Communications and Control Engineering Series Editors: A. Fettweis. 1. L. Massey 1. W Modestino M. Thoma

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2 Communications and Control Engineering Series Editors: A. Fettweis. 1. L. Massey 1. W Modestino M. Thoma

3 Scientific Fundamentals of Robotics 7 M.Vukobratovie B.Borovac D. Surla. D. Stokie Biped Locomotion Dynamics, Stability, Control and Application With 136 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

4 Professor MIOMIR VUKOBRATOVIC, D. Sc., Ph. D. Corr. member of Serbian Academy of Sciences and Arts Foreign member of Soviet Academy of Scienes Institute»Mihailo Pupin«, Beograd, Volgina 15, POB 15, Yugoslavia Assoc. professor BRANISLA V BOROVAC, Ph. D. Faculty of Technical Scienes University of Novi Sad, Yugoslavia Assoc. professor DUSAN SURLA, Ph. D. Mathematical Institute University of No vi Sad, Yugoslavia Assoc. professor DRAGAN STOKIC, Ph. D. Institute»Mihailo Pupin«, Beograd, Volgina 15, POB 15, Yugoslavia ISBN-13: e-isbn-13: DOl: / Library of Congress Cataloging-in-Publication Data Biped locomotion: dynamics, stability, control and application / M. Vukobratovii:... let al.]. (Scientific fundamental of robotics: 7) (Communications and control engineering series) Translated from the Serbo-Croatian (Cyrillic). Includes bibliographical references. ISBN-13 : I. Robots--Dynamics. 2. Human locomotion. I. Vukobratovii:, Miomir. II. Series. III. Series: Communications and control engineering series. T121l.4.B '92--dc This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms orin other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provision of the German Copyright Law of September9, 1965, in its version oflune 24, 1985,and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin Heidelberg 1990 Softcover reprint of the hardcover 1st edition 1990 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence ofa specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Printed on acid-free paper

5 Preface Fourteen years ago, the "Mihajlo Pupin" Institute published in English a comprehensive research monograph by M. Vukobratovic under the title LEGGED LOCOMOTION ROBOTS AND ANTHROP0110RPHIC MECHANISMS. In this monograph were presented results of the seven-year work of a small research group in the Biocybernetics Department of the Institute "Mihajlo Pupin", as well as results of researchers from several other centres in the world. In 1976, the monograph was published in Japanese and Russian, and in 1983 in the Chinese language. The period of vigorous research in the domain of mathematical modelling and motion synthesis of the legged locomotion mechanisms and machines which may be considered as the pre-manipulation era, was primarily concerned with the applications in the rehabilitation of disabled people and with the problems of specific (legged) transport. It is true, however, that in the "locomotion era" of robotics various manipulation systems had been developed, but only those aimed at the rehabilitation of severely disabled people. It should be noticed, for example, that the first journal articles published in English were devoted to the locomotion control, especially of anthropomorphic mechanisms. He shall mention here only two of them; one, "Contribution to the Synthesis of Biped Gait" by Vukobratovic M., and Juricic D., published in IEEE Trans. on Bio-J.iedical Engineering, Vol.. 16, No.1, 1969, the first paper on mathematical modelling of biped gait, and the other, "Some Considerations Relating to the Design of Autopilots for Legged Vehicles", by Frank A.A., and HcGhee R.B., published in Journal of Terramechanics, Vol. 6, No.1, In these and several other articles, cited in particular chapters of this book, a number of questions have been raised and some of them solved that were related to the essence of the activity which has later been established as the scientific-technical discipline under the general name ROBOTICS. Thus, in the course of studies of locomotion mechanisms, synthesis and stabilization of artificial gait, and the realization of rehabilitation devices, the appropriate mathematical procedures have been developed for computer-forming of mathematical models

6 VI of the dynamics of complex active spatial mechanisms which are usually encountered in the systems of anthropomorphic structure. The application of computer methods for automatic forming of locomotion mechanisms models have coincided with the beginning of the development of these methods for the needs of the open-chain manipulation mechanisms. In parallel with the computer-oriented procedures for forming dynamic models of active spatial mechanisms, the locomotion mechanisms, and the anthropomorphic ones in particular, have induced genuine needs for dynamic control, and this has also been extended on the domain of control of manipulation robots for the industrial and other purposes. Thus, with the aim of solving the problem of dynamic equilibrium of the gait in a perturbed regime of the initial condition type and of small parameters variation, the force feedback mechanisms based on dynamic reactions at the points of contact of the mechanism's foot and ground have been for the first time indicated (Vukobratovic M., Juricic D., Frank A.A., "On the Control and Stability of One Class of Biped Locomotion Systems, Trans. of the ASME, June, 1970) and introduced (Vukobratovic M., Stepanenko Yu., "On the Stability of Anthropomorphic Systems", Math. Biosciences, Vol. 15, Oct. 1972). Several years later, the load and force feedbacks have been introduced as a means of stabilization of manipulation robots, especially in the tasks of automatic assembly and constrained motion of the manipulator gripper in the tasks of mechanical metal working. Even today, the problem of the so-called hybrid control (positional and force control) attracts great attention of researchers in many research centres in the world. The following questions may be asked: Why has this book been written in the monograph series devoted to manipulation robots? i'lhat is new in this monograph in comparison to the first research monograph mentioned at the beginning of this preface? The answer to the first question is based on the fact that general problems of mathematical modelling, simulation, and stability, are common to all robotic systems, and that there are certain specificites in view of which the class of anthropomorphic mechanisms should be studied separately, what is actually the subject of the present monograph. As for the novelties in comparison to the previous monograph on anthropomorphic mechanisms, this monograph represents a step forward in dealing with the programming support for generating mathematical models of dynamics of an arbitrary anthropomorphic mechanism; the support is having now the form and properties of the customer's software package. In

7 VII contrast to the previous one, this monograph considers the problems of stability and stabilization of the biped gait in their fully resolved forms using the methods of large-scale systems stability that have been developed in the meantime, as well as relying upon the adopted strategy of the decentralized control structure, and including all the necessary feedback aiming at controlling the dynamics of anthropomorphic mechanisms in a disturbed regime, taking into account that the system possesses the uncontrolable degrees of freedom. If compared to the 1975 monograph, this monograph presents also the results on the new concept of modular active orthoses which have, to a great extent, replaced the first realizations of complete exoskeletons, and which at the time of their appearance, represented a technological and medico-biological challenge in the rehabilitation of the most severely disabled people with the insufficient or nonexistent motor activity of the lower extremities. These were the reasons why we, after a longer period of time, decided to prepare a substantially innovated text. The subject matter of thi.s monograph represents thus a sound background for mathematical modelling of the anthropomorphic bipedal gait, the investigation of its stability, as well as the synthesis of its dynamic control while taking into account the complete dynamic information of the system. The complexity attained in the modelling and synthesis of control of bipedal mechanical structure offers the possibility of its application in the synthesis of artificial biped gait of high anthropomorphic fidelity in its exoskeletal version of realization. If the results obtained in th,e meantime in several research centres a.broad are concerned, apart from some of them that have been repeated, this monograph encompasses new results from the USA (Hemami et al.) and in the USSR (aeletskii, Formal'skii, etc.) which are related to modelling and posture control of biped systems. It should be noticed that the contributions of the Soviet authors, and especially of Beletskii, h,ave been covered in more detail if compared to the English-speaking authors, as we are of the opinion that because of the language barrier, these results are less accessible. Th,e only t:;ext that has been taken from the previous monograph is the section entitled "Method based on Euler's angles". The reason for doing so is the fact that this method, developed by Jurici6 and Vukobratovi6 already in 1972, because of its high functionality in regard of modelling of anthropomorphic locomotion mechanisms, Was very close to the current computer-oriented procedure for forming a dynamic model in its symbolic form. For this reason the authors wanted to draw attention of

8 VIII the concerned researchers to adapt this form of the model for its automatic generation in symbolic form, which would be of great importance for the microprocessor implementation of the model in real time. This book consists of four chapters and two appendices. Chapter 1 considers the dynamics of biped gait. First, a survey is given of the results on mathematical modelling of biped locomotion mechanisms in regard of their complexity. Further, the synthesis of anthropomorphic gait is described that is based on the original method of the prescribed synergy, known in the literature as the semi-inverse method, and which has been adopted by practically all the researchers that, after the appearance of the fundamental works of the Belgrade school of locomotion robotics, have been systematically concerned with the dynamics of anthropomorphic gait. Appendix provides a detailed account of the method for modelling dynamics of the biped gait using Euler's angles. Chapter 2 is entirely devoted to the problems of synthesis of the nominal dynamics of biped mechanisms. A special programming package is used which has been developed on the basis of the mathematical model using the general theorems of mechanics; the locomotion mechanism is modelled as a set of branched chains of the open configuration in the single-support gait phase, while in the double-support phase the chain representing the mechanism's legs is closed. At the end of this chapter are given five examples of the nominal dynamics synthesis. In Example 1 we consider C\ mechanism having fourteen links, eight of them being powered, and involving the compensation by the trunk. ExC\mple 2 considers the same mechc\nism but with an additional degree of freedom at both the ankle joint and hip. The compensation in the frontal plane is achieved by the C\nkle joint, and in the sagittal plane by the trunk. In both exc\mples the mechanism's "arms" are fixed to the trunk, whereas in Example 3 they are considered as free passive pendulums. MechanicC\l configuration and the choice of joints for the synthesis of compensating movements are the same as in Example 2. In contrast to the previou,s three examples which are concerned with the nominal dynamics synthesis for C\ single-support phase, Example 4 includes al.so the double-support phase. ExC\mple 5 considers modelling of a two-link foot.

9 Chapter 3 is devoted to the control of biped motion and to the problems of its stability analysis. Its first section reviews the results in the field, including also some results in the domain of realization of the artificial gait. The concept of two-stage control synthesis is used which has been already announced in the earliest period of the biped gait synthesis (Vukobratovi6 M., Juri6i6 D., "Contribution to the Synthesis of Biped Gait", IEEE Trans. on Biomedical Eng. Vol. 16, No.1, 1969). In the first stage the control is synthesized which should ensure realization of the nominal (programmed) motion, while in the second stage (the stage of perturbed regimes) - the problems considered concerned with the gait realization under the conditions deviating from the nominal ones. A special method of control synthesis at the stage of perturbed regimes is developed in which accelerations of correctional movements of the mechanism are constrained in order to prevent significant disturbances of dynamic equilibrium of the system. In addition, the problem of global feedback with respect to the position of the mechanism's foot reaction is considered, whose purpose is to prevent overturning about the foot edge. All these considerations have been illustrated by the appropriate simulation results. The remaining part of the chapter is concerned with the stability analysis of a biped gait mechanism possesing also the unpowered degrees of of freedom (the angles between the foot and ground). The aggregation -decomposition method is applied which is based on the Lyapunov vector functions in the bounded regions of the state space. In view of the fact that this method has been developed for the mechanisms with powered joints, it has to be extended onto the specific case involving both the powered and unpowered mechanism degrees of freedom. 7he models of unpowered degrees of freedom are associated with the mathematical model of the one (chosen in advance) powered degree of mechanism freedom and their common stability is analyzed. In addition, the results are presented concerning the stability analysis for the case when the mathematical model of an unpowered degree of freedom is associated with mathematical model of the hip, or of the ankle joint and for different control structures employed. Chapter 4 presents a brief account on the fundamentals of biodynamics of locomotion and of biped gait in particular. The attention is paid to reviewing previous results in the field of realization of active devices in the form of exoskeleton for generating basic locomotor activity of the disabled with a special emphasis on the new results of modular design of active orthoses for a special class of insufficient

10 x motor activity of human extremities. Also, the most recent results are reviwed concerning the concept of hybrid actuator which uses the residual motor activity of the muscle system and which is additionally supplied with external energy from artificial actuators. This new volume of the monographic series is intended for researchers interested in the dynamics of biped gait and its stabilization. Having in mind that this is a rather narrow field of robotics, the book could not expect a wider readership. However, taking into account that the problems of modelling of complex kinematic chains, synthesis of anthropomorphic gait, stability analysis and dynamic control synthesis, have been tackled from the point of view of a general approach, the book might attract attention of a much wider audience outside the limited number of people involved in studying strictly the biped gait biodynamics, its synthesis, and exoskeletal realization of the artificial gait. In view of the above, the authors believe the book may be useful to students of general courses in robotics and biomechanics at both graduate and postgraduate level and of some special courses at technical faculties and faculties for physical culture and sport. The authors strongly believe in this statement because they also believe in the value of the results presented in the book, as far as their application is concerned, even in some new applications of mathematical modelling, simulation and dynamic control, for example in the biomechanics of athletics and other sports. At the end of the many-year period of studies in the field of bipedlocomotion and its application we feel obliged to m~ntion with great pleasure some people who gave highly valuable contribution to the results in this research. First of all, it is our great pleasure to name Professor Davor Juricic, who has not been involved in this research field for a number of years, but who with the first author of this book, twenty years ago, laid the foundations of mathematical modelling of biped gait and traced the way of its further development. The section devoted to the mathematical models of biped system via Euler's angles, written by D. Juricic in the previous research monograph is included as the appendix of Chapter 1 in this monograph. Also, we would like to express our gratitude to Dragan Hristic Ph.D.,

11 the unavoidable man in the field of rehabilitation robotics who, as the closest co-worker of Professor Vukobratovi6 ~or many years, has given substantial contribution both in the ideas and realization of artificial gait synthesis. He participated in both conceiving and writing of Chapter 4 of the present monograph. We are also indebted to Zora Konjovi6, M.Sc., for writing the appendix on the programming package for the gait dynamics modelling. Finally our thanks are due to Professor Luka Bjelica for translating the book into English, and to Miss Vera Cosic for her highly professional preparation of the typescript for publication. September 1989 Beograd Aut h 0 r s

12 Contents Chapter 1: Dynamics of Biped Locomotion 1.1. Introduction A brief survey of mathematical models used to study the effects arrising with locomotion systems Artificial gait synthesis - a method based on prescribed synergy References Single-support phase Double-support phase Appendix Description of the Mechanical Model Using Euler's Angles.... Chapter 2: Synthesis of Nominal Dynamics 2.1. Introduction Modelling the robotic system dynamics The method based on fundamental theorems of mechanics as a tool for forming dynamic equations of motion for open kinematic chain 2.4. Forming dynamic equations of motion for complex kinematic chains 2.5. Closed kinematic chains 2.6. Synthesis of functional movements The prescribed part of dynamics... 96

13 XIII 2.8. Numerical results References Appendix - Programme LOCDYN Chapter 3: Control and Stability Introduction Survey of results on biped posture and locomotion control and stability analysis Postural control Gait control Model of the system Control synthesis Some notes on biped control synthesis Synthesis of control with limited accelerations Synthesis of global control with respect to ZMP position Simulation of walk for a specific biped structure Stability analysis Modelling of composite subsystems Stability analysis Stability analysis for a specific biped structure Conclusion 309 References Chapter 4: Realizations of Anthropomorphic Mechanisms Introduction

14 XIV 4.2. Exoskeletal and orthotic systems Introduction Development of active exoskeletons for rehabilitation Hodular active orthosis - the "active suit" Other biped realizations Hybrid joint concept General remarks Efferent functional electrical stimulation model The hybrid system model Simulation results Legged robots References Subject index

Humanoid Robots and biped locomotion. Contact: Egidio Falotico

Humanoid Robots and biped locomotion Contact: Egidio Falotico e.falotico@sssup.it Outline What is a Humanoid? Why Develop Humanoids? Challenges in Humanoid robotics Active vs Passive Locomotion Active