincide (Fig. 1b), we obtain the mechanism

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Fig. 1. 4-DOF mechanism with a circular rail: (a) general structure; (b) structure that provides decoupled translation along the vertical direction.

with a decoupled translation. This allows us to use the central chain for reconfiguration and displace the spherical motion center. Paper (Laryushkin et al. 2022) also considers other design possibilities and provides a more rigorous mobility analysis using the screw theory.

3 Inverse Kinematics Inverse kinematics aims to determine coordinates of the joints (primarily, the driven ones) given the configuration of the output link. To formulate this problem for the considered mechanism, let us introduce some additional notations. First, we define stationary reference frame Oxyz, such that its Oz axis is collinear with the rail axis, and axes Ox and Oy have arbitrary directions (Fig. 2). We also attach reference frame Fx  y z to the moving plate: its origin coincides with point F, and its axes can have arbitrary directions. We can describe the orientation of frame Fx  y z and position of its origin F relative  T to Oxyz by rotation matrix R and position vector pF = 0 0 z . Angles θi , i = 1…3, which define the carriage positions (Fig. 2), and distance l between spherical joints M and N of the SPS chain represent the driven coordinates of the mechanism. With this notation, the inverse kinematics aims to find parameters θi and l for given R and z. To solve the inverse kinematics problem, let us first consider the RRRRR chains. We can find coordinates pCi of points C i relative to stationary frame Oxyz as follows: pCi = RrCi + pF ,

(1)

where vector rCi defines coordinates of point C i relative to frame Fx  y z . By mechanism design, axes of joints C i and Di intersect each other; therefore, parameter rCi is known and constant. In any RRRRR chain, points Ai , Bi , and C i always lie in a plane containing the Oz axis. So, the x and y coordinates of point C i are proportional to the same coordinates of Ai . Assuming that θi is measured counterclockwise from the Ox axis, we find: θi = atan2(yCi , xCi ),

(2)

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Fig. 2. Toward kinematic analysis of the considered mechanism.

where x Ci and yCi are the x and y components of vector pCi . The calculations for the SPS chain are also straightforward. First, we determine coordinates pN of point N relative to stationary frame Oxyz: pN = RrN + pF , where vector rN defines (known) coordinates of point N relative to frame Fx  y z . Next, we find the desired value of l from the equation below:  l = (pN −pM )2 ,

(3)

(4)

where vector pM defines (known) coordinates of point M relative to frame Oxyz.

4 Forward Kinematics The task of the forward kinematics is to determine output link orientation and position given the coordinates of the driven joints. For the discussed mechanism, we can formulate this problem as follows: find R and z for given θi , i = 1…3, and l. To solve the forward kinematics problem, let us consider the RRRRR chains first. In each chain, the plane through points Ai , Bi , and C i contains axis Oz, and the axes of the corresponding revolute joints are tangential to the circular rail. Therefore, the following equation exists: (pDi −pAi )T τˆ i = 0,

(5)

where vectors pAi and pDi define coordinates of points Ai and Di relative to frame Oxyz; τˆ i is a unit vector tangent to the circular rail (Fig. 2): T T   (6) τˆ i = τxi τyi τzi = − sin θi cos θi 0 . By mechanism design, vector pAi is orthogonal to τˆ i : the dot product of these vectors gives zero in (5). Vector pDi can be represented similar to (1): pDi = RrDi + pF ,

(7)

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where vector rDi defines (known) coordinates of point Di relative to frame Fx  y z .  T Now, if we substitute Eqs. (6) and (7), and pF = 0 0 z into (5) and expand the obtained relation, we will get the expression:  x    x  y y  y y x z τi r11 + τi r21 + rDi τxi r12 + τi r22 + rDi τi r13 + τi r23 = 0, (8) rDi y

x , r , and r z are components of vector r ; r where rDi Di mn is a component of rotation Di Di matrix R in its m-th row and n-th column. Notice that Eq. (8) does not depend on z. Since the orientation of any rigid body depends on three independent parameters at most, matrix R has only three independent components too. So, for i = 1…3, we have a system of three Eq. (8) in three unknowns (for example, three Euler angles). To solve this system, we could express matrix R in Euler angles, perform a tangent half-angle substitution for these angles, and follow the elimination techniques to get a univariate polynomial (Gosselin et al. 1994). The rational choice of moving plate frame Fx  y z makes some components of rDi equal to zero and simplifies Eq. (8). In this paper, however, we will apply an alternative method. First, let us represent  T rotation matrix R via unit quaternion qˆ = q0 q1 q2 q3 (see, e.g., Lynch and Park 2017, app. B.3): ⎡ 2 ⎤ q0 + q12 −q22 −q32 2(q1 q2 −q0 q3 ) 2(q0 q2 + q1 q3 ) R = ⎣ 2(q0 q3 + q1 q2 ) q02 −q12 + q22 −q32 2(q2 q3 −q0 q1 ) ⎦, (9) 2(q1 q3 −q0 q2 ) 2(q0 q1 + q2 q3 ) q02 −q12 −q22 + q32

with q02 + q12 + q22 + q32 = 1.

(10)

If we substitute components of (9) into (8), we will get a system of four quadratic equations, three (8) and one (10), with respect to four variables q0 …q3 . Bezout’s theorem (Garcia and Li 1980) dictates the system can have a maximum of 24 = 16 solutions. From Eqs. (8) and (10), we can easily see that if qˆ is a solution, then −qˆ is a solution too. But the structure of (9) shows that qˆ and −qˆ represent the same rotation matrix. Therefore, for given variables θi (and l), we can get at most eight diverse orientations of the moving plate. To solve Eqs. (8) and (10), we can apply the elimination method mentioned earlier; Rodriguez and Ruggiu (2013) also obtained a univariate polynomial for a similar system of equations by calculating a Dixon determinant. Instead, we will use a homotopy continuation approach (Sommese and Wampler 2005): in our case, this method will require no manipulations with the equations, opposed to the techniques above. The continuation approach follows the next idea. Suppose we have a system of equations f(x) = 0 to be solved; we can form an auxiliary system: h(x, λ) = (1−λ)g(x) + λf(x) = 0,

(11)

where g(x) is a system with the same number of solutions as f(x), and all these solutions are known; λ is a scalar parameter, which varies from 0 to 1.

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When λ = 0, h(x, 0) = g(x), and we know all the solutions. These solutions form an initial guess to solve a new version of h(x, λ) with a slightly increased value of λ by standard iterative algorithms. We repeat the process until λ = 1, when h(x, 1) = f(x). Thus, initial system g(x) = 0 with known solutions evolves toward desired system f(x) = 0. Having determined quaternion qˆ and its corresponding matrix R, we can find the  T remaining unknown variable z using Eq. (4). If we substitute (3) and pF = 0 0 z into (4), we will get a quadratic equation with respect to z: z 2 + 2(RrN −pM )z z + (RrN −pM )2 −l 2 = 0,

(12)

where (RrN – pM )z represents the z component of vector (RrN – pM ). In a general case, Eq. (12) generates two solutions for z. Therefore, given eight solutions for rotation matrix R, we can get 16 distinct solutions for the forward kinematics. To verify all the statements above, let us consider an example for the mechanism with the following parameters and values of the driven coordinates:  T  T (13) pM = 0 0 0.1 m, rN = 0 0 −0.25 m, T  rDi = 0.25 cos αi 0.25 sin αi −0.15 m with α1 = 0, α2 = 120◦ , α3 = 240◦ , θ1 = 0, θ2 = 120◦ , θ3 = 120◦ , l = 1 m.

(14)

We have solved the homotopy continuation problem using Bertini software (Bates et al. 2006), implemented in MATLAB package as BertiniLab interface ˆ the one (Bates et al. 2015). As a result, we got 16 real solutions for quaternion q, half of which were equal to another half up to a sign. Table 1 shows all these solutions together with the z values, calculated from Eq. (12). Figure 3 presents 16 distinct assembly modes of the mechanism, which correspond to these solutions. Note that one half of these modes is symmetrical to another half with respect to the rail plane. Table 1. Solutions to the forward kinematics (accurate to the thousandths).

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Fig. 3. Mechanism assembly modes corresponding to the solutions of the forward kinematics problem (Table 1).

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5 Conclusions The paper has considered inverse and forward kinematics of a recently proposed 4-DOF parallel mechanism with a circular rail. The inverse problem has a straightforward closed-form solution, while the forward one requires solving a system of coupled nonlinear equations, which structure depends on the variables used to describe output link orientation. We have chosen a quaternion to represent this orientation: such choice has led to four quadratic equations with eight distinct solutions for the output link orientation and sixteen different assembly modes for the entire mechanism. To solve the equations, we have applied the homotopy continuation approach and considered an example, which has verified the theoretical statements. The material presented in this study forms a basis for subsequent kinematic and dynamic analyses of the mechanism and its optimal design. The techniques we used to solve the forward kinematics can be applied for similar parallel mechanisms too. Acknowledgement. This research was supported by Russian Science Foundation (RSF) under grant no. 21-79-10409, https://rscf.ru/project/21-79-10409/.

References Asada, H., Granito, J.: Kinematic and static characterization of wrist joints and their optimal design. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 244–250 (1985) Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for numerical algebraic geometry. bertini.nd.edu (2006) Bates, D.J., Newell, A.J., Niemerg, M.: BertiniLab: a MATLAB interface for solving systems of polynomial equations. Numer. Algorithm. 71(1), 229–244 (2015) Baumann, R., Maeder, W., Glauser, D., Clavel, R.: The PantoScope: a spherical remote-centerof-motion parallel manipulator for force reflection. In: Proceedings of the IEEE International Conference on Robotics and Automation, vol. 1, pp. 718–723 (1997) Borras, J., Thomas, F., Ottaviano, E., Ceccarelli, M.: A reconfigurable 5-DoF 5-SPU parallel platform. In: Proceedings of the ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots, pp. 617–623 (2009) Fang, H.R., Chen, Z.H., Fang, Y.F.: A novel spherical parallel manipulator with circular guide. Appl. Mech. Mater. 325–326, 1014–1018 (2013) Furlong, T.J., Vance, J.M., Larochelle, P.M.: Spherical mechanism synthesis in virtual reality. J. Mech. Des. 121, 515–520 (1999) Garcia, C.B., Li, T.Y.: On the number of solutions to polynomial systems of equations. SIAM J. Numer. Anal. 17, 540–546 (1980) Gosselin, C.M., Sefrioui, J., Richard, M.J.: On the direct kinematics of spherical three-degree-offreedom parallel manipulators of general architecture. J. Mech. Des. 116, 594–598 (1994) Kong, X., Gosselin, C.M., Richard, P.: Type synthesis of parallel mechanisms with multiple operation modes. J. Mech. Des. 129, 595–601 (2007) Laryushkin, P., Antonov, A., Fomin, A., Glazunov, V.: Novel reconfigurable spherical parallel mechanisms with a circular rail. Robotics 11, 30 (2022)

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Li, T., Payandeh, S.: Design of spherical parallel mechanisms for application to laparoscopic surgery. Robotica 20, 133–138 (2002) Li, Q., Chen, Q., Wu, C., Hu, X.: Two novel spherical 3-DOF parallel manipulators with circular prismatic pairs. In: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 325–328 (2006) Lynch, K.M., Park, F.C.: Modern Robotics: Mechanics, Planning, and Control. Cambridge University Press, Cambridge (2017) Rodriguez, J., Ruggiu, M.: A novel method for the solution of the forward displacement problem of spherical parallel manipulators. ZAMM – J. Appl. Math. Mech. 93, 73–82 (2013) Saafi, H., Laribi, M.A., Zeghloul, S.: Forward kinematic model resolution of a special spherical parallel manipulator: comparison and real-time validation. Robotics 9, 62 (2020) Sommese, A.J., Wampler, C.W.: The Numerical Solution of Polynomials Arising in Engineering and Science. World Scientific, Singapore (2005) Staicu, S.: Recursive modelling in dynamics of Agile Wrist spherical parallel robot. Robot. Comput. Integrat. Manuf. 25, 409–416 (2009) Wampler, C.W.: Displacement analysis of spherical mechanisms having three or fewer loops. J. Mech. Des. 126, 93–100 (2004) Westwood, J.D.: Spherical mechanism analysis of a surgical robot for minimally invasive surgery– analytical and experimental approaches. Stud. Health Technol. Inf. 111, 422–428 (2005) Wu, G., Bai, S.: Design and kinematic analysis of a 3-RRR spherical parallel manipulator reconfigured with four–bar linkages. Robot. Comput. Integrat. Manuf. 56, 55–65 (2019) Wu, G., Caro, S., Wang, J.: Design and transmission analysis of an asymmetrical spherical parallel manipulator. Mech. Mach. Theory 94, 119–131 (2015) Xu, C.C., Xue, C., Duan, X.C.: A novel 2R parallel mechanism for alt-azimuth pedestal. IOP Conf. Ser. Mater. Sci. Eng. 428, 012053 (2018) Zhao, J., Feng, Z., Chu, F., Ma, N.: Kinematic synthesis of spatial mechanisms. In: Advanced Theory of Constraint and Motion Analysis for Robot Mechanisms, pp. 429–469 (2014)

Novel Gripper Design for Transporting of Biosample Tubes Artem A. Voloshkin1 , Larisa A. Rybak1(B) , Giuseppe Carbone1,2 , and Vladislav V. Cherkasov1 1 Belgorod State Technological University named after V.G. Shukhov, Belgorod, Russia

[email protected] 2 Department of Mechanical, Energy and Management Engineering, University of Calabria,

87036 Rende, Italy

Abstract. The article presents a novel design of the robot gripper for application as part of a robotic system for biomaterial aliquoting. The kinematic connection is made in the form of a worm having the form of a globoid with three uniformly executed screw surfaces and provides effective capture of test tubes and their transfer to the working area. The main characteristics are calculated and the results of mathematical modeling are presented. A 3D gripper model was created using the CAD system tools and its real prototype was made.

1 Introduction Currently, there is a shortage of biosamples, which is constantly increasing, and the requirements for their quality are becoming more stringent. This is due to an increase in medical research in epidemiology and microbiology. All this requires the introduction of new technological solutions see, e.g., Glazunov et al. (2009). The use of new robotic systems (RS) will reduce the human factor when working with biosamples and increase the repeatability of results. It is often claimed (see Noureddine et al. 2012, pp. 122–127, and Ding et al. 2000, for arguments) that there is a wide variety of kinematic structures of gripping devices (GD) of robots that can be used to solve aliquoting problems, as well as various options for gripping an object, which take into account the shape and material of the captured object (Morales et al. 2013; Hugo 2013; Carlos et al. 2013; Svinin et al. 2000; Luo 2013). Existing grips have a large number of movable elements, and as a result, backlashes occur in the joints, which leads to jerky movements and a decrease in the accuracy of the grip. The test tube, in which the biosample should be placed, has a simple geometry, and its reliable retention is ensured by the surface structure of the gripping jaws and the number of contact points. The proposed gripping device differs from analogs by the kinematic connection between the stepper motor and the gripping device. The connection is made like a worm, having the form of a globoid with four uniformly made helical surfaces with a step of 90°. The memory is made of a four-lever with the possibility of rotation on axes rigidly fixed in the housing through 90°. The spherical part of each lever is connected to one of the grippers of the globoid, while the jaws of the gripper levers are removable. The article considers a new type of © CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 255–262, 2022. https://doi.org/10.1007/978-3-031-06409-8_27

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modified GD, a method for calculating its main parameters. Thus, the requirements for the gripper device can be formulated as follows: the GD must ensure reliable retention of the test tube with biological fluid, have a minimum number of movable elements and have sufficient reliability.

2 Calculation Model of the Gripping Device The gripping device must provide high accuracy and reliability of gripping. Reliability is improved by ensuring smooth working movements of the gripping levers and providing three points of contact in the setting process. The accuracy of the convergence of the fingers of the gripper device is achieved through the use of a globoid transmission. The tubes used in the aliquotating process have a small diameter, the required accuracy of convergence of the fingers of the gripper device is 0.1 mm. The backlash that occurs in each of the moving parts of the mechanism has a negative impact on the accuracy of the mechanism, therefore, one of the ways to improve accuracy is to reduce the number of moving parts. There is a wide variety of such manipulators: a gripper with a crank-sliding mechanism see, e.g., Russo (2017), a gripper with a worm gear (Belyaev 2011) has a hinge for each lever (finger) of the gripper, which negatively affects the smoothness and reliability of the system. A new GD architecture is proposed, the diagram is shown in Fig. 1. The GD consists of the following components: base 7, globoid worm 6, lever 2, rotational hinge 3, spring 8. Compression and unclenching of fingers 2 is provided due to the radial motion trajectory 1. (Fig. 4a). The engine drives a four-way globoid worm 6. The spherical part of the lever 5 moves along the helical surface of the globoid and rotates the lever relative to the hinge 3 along the arc of a circle 1. The lever is fixed to the base 7 by means of a rotary hinge. The spherical part of the lever is pressed against the helical surface of the globoid due to the spring 8. The gripping device ensures the grip as a result of the radial movement of the lever along the trajectory 1 (Fig. 1a). Let us determine the main parameters of the proposed globoid worm in accordance with the scheme (Fig. 1b). Let us determine the radius of the pitch circle rA = ra2 − 2ha2 = S − d1 /2, where ra2 is the radius of rotation of the lever, ha2 is the height of the pitch diameter from the base of the helical surface, S is the center distance, d1 is the minimum pitch diameter of the globoid; C = 0.1ha2 – minimum radial clearance; Ra1 = S − 0.5da1 , the radius of the tops of the worm turns in the axial plane, where da1 is the minimum diameter of the globoid, da1 = d1 + 2ha2 ; Rf 1 = ra2 + C, the radius of the depressions of the globoid worm in the axial plane. The height of the globoid depends on the angle α (Fig. 2). The friction force occurs between the lever and the helical surface of the globoid, which has a significant impact on the performance of the grip and is calculated by the formula FμA = μNA cosβ, where μ is the coefficient of friction, NA is the force exerting pressure on the helical surface of the globoid NA = NB · rB /rA where rB is the radius (distance) of the spring fastening, and NB is determined by the formula  NB = −k

   (−rB − xP )2 + (rB − yP )2 − (−rB sinα − xP )2 + (rB cosα − yP )2 sina,

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Fig. 1. a - GD diagram. b - calculation scheme of globoid transmission

where k is the coefficient of elasticity of the spring, the coordinates of the point P relative to the center O. The spring in the mechanism is required to ensure a constant grip tension. Consider a gripper containing n levers. For the rotation of the globoid, it is necessary to provide a moment greater than the drag force. To do this, we determine the minimum moment M1 = (F μA nPg )/2π η where is η the efficiency, taken for the worm gear 0.8, Pg is the pitch of the helical surface of the globoid, which can be found by the formula Pg = 2π d tan β, where - d is the diameter of the globoid, which can be found by the formula d = 2(S − rA sin α), Due to the curvature of the surface, the globoid diameter decreases from the base to the center and increases from the center to the upper end. In order to maintain a constant step value, it is necessary to change the angle of the helical surface according to the following relationship: The tgβ = Pg /(2π 2(S − rA sin α)), motor torque, taking into account the elasticity of the springs, can be determined by the M1 = (F μA n2(S − rA sin α)tan β)/η. Formula nFμC = GC = mg, where mC is the mass of the tube with liquid. The friction force that occurs as a result of closing the fingers of the grip is determined by the formula FμC = μC NC cos γ , where is the rC radius of rotation of the finger, NC is the closing force of the gripping fingers, μC is the coefficient of friction. Let us express the angle of application of 1

the force in accordance with Fig. 2 cos γ = (1 − (S − 0.5D)/rC )2 ) 2 , where is D the diameter of the test tube. The force of closing the fingers of the grip is determined by the formula NC = (N M rA )/rC , where is NM the force acting on the lever because of the rotation of the globoid. Let us express the force required to hold the test tube. Let 1

2 2 NM = mgrC /nμC rA · (1 − (S − 0.5D)/r C ) ) , us determine the moment of the engine  for capturing the test tubes M = (2n NM + F μA (S −rA sin α)tan β)/η. The creation of the surface of a globoid depends on the height and radius of curvature. The curvature of the surface was determined based on the trajectory of the lever and amounted to 30 mm. Based on the results of the calculation, a 3D grip model was developed (Fig. 3a) and a layout was made. GD is supposed to be used as part of the RS for aliquoting biological

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material (Fig. 3b). The RS uses two modules of different topologies. The first is the DeLi 2 parallel manipulator based on the delta robot. The end effector is made in the form of a dispenser that provides aliquoting. The second is the Uni module, made based on a modification of a collaborative robot and equipped with a GD for transporting plates and test tubes. dispenser 3, which aliquots the biomaterial. A replaceable tip 4 is installed on it. The Uni 5 robot is installed on a fixed base 6 and ensures the capture of plates with test tubes 7 and movement to the working area 8.

Fig. 2. a – 3D model GD. b – calculation scheme of the globoid worm

Fig. 3. RS model

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The test tubes are filled with liquid to approximate the real mass (Fig. 4). Medical test tubes for blood samples were used for the experiment. To do this, they were half filled with liquid to create similar working conditions. A program was developed by which the robot moved test tubes from a tripod to a specified point. The output shaft torque is set when capturing a medical tube as a percentage of the rated engine power. The movement speed of the Universal UR3 robot was 100%. The gripper holds the test tubes at the calculated value, however, for the reliability of the hold, it was necessary to increase the power by 30%.

Fig. 4. Moving test tubes with liquid.

3 Simulation Results The dependence of the deflection angles of the lever α, as well as the angles of entry of the helical surface of the globoid, β on the required engine torque has been investigated. The initial parameters of the mechanism were selected to reveal the dependence of the lever deflection on the angle α when the globoid is rotated. The simulation was performed for the following mechanism parameters: S = 21 mm, rA = 12 mm, rB = 6 mm, rA /rB = 2/1, ha2 = 0.1 mm, Pg = 16 mm, xP = 3 mm, yP = 12 mm. Coefficient k = 5, μ = 0.3. The forces FμA and are calculated NA , as well as the moment M1 for different angles α. The simulation results are shown in Fig. 5. It can be seen from the figure that as the angle α increases, the pressure on the globoid also increases, thereby increasing the friction between the lever and the sliding surface of the globoid (Fig. 5a). This is due to the fact that with an increase in the angle α, the elongation of the spring also increases, thereby increasing the elastic force FμA . At the same time, at different angles β of the globoid surface, the friction force changes (Fig. 5b). It is shown that the friction force FμA varies depending on the deflection angle of the lever α.

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FricƟon force 120

10.000

100

8.000

80

6.000

60 4.000

40

2.000

20 0

0.000 0.0040.0100.0180.0280.0400.0530.0680.0840.1010.119

(N∙mm)∙10-3

a

Moment diagram 10.000

120 100 80 60 40 20 0

8.060 90 7.902 6.851 7.46880 6.161 70 5.479 60 4.857 4.314 50 3.855 40 30 20 10

8.000 6.000 4.000 2.000 0.000

0.004 0.010 0.018 0.028 0.040 0.053 0.068 0.084 0.101 0.119

(N mm)∙10-3 b Fig. 5. a - graphs of the friction force from angles α and β. b - moment diagram.

The moment M1 on the motor shaft, required for the movement of the mechanism, takes the values shown in Fig. 6. A feature of the proposed grip is that when the diameter of the object being grasped changes, the angle of the finger pressing force changes (Fig. 4). In accordance with Fig. 6, you can choose the optimal parameters of the object. For the calculated GD, the optimal diameter is 10 mm.

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5.00 4.50 4.00

M, N m

3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.001 0.0025 0.004 0.0055 0.007 0.0085 0.01 0.0115 0.013 0.0145

Diameter of object, m Fig. 6. The dependence of the engine torque.

4 Conclusion The proposed gripping device for working with test tubes is an important element of the robotic system, the use of which will eliminate the human factor in the process of working with biological fluid, as well as improve safety. The calculations made it possible to reveal the dependences of the forces acting in this configuration of the gripping device: the minimum moment necessary to set the system in motion, as well as the moment necessary to capture the test tube. The required torque on the gripper motor shaft is 0.137 N · mm. The proposed design contains fewer moving elements, which increases smoothness and accuracy, and an increase in the number of contact points allows for the required reliability of the tube capture. A gripping device layout was developed using 3D printing. The operability of the gripping device, which has a simple structure and can be used in various industries, has been experimentally proven. Acknowledgements. This work was supported by the state assignment of Ministry of Science and Higher Education of the Russian Federation under Grant FZWN-2020-0017.

References Glazunov, V.A., Lastochkin, A.B., Shalyukhin, K.A., et al.: To the analysis and classification of relative manipulation devices. In: Problems of Mechanical Engineering and Reliability of Machines, no. 4, pp. 81–85 (2009) Noureddine, R., Noureddine, F., Benamar, A.: In: ault Tolerant Gripper in Robotics/MecatronicsREM 2012, pp. 122–127 (2012) Ding, D., Liu, Y.-H., Wang, S.: The synthesis of 3-D form-closure grasps. Robotica 18, 51–58 (2000)

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Morales, A., Prats, M., Felip, J.: Sensors and methods for the evaluation of grasping. In: Grasping in Robotics, vol. 10, pp. 103–104 (2013) Hugo, P.O.: Industrial grippers: state-of-the-art and main design characteristics. In: Grasping in Robotics, vol. 10, pp. 107–132 (2013) Carlos, B., Antonio, G., Alberto, J.: Cabas Ramiro and Martinez de la Casa Santiago/A survey on different control techniques for grasping. In: Grasping in Robotics, vol. 10, pp. 223–246 (2013) Svinin, M., Ueda, K., Kaneko, M.: On the Liapunov stability of multi-finger grasps. Robotica 18, 59–70 (2000) Luo, M.: Orientation for robotic hands. Grasp. Robot. 10, 159–187 (2013) Russo, M.: Design and test of a gripper prototype for horticulture products. Robot. Comput. Integr. Manuf. 44, 266–275 (2017) Belyaev Alexey Nikolaevich [Gripper device]. Patent RF, no. 118579 (2011)

Techniques for Synthesis of Inherently Force Balanced Mechanisms Lorenzo Girgenti and Volkert van der Wijk(B) Delft University of Technology, Department of Precision and Microsystems Engineering, Delft, The Netherlands [email protected] Abstract. Compared to common dynamic balancing approaches, Inherent Dynamic Balancing aims at designing mechanisms with dynamic balance as a starting point. Then inherently balanced principal vector linkage architectures are used from which balanced linkage solutions are derived, ensuring the optimal kinematics for motions with a center of mass that is always stationary. This paper addresses various techniques for modifying principal vector linkage architectures in the synthesis process while maintaining the inherent balance. The techniques consist of constraining the mobility of links, shifting links to another position, and exchanging links with other machine elements such as sliders, gears, and belt and chain drives. To illustrate the potential, a synthesized 2-DoF inherently force balanced mechanism solution incorporating various techniques is presented and numerically evaluated for force balance.

1

Introduction

In order to prevent vibrations of the base of a machine, which can deteriorate the reliability and accuracy of mechanical systems in operation, shaking forces and shaking moments on the base need to be prevented (Lowen and Berkof, 1968). For this there are various approaches for dynamically balancing a certain linkage (Feng, 1991; Arakelian and Smith, 1999; Kochev, 2000; Gosselin et al., 2004) in which it is common to implement counter-masses for shaking force balance and counterrotations for shaking moment balance (Arakelian and Smith, 2005a,b). A disadvantage of these approaches is that they lead to a significant increase of mass, inertia, and design complexity of the initial mechanism (van der Wijk et al., 2009). An opposite approach called Inherent Dynamic Balancing (Van derWijk, 2014) has been developed to design balanced mechanisms by deriving them from inherently balanced linkage architectures. These basic architectures solely include the essential kinematic relations for balance and can be used as starting point in the design of an advantageously balanced mechanism for an intended task. It has been shown how a wide variety of normally constrained inherently balanced linkage solutions can be found from highly overconstrained inherently balanced linkage architectures by simply eliminating the undesired links (Van der Wijk, 2020). It is also possible to exchange links with other machine elements such as sliders and gears (Van der Wijk, 2014), however these design modifications have not been investigated extensively yet. c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 263–271, 2022. https://doi.org/10.1007/978-3-031-06409-8_28

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The goal of this paper is to present an overview of possibilities in the synthesis of inherently balanced mechanisms by introducing machine elements other than links. First, a brief description of a principal vector linkage is provided, followed by a summary of techniques for modification. Subsequently, multiple techniques are applied in an example of a synthesized 2-degree-of-freedom (2DoF) inherently force balanced mechanism of which the balance is numerically verified. The paper is based on the research presented in more detail in Girgenti (2021).

2

Principal Vector Linkages

e2

a21

a1

P1

P2 q q12 31 p12 B1 q32

A1 s1

S2 f2

p11 q11

A2

a23 p31 q13

p13

q44 p44

p1 S1 q 1

x1

S3 f3

a34

p21

A3

p42

q43

C1 p32

B2

a32 P3 e3 q21 q 42

B3

q22 p43 C2 p22 q14

p41

q41 P4 p4 q 4 S4

p14 S y1 x4 y4

a4 s4

y z

x

Fig. 1. 4-DoF principal vector linkage of 16 links connected with revolute pairs. Each link has a general CoM and the common CoM of the linkage is in S for all motions (Van der Wijk, 2014).

Fig. 1 shows a 4-DoF principal vector linkage consisting of 16 links which form 6 parallelograms with revolute pairs in each joint (Van der Wijk, 2014). Each link has a general mass distribution and the common CoM of all links is stationary in joint S for any of the 4-DoF motions about joint S. With S chosen as a joint with the base the principal vector linkage becomes a linkage that is force balanced with respect to the base. The exact dimensions of the links, their mass, and the precise locations of the link CoMs, defined with parameters p and q or e and f , follow from a set of equations, the balance conditions. Principal vector linkages are based on the method of principal vectors found by Fischer (1906). Principal vectors relate the motions of the element CoMs of a kinematic chain to their common CoM with the characteristic that they have a constant length and form parallelograms. Originally used for analysis of the motion of the center of mass of a system it allowed for describing the dynamics with insightful decoupled equations. Because of their constant length, the vectors can be transformed into real links with mass, thus obtaining principal vector linkages.

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3

265

Techniques to Modify Principal Vector Linkages

The kinematic property of the parallelogram structure is essential for inherent force balance of a principal vector linkage and determines the overall motions. Because of this, two observations can be made: • If the motions of a principal vector linkage are constrained, the parallelogram kinematic property is still maintained and therefore also the inherent balance capability. • When the parallelogram links are modified such that the parallelogram kinematic property is maintained, inherent balance capability is maintained for any modification. These observations led to the definition of three different types of techniques for modifying principal vector linkages while maintaining their essential kinematics. Where these kinematics guarantee the inherent balance capability, the exact parameter values of the links and masses are to be obtained from the corresponding force balance conditions. 3.1

Constraining Link Motions

The introduction of mechanical constraints to the motions of the principal vector linkage leads to a decrease of DoFs while not affecting the inherent balance capability. A link’s planar motions, i.e. two translations and one rotation, can be constraint relative to the base (absolute motion constraints), or to other links (relative motion constraints). Due to the reduction of DoFs, links may merge or become redundant and eliminated.

Fig. 2. (a) Absolute motion of link B3 P3 constrained with a sliding pair; (b) The kinematic property of parallelogram B2 P3 B3 C2 is maintained when link B3 P3 is shifted to K1 K2 .

Constraining Absolute Link Motions. A link can be constrained with the base in different ways. By an additional revolute joint, for example, the translations of a link can be constrained allowing the link to rotate only. By constraining a link with the base by means of a prismatic pair as illustrated in Fig. 2a, the

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link is restricted to solely translate for a determined orientation. When a link is completely fixed to the base, on the other hand, both its translations and rotation are constrained as immobile. When a slider with revolute pair with the base is used to constrain the motion of a link, then the link is constrained to rotate around the slider’s revolute joint and to translate along the slider trajectory. Constraining Relative Link Motions. Since a principal vector linkage consists of solely parallelogram linkages, the links have with respect to their connecting links relative rotations only. The relative rotations between links can be fixed by merging links for a certain relative angle. Also it is possible to constrain the relative rotations between links with a specific relation by a pair of gears or a belt or chain drive. These elements then are mounted on the links and the transmission ratios determine the relative motions between the links. 3.2

Modifying Link Positions

The kinematics of a parallelogram are not affected by the size and the exact positions of its links. Therefore also the inherent balance capability is not affected by the exact positions of the links in a principal vector linkage and links can be shifted to another location as illustrated in Fig. 2b. Multiple links may be shifted simultaneously, which however is not possible for all of them as the kinematic property of the parallelogram structure must be maintained. 3.3

Substituting Links

a)

b)

c) Fig. 3. Substitution of parallelogram links: (a) Link DC substituted with a belt or chain drive with a transmission ratio of 1; (b) Link DC substituted with a gear train with a transmission ratio of 1; (c) Links SC1 and SC2 substituted with circular sliders in C1 and C2 .

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The kinematic property of the parallelogram structure can also be maintained by replacing links with gears or belt or chain drives. Figure 3a shows how the link DC of the parallelogram can be replaced with a belt or chain drive with a transmission ratio of 1 for which the motions of links AD and BC remain equal. It is also possible to replace link DC with a set of gears with a transmission ratio of 1 as shown in Fig. 3b. The additional elements need to be mounted on the links and their mass must be included for balance. In addition, Fig. 3c shows how it is possible to replace links that are pivoted to the base with sliders that have a circular sliding trajectory. This is since any point in a link rotating about a fixed point in the base performs a circular trajectory. Here it is directly relevant for the links SC1 and SC2 in Fig. 1 which are connected in the fixed pivot in S and could be replaced with circular sliders in C1 and C2 .

4

Example of a 2-DoF Inherently Force Balanced Mechanism

Fig. 4. 4-DoF principal vector linkage of Fig. 1 with mass symmetric links in a specific pose with a base pivot in S.

To illustrate the potential of the modification techniques in the synthesis of a force balanced mechanism, an example is presented in this section. Starting from the principal vector linkage in Fig. 1, it can be redesigned as shown in Fig. 4. Here the links are considered mass symmetric (i.e. the link CoMs lay on the lines through the link joints) and the relative orientations are different.

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Fig. 5. 2-DoF inherently force balanced mechanism with end-effector in A4 obtained from Fig. 4 by introducing gears, a belt drive, shifting links and adding a second base pivot in A0 .

Then by implementing various modification techniques the 2-DoF inherently force balanced mechanism in Fig. 5 was obtained. By adding a second base pivot in A0 the 4-DoF motion in Fig. 4 is reduced to a 2-DoF motion indicated by θ1 and θ2 . Gear trains between P1 and A1 and between C1 and B2 replace links B1 P2 and SC2 , respectively. A belt drive between B2 and P3 has been introduced to replace link C2 B3 . Link B2 P3 has been shifted to the position K1 K2 . Due to the replacement of links SC2 and C2 B3 , link B2 C2 could be removed as no links remain connected in C2 . The two gears in B2 are fixed together with which they keep SC1 and B3 P3 parallel.

Fig. 6. Dynamic simulation to verify the force balance of the mechanism in Fig. 5, shown in the end pose with the initial pose in red dotted lines.

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m1 p1 = (m2 + m3 + m4 + m7 + m8 + m10 + m11 + mCenGear +2mExtGear + 2mP ulley )a1 + m6 p6 m2 p2 = −(m1 + mCenGear + 2mExtGear )a21 + (m3 + m4 + m10

(1)

+m11 + mP ulley )a23 − m5 p5 + m7 p7 m3 p3 = −(m1 + m2 + m5 + m7 + mCenGear + 2mExtGear )a32

(2)

+m4 a34 − m7 δ − m8 p8 − m9 p9 + m10 p10 m4 p4 = (m1 + m2 + m3 + m5 + m6 + m7 + m8 + m9 + 2mCenGear

(3)

+4mExtGear + 2mP ulley )a4 + m11 p11 + m12 p12

(4)

To verify the force balance after all the modifications, the mechanism in Fig. 5 was simulated with the values in Table 1 which were chosen and calculated from the four force balance conditions Eqs. (1-4) including all elements (see Girgenti, 2021). The dynamic simulation was performed with the Spacar software by applying torques of 0.3 Nm on θ1 and 2.0 Nm on θ2 for a time of 0.5 s with the initial pose and end pose as shown in Fig. 6. The position of the common CoM, initially in S and the origin of the reference frame, and the shaking forces in the base were measured and are shown in Fig. 7. Since the resulting values have a difference from 0 in the order of magnitude of the computational accuracy, they confirm that the mechanism is perfectly inherently force balanced. −12 Common

x 10

−12

CoM coordinates

1 x y

0.5

Reaction forces [N]

Coordinates [m]

1

0 −0.5 −1

0

0.1

0.2 0.3 Time[s]

0.4

0.5

x 10

Reaction Forces at Base fx fy

0.5 0 −0.5 −1

0

0.1

0.2 0.3 Time[s]

0.4

0.5

Fig. 7. The results of the simulations for (a) the X and Y coordinates of the common CoM and (b) the horizontal and vertical shaking forces in the base show that the linkage in Fig. 5 is perfectly force balanced.

5

Discussion and Conclusion

This paper presented various techniques for modifying a principal vector linkage while keeping the kinematic property of the parallelogram structure equal, which is essential for maintaining inherent force balance capability. The techniques consist of constraining the mobility of links - either absolutely with respect to the base as well as relatively with respect to other moving links, shifting links to other positions, and exchanging links with other machine elements such as sliders, gears, and belt or chain drives. The potential of these techniques was

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L. Girgenti and V. van der Wijk Table 1. Parameter values for simulation of the mechanism in Fig. 5. Mass values [g]

Principal dimensions [mm]

Mass parameters [mm]

m1 = 360

P1 A1 = a1 = 28

P1 S1 = p1 = 117.5222 RCenGear = 11

m2 = 270

A1 P2 = a21 = 90

P2 S2 = p2 = 77.1667

RExtGear = 3

m3 = 300

P2 A2 = a23 = 60

P3 S3 = p3 = 36.0333

RP ulley = 14

m4 = 450

A2 P3 = a32 = 40

P4 S4 = p4 = 67.6444

m5 = 165

P3 A3 = a34 = 100 B1 S5 = p5 = 45

m6 = 60

A3 P4 = a4 = 20

C1 S6 = p6 = 14

m7 = 105

K2 A2 = δ = 20

K1 S7 = p7 = 30

m8 = 105

A0 A1 = 200

B2 S8 = p8 = 30

m9 = 75

A1 A2 = 150

C1 S9 = p9 = 20

m10 = 180

A2 A3 = 140

B3 S10 = p10 = 50

m11 = 30

A3 A4 = 200

B3 S11 = p11 = 10

m12 = 30

Drive parameters [mm]

SS12 = p12 = 10

mCenGear = 9 mExtGear = 1 mP ulley = 15

illustrated by the presentation of a synthesized 2-DoF inherently force balanced mechanism solution incorporating various techniques. By simulation the force balance was verified including all modifications. While this paper considered solely inherent force balancing, the techniques can also be applied for inherent moment balancing, for which counter-rotating gears are useful and constraining link motions showed already a basic approach (Van der Wijk, 2014). Constraining link motions is generally also needed to obtain specific motions for an intended task, whereas modifications with different machine elements is useful to meet certain space requirements in the design. As can be observed from the synthesized mechanism, the more modifications are made, the more the result will appear different from the initial principal vector linkage. Therefore the modifications enhance the use of inherent balancing for a variety of applications where base stability or low base vibrations are important such as high-speed manipulators in the precision industry and manipulators mounted underneath drones or on moving platforms such as cable-driven robots and mobile robots.

References Arakelian, V.G., Smith, M.R.: Complete shaking force and shaking moment balancing of linkages. Mech. Mach. Theory 34, 1141–1153 (1999)

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Arakelian, V.H., Smith, M.R.: Shaking force and shaking moment balancing of mechanisms: a historical review with new examples. ASME J. Mech. Des. 127, 334–339 (2005) Arakelian, V.H., Smith, M.R.: Erratum: Shaking force and shaking moment balancing of mechanisms: a historical review with new examples. ASME J. Mech. Des. 127, 1035 (2005) Feng, G.: Complete shaking force and shaking moment balancing of 17 types of eightbar linkages only with revolute pairs. Mech. Mach. Theory 26(2), 197–206 (1991) Fischer, O.: Theoretische grundlagen f¨ ur eine Mechanik der lebenden K¨ orper. Teubner, Leipzig (1906) Girgenti, L.: Techniques to synthesize Inherently Force Balanced Mechanisms. MSc thesis, Delft (2021) Gosselin, C.M., Vollmer, F., Cˆ ot´e, G., Wu, Y.: Synthesis and design of reactionless three-degree-of-freedom parallel mechanisms. IEEE Tr. on Robot. Auto. 20(2), 191– 199 (2004) Kochev, I.S.: General theory of complete shaking moment balancing of planar linkages: a critical review. Mech. Mach. Theory 35, 1501–1514 (2000) Lowen, G.G., Berkof, R.S.: Survey of investigations into the balancing of linkages. Mechanisms 3, 221–231 (1968) Van der Wijk, V.: Methodology for analysis and synthesis of inherently force and moment-balanced mechanisms - theory and applications (dissertation). University of Twente (2014) Van der Wijk, V.: The grand 4R four-bar based inherently balanced linkage architecture for synthesis of shaking force balanced and gravity force balanced mechanisms. MMT 150, 103815 (2020) van der Wijk, V., Herder, J.L., Demeulenaere, B.: Comparison of various dynamic balancing principles regarding additional mass and additional inertia. ASME J. Mech. Rob. 1(4), 041006 (2009)

A Preliminary Study of Factors Influencing the Stiffness of Aerial Cable Towed Systems Vincenzo Di Paola1,2 , Edoardo Id` a3 , Matteo Zoppi1 , and St´ephane Caro4(B)

4

1 University of Genova - DIME, Genova, Italy [email protected], [email protected] 2 ´ Ecole Centrale de Nantes - LS2N, Nantes, France 3 University of Bologna - DIN, Bologna, Italy [email protected] CNRS, Laboratoire des Sciences du Num´erique de Nantes, Nantes, France [email protected]

Abstract. There is a growing attraction for robotic aerial systems in the academic world and in industry. Aerial Cable Towed Systems (ACTSs) are naturally subjected to various unknown external actions (wind, human, etc.). Accordingly, it is crucial to characterize their stiffness both for balancing external disturbances and for their design. Thus, in this paper, an ACTS with a point-mass end-effector, subjected to a generic force, is considered. The effect of the number of cables and the cable arrangement on the stiffness of the ACTSs is investigated. It turns out that the cable arrangement is the factor that enhances most the stiffness of the ACTSs.

1

Introduction

The last few decades have witnessed great advances in the domain of aerial robotics. These robots gained attention because of their potential applications. Initially, they were employed for mapping, surveillance and monitoring operations. Subsequently, interest in infrastructure maintenance operations, searchand-rescue (S&R), etc. led scientists to spend effort on the control and design of Aerial Cable Towed Systems (ACTSs) (Villa et al. 2018). Many of the abovementioned operations involve coping with external forces (i.e. physical interaction) (Tognon and Franchi 2020). Therefore, the study of the stiffness of these systems plays a central role. Indeed, the ACTSs can take infinite configurations and the strict relationship between the available wrench, the stiffness and their configuration is the reason behind this work. Recently, the first studies concerning the available wrench set to evaluate the robustness of the equilibrium of these robots were conducted in Erskine et al. (2019a). Despite the differences between ACTSs and Cable-Driven Parallel Robots (CDPRs), the wrench feasibility analysis of ACTS described in Erskine et al. (2019a) was based on the approach introduced in Bouchard et al. (2010) for the wrench feasibility of CDPRs. The importance of the wrench analysis lies in being able to assess the feasibility of a task. Hence, the c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 272–282, 2022. https://doi.org/10.1007/978-3-031-06409-8_29

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so-called capacity margin was defined in Ruiz et al. (2015). This capacity margin gave rise to a series of contributions in which the ACTS should accomplish a task while re-configuring. The optimal configuration was chosen with respect to the capacity margin (Erskine et al. 2019b) (Ya et al. 2021). So far, presented works assess the task feasibility by considering only the capacity margin (available wrench). However, the cables introduce compliance into the system and the overall stiffness should also be considered for optimal results. This paper aims to analyze the effect of the cable arrangement, the number of cables and the configuration parameters of the ACTS on its stiffness. The proposed approach focuses on the use of a directional stiffness index to quantify and compare the effects of cable arrangement and number. The paper is organised as follows. Section 2 recalls the static model of an ACTS. Section 3 provides some definitions of the stiffness and directional stiffness of an ACTS. Section 4 describes the methodology used to evaluate the effect of the cable arrangement and the number of cables on the directional stiffness of an ACTS. Conclusions and future work are drawn in Sect. 5.

2

System Modeling

Let’s consider a generic ACTS with n cables and a point-mass load as shown in Fig. 1. The symbols used to describe its configuration are summarised in Table 1. The static equilibrium for the point-mass load is given by: Wt + we = 0

(1)

where t is the cable tension vector containing each cable tension ti , we is the external wrench and W is the so-called wrench matrix, which can be expressed as:   (2) W = q1 q2 . . . qn

3

Stiffness Modeling

Based on previous CDPR stiffness modeling (Moradi 2013) (Behzadipour 2006), the stiffness of an ACTS can be defined as: K=−

δwe δxL

(3)

Table 1. Nomenclature. Symbols Physical meaning

Symbols Physical meaning

FO , FQi Inertial and ith drone frames

qi ∈ R3

O, Qi

Origin of inertial and ith drone frames fi ∈ R

m∈R

Mass of each drone

xL ∈ R3 Position of the load in FO

Unit vector of the ith cable in FO Thrust force of the ith drone

li ∈ R

ith cable length

g∈R

Gravity constant

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Fig. 1. Scheme of a generic ACTS with n cables and a suspended point-mass load.

where a small variation in the external wrench δwe generates a small displacement of the load δxL . Explicitly, substituting (1) into (3) gives:  δW δ  δt δW δt δl δW δt Wt = t+W = t+W = t + W WT (4) δxL δxL δxL δxL δl δxL δxL δl   where l = l1 l2 . . . ln . Defining δt δl = diag[k1 , k2 , . . . , kn ] leads n to the so-called passive stiffness matrix Kp = Wdiag[k1 , k2 , . . . , kn ]WT = i=1 ki qi qTi that is δW t= a function of cable arrangement and cable elasticity ki whereas Ka = δx L n ti T i=1 li (I3 − qi qi ) is the active stiffness matrix and depends on cable tensions ti and cable lengths li , i = 1, . . . , n. Thus, the stiffness matrix K is: K=

K=

n  ti i=1

li

(I3 − qi qTi ) +

n 

ki qi qTi = Ka + Kp

(5)

i=1

where I3 is the identity matrix of dimension 3. Remark 3.1. (Effects of ki , li and ti on the ACTS stiffness). Equation (5) states that K is directly proportional to ti and ki and inversely proportional to li . Therefore, increasing ti and ki leads to an increase in K and vice-versa for li . The tension values can not be chosen freely. They must have lower and upper bounds and depend on the ACTS configuration. Cable lengths are practically limited to guarantee quick changes in the cable directions during the operation of the ACTSs. Note that in series with the cable stiffness ki , the drone also contribute to the stiffness similarly to the winches of the CDPRs. The contribution of the drone stiffness will be the subject of future work.

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It is not always necessary to maximize the overall stiffness of a mechanism. For instance, for extinguishing a building fire as described in Khajepour (2020), the ACTS should remain almost static in front of the fire while water flows out from the nozzle suspended by the quadrotors and cables. Hence, depending on the task at hand, maximizing the stiffness in one direction may be sufficient. In this perspective, the directional stiffness index can be defined as (Moradi, 2013): kds =

||we || we −1 ||(K we )T ||w || e ||

(6)

which evaluates the stiffness along the direction of the external wrench we . The solution of Eq. (1) in terms of tensions can be written in a general form as: t = −W† we + Nλ †

T −1

(7)

where W = W (WW ) is the pseudo-inverse matrix of W, N ∈ R contains the vectors that span the kernel of W and λ ∈ R(n−3) . Let F be the n-dimensional box of feasible tensions: T

n×(n−3)

F := {t ∈ Rn | tmin,i ≤ ti ≤ tmax,i , i = 1, ..., n}

(8)

with tension limits tmin,i and tmax,i depending on the drone thrust limits fmin and fmax and their attitude as derived in Erskine et al. (2019a):  2 2 − 1) tmin,i = mgqz,i + fmin + m2 g 2 (qz,i (9)  tmax,i = mgqz,i +

2 − 1) 2 fmax + m2 g 2 (qz,i

(10)

Since the feasible tensions belong to a convex set due to the tension limits tmin,i and tmax,i , the lambda space L can be defined as: L := {λ ∈ R(n−3) | tmin ≤ −W† we + Nλ ≤ tmax }

(11)

which, in turn, is convex. Assume we want to minimize the displacement (i.e. maximize the stiffness) of the load δxL = K−1 δwe along the y direction, namely δxL,y . Then, the optimal λ vector can be obtained by solving the following optimization problem: min λ∈L

s.t.

|δxL,y | tmin,i ≤ ti ≤ tmax,i

(12) i = 1, ..., n

Thus, as long as dim(L) ≥ 1 (i.e. n > 3), it is possible to search for a set of cable tensions such that the displacement along a given direction is the smallest one. There exist different methods to solve this problem: the Stiffness Oriented Tension Distribution Algorithm (SOTDA) developed in Picard et al. (2021) is the one used here.

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Effect of the Number of Cables and Their Arrangement on the ACTS Stiffness

This section attempts to quantify the influence of the number of cables and their arrangement on the stiffness of the ACTSs. However, since the arrangement and number of cables are coupled inside K, it is necessary to develop a methodology that distinguishes their effects before being able to quantify them. Equation (5) can be rewritten as follows: K=

3 

ki qi qTi +

i=1

+

n 

3  ti i=1

kj

j>3

where: qj =

3 

li

(I3 − qi qTi )

a2ij qi qTi +

i=1 3 

n  tj j>3

lj

(I3 −

3 

(13) a2ij qi qTi )

i=1

aij ∈ R, j = 4, . . . , n

aij qi

(14)

i=1

is expressed as linear combination of the vectors qi which, in turn, generate the base B (Lang, 2002): B := {qi | i = 1, 2, 3}

 = 1, . . . , ∞

(15)

where subscript  identify one base among infinity. Table 2. ACTS parameters. Symbols

Value

Symbols Value

m

15 kg

fmin,i

150 N

l

2m

fmax,i

450 N

dcable (diameter)

0.7 mm we

(0, –10, 0)T N

Ecable (Young modulus) 4 GPa

Equation (13) provides some insight regarding the dependency of the matrix K on the system parameters described in Remarks 3.1. Assume that k1 = · · · = kn = k, l1 = · · · = ln = l and that each cable direction and basis B are randomly generated by using a continuous uniform distribution U (Weisstein, 1999). In particular, each component of qi is generated as U([0, 1]). This makes the procedure general and systematic allowing each cable to take any direction in space. Hence, the variables that play an important role in the stiffness are, B , aij , ti , k, l. Moreover: • assign B and assume n > 3, then according Behzadipour (2006): ∀ aij ∈ R s.t. ti > 0 ∀ i ⇒ qi qTi ≥ 0 ⇒ the larger n, the higher kds

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• assign n ≥ 3 and assume the existence of a finite number of possible configurations: it becomes possible to estimate the mean k˜ds and standard deviation kˆds of kds as (Weisstein, 1999):

kˆds

kds,max + kds,min k˜ds = 2 (kds,max + kds,min )2 nc + 1 = 12 nc − 1

(16)

(17)

where kds,max and kds,min represent the extreme values of the interval I = [kds,min , kds,max ] to which kds belongs, whereas I is discretized into nc elements. To assess the effects of number of cables and cable arrangement, some parameters are set and given in Table 2. The directional stiffness index were computed neval = 105 times using the fixed set of data. Results are collected in Tables 3 and 4 and depicted in Figs. 2. The mean value k˜ds is representative of the directional stiffness of all the configurations; independently from the cable arrangements. In parallel, the standard variation kˆds , is an indicator of the influence of cable directions: it underlines the influence of changing B . These properties are inherent in the stiffness of the system modelled by equation (13): this is supported by the strong correlation between the results obtained from arbitrary simulations; as depicted in Figs. 2. Table 3. 1st Simulation data. n

˜ds k

3 4

ˆds k

Table 4. 2nd Simulation data. n

˜ds k

ˆds k

304 N/m 1239 N/m

3

310 N/m 2436 N/m

301 N/m

208 N/m

4

301 N/m

204 N/m

5

393 N/m

245 N/m

5

395 N/m

245 N/m

6

497 N/m

281 N/m

6

499 N/m

282 N/m

7

601 N/m

311 N/m

7

602 N/m

311 N/m

8

710 N/m

340 N/m

8

708 N/m

338 N/m

9

815 N/m

362 N/m

9

813 N/m

363 N/m

10 923 N/m

387 N/m

10 924 N/m

386 N/m

Collected data shown that kˆds drop down rapidly in the transition between n = 3 and n = 4. Subsequent values of kˆds (from n ≥ 5) decrease as the number of cables rises. Meanwhile, for k˜ds , adding one cable leads to an increase in stiffness. In other words, the k˜ds variations due to a change of cable number become almost constant for n ≥ 5.

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Thus, from n ≥ 5 the net difference between the effects of the number and direction of the cables on the stiffness becomes apparent. These numerical results pave the way for a more general result: Theorem 4.1. Consider an ACTS with n cables and a point-mass end-effector. Then if n → ∞, kˆds becomes constant. Proof. Let’s pick two different bases B := {qi | i = 1, 2, 3} and B∗ := {q∗i | i = 1, 2, 3} and assume that k1 = · · · = k∞ = k, l1 = · · · = l∞ = l.

Fig. 2. Directional stiffness (mean and standard deviation) as a function of the number of ACTS configurations.

It is straightforward to see that, given an external wrench we ∈ R3 , t = −W† we and t∗ = −W∗† we are equal because both W† and W∗† contains all the direction vectors possible. Therefore, the new stiffness matrix K∗ is: ∗

K =

3 

ki q∗i q∗T i

+

i=1

3  ti i=1

li

q∗i q∗T i )

(I3 −

+

n 

kj

j>3

+

n  tj j>3

lj

3 

∗ ∗T a∗2 ij qi qi

i=1

(I3 −

3 

(18)

∗ ∗T a∗2 ij qi qi )

i=1

a∗ij

∈ R. with As before, using the concept of linear combination yields: q∗i =

3 

b k qk

bk ∈ R, qk ∈ B

(19)

k=1

which substituted in (18), leads to: K∗ =

3 

ki

i=1

+

n 

3 

bik qk qTk +

i=1

k=1

kj

j>3

with bik ∈ R.

3  i=1

3  ti

a∗2 ij

3  k=1

li

(I3 −

bik qk qTk +

3 

bik qk qTk )

k=1 n  tj j>3

lj

(I3 −

3  i=1

a∗2 ij

3  k=1

(20) bik qk qTk )

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Subsequently, recalling that kˆds = f (B, aij ) and using the arbitrariness of the coefficients of the linear combination, gives us the results.

Remark 4.2. (Numerical trend of ΔIˆkds ). To shed light on Theorem 4 the trend of ΔIˆkds = Iˆkds ,s+1 − Iˆkds ,s is depicted in Fig. 3. Each step-s implies an increase of 10 cables in the configuration. It can be seen that as the number of cables n increases, ΔIˆkds decreases. Consequently when n → ∞ then ΔIˆkds → 0 and kˆds = const. Corollary 4.3. The arbitrariness of the coefficients of the linear combination leads to K = K∗ . Proof. Once again, using equation (20), it is straightforward to see that there exists a set of coefficients for the linear combination (19) such that the two matrices K = K∗ are identical.

Corollary 4.4. The arbitrariness of the coefficients of the linear combination ∗ leads to kˆds = kˆds . Proof. Relying on Theorem 4, Corollary 4.3 and Remark 4.2, when n → ∞ and ΔIˆkds → 0 then all the kˆds are constant and then equal one another kˆds = ∗ kˆds .

Remark 4.5. (Empirical Design Rule). The results of Corollary 4.4 involves design aspects of the ACTSs. It points out that for a high number of cables the effect of the configuration vanishes. Technically speaking, the limit on the number of cables can be set at 10. Indeed, Tables 2 and 3 show that by varying the number of cables in this range, ΔIˆkds ≈ 20 N/m < k˜ds ≈ 100 N/m. In order to fully consider the influence of cable directions on stiffness, it is necessary to evaluate the lambda space L: it provides the possibility of picking a set of tensions that increases stiffness. The shape of L and thus the tensions applicable to the cables depend on the configuration. Hereby, we compare the directional stiffness of a classical ACTS with a hybrid one, both with n = 5. The classic ACTS is characterized by cables whose directions point upwards. In the hybrid, the 4th and 5th cables are attached to the ground as in Jamshidifar and Khajepour (2020), whereas the other three cables have the same directions as in the classical one shown in Fig. 4. In this example the directional stiffness index and the load-displacement δxL,y are used for comparison. Figure 5 depicts the load-displacement within the available wrench set W for the classical and hybrid ACTSs. In particular, the available wrench set is projected onto the cartesian planes and discretized, then the load-displacement δxL,y is computed for all the points of the grid. The wrench sets generated are different for the two ACTSs: the relevant parameters that distinguish the two ACTS are summarized in Table 5. For the scope of this section, the important aspect of this analysis is that kds,Classical = 467 N/m and kds,Hybrid = 470 N/m are in accordance with the data collected in the Table 3 for n = 5. Indeed, the generated base B is common for both the

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Fig. 3. Decreasing trend of ΔIˆkds due to the increase in the number of cables. Table 5. Features of classical and hybrid ACTS. Wxy ACTS

Wyz

Wxz

VolumeW δxL,ymin ΔδxL,y δxL,ymin ΔδxL,y δxL,ymin ΔδxL,y

Classical 1.2 e8

0.021

0.0004

0.012

0.008

0.012

0.008

3.0 e7

0.018

0.005

0.018

0.006

0.018

0.004

Hybrid

Δ(·) defines the difference between the maximum and the minimum δxL,y values.

ACTS and the directional stiffnesses are similar between them and w.r.t. the mean value of k˜ds for n = 5. However, as the tensions depend on the configuration, the δxL,y can be recomputed and eventually minimised using the SOTDA procedure. Figure 6 depicts the lambda space L for the hybrid ACTSs only. Indeed, the lambda space L of the classical ACTS degenerates in a point (i.e. λ = 0). This crucial difference is due to the change in the cable arrangement. As a consequence, the proper choice of λ can guarantee a reduction of the displacement up to ≈ 2 times. Indeed, the new kds,Hybrid = 1083 N/m: it is higher than the maximum value gived in Table 3 because here the SOTDA is exploited.

Fig. 4. Classical and hybrid configurations of an ACTS with n = 5.

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Fig. 5. Estimated load-displacement δxL,y within the 2D available wrench set for the classical (a) and hybrid (b) ACTS. The load is subjected to a wrench we = (0, −10, 0)T and its position is xL = (0, 0, 1)T .

Fig. 6. Estimated load-displacement δxL,y over the lambda space L for the hybrid ACTS.

5

Conclusion and Future Works

A preliminary study of the factors influencing the stiffness of ACTS was conducted in this paper. The results show that the cable arrangement of an ACTS has a major impact on its stiffness. Indeed, the directional stiffness index is twice as important with the hybrid configuration of the ACTS than with the suspended configuration. Future work aims to generalize these results for systems with rigid

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body end-effector and to carry out the design of a controller capable of guiding the system to achieve the maximum index of directional stiffness. The theoretical results will be validated experimentally.

References Behzadipour, S.: Stiffness of cable-based parallel manipulators with application to stability analysis. J. Mech. Des. 128(1), 303–310 (2006) Bouchard, S., Gosselin, C., Moore, B.: On the ability of a cable-driven robot to generate a prescribed set of wrenches. J. Mech. Rob. 2, 011010 (2010) Erskine, J., Chriette, A., Caro, S.: Wrench analysis of cable-suspended parallel robots actuated by quadrotor unmanned aerial vehicles. J. Mech. Rob. 11, 020909 (2019) Erskine, J., Chriette, A., Caro, S.: Control and configuration planning of an aerial cable towed system. In: Proceedings - IEEE International Conference on Robotics and Automation (2019) Jamshidifar, H., Khajepour, A.: Static workspace optimization of aerial cable towed robots with land-fixed winches. IEEE Trans. Rob. 36, 1603–1610 (2020) Lang, S.: Algebra. Springer Graduate Texts in Mathematics (2002). https://doi.org/ 10.1007/978-1-4613-0041-0 Moradi, A.: Stiffness analysis of cable-driven parallel robots. Ph.D. thesis at Queen’s University Kingston, Ontario, Canada (2013) Picard, E., Caro, S., Plestan, F., Claveau, F.: Stiffness oriented tension distribution algorithm for cable-driven parallel robots. In: Lenarˇciˇc, J., Siciliano, B. (eds.) ARK 2020. SPAR, vol. 15, pp. 209–217. Springer, Cham (2021). https://doi.org/10.1007/ 978-3-030-50975-0 26 Ruiz, A.L.C., Caro, S., Cardou, P., Guay, F.: Arachnis: analysis of robots actuated by cables with handy and neat interface software. In: 2nd International Conference on Cable-Driven Parallel Robots (2015) Tognon, M., Franchi, A.: Theory and Applications for Control of Aerial Robots in Physical Interaction Through Tethers. STAR, vol. 140. Springer, Cham (2021). https:// doi.org/10.1007/978-3-030-48659-4 Villa, D.K.D., Brand˜ ao, A. S., Sarcinelli-Filho, M.: Load transportation using quadrotors: a survey of experimental results. In: International Conference on Unmanned Aircraft Systems, ICUAS (2018) Eric Weisstein, C.: Discrete uniform distribution. MathWorld, A Wolfram Web Resource (1999). https://mathworld.wolfram.com/DiscreteUniformDistribution. html Ya, L., Fan, Z., Panfeng, H., Xiaozhen, Z.: Analysis, planning and control for cooperative transportation of tethered multi-rotor UAVS. Aerosp. Sci. Technol. 113, 106673 (2021)

Design Optimization of a 6-DOF Cable-Driven Parallel Robot for Complex Pick-and-Place Tasks Luca Guagliumi1 , Alessandro Berti2 , Eros Monti2 , and Marco Carricato1(B) 1

Department of Industrial Engineering, University of Bologna, Bologna, Italy {luca.guagliumi3,marco.carricato}@unibo.it 2 Marchesini Group S.p.A., Pianoro, Bologna, Italy

Abstract. In this paper, an overconstrained cable-driven parallel robot with 8 cables and 6 DOFs is used for the execution of 3D-picking tasks with high dynamics. The main objective of the manipulator is the feeding of an automatic packaging machine by picking low-weight products from a pile. In particular, here, we study the optimal geometric design of the robot platform based on its task and using a genetic algorithm.

1

Introduction

Parallel robots are widely used for the execution of high-dynamics tasks. In Cable-Driven Parallel Robots (CDPRs), wires replace the rigid links that form the legs of standard parallel manipulators (Pott 2018). Using cables gives CDPRs a bigger workspace and lower moving masses compared to other robots; moreover, it is easy to design robots with simple reconfigurability. On the contrary, cables can only provide tensile forces, and it is necessary to use at least m = n+1 actuated cables to fully control the n Degrees Of Freedom (DOFs) of the endeffector (EE). A typical industrial problem characterized by high accelerations and velocities is pick-and-place. For classical pick-and-place tasks (i.e. pure translations in 3D space), robots with 3 DOFs are sufficient, but for more complex 3D-picking (or bin-picking) operations, it is necessary to control more DOFs of the EE. As a matter of fact, in this situation, the products moved by the manipulator can have a general 6-DOF pose in space. Moreover, the robot must not interfere with the vision system necessary to control the EE movements. For these reasons, serial manipulators are commonly used to perform these tasks, even if this implies a reduction of maximum velocities and accelerations. We propose to develop a CDPR for the execution of high dynamics 3Dpicking applications. In particular, the robot target is to feed an automatic packaging machine by picking light-weight products from a pile. A CDPR seems to be appropriate for this task since it can guarantee a light-weight structure, reconfigurability, fast movements, and a small footprint in the pick-up area, where the vision system acquires images from above. CDPRs have already been c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 283–291, 2022. https://doi.org/10.1007/978-3-031-06409-8_30

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proposed for classical pick-and-place operations (Zhang et al. 2020b), but not yet for 3D-picking tasks. This paper focuses on the optimal geometric design of the robot by taking into account the special task it has to accomplish. The main goal is the identification of the cable attachment points to the EE (the attachments to the frame are almost completely determined based on the robot maximum sizes and the overall dimensions of its mechanical parts). These are one of the main parameters that influence the robot behavior and its workspace (Ennaiem et al. 2020 and Kumar et al. 2018). We require an easy reconfiguration of the manipulator through a simple modification of the optimization problem input data. The topic of CDPR optimal design was analyzed in the literature for different fields compared to the one studied here and with different approaches; for the sake of brevity, only some examples are reported: – – – – – –

2

warehousing (Zhang et al., 2020a and Reichert and Bruckmann, 2019); rehabilitation (Ben Hamida et al., 2021 and Lamine et al., 2017); construction and maintenance of building facades (Hussein et al., 2018); load lifting (Pusey et al., 2004 and Gouttefarde et al., 2015a); classical 3-DOF pick-and-place (Zhang et al., 2020b); generic CDPRs without a specific task (Ouyang and Shang, 2014 and Nasr and Moosavian, 2015).

Design, Kinematics and Statics

To perform high-dynamics tasks it is necessary to use an overconstrained robot architecture. To guarantee high stiffness and mobility in every direction, the moving platform must be connected to the frame with cables coming from both the ceiling and the bottom of the workspace. Moreover, 6 DOFs are necessary to perform 3D-picking tasks. For this reason, a design with m = 8 cables was chosen (see Fig. 1). Dyneema wires with a 3mm diameter (d) were selected, since the breaking load of these cables is high. The cables must be guided by pulleys from the actuation system into the workspace, to reduce their wear and friction phenomena. The last pulleys before the moving platform must have a pivoting axis such that they can follow the EE during its motion. The main objective of this work is to define the attachment points of the cables to the moving platform (points Bi ). We must take into account the presence of the swivel pulleys on the robot frame in the kinematic model. From Fig. 2, we can obtain, for the i-th cable, the following closure equation: ρi = a0,i + ac,i + at,i − Rbi − p

(1)

where p is a position vector and R is the rotation matrix describing the EE orientation. Reference (Id` a et al., 2019) shows a complete description of the kinematic model of a robot with swivel pulleys. A complex topic related to overconstrained architectures is the cable-tension computation. The equilibrium of the EE in space is described by 6 equations.

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285

Fig. 1. Robot model in the initial guess configuration. The dashed lines represent the task workspace.

Fig. 2. Swivel-pulley kinematics

Having 8 > 6 cables attached to the EE means that there are ∞2 solutions to the inverse static problem: (2) Js t + w = 0 where Js is the structure matrix that depends on the EE pose (p, R), t = [t1 . . . t8 ]T is the cable tensions array, and w is the external wrench applied to the EE. Not all the ∞2 solutions are acceptable, since only the sets with all positive components are feasible. The set of correct solutions can be further reduced by accepting only the ones in which all components are included between a minimum (tmin ) and a maximum (tmax ) value. These values are established on the necessity of maintaining the cables always in tension to obtain a stiff and precise manipulator, but without exceeding the cable breaking load or the motor maximum torque. In the following, we assume tmin = 30N and tmax = 350N . To find the solution of Eq. (2) we use the procedure explained in (Gouttefarde et al., 2015b) to obtain the 2−norm minimum tension distribution. Based on the previous considerations, one can define the Total Orientation Wrench Feasible Workspace (WT O ) as the set of positions in which Eq. (2)

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is verified under the assumptions tmin < ti < tmax , for i = 1 . . . 8, for all the orientations belonging to a given set RO . Moreover, the cables must not interfere with each other. Accordingly, a position p belongs to WT O if the following two conditions hold: Js t + w = 0 with tmin < ti < tmax for i = 1 . . . m ∀ R ∈ RO djk > d for j = 1 . . . m \ {k} ,

k = 1 . . . m \ {j}

(3) (4)

where, djk represents the distance between cables j and k supposed as straight lines.

3

Design Optimization

The maximum dimensions of the robot frame define its geometrical design. Here, we consider a parallelepiped of 1200 × 2200 x 1600 mm. The positions of the swivel pulleys are chosen based on the mechanical dimensions of the robot components and the environment (bins, conveyor belt, etc.). The only free parameter is the vertical assembly coordinate y of the pulleys at the top (u8 in Fig. 1). On the contrary, we can freely choose the design of the manipulator EE. By using the coordinates system of the mobile platform O − x y  z  (see Fig. 3), the main goal is to maximize WT O by neglecting the rotation of the EE around the y  direction. This is the less important DOF for picking products from a pile: if we imagine to approximate the pile with a pyramid, the rotations around x and z  are necessary to bring the EE in the correct pose to take products from the faces of the pyramid. In practice, we want to maximize the semi-aperture angle θ of the cone that the EE can describe in every position of the task workspace (see Fig. 3). The whole cone is described by varying the angle γ from 0o to 360o . We assume that the cable are arranged on the platform as shown in Fig. 1. This

Fig. 3. Platform geometrical parameters in the initial guess configuration and cone described by the EE to define WT O .

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seems to be a good choice, as explained in (Kamali et al. 2018), also because, with more complex arrangements, it is difficult to avoid cables intersections. The platform geometry is completely defined by the dimensions shown in Fig. 3, which determine the cable attachments points on the EE. These, together with the vertical mounting dimension of the pulleys at the top of the workspace (see Fig. 1), form the vector u = [u1 u2 u3 u4 u5 u6 u7 u8 ] of the unknowns of the optimization problem. The fitness function is built with a set of points (Pk with k = 1 . . . r) defined in the task workspace. We associate to every point a weight (ck ) based on the robot task: the most important points (with a higher weight) are the ones in the pick-up area (where the bins are placed in Fig. 1), since here it is more important to have a bigger θ. The exact numerical values of the weights ck are empirically designed after some running tests of the optimization problem and by evaluating the workspace of the manipulator optimized configurations: they are a free parameter to be chosen by the designer. The fitness function f (u) is computed as follows: f (u) =

r 

(90 − θmax,k (u)) · ck

(5)

k=1

where θmax,k is an approximate estimation of the maximum semi-aperture angle θ (in degrees) of the cone that the EE can describe in Pk such that Pk belongs to the robot workspace for every orientation in which θ ≤ θmax,k . In practice, Eq. (5) represents the weighted sum of angles complementary to the maximum θ in every point Pk ; the complementary value of θmax,k is used only for simplicity since it is common to deal with fitness functions to be minimized (not to be maximized). To apply Eq. (5), the mass and moment of inertia of the platform are estimated for the current set of parameters (u); then, the following operations are executed for every point Pk : – the EE is placed in Pk ; – if conditions (3) and (4) are not verified with R = I3 (i.e. Pk do not belong to the pure translational workspace of the robot), we impose θmax,k = −90o and ck = maxk (ck ); in this situation, the k−th term of the summation in Eq. (5) is equal to 180o · maxk (ck ), which is a very high value compared to the terms corresponding to points that belong to the pure translational workspace, even if the point is defined with a low weight; as a matter of fact, it is important to obtain a platform that guarantees to reach all points Pk at least with R = I3 ; – otherwise, we can approximately estimate the maximum cone angle reachable by the platform in Pk by repeating the check on conditions (3) and (4) for incremental values of θ; for each θ, such conditions are verified for sampled values of γ from 0o to 360o ; when one of the two conditions is no longer valid, we assume as θmax,k the maximum value of θ for which they are valid for all sampled values of γ. To obtain an optimized solution, we minimize the fitness function (5) by using a Genetic Algorithm (GA). A GA can work even if the fitness function is

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u2

u3

u4

u5

u6

u7

Lower bound 45

45

30

30

45

45

60

u8 1150

Upper bound 100 100 150 150 100 100 200

2000

Initial guess

60

1450

uopt

88.3 88.3 31.5 31.5 54.6 70.8 127.8 1640

60

50

50

60

60

100

highly non-linear and discontinuous, when gradient-based algorithms fail; GAs were already used for CDPR optimization (Ben Hamida et al., 2021, Reichert and Bruckmann, 2019 and Nasr and Moosavian, 2015). For our study, we used the GA defined in MATLAB through the function ga. Since the GA works by creating an initial random population and by making random changes to some individuals at every new population, we execute the same algorithm many times with the same input parameters. This is because the results of the random choices made by the algorithm are different sets of optimal parameters (even if all the optimal solutions are similar to each other). Moreover, we test different input parameters (e.g. different sets of input points and their weights). The parameters settled on Matlab to manage the GA are the following: – maximum number of generations (500); – function tolerance (10−3 ) and maximum number of stall generations (15) that together define the stopping criteria of the optimization; – crossover fraction (0.7), which defines the percentage value of individuals created for the next generation by crossover; – initial population Matrix (Initial Guess in Table 1), which defines the first individual for a constrained minimization problem; – lower and upper bound (shown in Table 1) which constraint the searching area. In the following, we consider only the best solution obtained with the GA.

4

Results

The optimized parameters (uopt ) obtained through the GA are shown in Table 1. In particular, it is interesting to notice that an asymmetry is introduced in the EE, since u5 = u6 . As a matter of fact, the task itself is not symmetric along the z-coordinate because of the presence of the bins (pick-up area). Here, the total orientation of the EE is more important, to allow a correct grip of the products. For this reason, a higher weight is associated with the points Pk in this area. If the task is different (e.g. different location of bins or conveyor belt), by simply changing the position and the weights of the points given in input to the fitness function, one can obtain another moving platform with a different symmetry. We find the manipulator workspace through a discretization method (Pott, 2018), by sampling the task workspace shown in Fig. 1 with a dense grid of points.

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Fig. 4. Discretized points belonging to the WT O for θ = 35o in the optimized configuration. On the right, the top view of the task workspace is shown: from here it is clear the asymmetric behavior of the robot.

We estimate the workspace volume through the number of sampled points that belong to WT O (Pusey et al., 2004 and Ouyang and Shang, 2014). We give the result as a percentage value of the total number of points. We repeat the procedure for different values of the cone angle θ. The case θ = 0o represents the pure translational workspace. The results are shown in Table 2 for the initial guess and the optimized solution. In Fig. 4 the workspace of the optimized configuration for θ = 35o is shown. We can notice that the workspace is asymmetric due to the asymmetric EE. This reflects the robot task and the fitness function designed on it. Since the aim of the optimization is only to maximize WT O , there is no control on the tensions required to maintain the platform in equilibrium. The robot geometry influences its statics, and so in many works related to CDPR optimization, one of the main objectives is to minimize the maximum cable tensions. This is important to reduce the motor size. To verify the acceptability of the proposed solution, a pick-and-place trajectory with very high accelerations (with maximum value approximately equal to 280m/s2 ) is simulated (see Fig. 5). We compute the evolution in time of the cable tensions for both the initial guess and the optimized solution. We can notice higher forces required for the optimized solution, but the increase is small, and for most of the motion the power required to motors is lower than the nominal one of the servomotor chosen (BECKHOFF AM8041). Finally, we evaluate the maximum cable-tension average value in the pure translational workspace of the robot. This parameter reaches 124.17N for the initial guess solution and 111.07N for the optimized one. These values validate the usability of the optimized platform to execute the task.

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L. Guagliumi et al. Table 2. Percentage volumes of WT O compared to the task workspace θ

0o

10o

20o

30o

35o

Initial guess 93.1% 80.2% 67.6% 39.0% 6.5% uopt

40o 0.0%

96.8% 90.5% 79.8% 60.2% 45.4% 19.1%

Fig. 5. Optimized configuration and trajectory example. On the right, it is shown the evolution in time of the cable tensions during the trajectory. The solid lines refer to the optimized geometry, the dashed ones to the initial guess.

Since it is difficult to foresee the orientations necessary to take products from every point of the pick-up area, in this first design stage the cable tensions are evaluated only for pure translational motions. In the future, the effective orientation capability of the platform will be evaluated in practice on a robot prototype.

5

Conclusions and Future Work

The main objective of this work is the optimization of a CDPR with 6 DOFs and 8 cables for 3D-picking operations. By taking into account the task (feeding of a conveyor belt with low-weight products picked from a pile), the frame design is almost completely defined. We thus focused on the end-effector geometry, found through a genetic algorithm to maximize the Total Orientation Wrench Feasible Workspace in the pick-up area. The optimal solution represents an EE asymmetric along the direction in which the task is not symmetric. This suggests that changes in the robot operations make it beneficial to execute a reconfiguration of the platform through a new task-based optimization. The method allows obtaining good results in terms of workspace without excessively increasing the maximum tensions required to robot actuators. In the future, we will develop a prototype of the robot.

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Acknowledgments. This work was supported by Marchesini Group S.p.A., a leading company in the field of automatic packaging machines, especially for pharmaceutical and cosmetic products.

References Ben Hamida, I., Laribi, M.A., Mlika, A., Romdhane, L., Zeghloul, S., Carbone, G.: Multi-objective optimal design of a cable driven parallel robot for rehabilitation tasks. Mech. Mach. Theory 156, 104141 (2021) Ennaiem, F., et al.: Sensitivity based selection of an optimal cable-driven parallel robot design for rehabilitation purposes. Robotics 10, 7 (2020) Gouttefarde, M., Collard, J.F., Riehl, N., Baradat, C.: Geometry selection of a redundantly actuated cable-suspended parallel robot. IEEE Trans. Rob. 31(2), 501–510 (2015) Gouttefarde, M., Lamaury, J., Reichert, C., Bruckmann, T.: A versatile tension distribution algorithm for n-DOF parallel robots driven by n+2 cables. IEEE Trans. Rob. 31(6), 1444–1457 (2015) Hussein, H., Santos, J.C., Gouttefarde, M.: Geometric optimization of a large scale cdpr operating on a building facade. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS, pp. 5117–5124 (2018) Id` a, E., Bruckmann, T., Carricato, M.: Rest-to-rest trajectory planning for underactuated cable-driven parallel robots. IEEE Trans. Rob. 35(6), 1338–1351 (2019) Kamali, K., Joubair, A., Bonev, I.A.: Optimizing cable arrangement in cable-driven parallel robots to improve the range of available wrenches. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 5B, 42nd Mechanisms and Robotics Conference (2018) Kumar, A.A., Antoine, J.F., Zattarin, P., Abba, G.: Influence of payload and platform dimensions on the static workspace of a 4-cable driven parallel robot. In: 2nd Robotix-Academy Conference for Industrial Robotics, RACIR, vol. 6, p. 103693. University of Luxembourg. Luxembourg (2018) Lamine, H., Laribi, M.A., Bennour, S., Romdhane, L., Zeghloul, S.: Design study of a cable-based gait training machine. J. Bionic Eng. 14(2), 232–244 (2017). https:// doi.org/10.1016/S1672-6529(16)60394-3 Nasr, A., Ali, S., Moosavian, A.: Multi-criteria design of 6-dof fully-constrained cable driven redundant parallel manipulator. In: 2015 3rd RSI International Conference on Robotics and Mechatronics, ICROM, pp. 001–006 (2015) Ouyang, B., Shang, W.: Wrench-feasible workspace based optimization of the fixed and moving platforms for cable-driven parallel manipulators. Robo. Comput. Int. Manufac. 30(6), 629–635 (2014) Pott, A.: Cable-Driven Parallel Robots: Theory and Application, May 2018 Pusey, J., Fattah, A., Agrawal, S., Messina, E.: Design and workspace analysis of a 6–6 cable-suspended parallel robot. Mech. Mach. Theory 39(7), 761–778 (2004) Reichert, C., Bruckmann, T.: Optimization of the Geometry of a Cable-Driven Storage and Retrieval System, pp. 225–237, July 2019 Zhang, F., Shang, W., Zhang, B., Cong, S.: Design optimization of redundantly actuated cable-driven parallel robots for automated warehouse system. IEEE Access 8, 56867–56879 (2020) Zhang, Z., Shao, Z., Wang, L.: Optimization and implementation of a high-speed 3DOFS translational cable-driven parallel robot. Mech. Mach. Theory 145, 103693 (2020)

Improved Resolution of a Forward Kinematic Model of Parallel Manipulators Using a Serial Approach Majdi Meskini1,2 , Houssem Saafi1,3 , Abdelfattah Mlika1 , Marc Arsicault2 , Saïd Zeghloul2 , and Med Amine Laribi2(B) 1 Mechanical Laboratory of Sousse (LMS), National Engineering School of Sousse,

University of Sousse, 4000 Sousse, Tunisia 2 Department of GMSC, Pprime Institute, CNRS - University of Poitiers - ENSMA - UPR 3346,

Poitiers, France [email protected] 3 Preparatory Institute for Engineering Studies of Gafsa, University of Gafsa, 2000 Gafsa, Tunisia

Abstract. This paper presents an approach to improve the resolution of the forward kinematic model (FKM) of parallel manipulators. This approach is based on placing sensors on one leg of the manipulator rather than placing them on the actuated joints at the base. Using this method, we aim to cope with the complexity of the FKM of parallel manipulators and to improve the computation time of the FKM. The proposed method is applied on two different robots, i.e.: the 3-UPU and RAF manipulators which are Translational Parallel Manipulators (TPMs). Simulations were carried out to validate this method. These simulations show how the serial approach improve the accuracy, overcome singular configurations inside the workspace and speed up the resolution of the FKM.

1 Introduction For several years great effort has been devoted to the study of parallel manipulators due to their several advantages [1]. In facts, TPMs, are ones of the most popular robots and have a very wide range of applications. However, these manipulators have complex kinematic and dynamic models and suffer from singularities in their workspace. In facts, L. Tancredi et al. in [2] indicate that it is difficult to solve the forward kinematic model (FKM) of parallel manipulators and this is due to two majors problems: the first one is the determination of the admissible solutions and the second one is how to find the acceptable solution„ among all the others, that describes the exact posture of the mobile platform. Also note that the determination of the FKM involves the resolution of a nonlinear equations whose solution may not be unique. Numerical method may be an effective way to guarantee that we have one solution for the FKM, namely, the Newton iterative method used by Bai et al. in [3]. The problem with this approach is that it did not give any guarantee that the solution found is the closest to the effective one and it requires an estimation of the initial posture of the robot to reduce the computation © CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 292–300, 2022. https://doi.org/10.1007/978-3-031-06409-8_31

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time. The work of Gosselin et al. in [4] deals with a method that overcome the FKM problem by resolving of an eight-degree polynomial. Moreover, parallel manipulators suffer from singularities in their workspace which affect the accuracy of the FKM. A latest experimental solution for resolving the FKM using a serial method developed by Saafi et al. as described in [5] and [6] where sensors where placed on one leg of a spherical parallel manipulator. This method was used on the DELTA robot architecture intended to be used in medical task as described in [7–11]. Using this serial method, the resolution of the FKM has been improved. According to that we propose in this paper to extend this approach to other TPMs which are the 3-UPU robot and the RAF robot. This paper is organized as follows: Sect. 2 outlines the kinematic study of the proposed TPMs using the classical approach and the serial one. Section 3 is devoted to the comparison based on the accuracy of each method. Finally, Sect. 4 gives a conclusion and perspective of this study.

2 Kinematic Study of TPMs 2.1 3-UPU Robot The 3-UPU is a parallel manipulator proposed by Tsai [12]. This manipulator is composed also of three identical limbs equipped with three linear actuators. Each leg is attached to the base by a universal joint then a prismatic joint and finally attached to the mobile platform by a universal joint as illustrated in [13]. • Classical approach The kinematic study of this manipulator and the classical approach for the resolution of the FKM have been detailed by I. Ben Hamida et al. in [13]. • Serial approach To solve the FKM of the 3-UPU robot using the serial approach, the sensors are placed on one leg. These sensors measure θ1i (rotation around X), θ2i (rotation around Y2) and L1i . To determine the location of the mobile platform, the vector OP is expressed in the frame R0 . The geometric parameters of the manipulator presented in Fig. 1 are, Rb , Rp, θ1i , θ2i , αi and L1i . α1 , α2 , α3 are the intersection angles between OBi and the X axis. The forward kinematic model of the robot, using the serial approach computes the coordinates of the point P written as follows: ⎧ ⎨ XP = Rb cosαi + L1i cosθ1i sinθ2i − Rp cosαi (1) Y = Rb sinαi + L1i sinθ1i − Rp sinαi ⎩ P ZP = L1i cosθ1i cosθ2i θ1i , θ2i and L1i are given by the sensors (i = 1,2,3). The inverse kinematic model computes the angles θ1 , θ2 and the length L1 for a given location of the mobile platform.

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% Z

Z

B

Z2

Y

Base

B X

O

B

X Y2

Y

Fig. 1. 3-UPU robot parameters.

From the Eq. (1) we can express θ2 as follows:    −1 XP − cos(αi ). Rb − Rp θ2i = tan ZP

(2)

The quadratic sum of the Eq. (1) gives the following expression of L1 : 

  

XP + Rp − Rb .cos(αi ) 2    2   −1  + ( YP + Rp − Rb sin(αi ) L1i = sin tan (N ) With:

(3)

  XP − cos(αi ). Rb − Rp N= ZP The solution of the angle θ1i can be obtained easily after the determination of θ2i and L1i . 2.2 RAF Robot The RAF robot is a translational manipulator designed by Romdhane et al. [14]. The mobile platform of this robot is connected to the base through two different types of chains: the first chain is an active one composed of two spherical joints and an actuated joint, similar to the 3-UPU robot. The second chain is a passive one which is similar to the delta robot chain architecture. The design parameters of the RAF robot are presented in [13].

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• Classical approach Due to the similarity between the active chain of the RAF and the 3-UPU the kinematic model using the classical method of the RAF can be written as the same as the 3-UPU. • Serial approach For solving the FKM of the RAF robot using the serial approach, sensors were placed on the passive chain where they measure 1i , 2i , 3i as described in Fig. 3. This chain has the same architecture as the DELTA robot. The design parameters are presented in Fig. 2. The active leg that gives the input motion is designed by Li (i = 1,2,3). Rp , Rb are the radius of the mobile platform and the base, respectively. α1 , α2 , α3 are the intersection angles between OAi and the X axis. For the passive chain, L1i and L2i are the length of the arms and RS , Rm are the radius of the mobile platform and the base, respectively, β1 , β2 , β3 are the intersection angles between OBi and the X axis. The kinematic study using the serial approach of the passive chain is detailed in [7]. The serial kinematic model is written as follows (i = 1,2,3). 1i , 2i , 3i are described in [7]: ⎧ ⎨ XP = Rm cosβi − Rs cosβi + L1i cos1i + L2i cos3i sin2i (4) Y = Rm sinβi − L1i sin3i − Rs sinβi ⎩ P ZP = L2i sin1i + L1i cos3i cos2i

O

Y

Z

o

Y

X

Base

X

Fig. 2. RAF robot parameters

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Fig. 3. RAF robot passive chain parameters

3 Comparison Between the Serial and the Classical Approach for Solving the FKM In order to verify the accuracy of the serial approach for solving the FKM of the TPMs presented in the previous section we carried out several simulations. In fact, the algorithm described in the Fig. 4 allows the calculation of the error between a prescribed trajectory and the obtained one using the classical method and the serial approach. The algorithm is presented as follows: 1. Generation of a known trajoctory P c = [Xc, Yc, Zc] near the singularity of the robot. 2. Calculation of the input parameters using the IKM. 3. Adding a random values d  representing the sensors sensitivity (normal distribution with σ = 0.04).

4. Calculation of the trajoctory defined by P p = Xp, Yp, Zp using the FKM. 5. Calculation of the error ε between the initial trajectory and the calculated one.

Fig. 4. Algorithm for error calculation

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The error is calculated between a known configuration of the mobile platform designed by P c and the calculated one designed by P p using the equation below:   2  2  2 k ε = XCk − Xpk + YCk − Ypk + ZCk − Zpk , k = 1..n (5) In order to validate the accuracy of the FKM using the serial approach on the 3-UPU robot, we compare the two models: classical and serial FKM. The proposed trajectory starts from the interieur to edges of the workspace. Figure 5.a shows the singular configurations in the trajectory where dexterity is null. Near these positions we observe error amplification in the calculation of the FKM of 3-UPU robot using the classical approach. By applying the serial approach we manage to overcome this problem (Fig. 5.b). The serial method is an effective way to reduce the error values. In fact, Fig. 6 shows that Initial trajectory

Calculated trajectory

Singular configurations

Y[mm] X[mm]

(b)

(a)

Fig. 5. Evolution trajectory of the 3-UPU robot using: (a): classical method; (b): serial method

Configurations

Configurations

Fig. 6. Error calculation of the 3-UPU robot

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the error is less than 0.027 mm and there is no error amplification along the chosen trajectory. The distribution of the dexterity is being used in order to prove the existence of the singular configurations along the chosen trajectory. The dexterity is measured using the inverse conditioning number of the Jacobian matrix. Its expression is as follows: μ(J ) =

1 cond (J )

(6)

For the RAF manipulator, Fig. 7.a and Fig. 7.b plot the initial configuration and the calculated ones using the classical and the serial approach starting from the interieur to the edges of the workspace. It can be seen that the serial approach has a better accuracy. Moreover, the calculation error for the RAF robot is less than 0.1 mm using the serial approach (Fig. 8), contrary to the classical method where the error has amplifications near the singular configuration as shown in Fig. 8.

Fig. 7. Evolution trajectory of the RAF robot using: (a): classical method; (b): serial method

One of the big advantages of the serial approach for solving the FKM is that it speeds up the computation time of the FKM and even make it possible for real time application where the time is crucial. Table 1 shows the computing time using the two methods for the two TPMs implemented on an INTEL i7 processor up to 3.6 GHz using MATLAB.

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Fig. 8. Error calculation of the RAF robot

Table 1. FKM computing time 3-UPU

RAF

Method

Time

Classical

0.1 s

0.1 s

Serial

0.03 s

0.002

4 Conclusion In this paper we use an approach based on placing sensors on one leg of a parallel manipulator rather than placing them on the actuated joint. The proposed approach improved the resolution of the forward kinematic model of parallel manipulators which suffers from the complex kinematic and the presence of singularities in their workspace. A comparison of the calculation time of the FKM of the TPMs using the classical and the serial approach is being introduced in this work. It is clear that through the simulations the serial approach presented an effective and robust solution and specially adapted in the vicinity of singular configurations. By this approach we manage to overcome the singularity problem and improve the accuracy of the model and speed up the computing time. Experiments could be performed on real prototypes which could be addressed as future work.

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References 1. Carricato, M., Parenti-Castelli, V.: Singularity-free fully-isotropic translational parallel mechanisms. Int. J. Rob. Res. 21(2), 161–174 (2002) 2. Tancredi, L., Teillaud, M.: Application de la géométrie synthétique au probleÁme de mode lisation géométrique directe des robots parallèles. Mech. Mach. Theory 34, 255–269 (1999) 3. Bai, S., Hansen, M.R., Angeles, J.: A robust forward-displacement analysis of spherical parallel robots. Mech. Mach. Theory 44(12), 2204–2216 (2009) 4. Gosselin, C.M., Sefrioui, J., Richard, M.J.: On the direct kinematics of spherical three-degreeof-freedom parallel manipulators of general architecture. J. Mech. Des. Trans. ASME 116(2), 594–598 (1994) 5. Saafi, H., Vulliez, M., Zeghloul, S., Laribi, M.A.: A new serial approach of the forward kinematic model of spherical parallel manipulators for real-time applications. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 232(4), 677–684 (2018) 6. Saafi, H., Laribi, M.A., Zeghloul, S.: Forward kinematic model resolution of a special spherical parallel manipulator: comparison and real-time validation+. Robotics 9(3), 62 (2020) 7. Meskini, M., Saafi, H., Laribi, M.A., Mlika, A., Arsicault, M., Zegloul, S.: Serial approach for solving the forward kinematic model of the DELTA robot. In: Zeghloul, S., Laribi, M.A., Arsicault, M. (eds.) MEDER 2021. MMS, vol. 103, pp. 305–312. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75271-2_32 8. Preaúlt, C., Saafi, H., Laribi, M.A., Zeghloul, S.: Optimal design and evaluation of a dexterous 4 DoFs haptic device based on delta architecture. Robotica 37(7), 1267–1288 (2019) 9. Saafi, H., Laribi, M.A., Zeghloul, S., Ibrahim, M.Y.: Tele-operation control system for nonhomothetic master/slave kinematics for minimally-invasive surgery (2013) 10. Saafi, H., Laribi, M.A., Zeghloul, S.: Optimal haptic control of a redundant 3-RRR spherical parallel manipulator. IEEE Int. Conf. Intell. Robot. Syst. 2015, 2591–2596 (2015) 11. Saafi, H., Laribi, M.A., Zeghloul, S., Arsicault, M.: On the development of a new master device used for medical tasks. J. Mech. Robot. 10(4), 8–13 (2018) 12. Tsai, L.-W.: Kinematics of a Three-Dof platform with three extensible limbs. Recent Adv. Robot Kinemat. 1962, 401–410 (1996) 13. Ben Hamida, I., Laribi, M.A., Mlika, A., Romdhane, L., Zeghloul, S.: Dimensional synthesis and performance evaluation of four translational parallel manipulators. Robotica 39(2), 233– 249 (2021) 14. Romdhane, L., Affi, Z., Fayet, M.: Design and singularity analysis of a 3-translational-DOF in-parallel manipulator. J. Mech. Des. Trans. ASME 124(3), 419–426 (2002)

Artificial Neural Network Modelling of Cable Robots Leila Notash(B) Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON, Canada [email protected]

Abstract. In this paper, artificial neural network-based cable models are developed for cable-driven parallel robots. A variety of scenarios, including the transfer learning-based ANN, are examined to predict the deflections while improving the efficiency and performance of ANN. The predicted deflections for the previously unseen poses from the same region, as well as the adjacent regions, are highly satisfactory and comparable to the results obtained by a nonlinear optimization method. In addition, ANN models could predict the deflections for poses that the nonlinear optimization methods may not.

1 Introduction Artificial neural network (ANN) is a machine learning method where each input and output is represented with a neuron (node or perceptron). The ANN model parameters (weights and biases), which specify how to transform the input data into the desired output, are identified from the data during training. After initialization of weights and biases, the input data are fed to the neurons in the input layer. Using a feedforward ANN model, the input x of each layer is multiplied with its weight W and summed by the bias b to generate the output of that layer Wx + b. Then, the output of the layer is passed into a transfer/activation function (e.g., sigmoid or hyperbolic tangent function) before being fed to the next layer. Because of the multiple hidden layers of deep neural networks, the complex nonlinearity in data can be represented, e.g., Nielsen (2015). In transfer learning (TL), the knowledge learned from one system/application is transferred between similar systems/applications for training in order to develop their predictive models. In the ANN-based TL, the purpose of TL is to improve the performance of modelling by reusing existing models and/or data on related neural network models (e.g., Lu et al. 2015; Weiss et al. 2016; Tang and Notash 2021a, b). Application of ANN for modelling cable-driven robots has been very limited, e.g., solving the forward displacement problem (Ghasemi et al. 2010; Schmidt et al. 2014), robot deflections (Notash 2020b) and cable tensions (Chawla et al. 2021). In this paper, modelling of cable robots and their deflections are investigated. The pre-trained ANN model for a region of the workspace is used to predict the deflection maps for previously unseen poses within that region, as well as the adjacent regions. The effectiveness of transfer learning on improving the efficiency and performance to develop the cable robot models is examined. © CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 301–308, 2022. https://doi.org/10.1007/978-3-031-06409-8_32

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2 Platform Deflections The catenary model of a cable, including its mass, for a cable-driven robot is illustrated in Fig. 1. Considering the cable forces fBj acting on the platform at the cable attachment T  points Bj , j = 1, . . . , n, the cable force vector τ = τ1 . . . τn and the platform T T T wrench F = [f related by the transposed Jacobian matrix JT as n; m T] are linearly T T F = J τ = j=1 Jj τj , where Jj includes the Plücker line coordinates of the axis of force fBj that cable j exerts on the platform, and τj = fBj = fBj 2 . When the cable mass is negligible (linear model), JjT represents the axes of both pertinent cable and its force, i.e., the cable force is aligned with the cable axis. Thus, for a given platform pose, column JjT is known for the linear model and the cable forces at the two ends Aj and Bj have the same magnitude, i.e., τj = fBj = fAj . When the cable mass is included, the cable force is tangent to the catenary profile (nonlinear model), and both the magnitude and orientation of cable force vary along the cable. For this case, the horizontal components of fAj and fBj have the same value, refer to Fig. 1(b), while the difference in their vertical components (z-direction here) is the cable weight.

A4 A2

τ4 τ2 Platform

τ1 A1

τ3 Z

Y

(a)

A3 X

(b)

Fig. 1. (a) Cable robot (b) Cable forces.

For the 3 DOF four-cable spatial robot of Fig. 1 with point mass platform P, using the catenary cable profile, the cable end forces and the coordinates of the cable attachment point on the platform Bj = P are related as (Notash 2020a).      l0 fA + fAz 1 ln bxj = axj + nxj fBvj cosαBvj + (1) EA0 ρg fB + f Bz j      fA + fAz l0 1 byj = ayj + nyj fBvj sinαBvj ln + (2) EA0 ρg fB + f Bz j

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 bzj = azj −

ρgl02 fBz l0 1 (fA − fB ) + + EA0 2EA0 ρg

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(3) j

where lo , Ao , E and ρ are respectively the unstrained length, cross-sectional area, modulus of elasticity   and mass per unit length of the cable; axj , ayj , azj are the coordinates of anchor Aj ; fBvj  is the absolute value of tension force fBvj (horizontal component of cable force) and tanαBvj = fByj /fBxj . The values of nx and ny depend on the dimension of the taskspace (Notash 2020a, b). To formulate the compliance matrix of each cable, Eqs. (1) through (3) may be differentiated with respect to the respective cable forces, which will result in the 3 × 3 symmetric cable compliance matrix. Alternatively, to reduce the formulations of partial derivatives, the coordinates of the attachment points Bj can be expressed in the cable reference frame xyz, with the cable profile lying on the xz-plane. Then, the partial derivatives of these coordinates relative to the respective cable forces will generate the 2 × 2 symmetric cable compliance matrix Bj Cj (Notash 2020a). The leading superscript Bj is used to emphasize the reference frame is the cable frame with origin at Bj . Next, the compliance matrix of each cable is expressed relative to a reference frame that its origin is at P but the orientation of its axes are the same as that of the base frame of the robot, i.e., frame XYZ, using the rotation matrix RP0,Bj from each cable frame xyz to robot frame XYZ. Then, applying the compliance matrices of catenary cables, the deflection of operation point P is formulated as ⎛ δp = ⎝

n

⎞−1  P0 −1 ⎠ C f j

(4)

j=1

3 ANN Modelling and Prediction In this section, the deflections δp = δp2 and the pertinent components δpx , δpy and δpz (in mm) of the operation point P are calculated for each pose. ANN models are designed for the cable robot using the catenary cable model. The developed models are compared to the pertinent deflection maps generated with the fmincon function of Matlab. The elasticity, mass per unit length and radius of the cables are respectively E = 199 GPa, ρ = 0.067 kg/m and rc = 2 mm. The minimum and maximum acceptable cable tensions are chosen as fmin = 1 and fmax = 100 N for generating the wrench-feasible workspace. To represent the effect of deflection limit on the workspace size/shape, no deflection limit is set when generating the maps. Anchors Aj are arranged at (–2.02, –2.02, –2), (2.02, –2.02, 2), (2.02, 2.02, –2), (–2.02, 2.02, 2) m, and m = 2 kg for the point mass platform. The horizontal and vertical positions of the examined poses are within the space defined by anchors, Aj , j = 1, . . . , 4. For each examined vertical (z) elevation, 6561 data are generated using increments of 0.05 m. For the nonlinear optimization, using the fmincon function, the unstrained cable lengths and cable forces (magnitude and direction) are obtained by minimizing the 2norm of the non-negative cable tensions. These unknowns are identified subject to the

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nonlinear equality constraints (1) through (3) for each cable with Bj = P, the nonlinear equality constraints for the platform force balance, and the inequality constraints on the minimum and maximum allowable cable force. The minimum 2-norm non-negative cable tension of the linear model (Notash 2013), with fAj = f Bj , are used as the initial guess to identify the cable tensions for the catenary profile. For the plots of Fig. 2, the wrench-feasible workspace at three elevations (z = −1.56, −1.51, and −1.46 m) is characterized by a total of 9083 poses (slightly over 46% of the original data of 19683). As it is illustrated, for this cable robot, the deflection maps are symmetric and bounded at these elevations, with the maximum deflection occurring along the workspace borders.

(a)

(b)

(c)

(d)

Fig. 2. Deflection maps using fmincon at z = −1.56, −1.51, and −1.46 m (in mm).

For the ANN modelling of the spatial robot, using the Levenberg-Marquardt algorithm of Malab, the cost function is the mean squared error of the difference between the deflections calculated using fmincon and ANN. The input to the ANN is the three coordinates of the operation point P and the output is the corresponding deflections. Linear and sigmoid activation functions are used respectively for the output layer and the hidden layers. To design an ANN for the whole workspace, a large number of neurons and/or hidden layers will be required. As a result, it will take exceedingly long time to train the pertinent weights and biases. Hence, alternatives for improving the efficiency and performance of ANN are examined here. With three ANN inputs and three outputs, after many trials, four hidden layers with 40, 35, 35 and 30 neurons respectively are considered, i.e., 3-40-35-35-30-3 architecture. The weights and biases are initialized randomly unless otherwise noted. First, a single elevation is considered. To train the ANN, every third generated data are used for z = −1.51 m, i.e., 33% of data at that elevation. The predicted deflection maps for 100% of the data are depicted in the plots of Fig. 3.

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(a)

(b)

(c)

(d)

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Fig. 3. ANN predicted deflections at z = −1.51 m (in mm).

Fig. 4. Difference of ANN predicted deflections with fmincon at z = −1.51 m.

The difference of deflections identified by fmincon (middle elevation in Fig. 2) and those predicted by ANN in Fig. 3 (in mm) are displayed in Figs. 4 and 5, where only 33% of data were accessible for the training of ANN. Because of the symmetric layout of the cable attachments to the base, as it is verified in Figs. 2 and 3, the deflection maps δpx and δpy are analogous when no horizontal force is applied on the platform. Thus, only the performance the plot of δpx is included in Fig. 4. To illustrate  of the ANN model,  the poses with the total deflection difference δpfmincon − δpANN  larger than 1 mm and 2 mm are displayed in the plots of Fig. 5. Due to space limitation, in the following figures, the total deflections of the operation point are displayed. In Fig. 2, the displayed voids within the workplace at the three elevations, as well as along the workspace borders, are due to the high nonlinearity in the relations used in the optimization function fmincon. Another example case for z = −1.41 and −1.61 m is illustrated in Fig. 6, in which the ANN predicts continuous map for the deflections even when the optimization function does not converge for a relatively large number of

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Fig. 5. Poses with large deflection difference at z = −1.51 m (in mm).

poses. As noted earlier, the voids in Fig. 6(a) correspond to poses that fmincon did not identify a solution for the nonlinear Eqs. (1) to (3). The continuous map predicted by the ANN, using 33% of the data of fmincon, is illustrated in Fig. 6(b).

Fig. 6. Deflection maps (a) fmincon, (b) ANN predictions, at z = −1.61, −1.41 m.

Next, a network was designed for two elevations z = −1.56 and −1.46 m (the top and bottom elevations in Fig. 2; 100 mm apart) using 33% of data. The difference of ANN predicted deflections with fmincon is displayed in Fig. 7, which indicates ANN can predict the deflections for the previously unseen 67% of data very precisely. Then, the designed ANN for these two elevations is used to predict the deflections at z = −1.51 m. As it is evidenced from Fig. 8, the ANN can very well predict the deflections for the majority of the unseen 100% of data at this elevation. The precision of prediction is further improved if the distance between the two elevations is reduced, e.g., to 60 or 40 mm, from the reported 100 mm. As another case, an ANN is designed by transfer learning for two elevations z = −1.56 and −1.46m (using 33% of data), and initialized with the designed parameters (weights and bias) of an already designed ANN for z = −1.51 m (source). The designed ANN (target) was used to predict the deflections at z = −1.51. As it is illustrated in Fig. 9, and compared with Fig. 8, the TL-based ANN has further improved the accuracy in predicting the deflections.

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Fig. 7. Difference of ANN predicted deflections with fmincon at z = −1.56, −1.46 m.

Fig. 8. Difference of ANN predicted deflections with fmincon, using ANN of z −1.56, −1.46 m at z = −1.51 m.

=

Fig. 9. Difference of TL-based ANN predicted deflections with fmincon at z = −1.51 m.

4 Discussion and Conclusions In this work, the artificial neural network models of cable-driven robots were developed to predict their deflection maps. It was presented that the ANN models could have a very good prediction of the deflections, even if a relatively small number of poses are used for the training of ANN. While the ANN modelling was used to predict the deflections of cable-driven robots in this paper, considering the simulation results for cable forces and lengths, it can be concluded that using the learning methods, a relatively small number of input-output datasets could be generated/collected across the workspace in order to develop robot models.

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Nonlinear optimization methods identify the deflection at each discrete pose within the workspace. They may produce disconnected info on the model, and would require interpolation for the poses and/or parameter variations that are not investigated. Whereas, learning methods such as ANN are very powerful when the nonlinear optimization methods cannot produce a solution for some robot poses. As well, the ANN model can precisely predict the deflection for the unseen poses within the trained workspace. Furthermore, the designed ANN for a region of workspace can produce accurate prediction of deflections for adjacent regions with no prior info.

References Chawla, I., Pathak, P.M., Notash, L., Samantaray, A.K., Li, Q., Sharma, U.K.: Neural networkbased inverse kineto-static analysis of cable-driven parallel robot considering cable mass and elasticity. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 50–62. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2_5 Ghasemi, A., Eghtesad, M., Farid, M.: Neural network solution for forward kinematics problem of cable robots. J. Intell. Robot. Syst. 60(2), 201–215 (2010) Lu, J., Behbood, V., Hao, P., Zuo, H., Xue, S., Zhang, G.: Transfer learning using computational intelligence: a survey. Knowl.-Based Syst. 80, 14–23 (2015) Nielsen, M.A.: Neural Networks and Deep Learning. Determination Press, San Francisco (2015) Notash, L.: Deflection maps of planar elastic catenary cable-driven robots. ASME J. Mech. Robot. 12(2), 021103 (2020) Notash, L.: Artificial neural network prediction of deflection maps for cable-driven robots. ASME Paper No. DETC2020-17859 (2020b) Notash, L.: Designing positive tension for wire-actuated parallel manipulators. In: Kumar, V., Schmiedeler, J., Sreenivasan, S.V., Su, H.-J. (eds.) Advances in Mechanisms, Robotics and Design Education and Research, pp. 251–263. Springer, Cham (2013). https://doi.org/10.1007/ 978-3-319-00398-6_20 Schmidt, V., Müller, B., Pott, A.: Solving the forward kinematics of cable-driven parallel robots with neural networks and interval arithmetic. In: Computational Kinematics, pp. 103–110. Springer, Cham (2014). https://doi.org/10.1007/978-94-007-7214-4_12 Tang, H., Notash, L.: Neural network based transfer learning of manipulator inverse displacement analysis. ASME J. Mech. Robot. 13(3), 035004 (2021) Tang, H., Notash, L.: Neural network based transfer learning for robot path generation. ASME Paper No. DETC2021-69006 (2021b) Weiss, K., Khoshgoftaar, T.M., Wang, D.: A survey of transfer learning. J. Big Data 3(1), 1–40 (2016). https://doi.org/10.1186/s40537-016-0043-6

Part VI: Technological Applications and Teaching Aspects

Design of a Robot for the Automatic Charging of an Electric Car Damien Chablat1,2(B) , Riccardo Mattacchione2 , and Erika Ottaviano2 1

DICeM, University of Cassino, G. Di Biasio 43, 03043 Cassino, Frosinone, Italy 2 LS2N, UMR CNRS 6004, 1 rue de la No¨e, 44321 Nantes, France [email protected]

Abstract. In this paper, a robot with parallel architecture is proposed for charging an electric vehicle having the charging socket on its front side. Kinematic models are developed to design the robot for a given workspace that corresponds to the car’s plug placements. A demonstrator composed by commercial components is shown.

1

Introduction

Nowadays, the automotive industry develops electric vehicles with associated tools for a broad range of customers and users. To charge the batteries, most users still have to manually connect the car to a charging station or home outlet. In addition, to enhance the range of electric cars, the batteries are equipped with a larger capacity and require a high charging intensity, which must be performed by large cables. Approximately one meter of cable consisting of four wires with a section of 25 mm2 each has a mass of approximately 0.9 Kg. Therefore, some pioneering studies have been developed to explore the possibility of using a robot that could relieve the user from the task of log in and out the plug. However, among the other possible industrial applications in the automotive industry, only a few robots are designed exclusively for recharging electric cars. In August 2015, the car manufacturer Tesla presented a snake robot for charging Tesla Model S cars. This robot has the morphology of a snake and can detect the vehicle’s charging port and connect to it autonomously. The charging cover opens automatically when the parked vehicle is ready to be charged. Once the connection between the robot and the vehicle is established, the charging process can be started. The idea behind this concept is that the driver does not need to get out of the car to charge the vehicle (Walzel et al. 2016). In July 2017, Wolkswagen company unveiled discreetly the prototype of the future Gen.E. A robotic articulated arm equipped with a combo plug is installed on the side of the vehicle. The robot is designed for underground and multi-store garages for autonomous charging of the vehicle. Except the snake robot, whose architecture is unique, the mechanical architecture of the robot for charging the Gen.E is the KUKA serial architecture. The end-effector is adapted for the insertion of the power outlet into the electric vehicle. The DC quick charging process starts with communication between the vehicle and the electric filling station. c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 311–321, 2022. https://doi.org/10.1007/978-3-031-06409-8_33

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The vehicle transmits its data to the charging station, which gives back the target position for the automated parking (Walzel et al. 2016). The charging socket of the vehicle has to be in a target area of 200 by 200 mm. Afterwards, a camera on the robot detects the exact position of the charging socket, with an accuracy of mm. After, the gripper picks up the DC-Connector and connects with the charging socket of the vehicle. After having linked the DC-Connector, the charging process starts. Once the battery is fully charged, the robot automatically unplugs the DC-Connector. Another automatic recharge system is the NRG-X, it can be adapted to any EV or PHEV and is capable of fast charging. Furthermore, it has a tolerance for inaccurate parking positions. The NRG-X system is based on a combination of conductive and inductive charging beneath the vehicle, thus an adapter for the vehicle is necessary (Miseikis et al. 2017). It has to be noted that although industrial robots can very well adapt to several tasks, their design and production remain very expensive for the application. For public places, such as car parks, shared solutions have been proposed (Walzel et al. 2016). However, this solution is expensive in terms of infrastructure and cannot be used for individual houses. The price of an industrial robot is about 20–40% of the price of a car. Research projects in this context aim to develop specific robots by reducing the number of actuators (Yuan et al. 2020) or overcoming the effects of gravity on actuators (Gao et al. 2016) and (Ketfi-C. et al. 2018), these are the main objectives on which this article is based. This is the main reason why the French company Renault decided in 2018–2019 to develop a less expensive charging robot for its upcoming electric cars (Kamga 2017). The specifications for the robot indicate that it has to be mounted on the garage wall and will recharge the battery by facing the car. The robot must be small, precise, and stiff. It must allow the plug to be inserted in safely and properly way. The main requirements are: (a) Insertion and withdrawal forces: 70 N, ±10 N (b) Depth of insertion: 40 mm, ±20 mm (c) Distance between plug and socket: 20 mm, +10 mm (d) Workspace: 200 ×200 ×20 mm3 (e) Degree of freedom: 3 (f) Location accuracy: 0.5 mm, ±0.1 mm (g) Length of electric cables: 1500 mm, +500 mm (h) Weigh of electric cables: 1.4 kg,+0.45 kg (i) Plug diameter: 65 mm, ±5 mm (j) Total weigh of the robot: 10 kg, +5 kg, and (k) Occupied space 1500 ×200 mm2 .

2

Design Requirements of a Robot for the Automatic Charging of an Electric Car

Our work focuses on the Renault Zoe, for this car, the plug is placed on the front and is slightly inclined. The mobility required for the robot could be at least three translations if the car is on a horizontal floor, plus a translation along the insertion axis for the electric plug. By using the flexibility of the insertion tool, it is possible to avoid using a wrist or spatial mechanisms (Figliolini et al. 2012), (Figliolini et al. 2016).

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To reduce the cost of the robot, it is possible to mix the vertical movement with the insertion direction. Figure 1 depicts the car and the dexterous workspace of the robot. Several robot architectures were analyzed before settling on a parallel P-Pi-R-P type robot for planar motion (Rea et al. 2013) put in series with a linear actuator for insertion movement where P stands for prismatic actuated joints, Pi for a four-bar mechanism similar to an RR mechanism but with opposite bars parallel two by two, and R for revolute joint. The main advantages of this architecture are, simplified design (Wenger et al. 2001; Ottaviano and Rea 2013), and the forces applied to the actuators decrease as the end-effector moves away from the base. This will be the case when inserting the plug.

Plug 10 cm Dextrous workspace 10 cm 20 cm 60 cm Dextrous Plug workspace Insertion 1 cm axis 35° 20 cm

1 cm 60 cm

Fig. 1. Workspace of robot and the Zoe car

3

Mechanism Understudy

The kinematics of the robot can be written in three stages (i) the α mechanism which allows translations of the end-effector in one plane (ii) the β mechanism which does the insertion of the plug, and (iii) the γ mechanism which is the assembly of the first two. 3.1

The α Mechanism: P-Pi-R-P Planar Parallel Robot

The first mechanism aims to achieve a planar displacement. To choose this architecture, the constraint of storage on a wall and the reduction of on-board weight lead to the choice of prismatic joints. The design parameters of the α mechanism are the following:

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L: the length of the two arms; L2 : the length of the robot’s base; L3 : the distance between B and C ρ1 and ρ2 : the length of the left and right linear actuator; [x, y]: the coordinate of point V located in the middle of (B, C); [0, epy ]: the offset from V to F ; [e1x + ρ1 , e1y ]: the coordinate of A; [L2 − ρ2 − e2x , e2y ]: the coordinate of E;

Fig. 2. Kinematic scheme of the α mechanism

According to the kinematic scheme in Fig. 2, the constraint equations that can be obtained are the following (x − e1x − ρ1 − L3 /2)2 + (y − e1y )2 = L2 (L2 − x − L3 /2 − e2x − ρ2 ) + (y − e2y ) = L 2

2

2

with Eqs. 1–2, the inverse kinematic equations can be obtained as  ρ1 = x − e1x − L3 /2 − 2L2 − (y − e1y )2  ρ2 = L2 − x − L3 /2 − e2x − 2L2 − (y − e1y )2

(1) (2)

(3) (4)

Solving Eqs. 1–2 for the Cartesian coordinates x and y allows the evaluation of two assembly modes. However, only one will be accessible, thanks to the joint limits (Wenger et al. 2001).

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To avoid the singularity of the Pi joint in the folded position, the offset epy is defined. The inclined placement of the parallelogram allows sufficient stiffness to be maintained when the robot is close to its base in the folded position. Indeed, the stiffness of a parallelogram is proportional to the area formed by the four bars (Majou et al. 2002). The placement of the bar (DF ) does no influence on the kinematics, but it does influence the stiffness of the robot. In Fig. 3, two singular postures are depicted to define the boundaries of the workspace and to separate into two aspects by the parallel singularity. The IIMtype singularity of the four-bar mechanism defined in (Zlatanov et al. 1993) is avoided thanks to the offset epy .

Fig. 3. Parallel and serial singular configuration of the α mechanism

3.2

The β Mechanism: Serial Robot

For this mechanism, the constraint equations define P with respect to the Cartesian frame z  -y  . According to Fig. 4, it is possible to note that: • ρ3 : is the length of the linear actuator that defines the coordinate of H with respect to G; • [e3y , e3z ]: is the coordinate of H with respect to P ; • [e4y , e4z ]: is the coordinate of V in the reference frame z  -y  . • [x , y  ]: is the coordinate of P in the reference frame z  -y  .

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Fig. 4. Kinematic scheme of the β mechanism.

According to the kinematic scheme in Fig. 4, the constraint equations that can be obtained are the following z  = e3z + ρ3 cos(θ) y  = e3y + ρ3 sin(θ) and, its inverse kinematic as ρ3 =



(z  − e3z )2 + (y  − e3y )2

(5) (6)

(7)

The offset actuator reduces the volume of the robot when it is folded1 . 3.3

The γ Mechanism

The merging of the mechanisms described in earlier subsections gives a 3 d.o.f. robot, the γ mechanism. For verifying the operation of the robot prototype, an Arduino micro-controller can be used. Figure 5 shows a CAD model in CATIA V5 with components from Makeblock (2022).

Fig. 5. CAD mode of robot 1

See movie: https://youtu.be/lCRXJ5tpTMM.

Fig. 6. Prototype of the robot

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4

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Workspace Analysis

The workspace of a robot can be defined as the set of points that can be reached by its end-effector (Khalil and Dombre 2000). This is the space where the robot can operate and carry out tasks. In this work, the workspace is evaluated according to the reference point V of the α mechanism. It is important to know the positions V can reach to avoid issues such as singularity configurations, interference, collisions. Workspace is assessed as an intersection of two sets of points2 . Following distances have to be defined: • e5x : is the distance between the y-axis and the a joint, when the left actuator is at the end of its stroke; • e6x : is the distance between the y-axis and the e joint, when the right actuator is at the end of its stroke; • e7x : is the distance between the y-axis and the e joint when the right actuator is at the end of his stroke. The complete workspace is displayed in Fig. 7. The workspace boundaries can be described analytically using five equations that can be used to verify whether the end-effector can reach a specified position or not.

Fig. 7. Workspace of the α mechanism.



L3 − e1x )2 + (y − e1y )2 ≥ L, (x − 2  L3 − e5x )2 + (y − e1y )2 ≤ L, (x − 2

2



L3 − e7x )2 + (y − e2y )2 ≥ L 2  L3 − e6x )2 + (y − e2y )2 ≤ L (x + 2 (x +

See the movie: https://youtu.be/Y0uyeXuP5Eg.

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To analyse the position of the end-effector, a regular workspace shape is considered. In our case, we evaluate the largest square that gives an inner tangent to the boundaries of the workspace (Chablat et al. 2004). The side length of the square is given by lb and, for algebraic computation, a Lame curve is used, (x − xc )n + (y − yc )n = lbn

(8)

with n a positive even integer and (xc , yc ) are the center of the workspace. To limit the computation cost, n is set at 6 as a first approximation and then to 12 to better approximate the square shape. To find the position of the regular workspace, the methodology defined by Chablat et al. (2012) is used. The optimisation problem is written as the intersection of the robot workspace boundary with the boundary of the desired regular workspace whose placement depends on (xc , yc ) . If the set of solutions obtained is non-zero, then we have found a candidate solution for the robot realization. This calculation was carried out using the Siropa library, whose functions are presented in (Jha et al. 2018). To simplify the design, only commercial components from (Makeblock 2022) have been used, without the need for machining. Several bar sizes were tested. This demonstrates that this robot can easily be built. The main sizes are L = 532 mm, L2 = 1300 mm, and L3 = 160 mm. For this set of design parameters, a set of possible placements for the regular workspace was found by using Cylindrical Algebraic Decomposition Moroz et al. (2010). Figure 8 presents several regions where there is no intersection between the regular workspace and the workspace of the robot. However, only one region is included in the workspace of the robot. If the robot can reach the desired workspace, we must be checked that the stiffness and workspace properties are under the specifications. The stiffness of the mechanism is therefore analysed in nine different configurations, which correspond to the precision points from 1 to 9, as shown in Fig. 9.

Fig. 8. Possible placements of the regular workspace

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Fig. 9. On the left, the regular square inside the regular workspace, on the right, the nine test points for the stiffness analysis with xc = 570 and yc = 335.

5

Stiffness Analysis

To evaluate the stiffness of the robot to meet the specifications, simulations in CATIA were performed. The robot’s stiffness is analysed in its working space. Two types of simulations have been taken into account: (i) when ρ1 and ρ2 move, i.e. the α mechanism, and (ii) when ρ3 moves, i.e. the β mechanism. The materials used are mainly aluminium, except for the shafts, the bearings, and slide guides, which are made of steel. In all simulations, the self-weight of the structure is taken into account. As ball bearings are used, there is little backlash in the robot. The bending of the end-effector is due to the bending of the arms. The finite element analysis gives us the following results for case (i), Table 1 and case (ii), Table 2. However, as we are using an external sensor-based controller, an embedded camera, the most important issue is to have a constant deformation. Table 1. Maximum displacements throughout the workspace

Table 2. Maximum displacements for the β mechanism

Position Max displacement [mm]

ρ3 [mm] Max displacement [mm]

1 2 3 4 5 6 7 8 9

50 75 100 125 150

0.995 0.993 0.991 0.990 0.996 0.989 1.030 1.010 1.030

1.080 0.996 0.936 0.907 0.958

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Fig. 10. Boundaries conditions and gravity load in the CAD modelling

6

Hardware and Software

The control is realized using a Rasberry Pi 4 board, which allows the use of a touch screen, a camera to detect the position of the socket, and a serial link to an Arduino board (Microcontroller Me Auriga MakeBlock). Two stepper motors are used (42BYG Geared Stepper Motor) with two belts and four pulleys for the α mechanism, and a DC motor and a ball screw for the β mechanism (Makeblock DC25 with sensor, 86 rpm). MakeBlock modules are used for the input-output connection with the Arduino board. A Python program manages a user interface, the camera with OpenCV functions, and the communication with the Arduino board. To locate the plug on the car, we use a QR code stuck next to the plug. When the robot moves, the QR code seen by the camera is used to adjust the trajectory before starting the insertion of the plug3 .

7

Conclusions

In this article, we presented a robot that allows the insertion of a plug into an electric car for charging. This robot is designed for the Zoe car but could be adapted to other vehicles having a front charging system, like the Nissan Leaf. To reduce manufacturing costs, off-the-shelf components were used in combination with hybrid kinematics with a simple kinematic model. A prototype was realized as a proof of concept associated with the patent application. A GitHub repository will soon be created to make the CAD and source code of the control available. Acknowledgements. This work was supported by the project Chair between Renault and Centrale Nantes about performances of electric vehicles propulsion.

References Chablat, D., Wenger, Ph., Majou, F., Merlet, J.-P.: An interval analysis based study for the design and the comparison of three-degrees-of-freedom parallel kinematic machines. Int. J. Rob. Res. 23(6), 615–624 (2004) 3

See the movie: https://youtu.be/P5wCgRqSyDQ.

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Chablat, D., Moroz, G., Arakelian, V., Briot, S., Wenger, P.: Solution regions in the parameter space of a 3-rrr decoupled robot for a prescribed workspace. In: Latest Advances in Robot Kinematics, pp. 357–364. Springer (2012). https://doi.org/10. 1007/978-94-007-4620-6 45 Figliolini, G., Rea, P., Angeles, J.: The synthesis of the axodes of spatial four-bar linkages. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 45035, pp. 1597–1605. American Society of Mechanical Engineers (2012) Figliolini, G., Rea, P., Angeles, J.: The synthesis of the axodes of RCCC linkages. J. Mech. Rob. 8(2), 021011 (2016) Gao, D., et al.: Robotically operated vehicle charging station. US Patent 9,266,440, 23 February 2016 Jha, R., Chablat, D., Baron, L., Rouillier, F., Moroz, G.: Workspace, joint space and singularities of a family of delta-like robot. Mech. Mach. Theory 127, 73–95 (2018) Kamga, G.S.: Robotisation de la recharge ` a domicile d’une Renault Zo´e. Master’s thesis, Ecole Centrale de Nantes (2017) Ahmed Ketfi-C., Chablat, D., Wenger, P.h., Ghanes, M.: Dispositif automatise de charge d’un v´ehicule ´electrique, FR3078660, March 2018 Khalil, W., Dombre, E.: Modeling, identification and control of robots, kp science (2000) Majou, F., Wenger, P., Chablat, D.: Design of a 3 axis parallel machine tool for high speed machining: The orthoglide. In: IDMME, 4`eme Conf´erence Internationale sur la Conception et la Fabrication int´egr´ees en m´ecanique, pp. 1–10. AIP-Prim´eca (2002) R official store, robot kits, stem toys for kids & makers Makeblock. MakeblockUS (2022). https://www.makeblock.com/, accessed 21 Febuary 2016 Miseikis, J., Ruther, M., Walzel, B., M Hirz, M., Brunner, H.: 3d vision guided robotic charging station for electric and plug-in hybrid vehicles (2017). arXiv preprint arXiv:1703.05381 Moroz, G., Chablat, D., Wenger, P., Rouiller, F.: Cusp points in the parameter space of rpr-2prr parallel manipulators. In: New Trends in Mechanism Science, pp. 29–37. Springer (2010). https://doi.org/10.48550/arXiv.1012.2662 Ottaviano, E., Rea, P.: Design and operation of a 2-dof leg-wheel hybrid robot. Robotica 31(8), 1319–1325 (2013) Rea, P., Ottaviano, E., Conte, M., D’Aguanno, A., De Carolis, D.: The design of a novel tilt seat for inversion therapy. Int. J. Imag. Rob. 11(3), 1–10 (2013) Walzel, B., Sturm, C., Fabian, J., Hirz, M.: Automated robot-based charging system for electric vehicles. In: 16. Internationales Stuttgarter Symposium, pp. 937–949. Springer (2016). https://doi.org/10.1007/978-3-658-13255-2 70 Wenger, P., Gosselin, C., Chablat, D.: A comparative study of parallel kinematic architectures for machining applications. Elect. J. Comput. Kinem. 1(1), 23 (2001) Yuan, H., Qiong, W., Zhou, L.: Concept design and load capacity analysis of a novel serial-parallel robot for the automatic charging of electric vehicles. Electronics 9(6), 956 (2020) Zlatanov, D., Fenton, R.G., Benhabib, B.: Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics. In: IEEE International Conference on Robotics and Automation, vol. 1(2), pp. 986–986. IEEE (1993)

A New Parameter Selection Method for Power Skiving Tools Enea Olivoni(B) , Rocco Vertechy, and Vincenzo Parenti-Castelli Department of Industrial Engineering, University of Bologna, Bologna, Italy [email protected]

Abstract. This paper presents a new selection method of the teeth number and the helix angle of a power skiving tool, which are basic parameters for machining internal gears. It is based on screw theory, whose use for power skiving has not been yet documented in the literature. The tool parameters are selected based on an optimization procedure of the machining process kinematics, differently from the common practice that defines them by experience. In addition, specific power skiving requirements are also considered in the selection. A working example that shows the effectiveness of the method is provided, and the optimal tool parameters are found. In particular, the results show that two different types of tool can machine the same internal gear.

1 Introduction Power skiving is a machining process for gear production (Fig. 1a). It offers significant advantages over common gear manufacturing processes in terms of productivity and flexibility, that allowed it to spread widely among the gear companies. However, there are many issues of power skiving that still need to be investigated. Among these, the optimal selection of the tool teeth number and its helix angle relatively to the workpiece to cut still deserves further investigations, since they heavily affect the process. Several authors have explored their effects on power skiving. Guo et al. [1] explored their impact on the process in terms of rake angle values and lead deviations on the machined workpiece. Klocke et al. [2] investigated how different tool teeth number and helix angles, among other parameters, affected chip thickness, local rake angle and cutting and sliding velocities. The effect that different tool helix angles had on power skiving were also studied by Uriu et al. in [3]. Interestingly they suggested the possibility of employing a tool with opposite helix with respect to the workpiece helix. However, the definition of the two parameters, which are addressed as design parameters in this paper, is usually based on practical experience and analytical methods to select them are still lacking in the literature. The objective of this paper is to define a selection method to determine the tool design parameters, given an internal gear to cut, considering the machining process kinematics and aiming at fulfilling specific power skiving process requirements. The Authors would like to express their sincere gratitude to Metalcastello S.p.A. for supporting Olivoni’s PhD program on power skiving manufacturing process. © CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 322–331, 2022. https://doi.org/10.1007/978-3-031-06409-8_34

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Fig. 1. a) Power skiving process; b) schematic of a skew axis gearset.

The proposed approach lays foundation on screw theory, which is widely used in the literature related to gear design [4–6]. However, to the authors knowledge, screw theory has not yet exploited in the study of power skiving. In this work, both the pitch of relative motion and the departure from the instantaneous screw axis for the tooth generation are employed. Hence, this approach represents a novelty in the power skiving arena in both the selection method for the tool parameters and the kinematic concepts it is based on. In particular, the advantage of using the proposed approach based on an analytical procedure is that the resulting parameters can fulfil specific power skiving requirements such as a good surface finishing among other defined in Sect. 3, that a parameter selection based on practical experience cannot consider. The paper is structured as follows. Section 2 introduces the kinematics of helical gears with skew axes and the concept of infinitesimal teeth. Section 3 focuses on the selection method of the design parameters and proposes a kinematic index representative of the kinematics during machining. A working example is provided in Sect. 4 that shows the feasibility of the method. Conclusions are drawn in the last Section.

2 Kinematics of Helical Gears with Skew Axes This section briefly reviews a few basic concepts from [4, 5] that can be useful to the reader. The schematic of a gearset consisting of two meshing gears 1 and 2 with skew axes and the frame 3 is depicted in Fig. 1b, where only the axes of the two gears are represented for clarity. Here ω1 and ω2 are the shaft absolute angular velocities. The shapes of the gear teeth surfaces are irrelevant for the considerations that follow. The relative position of the two shafts is described by their shortest distance d , defined by the points A1 and A2 , and the shaft angle θ between them. For a given gearset the relative motion between the two gears, can be described by the instantaneous screw axis (ISA) that results from the linear combination of the two screws of zero pitch of the two gears. The relative motion is a skew motion performed about the ISA with pitch p12 . The ISA lies on the plane π orthogonal to A1 A2 and passing through the point A. Its pose is described by the distance r1 from the shaft of gear 1 to point A and by the angle ϕ2

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between the shaft of gear 1 and the ISA (see Fig. 1b). Selecting the triplet (τ, d , θ ), where τ is the gear ratio, the gearset kinematics is fully determined. Indeed, the pose of the ISA and its relative pitch p12 depend uniquely on (τ, d , θ ). 2.1 The Infinitesimal Teeth Revolving the ISA about each gear shaft generates two hyperboloids, which are the axodes of the gearset (Fig. 2a). The two hyperboloids, in line contact, roll and slide over each other about and along the ISA. After selection of the normal module, an integer number of equidistant lines can be drawn on the axodes (Fig. 2a). These lines can be depicted as straight small teeth of infinitesimal depth and thickness called infinitesimal teeth. In this scenario, the two axodes can be viewed as solid surfaces rotating about their shaft axes, transmitting motion one to another with gear ratio τ , while being in line contact along the ISA. The relative velocity magnitude of the points along the ISA is given by: vt = p12 ω12

(1)

where ω12 is the modulus of the gear relative angular velocity, ω12 = ω1 −ω2 . Differently from the case of parallel or intersecting axes, for skew axes helical gears it is possible to move the infinitesimal teeth passing from the ISA at point A, to another point B at an arbitrarily chosen distance apart dB on the common perpendicular (Fig. 2b), while leaving unchanged the triplet (τ, d , θ ). The distance dB between A and B is addressed as departure. To maintain the meshing of the gearset, while an infinitesimal tooth is moved from A to B, its direction must be twisted of an angle ϕ with respect to the ISA, according to the following relation: dB = p12 tan ϕ

(2)

the ISA and the hyperboloids remain the same, because the triplet (τ, d , θ ) is not changed. However, the departure from A involves several issues. It is worth mentioning that the shapes of the solid surfaces change and become cylindrical. The resulting cylinders are

Fig. 2. a) Gearset axodes; b) schematic of an infinitesimal tooth moved from the ISA.

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in point contact at B, where the relative velocity increases in magnitude according to the following relation:  (3) vt = p12 ω12 1 + tan ϕ 2 such cylinders are the well-known operating pitch surfaces [7] and the angle that the departed infinitesimal tooth at B forms with the shaft axis is the helix angle of the real gear. The departed infinitesimal teeth are indicative of the relative motion between the real teeth of the gear and the tool nearby point B. Hence, because the kinematics of the axodes can be exploited for gear synthesis as reported in [8], the same can be done for the kinematics of the departed teeth on the operating pitch cylinders. 2.2 The Case of Power Skiving Referring to the gearset of Fig. 1b, as representative of the power skiving process of Fig. 1a, the tool can be viewed as the gear 1 and the workpiece as the gear 2. In practice, the tool realizes its cutting action with multiple passes, by varying the distance among shafts at each pass. The study that follows is conducted by referring to the last pass with the tool and the workpiece set at a distance d . This choice does not affect the design parameter selection. Therefore, the kinematics concepts presented in the previous sections can be used to describe the power skiving process. Indeed, given a workpiece to cut, the selection of the tool design parameters (β2 , z2 ) can be made based on kinematic considerations done for the departed teeth of the gearset of Fig. 1, consisting of the workpiece and the tool. However, the kinematic requirements of the power skiving process differ from those of gearsets designed for motion transmission, where the relative velocity among the teeth must be kept low. Indeed, in the case of power skiving, the relative velocity should be kept high along the teeth longitudinal direction, where the cutting action takes place, whereas a minimal component along the teeth traversal direction is desired to reduce the tool wear [2]. This traversal component occurs when the contact point lies outside the line of the shortest distance and it is as bigger as higher the relative angular velocity and the twist ϕ are. Referring to Eq. (3), it can be concluded that gearsets characterized by high pitch, moderate relative angular velocity, and small value of the departure dB from their ISA are the most apt to fulfil the kinematic requirements of power skiving.

3 Tool Parameters Selection Method The problem of selecting a suitable tool for machining a given workpiece, begins with the selection of the tool design parameters β2 and z2 . With reference to Fig. 1b, regarded as a scheme of the given workspace and the tool, their normal module mn must be the same, then the following relation must hold: mn =

2r2 cos β2 2r3 cos β3 = z2 z3

(4)

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where r, β, z are the pitch radius, the helix angle and the teeth number respectively, whereas the subscript 2 and 3 are relative to the tool and the workpiece, respectively. Considering Eq. (4) and the following standard formulas for internal gears: τ=

z2 z3

d = r3 − r2 θ = |β2 − εβ3 |

(5)

only two parameters of the triplet (τ, d , θ ) are independent1 . Given the workpiece geometry and considering Eqs. (4) and (5), by selecting two of the three parameters of the triplet, the whole triplet is determined as well as the tool design parameters (β2 , z2 ). Hence, the gearset along with its kinematics is also defined. For convenience, the study is carried out analysing all the possible options of the couple (τ, θ ) for the given workpiece. These options constitute a discrete domain of points (τ, θ ) since the combinations for the transmission ratio are limited by the condition to have an integer number of teeth. Each point on the domain (τ, θ ) corresponds to a different gearset made up of the given workpiece and a tool characterized by its specific design parameters (β2 , z2 ). The kinematic entities of Eq. (3) are evaluated for each gearset on a such domain. In order to establish which one, among all the gearsets, better fulfil the kinematic requirements previously discussed, a kinematic index f is defined: k1 k2 k3 p12n ω12n tan ϕnk4 f = vtn

(6)

where ki are weights that can be chosen in the range, [−1, 1], and the subscript n indicates that each entity (·) in Eq. (6) has been normalized as (·) = ((·) − (·)min )/((·)max − (·)min ). The values (·)max and (·)min correspond to the maximum and the minimum value of the entity (·) on the domain (τ, θ ), respectively. The weights k1 and k2 are set as negative, whereas k3 and k4 as positive, to promote the kinematic characteristics highlighted at the end of Sect. 2.2. The gearsets that better fulfil the kinematic requirements for power skiving are those in the neighbourhood of the lowest value of the kinematic index f . Hence, the design method is to evaluate the minimum values of f over the discrete domain (τ, θ ). The corresponding values (β2 , z2 ) are considered the optimal tool design parameters for machining the given workpiece. In addition to the considerations done for the kinematics of machining, the proposed approach aims at considering additional process requirements such as good surface finishing, low cutting-force generation and tool-workpiece collision avoidance. These requirements are modelled as process parameters and computed over the domain (τ, θ ). By setting limit values for such process parameters, the available options for the tool design parameters are limited. In the following section the specific process requirements are presented. 3.1 Requirement of Good Surface Finishing Considering the tool-workpiece engagement, the trace left from the tool tip on the workpiece root at each pass can be approximated as an ellipse in the plane passing through the gear axis and the tool tip at zero feed. A section of the gear on this plane is shown 1 Here ε = 1 when the helices are in the same direction and ε = −1 for opposite helices.

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in Fig. 3. With the same feed and infeed, the crest height  left on the workpiece root in the section plane, depends on the design parameters. The higher the crest height the worse the gear surface finishing is (Fig. 3a). The value of the crest height  is computed over the domain (τ, θ ). 3.2 Requirement of Low Cutting Forces An inherent characteristic of power skiving is the variation of the instantaneous rake angle γ (Fig. 3b) on different portions of the tool profile during cutting. It is known that negative values of the instantaneous rake angle lead to high cutting forces and consequently to a rapid tool wear. As a representative value for the instantaneous rake angle, its value on the tool tip can be taken [1]. With the same feed and infeed of the tool and for a fixed tool rake angle γ0 , the instantaneous rake angle γ on the tip varies along the ellipse traced by the tool tip during motion (Fig. 3b). The way it varies and the values it assumes, changes for each gearset and depends on the design parameters. The minimum value of the instantaneous rake angle on the tool tip is computed over the domain (τ, θ ).

Fig. 3. a) Crests left on the workpiece root; b) rake angle during cutting

3.3 Constraint for Collision Avoidance A well-known problem in power skiving of internal gears is the collision that may occur between the tool holder and the machined internal gear. This sets an upper limit for the permissible value of the shaft angle θmax when machining an internal gear (Fig. 4a). To avoid collision, the value θmax for the gearset must be computed and only tools that based on Eq. (5) realize θ < θmax must be selected. In order to determine the value θmax , a normal section of the gear on the plane orthogonal to the gear axis and passing through the collision point is taken, (Fig. 4a). In the plane section, the tool holder, which is sunk on the toothed portion of the gear of a height H , is an ellipse, whereas the gear is a circle with radius equal to the gear tip radius (Figs. 4b and 4c). Given the radius Rm of the tool holder and the height H , the equation of the ellipse written in the reference frame (x, y) depicted in Fig. 4b is: (x − H tan θ )2 cos θ 2 + (y − d )2 = R2m

(7)

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the dimension of the ellipse and its centre’s abscissa depend on the shaft angle, while the coordinate of the ellipse centre on the y-axis is the distance d . Imposing the tangency of the ellipse with the gear tip circle, a relation between θ and d can be found. For each value of θ a limit value for d exists, above which the tool holder collides with the gear when the tool is sunk of the height H (see Fig. 4c). From Eqs. (4) and (5), stems that for each value of θ exists a minimum value τmin for the gear ratio under which collisions arise. Furthermore, since the distance d must be positive, for each value of θ also a maximum value τmax for the gear ratio exists. Therefore, two limiting curves are generated for collision avoidance on the domain (τ, θ ). An example is shown in the next Section.

Fig. 4. A) Example of collision; b) normal section of the gear and of the tool-holder; c) condition of tangency between the gear tip circle and the tool-holder ellipse

4 Working Example The data relative to this example are reported in Table 1. The workpiece was chosen as a left-hand internal gear with involute teeth, but the method is applicable also to different tooth shapes. The goal is to find the minimum values of the kinematic index f on the domain (τ, θ ) which is restricted by limits set on the process requirements. Table 1. Data employed in the example mn 3

z3 67

β3

r3

H

Rm

γ0

Feed

Infeed

deg

mm

mm

mm

deg

mm/ rev

mm

23

109.2

120

60

5

0.5

0.3

k1 , k2

k3 , k4

−1

1

From Eq. (7), the maximum value for the shaft angle that avoids collision is θmax = 25.4◦ . By referring to Fig. 4c, this case occurs when d = 0 and the tool holder ellipse is tangent to the gear tip circle. Hence, the available gearsets are studied in the domain (τ, θ ) with τ ∈ [0, 1] and θ ∈ [−θmax , θmax ]. It is worth mentioning that points in the northwest region of the domain correspond to negative values of d and are not realizable. These

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cases are excluded from the calculations as shown in the next figures. With reference to Fig. 5a, positive values of θ are relative to tools inclined as the workpiece helix. These tools are left-hand (LH) and have a mild helix angle for θ ≤ β3 , that is the most used option by designers; whereas for θ > β3 they result right-hand (RH). Negative values of θ are relative to LH tools with high helix angle that are inclined as in Fig. 5a. The kinematic entities of Eq. (3) and the process parameters  and γ are computed for each gearset. Figure 5b shows the values of the minimum rake angle on the domain (τ, θ ). To ensure low tool wear and high quality of the machined gear, limits for the minimum acceptable rake angle and maximum crest height are set as γmin = 5◦ and max = 6, 4 μm respectively. Referring to Fig. 5b, this is equivalent to intersect the depicted surface with a horizontal plane at the height of the set limit γmin . The same reasoning is applied for the crests. The resulting intersection curves, addressed as limiting curves, divide the domain (τ, θ ) in regions where the limits for the process parameters are not respected and the corresponding gearsets must be discarded from the available options. In addition, using Eq. (7) the limiting curves for collision avoidance are computed. Then, the kinematic index f is evaluated on the domain shown in Fig. 6a. By considering the limiting curves, the gearsets that do not fulfil the process requirements are discarded and the available domain, corresponding to the coloured regions in Fig. 6b, is reduced. Since a precise choice of the weights in Eq. (6) was not the objective of this work, it is reasonable to look for regions where f is low rather than seeking a local minimum. Blue areas in the right and left part of Fig. 6b are those where f is close to its lowest value and the corresponding design parameters are considered optimal. Referring to Fig. 5a such areas correspond to two different kinds of tools. In particular, the tool kind in the right region is the one commonly adopted in practice, whereas that in the left region is the one studied in [3], where its good performances were highlighted. For completeness, the design parameters relative to one tool of each kind are reported in Table 2. They correspond to the points highlighted with crosses in Fig. 6b.

Fig. 5. A) Different tool kinds for machining the same workpiece; b) evaluation of the rake angle γ on the domain (τ, θ).

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Fig. 6. Kinematic index f evaluated on a) the entire domain; b) the available domain reduced by the limiting curves of τmin , τmax , γ, and . Table 2. Examples of optimal tool of the two kinds. τ

z2

θ

β2

r2

Tool 1 (LH)

0.63

42

−13°

36°

77.87 mm

Tool 2 (LH)

0.66

44

16°

7°

66.50 mm

5 Conclusions This paper presented a new selection method of the number of teeth and the helix angle of a power skiving tool for machining internal gears. The method is applicable to different tooth-profile shapes and it is based on kinematic considerations from screw theory, which was not yet employed for the study of power skiving. It relies upon an analytical procedure that aims at satisfying several process requirements, thus overcoming the limitations of the common practice that selects the parameters by experience. A working example was presented. The results shown that two different kinds of tools can be chosen to machine the same workpiece. This is in accordance with the literature and the common practice, highlighting the effectiveness of the proposed method. In the future, the method will be extended also to non-standard gearsets. Furthermore, machining simulations of a tool with the design parameters selected by this method will be carried out.

References 1. Guo, E., Hong, R., Huang, X., Fang, C.: Research on the cutting mechanism of cylindrical gear power skiving. Int. J. Adv. Manuf. Technol. 79, 541–550 (2015) 2. Klocke, F., Brecher, C., Löpenhaus, C., Ganser, P., Staudt, J., Krömer, M.: Technological and simulative analysis of power skiving. Procedia CIRP 50, 773–778 (2016) 3. Uriu, K., et al.: Effects of shaft angle on cutting tool parameters in internal gear skiving. J. Mech. Sci. Technol. 31(12), 5665–5673 (2017) 4. Phillips, J.: Freedom in Machinery. Cambridge University Press, Cambridge (2007)

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5. Phillips, J.: General Spatial Involute Gearing. Springer, New York (2003). https://doi.org/10. 1007/978-3-662-05302-7 6. Dooner, D.B.: Kinematic Geometry of Gearing (2012) 7. Litvin, F.L.: Theory of Gearing. Springer, Heidelberg (1989). https://doi.org/10.1007/978-3662-48487-6_16 8. Figliolini, G., Angeles, J.: The synthesis of the pitch surfaces of internal and external skew-gears and their racks. J. Mech. Des. Trans. ASME 128(4), 794–802 (2006)

Low Cost Delta Robot for the Experimental Validation of Frame Vibration Reduction Methods Christian Mirz1(B) , Mathias H¨ using1 , Yukio Takeda2 , and Burkhard Corves1 1

2

Institute for Mechanism Theory, Machine Dynamics and Robotics, RWTH-Aachen University, Aachen, Germany [email protected] Mechanical Systems Design Laboratory, Tokyo Institute of Technology, Tokyo, Japan

Abstract. Dynamic manipulation tasks require lightweight yet stiff robotic systems, to achieve high efficiency. The most widely used parallel robot designed for these tasks is the Delta robot. However, in many industrial applications these robots must be operated at reduced speed to allow frame vibrations to decay and hereby, prevent unacceptably high vibrations. This paper presents a low cost Delta robot test bench designed and build to test vibration reduction methods in an industrial pick and place scenario.

1

Introduction

One of the main applications of industrial robots is in the handling of objects, especially in pick and place tasks. To enable economical operation, short cycle times and large handling capacities are targeted. A typical application is the loading and unloading of conveyors. Such highly dynamic handling tasks require a high payload to moving mass ratio, high positioning accuracy and excellent stiffness properties. Delta robots meet these requirements due to their parallel architecture with actuators stationary mounted on the base and thus low moving masses (Brinker 2018). Since the invention of the Delta robot by Clavel, a variety of optimization strategies have been proposed for the dimensional synthesis, pushing the cycle times to a minimum (Brinker 2018). However, in many industrial applications Delta robots must be operated at reduced speed or dwell times have to be included in the motion planning, to allow frame vibrations to decay and hereby, prevent unacceptably high vibrations to build up (Van der Wijk and Herder 2009). As a result, the full potential of Delta robots cannot be exploited. These vibrations can be explained by the high accelerations required to achieve the desired short cycle times, causing high inertia forces and torques. As illustrated by the following example, their effect is exacerbated by the mostly slim and This work is supported by the DAAD with funds from the German Foreign Office. c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 332–339, 2022. https://doi.org/10.1007/978-3-031-06409-8_35

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tall structure of the frame. Figure 1 shows the first four eigenmodes and natural frequencies of a standard robot frame with a footprint of 1.4 m × 1.4 m and a height of 2 m. In industrial applications the frames are typically bolted to a thick steel base plate, which in turn is fixed to the foundation using chemical anchor bolts. Thus the frame can be assumed to be fixed ideally stiff to the foundation.

1. Eigenmode 2. Eigenmode 3. Eigenmode 4. Eigenmode 132.9 Hz 31.7 Hz 31.8 Hz 49.2 Hz

Fig. 1. Eigenmodes and natural frequencies of a typical Delta robot frame

As expected, the natural frequencies in the horizontal plane are much lower than the natural frequencies in the vertical direction. In combination with the mostly horizontal robot motions, especially in pick and place tasks, these characteristics render the frame prone to lateral vibrations. Nevertheless, such a structure is necessary, since Delta robots are generally suspended over conveyor belts. Furthermore, there are limits to the stiffening of the frames, given installation space constraints. To address this problem, vibration reduction methods such as dynamic balancing or active vibration damping shall be compared and validated on basis of experiments. To this end, a low cost Delta robot called the Suisui Bot has been designed, which will be presented following a brief overview of the most common vibration reduction methods. Finally, experiments to access the characteristics of the Suisui Bot will be shown.

2

Vibration Reduction and Its Application to Delta Robots

Methods for the reduction of frame vibrations may be classified as software, hardware and hybrid approaches. Software based approaches comprise trajectory planning methods which aim either for the reduction of frame forces, minimizing the excitation of natural frequencies and optimal damping of frame vibrations. Dynamic mass balancing represents a hardware based approach, whereas active mass balancing may be considered as a hybrid approach. Starting with software based approaches, a brief overview of the most relevant methods shall be given below. Typical pick and place trajectories involve a vertical upstroke at the pick position, a horizontal translation and a vertical downstroke at the place position.

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Instantaneous transition from the vertical to the horizontal motion would lead to acceleration discontinuities, which results in mechanical vibrations (Barnard et al. 2012). Therefore, it is desirable to generate trajectories with continuous acceleration profiles and limited jerk (Gasparetto et al. 2015). To this end, Clothoids, Lam´e curves or Splines are used to smooth the transition from the vertical to the horizontal motion in the geometric path (Barnard et al. 2012; Gauthier et al. 2008). In general, these approaches do not take into account the properties of the frame. Cheng and Li (2018) present a method to include the frequency response of the robot and its frame into a trajectory optimization. Here, the goal is to minimize the cycle time and maximum frame acceleration by optimizing the parameters of a jerk-limited acceleration profile to avoid the excitation of natural frequencies. Strategies to obtain optimal damping of frame vibrations comprise e.g., input shaping and optimal control based path planning (Auer et al. 2021). Trajectory planning methods aimed at the minimization of frame forces, rely on the optimal control of the robot’s centre of gravity (COG), e.g., by minimization of the COG acceleration as in (Briot et al. 2012). The reaction null space control presented by Nenchev et al. (1999) represents a generalization of this concept, which allows base vibration suppression, minimal vibration induction and end effector control in the presence of frame vibrations. However, a necessary condition to achieve both end effector control and vibration suppression is the existence of kinematic or dynamic redundancy, which is not met by Delta robots. Due to the lack of redundancy, the COG and end effector position cannot be controlled independently. Thus, trajectory planning methods achieve vibration minimization at best. Only dynamic mass balancing can achieve vibration prevention. Van der Wijk and Herder (2009) present a dynamically balanced Delta robot with respect to inertial forces, by adding three balancing masses and two additional links. As a result the mass of the robot increases approximately threefold. This is accompanied by an increase of the driving torques. The kinematics and the balancing masses have to be designed application-specific and cannot be adapted during the operation. Here, the use of an active balancing mechanism offers the advantage that it can be adapted in software to meet a wide range of tasks. In addition, the handling object’s mass can be taken into account, which offers advantages especially for handling tasks with heavy objects.

3

The Suisui Bot

The Suisui Bot (Fig. 2) is a classical Delta robot with three degrees of freedom, designed to allow software and hardware based vibration reduction methods to be readily implemented. It consists of three identical R(SS)2 kinematic chains, connecting the moving platform with the basis. Here, R denotes a revolute joint and S a spherical joint. It has a cylindrical workspace of 300 mm in diameter and a height of 100 mm. The geometry of the robot was determined using a kinematics and dynamics optimization proposed by the authors in (Mirz et al. 2021). The robot is driven by SCA5618L2804 stepper motors from Nanotec Electronic

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equipped with planetary gearboxes. To prevent step losses the controller run in closed loop operation using rotary encoders.

Spring Steel Rods Proximal Link Distal Link

Rigid Frame Base Camera Vacuum Suction Cup Turntable

Fig. 2. Setup of the Suisui Bot Delta Robot

To simulate an industrial pick and place scenario, a turntable is placed underneath the robot. Orange steel discs of different masses are used as handling objects. The picking is performed using a vacuum suction gripper. A Logitech C930 webcam installed next to the robot is used to detect handling objects on the turntable. The robot is mounted on a rigid frame base made of aluminum profiles, which in turn is connected to the foundation by four supports made of spring steel rods. Qualitatively, this design leads to the same vibration behavior as industrial robot frames, with low natural frequencies in the horizontal plane and high natural frequencies in the vertical direction, but at the same time it amplifies the vibration amplitudes, rendering it an ideal platform for the analysis vibration reduction methods. To obtain a realistic operation scenario, emulating the unloading of a conveyor belt, three areas are defined on the turntable (see Fig. 3(b)); a picking, a placement and an object detection area. These areas are characterized by polar coordinates, represented by their inner (rpl,i , rpi,i ) and outer (rpl,o , rpi,o ) radii and angles (α, β). Within these areas the pick and place positions and the motion time are varied using the quasi-randomized Sobol sequencer. The trajectories are based on the Adept-Cycle with 25 mm up- and down-stroke. The horizontal translation is defined by the pick and place positions. Following Gauthier et al. (2008) the geometrical path is smoothed at the transitions using Lam´e curves and combined with a seventh-order polynomial motion profile. The control system of the robot consists of the camera for object detection, a computer on which all calculations are performed and two microcontrollers for the rotary table and the robot. It is implemented using the Robot Operating System (ROS) 2 environment, giving access to most C++ libraries and therefore facilitating the implementation of software based vibration reduction methods. Four ROS nodes are implemented each for one of the following tasks; communication between the micro-controllers and the computer, trajectory planning and object detection. As the most relevant node, the latter shall now be presented in more detail.

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(b) Sectioning of the Turntable

Fig. 3. (a) shows two handling objects (orange) and their centroid position as identified by the object detection algorithm. (b) shows the sectioning of the turntable as used for the trajectory planning.

The object detection is realized based on the OpenCV library. Since the camera captures not only the turntable, but some of the surrounding area as well (Fig. 3(a)), the object detection has to be sufficiently stable to be unaffected by possible artifacts, motions or other objects that appear on the images. There are several methods to detect objects, which may be divided into neural networkbased or non-neural approaches. Traditional non-neural approaches consist of three main steps, the region selection, feature extraction and the classification (Zhao et al. 2019). In contrast to industrial applications, in the presented scenario there is no need to detect objects with multiple colors and complex shapes. The handling object’s color and shape can be adapted to facilitate the object detection. Thus, a bright orange color is chosen, which contrasts strongly with the surroundings. Hereby, the object detection can be performed purely colorbased and further region selection, feature extraction and classification become obsolete. By applying a color filter a binary image is obtained. Subsequently, morphological noise removal is performed to remove any artifacts in the picture. Retrieving the contours of the handling objects in the binary image is done using the contour finding method provided by the open source computer vision library OpenCV, which is based on the algorithm proposed by Suzuki and Abe (1985). If still smaller artifacts have not yet been removed by the morphological processing, these are sorted out directly by the size of their associated edges. Finally, the handling objects centroid position is calculated from the recognized contours. Using a projection through the pinhole camera model, the centroid positions in the camera coordinate system are transferred into the coordinate system of the robot.

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Experimental Results

ZTCP [mm] YTCP [mm]

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Experiments have been performed to analyze the Suisui Bot’s characteristics. The object detection system consistently detects all discs on the turntable and locates them within one millimeter precision. The minimal motion time that can be achieved in a pick and place operation with an up- and downstroke of 25 mm and 250 mm translation is 0.25 s, limited only by the latency of the vacuum gripper. This latency is caused by the evacuation and ventilation process of the vacuum hose. Without gripping, the minimal motion time that has been successfully tested in the same scenario is 0.2 s. To access the accuracy of the robot, measurements have been performed using a Nikon K600 optical coordinate measuring machine (CMM) with a volumetric accuracy of 90 µm. Its capability to measure multiple 3D targets, marked by infrared LED markers, allows to measure the motion of the tool center point (TCP) relatively to the rigid frame base, thus eliminating the effects of the relatively compliant supports made of spring steel rods. 0 −20 −40 −60 100 50 0 −50 −100 −300 −310 −320 −330 −340

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In total, nine measurements have been performed with pick and place positions defined by a Sobol sequence and cycle times between 3.25 s to 3.35 s. To evaluate the accuracy after multiple runs, one these measurements was taken 50 and another one 100 pick and place cycles after the last homing. A loss in accuracy could not be observed. The average error over all trajectories is 0.78 mm in the X-axis, 0.96 mm in the Y-axis and 0.80 mm in the Z-axis. The average path following error is 1.71 mm. As it can be seen from the exemplary measurement depicted in Fig. 4, especially the transitions from the vertical to the horizontal motion and vice versa, are subject to errors. The prominent points like the pick

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and place positions are met at much higher precision. It should be noted, that the resolution of the stepper motors is limiting the maximal accuracy which can be achieved (about ±0.5 mm, depending on the position of the TCP). Reliable operation is achieved in scenarios, with one or up to three handling objects. In the more complex operation with multiple handling objects, the control system is programmed to react dynamically to situations, where another object is at the place position defined by the Sobol sequence or where the pick position can no longer be reached in time. In both cases the pick and place positions are adjusted and the operation can be maintained.

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Conclusion

This paper presents a low cost Delta robot test bench designed to compare and validate vibration reduction methods in an industrial pick and place scenario, emulating the unloading of a conveyor. Handling objects, represented by steel discs are reliably detected by a camera based object detection and handled using a vacuum gripper. Measurements and experiments have shown, that a reliable operation of the overall system can be achieved. As intended, the frame vibrates prominently visible during operation.

References Auer, T., Kronthaler, P., Woittennek, F.: Optimal control approach for an elastically coupled gantry system with experimental results. In: IFToMM D-A-CH, Universit¨ at Duisburg-Essen (2021). https://doi.org/10.17185/duepublico/74048 Barnard, C.J., Briot, S., Caro, S.: Trajectory generation for high speed pick and place robots. In: Proceedings of the ASME 11th Biennial Conference on Engineering Systems Design and Analysis, Nantes, France, pp. 165–174 (2012) Brinker, J.: Optimal design of functionally extended parallel robots with Delta-like architecture. Dissertation, RWTH Aachen University (2018) Briot, S., Arakelian, V., Le Baron, J.-P.: Shaking force minimization of high-speed robots via centre of mass acceleration control. Mech. Mach. Theory 57, 1–12 (2012). ISSN 0094114X Cheng, H., Li, W.: Reducing the frame vibration of delta robot in pick and place application: an acceleration profile optimization approach. Shock Vibr. 2018(3), 1– 15 (2018). ISSN 1070–9622 Gasparetto, A., Boscariol, P., Lanzutti, A., Vidoni, R.: Path planning and trajectory planning algorithms: a general overview. In: Carbone, G., Gomez-Bravo, F. (eds.) Motion and Operation Planning of Robotic Systems. MMS, vol. 29, pp. 3–27. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-14705-5 1 Gauthier, J.F., Angeles, J., Nokleby, S.: Optimization of a test trajectory for SCARA systems. In: Lenarcic, J., Wenger, P. (eds.) Advances in Robot Kinematics: Analysis and Design, vol. 18, pp. 225–234. Springer, Dordrecht (2008). https://doi.org/10. 1007/978-1-4020-8600-7 24. ISBN 978-1-4020-8599-4 Mirz, C., Uzsynski, O., Angeles, J., Takeda, Y., Corves, B.: Stiffness optimization of delta robots. In: Venture, G., Solis, J., Takeda, Y., Konno, A. (eds.) ROMANSY 2020. CICMS, vol. 601, pp. 396–404. Springer, Cham (2021). https://doi.org/10. 1007/978-3-030-58380-4 48

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Nenchev, D.N., Yoshida, K., Vichitkulsawat, P., Uchiyama, M.: Reaction null-space control of flexible structure mounted manipulator systems. IEEE Trans. Robot. Autom. 15(6), 1011–1023 (1999) Suzuki, S., Abe, K.: Topological structural analysis of digitized binary images by border following. Comput. Vis. Graph. Image Process. 30, 32–46 (1985). https://doi.org/ 10.1016/0734-189X(85)90016-7 van der Wijk, V., Herder, J.L.: Dynamic balancing of Clavel’s delta robot. In: Kecskem´ethy, A., M¨ uller, A. (eds.) Computational Kinematics, vol. 3, pp. 315–322. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01947-0 39 Zhao, Z.-Q., Zheng, P., Xu, S.-T., Wu, X.: Object detection with deep learning: a review. IEEE Trans. Neural Netw. Learn. Syst. 30(11), 3212–3232 (2019)

Influence of the Receiving Timing of an Online Lecture in Actual Practical Mechanical Engineering Training Koji Makino1(B) , Yasutake Haramiishi1 , Chaofeng Zhang2 , and Hiroshi Hashimoto2 1

2

University of Yamanashi, Kofu, Yamanashi, Japan [email protected] Advanced Institute of Industrial Technology, Shinagawa, Tokyo, Japan

Abstract. It is important to realize an online education format in which participants can attend a lecture at any time and in any place. One of the reasons for the need for this type of education is that during the COVID-19 pandemic, it has become necessary to avoid the gathering of participants. It is easy to apply digital transformation (DX) to online education. There are problems in realizing the online education of practical mechanical engineering training, such as milling machines. It may be difficult for participants to understand the purpose of mechanical tools and machines if they do not have experience with them. This paper describes the influence of the timing of online education in practical mechanical engineering training. To investigate this topic, students are separated into two groups: one group receives the online lecture before actual practical training, while the other group receives the lecture after actual practical training. The influence of the timing of the online lecture is considered using a questionnaire. As a result, it is clear that online lectures are important and efficient if students receive online lectures before, rather than after, actual practical training. We confirm that the difficulty and length of such training should be carefully planned. Keywords: Engineering education

1

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Introduction

Online education has become an important focus since it allows for participants, such as students, to receive lectures in any place and at any time [1–3]. In addition, the importance of online education has become stronger during the COVID-19 pandemic to avoid the gathering of participants [4–6]. Furthermore, digital transformation (DX), which adopts disruptive technologies to increase productivity, value creation, and social welfare [7,8], has been applied to online education [9]. Many online education methods have been proposed, studied and developed [10]. Online education formats, such as experiments, practical training, and exercises, which can be used in various areas, has become increasingly demanded; c CISM International Centre for Mechanical Sciences 2022  A. Kecskem´ ethy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 340–349, 2022. https://doi.org/10.1007/978-3-031-06409-8_36

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however, text-based knowledge is dealt with in almost all online education. Moreover, in practical mechanical engineering training, such as milling machines, metal casting (the melted metal is inserted into the mold), and forging (the heated metal is hit to transform the shape), experience using real tools and machines is necessary to understand such training since senses, such as touch, are important to learn how to use these new tools [11,12]. Therefore, it is difficult to provide online education using mechanical tools and machines. The functions of mechanical tools and machines, their method of usage and the practical training procedure are often explained before participants perform practical training. Part of this explanation of practical mechanical engineering training should be converted to an online format [13–17]. It is generally desirable for participants to receive online education before they receive practical training. However, it is difficult for them to understand the ideas behind mechanical tools and machines in most cases if they do not have experience with them because they are not able to imagine the tools and machines. It is necessary to investigate the influence of whether participants receive online education before or after practical training. This paper describes the influence of the timing of online education in practical mechanical engineering training. We separate students who attend the practical training in our university department into two groups: one group receives the online lecture before actual practical training, while the other group receives the online lecture after actual practical training. In Sect. 2, the concept of practical training is described. The questionnaire is administered to investigate the influence of the receiving timing. In Sect. 3, the results of the questionnaire are considered. An improved method using 3D modeling is introduced in Sect. 4. Finally, Sect. 5 concludes the paper.

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Practical Training

Prior to the COVID-19 pandemic, practical training in our university department did not utilize online lectures. Under COVID-19, online lectures have become necessary to avoid the gathering of students. In this section, the differences in practical training before and after the COVID-19 pandemic are described. Practical training was provided to students of the Department of Mechanical Engineering at the University of Yamanashi, who were working toward their 2nd degrees, and was composed of 11 themes (turning machine (lathe), milling machine, CAD/CAM&MC, electric discharge machining (EDM), welding, forging, grinding machine, polishing, metal casting, cutting & drilling, and hand finishing). The number of practical training sessions for each theme was different (1, 2, or 4). Students were classified into 12 groups, and the number of students in each group was approximately 5. In all themes except the grinding machine and polishing, students in both groups attended the same practical training together prior to COVID-19. Therefore, in 2016, the number of students in each theme, except the abovementioned two themes, was 10, as shown in Fig. 1. To avoid gatherings, the number of students who attended the same theme was changed to less than 5 in 2020 under COVID-19, as shown in Fig. 1.

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Therefore, at the first day, five students received actual training, while the other five students received the online lecture instead of actual training. And second day, the role was changed. The five students who received actual training at first day received the online lecture at second day.

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Fig. 1. Practical training in 2016 and 2021.

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Influence of Receiving Timing

The online lectures for practical training emerged as an alternative to actual practical training since the number of participants had to be reduced to avoid the gathering of students. In this section, the influence of the receiving timing of the online lecture is considered using the questionnaire. The questionnaire items are as follows: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15

Interest in actual practical training Enjoyment from actual practical training Positive view of actual practical training Understanding actual practical training Satisfaction from actual practical training Use of an online lecture Concentrating in the online lecture Problem solution in the online lecture Understanding the online lecture Effectiveness of the online lecture Time of the online lecture Satisfaction from the online lecture Receiving timing Repeat watching of the online lecture Satisfaction of the practical training including the online lecture

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Questionnaire items Q1 to Q5 are questions related to actual practical training, Q6 to Q12 are questions related to the online lecture, and Q13 to Q15 are other questions. Students answered each item with a score ranging from 1 to 5. Scores of 1, 3 and 5 indicate no satisfaction, normal satisfaction and satisfaction, respectively. The questionnaire was completed on the internet using Google Forms on July 10–16, 2021. The number of students in practical training in 2020 was 60, and the number of responses received was 50 (83 %). Students were separated into two groups; the group of the students who received the online lecture after actual practical training (SA), the other group of the students who received online lecture before the actual practical training (SB). First, the average values of each item are shown in Fig. 2. The horizontal and vertical axes indicate each item of the questionnaire and the average value, respectively. The gray and black legends depict the values obtained by the SA and SB, respectively. The satisfaction of the items (Q2, Q3, and Q4) related to the actual practical training of SB was higher than that of SA, while the other items were almost same. It is clear that satisfaction increases if the explanation of the practical training is received before it is actually carried out.

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Next, the correlation between each item is shown in Fig. 3. Figures 3(a) and (b) show the correlation of the SA and SB answers, respectively. Cells with correlation values greater than 0.7 and 0.4 are drawn in red and yellow, respectively. Values greater than 0.7 and 0.4 depict that there is strong and weak correlation, respectively. The correlations of the items from Q1 to Q5, which are related to actual practical training, range from almost weak and strongly correlated. More than half of the items from Q6 to Q12, which are related to the online lecture, are weakly correlated. There are few correlations between the items related to actual practical training and the online lecture. These trends are the same for SA and SB. Here, the correlation of the Q15 column in the correlation table is addressed. The correlation between the items related to actual practical training (Q1, Q2,

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Q3, Q4, and Q5) and that between those related to satisfaction from practical training including the online lecture (Q15) of SA and SB are different. In the SA correlation table, as shown in Fig. 3(b), the correlation is strong since its value is greater than 0.7. Conversely, the correlation values are almost less than 0.4 in the SB correlation table, as shown in Fig. 3(a). The reason for these different results is discussed. The satisfaction of SA depends on actual practical training and is not related to the online lecture. Therefore, we hypothesize that the online lecture may not be very useful for SA. Moreover, the item of the satisfaction (Q15) of SB is not related to almost all items. We hypothesize that the satisfaction of SB influences both actual practical training and the online lecture. As a result, it is desirable for the online lecture to be received before actual practical training, and prior knowledge from the online lecture is helpful to students for understanding actual practical training.

(a) SA (after actual practical training)

(b) SB (before actual practical training)

Fig. 3. Correlation values of items.

Finally, the responses to the questionnaire were analyzed using portfolio analysis, as shown in Fig. 4. Here, the idea behind portfolio analysis is explained briefly. In portfolio analysis, the horizontal axis denotes importance, and the vertical axis denotes satisfaction. The importance is calculated from the

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correlation between the satisfaction level of each item and the overall satisfaction level. And the satisfaction is the ratio of the sum of very good (score 5) and good (score 4). Moreover, both values are converted to adjusted standard deviation score. Each quadrant in the graph has a different meaning. The first and second quadrants are priority maintenance items and maintenance items, respectively. It is unnecessary to improve these items; however, the items in the first quadrants are important for respondents in completing the questionnaire. Therefore, the quality of the lecture or practical training should be maintained, although the item does not need to be improved initially. Conversely, the third and fourth quadrants are improvement and priority improvement items, respectively. Items in the fourth quadrant should be improved as soon as possible. In particular, the item for which its length from the origin in the fourth quadrant is farthest should be improved as a priority. Then, the results are discussed below. The priority maintenance items placed in the 1st quadrant are important in portfolio analysis, and in both figures, they are Q1, Q3, and Q5, which are related to actual practical training. Q2 and Q3 are placed in the 2nd quadrant, which means that these items are maintained. Therefore, actual practical training exhibits good performance, and there are no problems of quality. Moreover, the priority improvement items placed in the 4th quadrant are addressed. In Fig. 4(a), which shows the result of SA, there are no priority improvement items placed in the 4th quadrant. This result is desirable since there are no items that should be improved as a priority. In Fig. 4(b), which shows the result of SB, Q9, Q10, and Q12 are placed in the 4th quadrant and are related to the online lecture. In particular, Q9 and Q10 are items related to the understanding of the online lecture, and Q12 is an item related to satisfaction with the online lecture. We hypothesize that SB is not satisfied with the online lecture since it is difficult to understand. Therefore, it is important to improve the online lecture more easily to allow for it to be better understood. Furthermore, Q11 is placed in the 3rd quadrant in both figures, and satisfaction is lowest among all items. Q11 is related to length of the online lecture. It is desirable that the length of the online lecture be shorter if possible. It is clear that online lectures are important and efficient if students receive them before actual practical training. However, their difficulty and length should be carefully considered.

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Teaching Material to Aid in Understanding

From the above section, it is clear that teaching materials are necessary to aid in the understanding of actual practical training. It is difficult for students to imagine the shape of the tools used in actual practical training since many tools are not general and are not present in normal life. For example, shovels are addressed in this section. In fact, shovels are used in practical training for forging and welding; however, there are various shovel shapes, as shown in Fig. 5.

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Fig. 5. Shape of shovel used in practical training for welding and forging.

We make a 3D model of tools to aid students in understanding them. Figure 6 shows the 3D model of the shovel used in practical training for forging. Students can move the 3D image using the mouse or tapping on the screen.

Fig. 6. 3D model of the shovel used in practical training for forging.

However, the above example may not be effective since a shovel has a very fundamental shape. Complex tools are used in practical training, such as turning and milling machines as shown in Fig. 7.

(a) turning machine

(b) milling machine

Fig. 7. Complex tools used in practical training.

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Conclusions

This paper describes the influence of the timing of online lectures, which are performed either before or after actual practical training. In our university department, because of the COVID-19 pandemic, the number of students taking part in practical training had to be decreased to expand the distance between students. Therefore, students were separated into two groups; the group of the students who received the online lecture after actual practical training (SA), the other group of the students who received online lecture before the actual practical training (SB). The concept of practical training was described. Then, the results of the questionnaires were considered. Moreover, the questionnaire items that were scored as 5 are discussed below. The items are classified into three categories: questionnaires for actual practical training, questionnaires for online lectures, and questionnaires for practical training. First, the average of each group was considered. The scores of the items related to the actual practical training of SB were greater than those of SA. Moreover, the correlation values were considered. As a result, it is desirable for the online lecture to be received before actual practical training, and prior knowledge from the online lecture is helpful for aiding students in understanding actual practical training. In addition, through portfolio analysis, it is clear that it is difficult for SB to understand online lectures. Consequently, we confirm that it is important to improve the online lecture, particularly by shortening its length. It is clear that online lectures are important and efficient if students receive them before practical training. However, their difficulty and length should be considered carefully. Finally, a 3D model for helping students understand the online lecture is constructed since the tool used in actual practical training is not general, and thus, it is difficult for students to imagine its shape. We hypothesize that it is easier for the students to understand the online lecture if it includes a 3D model. We will improve the online lecture using DX.

References 1. Britt, R.: Online education: a survey of faculty and students. Radiol. Technol. 77(3), 183–190 (2006) 2. Allen, I.E., Seaman, J.: Changing Course: Ten Years of Tracking Online Education in the United States, Babson Survey Research Group (2013) 3. Begum, D.: Re-engineering education through digitalization. In: Proceedings of International Conference on Future Dimensions in Higher Education: A Quantum Leap in Technology, pp. 68–76 (2020) 4. Demuyakor, J.: Coronavirus (COVID-19) and online learning in higher institutions of education: a survey of the perceptions of Ghanaian international students in China. Online J. Commun. Media Technol. 10(3) (2020) 5. Chaturvedi, K., Vishwakarma, D.K., Singh, N.: COVID-19 and its impact on education, social life and mental health of students: a survey. Child. Youth Serv. Rev. 121, 105866 (2021)

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6. Tantawi, K.H., Fidan, I., Chitiyo, G., Cossette, M.: Offering hands-on manufacturing workshops through distance learning. In: 2021 ASEE Virtual Annual Conference, Paper ID #34923, July 2021 7. Ebert, C., Henrique, C., Duarte, C.: Digital transformation. IEEE Softw. 35(4), 16–21 (2018) 8. Verhoef, P.C., et al.: Digital transformation: a multidisciplinary reflection and research agenda. J. Bus. Res. 122, 889–901 (2021) 9. Bogdandy, B., Tamas, J., Toth, Z.: Digital transformation in education during COVID-19: a case study. In: 11th IEEE International Conference on Cognitive Infocommunications (CogInfoCom), pp. 173–178, September 2020 10. Harasim, L.: Shift happens: online education as a new paradigm in learning. Internet High. Educ. 3(1), 41–61 (2001) 11. Gyllen, J., Stahovich, T., Mayer, R.: How students read an e-textbook in an engineering course. J. Comput. Assist. Learn. 34(6), 701–712 (2018) 12. Villegas, D.F., Sanchez, H.G., Riberos, C.A.: Strategies for the teaching-learning experiences in the engineering dynamics course based on the information and communication technologies. J. Phys. Conf. Ser. 1161, 012011 (2019) 13. Yang, D., Pakala, K.: Building an effective online thermodynamics course for undergraduate engineering students. In: ASEE Annual Conference & Exposition, Paper ID #20333 (2017) 14. Tian, M.: Research and development of NC intelligent online teaching system. In: 2018 International Symposium on Computer, Consumer and Control (IS3C), pp. 404–407, December 2018 15. Libre, N.A.: Affordable learning solutions and interactive content in engineering mechanics. In: ASEE Annual Conference & Exposition, Paper ID #25644, June 2019 16. Garc´ıa-Alberti, M., Su´ arez, F., Chiy´ 0n, I., Feijoo, J.C.M.: Challenges and experiences of online evaluation in courses of civil engineering during the lockdown learning due to the COVID-19 pandemic. Educ. Sci. 11, 59 (2021) 17. Zhang, P.: Study on inquiry blended teaching mode in manufacturing technology foundation course. In: 2nd Information Communication Technologies Conference (ICTC), pp. 329–332. IEEE, May 2021

Author Index

A Agostini, Lorenzo, 142, 151 Allotta, Benedetto, 70 Andras, Iulia, 115 Antonov, Anton, 246 Aoustin, Yannick, 199 Arakelian, Vigen, 199 Arsicault, Marc, 292 Avallone, Giulia, 142, 151 B Bagneschi, Tommaso, 106 Baldassarri, Alberto, 220 Bartalucci, Lorenzo, 70 Berti, Alessandro, 283 Botta, Andrea, 53 Brogi, Chiara, 70 Bruckmann, Tobias, 220 C Cao, Lei, 91 Carbone, Giuseppe, 255 Caro, Stéphane, 272 Carricato, Marco, 220, 283 Chablat, Damien, 311 Chen, Yi, 151 Cherkasov, Vladislav V., 255 Chiaradia, Domenico, 106 Chumichev, Mikhail, 99 Coelho, Joana, 191 Conconi, Michele, 142 Corves, Burkhard, 332

D D’Angelo, Michele, 238 Di Paola, Vincenzo, 272 Dias, Bruno, 191 Diepenbruck, Jens, 220 Dobashi, Yuta, 43 E Ebel, Henrik, 163 Eberhard, Peter, 163 Eugster, Manuela, 37 F Fahse, Daniel Niklas, 163 Fasel, Lorin, 37 Figliolini, Giorgio, 238 Flores, Paulo, 79, 191 Fomin, Alexey, 246 Frisoli, Antonio, 106 G Gattringer, Hubert, 210 Gerig, Nicolas, 37 Gherman, Bogdan, 115 Ghiassi, Mehdi, 133 Girgenti, Lorenzo, 263 Glazunov, Victor, 246 Gonçalves, Fernando, 79 Gosselin, Clément, 3 Grube, Malte, 125 Guagliumi, Luca, 283

© CISM International Centre for Mechanical Sciences 2022 A. Kecskeméthy and V. Parenti-Castelli (Eds.): ROMANSY 2022, CISM 606, pp. 351–352, 2022. https://doi.org/10.1007/978-3-031-06409-8

352 H Haramiishi, Yasutake, 340 Hashimoto, Hiroshi, 340 Hüsing, Mathias, 332 I Idà, Edoardo, 272 Irlinger, Franz, 12 K Kahraman, Oguz, 62 Karaki, Masato, 43 Kawakami, Yasuo, 231 Kecskeméthy, Andrés, 133 Klein, Corina, 182 Kuroiwa, Yuji, 231 L Lanni, Chiara, 238 Laribi, Med Amine, 292 Laryushkin, Pavel, 246 Leonardis, Daniele, 106 Lim, Hun-ok, 231 Liu, Ying-Chi, 53 Lopes, Gil, 79, 191 Lu, Yaodong, 199 Lueth, Tim, 12 M Maibaum, Joey, 133 Makino, Koji, 43, 340 Mattacchione, Riccardo, 311 Meskini, Majdi, 292 Mineshita, Hiroki, 231 Mirz, Christian, 332 Mlika, Abdelfattah, 292 Monari, Eugenio, 151 Monti, Eros, 283 Morara, Giuliano, 220 Müller, Andreas, 210

Author Index P Pagliai, Marco, 70 Parenti-Castelli, Vincenzo, 91, 142, 322 Petinari, Andrea, 91 Pisla, Doina, 115 Q Quaglia, Giuseppe, 53 R Rauter, Georg, 37 Ribeiro, António Fernando, 79 Ribeiro, Fernando, 191 Ribeiro, Tiago, 79 Ridolfi, Alessandro, 70 Rindi, Andrea, 70 Rixen, Daniel, 182 Rosenfelder, Mario, 163 Rybak, Larisa A., 255 S Saafi, Houssem, 292 Sakaguchi, Masanori, 231 Sancisi, Nicola, 91, 142, 151 Secciani, Nicola, 70 Seifried, Robert, 125 Sun, Xiao, 43 T Takanishi, Atsuo, 231 Takeda, Yukio, 53, 91, 332 Terada, Hidetsugu, 43 Tucan, Paul, 115 V Vaida, Calin, 115 van der Wijk, Volkert, 263 Vertechy, Rocco, 142, 151, 322 Voloshkin, Artem A., 255 Vuocolo, Antonello, 91

N Notash, Leila, 301

W Wittmann, Jonas, 182 Wojtyra, Marek, 171 Woli´nski, Łukasz, 171

O Olivoni, Enea, 322 Orlov, Igor, 99 Otani, Takuya, 231 Ottaviano, Erika, 311

Z Zeghloul, Saïd, 292 Zhang, Chaofeng, 340 Zielinska, Teresa, 62 Zoppi, Matteo, 272