Hybrid Control and Motion Planning of Dynamical Legged Locomotion

275 Pages·2012·6.47 MB·English

by  Nasser Sadati, Guy A. Dumont, Kaveh Akbari Hamed, William A. Gruver(auth.)

HYBRID CONTROL AND
MOTION PLANNING
OF DYNAMICAL LEGGED
LOCOMOTION
IEEE Press
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HYBRID CONTROL AND
MOTION PLANNING
OF DYNAMICAL LEGGED
LOCOMOTION
Nasser Sadati
Guy A. Dumont
Kaveh Akbari Hamed
William A. Gruver
Cover Image: Courtesy of the authors
Copyright © 2012 by the Institute of Electrical and Electronics Engineers, Inc.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Hybrid control and motion planning of dynamical legged locomotion / Nasser Sadati. . . [et al.].
p. cm.
ISBN 978-1-118-31707-5 (hardback)
1. Mobile robots. 2. Robots–Motion. 3. Walking. I. Sadati, Nasser.
TJ211.415.H93 2012
′
629.8 932–dc23
2012002035
ISBN: 978-1-118-31707-5
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
CONTENTS
Preface ix
1. Introduction 1
1.1 Objectives of Legged Locomotion and Challenges in Controlling
Dynamic Walking and Running 1
1.2 Literature Overview 4
1.2.1 Tracking of Time Trajectories 4
1.2.2 Poincare´ Return Map and Hybrid Zero Dynamics 5
1.3 The Objective of the Book 7
1.3.1 Hybrid Zero Dynamics in Walking with Double Support
Phase 7
1.3.2 Hybrid Zero Dynamics in Running with an Online Motion
Planning Algorithm 8
1.3.3 Online Motion Planning Algorithms for Flight Phases
of Running 9
1.3.4 Hybrid Zero Dynamics in 3D Running 10
1.3.5 Hybrid Zero Dynamics in Walking with Passive Knees 11
1.3.6 Hybrid Zero Dynamics with Continuous-Time Update Laws 12
2. Preliminaries in Hybrid Systems 13
2.1 Basic Definitions 13
2.2 Poincare´ Return Map for Hybrid Systems 16
2.3 Low-Dimensional Stability Analysis 23
2.4 Stabilization Problem 28
3. Asymptotic Stabilization of Periodic Orbits for Walking with Double
Support Phase 35
3.1 Introduction 35
3.2 Mechanical Model of a Biped Walker 37
3.2.1 The Biped Robot 37
3.2.2 Dynamics of the Flight Phase 37
3.2.3 Dynamics of the Single Support Phase 39
3.2.4 Dynamics of the Double Support Phase 40
3.2.5 Impact Model 43
v
vi CONTENTS
3.2.6 Transition from the Double Support Phase to the Single
Support Phase 45
3.2.7 Hybrid Model of Walking 45
3.3 Control Laws for the Single and Double Support Phases 46
3.3.1 Single Support Phase Control Law 46
3.3.2 Double Support Phase Control Law 49
3.4 Hybrid Zero Dynamics (HZD) 54
3.4.1 Analysis of HZD in the Single Support Phase 55
3.4.2 Analysis of HZD in the Double Support Phase 57
3.4.3 Restricted Poincare´ Return Map 58
3.5 Design of an HZD Containing a Prespecified Periodic Solution 60
3.5.1 Design of the Output Functions 60
3.5.2 Design of u1d and u2d 62
3.6 Stabilization of the Periodic Orbit 67
3.7 Motion Planning Algorithm 71
3.7.1 Motion Planning Algorithm for the Single Support Phase 72
3.7.2 Motion Planning Algorithm for the Double Support Phase 73
3.7.3 Constructing a Period-One Orbit for the Open-Loop Hybrid
Model of Walking 76
3.8 Numerical Example for the Motion Planning Algorithm 77
3.9 Simulation Results of the Closed-Loop Hybrid System 82
3.9.1 Effect of Double Support Phase on Angular Momentum
Transfer and Stabilization 82
3.9.2 Effect of Event-Based Update Laws on Momentum Transfer
and Stabilization 92
4. Asymptotic Stabilization of Periodic Orbits for Planar Monopedal
Running 95
4.1 Introduction 95
4.2 Mechanical Model of a Monopedal Runner 97
4.2.1 The Monopedal Runner 97
4.2.2 Dynamics of the Flight Phase 97
4.2.3 Dynamics of the Stance Phase 98
4.2.4 Open-Loop Hybrid Model of Running 99
4.3 Reconfiguration Algorithm for the Flight Phase 99
4.3.1 Determination of the Reachable Set 103
4.4 Control Laws for Stance and Flight Phases 120
4.4.1 Stance Phase Control Law 121
4.4.2 Flight Phase Control Law 122
4.4.3 Event-Based Update Law 124
4.5 Hybrid Zero Dynamics and Stabilization 125
4.6 Numerical Results 127
CONTENTS vii
5. Online Generation of Joint Motions During Flight Phases of Planar
Running 137
5.1 Introduction 137
5.2 Mechanical Model of a Planar Open Kinematic Chain 138
5.3 Motion Planning Algorithm to Generate Continuous Joint Motions 140
5.3.1 Determining the Reachable Set from the Origin 143
5.3.2 Motion Planning Algorithm 150
5.4 Motion Planning Algorithm to Generate Continuously
Differentiable Joint Motions 152
6. Stabilization of Periodic Orbits for 3D Monopedal Running 159
6.1 Introduction 159
6.2 Open-Loop Hybrid Model of a 3D Running 160
6.2.1 Dynamics of the Flight Phase 162
6.2.2 Dynamics of the Stance Phase 163
6.2.3 Transition Maps 164
6.2.4 Hybrid Model 166
6.3 Design of a Period-One Solution for the Open-Loop
Model of Running 167
6.4 Numerical Example 172
6.5 Within-Stride Controllers 175
6.5.1 Stance Phase Control Law 175
6.5.2 Flight Phase Control Law 178
6.6 Event-Based Update Laws for Hybrid Invariance 181
6.6.1 Takeoff Update Laws 184
6.6.2 Impact Update Laws 185
6.7 Stabilization Problem 186
6.8 Simulation Results 189
7. Stabilization of Periodic Orbits for Walking with Passive Knees 193
7.1 Introduction 193
7.2 Open-Loop Model of Walking 194
7.2.1 Mechanical Model of the Planar Bipedal Robot 194
7.2.2 Dynamics of the Single Support Phase 195
7.2.3 Impact Map 195
7.2.4 Open-Loop Impulsive Model of Walking 196
7.3 Motion Planning Algorithm 197
7.4 Numerical Example 200
7.5 Continuous-Times Controllers 202
7.6 Event-Based Controllers 209
7.6.1 Hybrid Invariance 209
7.6.2 Continuity of the Continuous-Time Controllers During
the Within-Stride Transitions 212
viii CONTENTS
7.7 Stabilization Problem 213
7.8 Simulation of the Closed-Loop Hybrid System 217
8. Continuous-Time Update Laws During Continuous Phases
of Locomotion 221
8.1 Introduction 221
8.2 Invariance of the Exponential Stability Behavior for a Class
of Impulsive Systems 222
8.3 Outline of the Proof of Theorem 8.1 224
8.4 Application to Legged Locomotion 227
A. Proofs Associated with Chapter 3 229
A.1 Proof of Lemma 3.3 229
A.2 Proof of Lemma 3.4 230
A.3 Proof of Lemma 3.7 230
B. Proofs Associated with Chapter 4 233
B.1 Proof of Lemma 4.2 233
B.2 Proof of Theorem 4.2 234
C. Proofs Associated with Chapter 6 237
C.1 Proof of Lemma 6.1 237
C.2 Proof of Lemma 6.2 238
C.3 Invertibility of the Stance Phase Decoupling Matrix on the
Periodic Orbit 240
Bibliography 241
Index 249
PREFACE
During the last three decades, enormous advances have occurred in robot control of
dynamical legged locomotion. The desire to study legged locomotion has been mo-
tivated by the desire to assist people with disabilities to walk and replace humans
in hazardous environments. The control of dynamical locomotion is complicated by
(i) limb coordination, (ii) hybrid nature of walking and running due to presence of
impact and takeoff, (iii) underactuation, (iv) overactuation, (v) inability to apply the
Zero Moment Point criterion during dynamic walking and running, (vi) lack of algo-
rithms to achieve feasible period-one orbits, and (vii) conservation of angular momen-
tum about the robot’s center of mass during flight phases of running. New applications
of complex legged robots also require the use of system engineering approaches to
resolve these issues that are beyond any single traditional engineering discipline. As
new problems in legged locomotion require multidisciplinary methodologies, there is
a critical need for a comprehensive book covering motion planning algorithms and hy-
brid control. This book fills that gap for researchers, professionals, and students who
are versed in robotics and control theory. This book serves as a reference and essential
guide for researchers and engineers to perform future research and development in
order to advance various topics of hybrid control of legged locomotion. This volume
also provides a comprehensive overview of hybrid models describing the evolution
of planar and 3D legged robots during dynamical legged locomotion, and hybrid
control schemes to asymptotically stabilize periodic orbits for the resulting closed-
loop systems. The major topics of this book include hybrid systems, systems with
impulse effects, offline and online motion planning algorithms to generate periodic
walking and running motions and two-level control schemes including within-stride
feedback laws to reduce the dimension of hybrid systems, continuous-time update
laws for online minimization of a general cost function, and event-based update laws
to asymptotically stabilize the generated desired orbits. This volume can be viewed
as a handbook in this important field, as well as a reference book for researchers and
practicing engineers.
Chapter 2 introduces basic ideas, definitions, and results from the literature of
hybrid systems. Chapter 3 shows how to design a continuous-time-invariant feedback
law that asymptotically stabilizes a feasible periodic trajectory using an extension of
hybrid zero dynamics for a hybrid model of walking. The main objective is to develop
a continuous-time-invariant control law for walking of a planar biped robot during
the double support phase.
A number of control problems for reconfiguration of a planar multilink robot
during flight phases have been considered in the literature. However, these methods
ix
x PREFACE
cannot be employed online to solve the reconfiguration problem for monopedal run-
ning. For this reason, Chapters 4 and 5 present online reconfiguration algorithms that
provide a solution to this latter problem for given flight times and angular momenta.
The algorithms proposed in this book are expressed using the methodology of reach-
ability and optimal control for time-varying linear systems with input and state
constraints. In addition, a two-level control scheme based on the online reconfigura-
tion algorithms and hybrid zero dynamics is proposed in Chapter 4 to asymptotically
stabilize a desired period-one orbit for a hybrid model describing running by planar
monopedal robots. Chapter 6 presents a time-invariant control scheme to asymptoti-
cally stabilize a desired feasible periodic orbit for running by a 3D legged robot along
a straight line. A systematic algorithm to generate desired feasible periodic orbits for
3D running is also presented. Chapter 6 extends the results of Chapters 4 and 5 to 3D
running robots.
In order to reduce the number of actuated joints for walking on a flat surface and
restore walking motion for persons with disabilities, a motion planning algorithm is
developed in Chapter 7 for walking with passive knees. In addition, a time-invariant
two-level control scheme is presented to stabilize the desired motions that are gen-
erated. In Chapter 8, an analytical approach for designing a class of continuous-time
update laws to update the parameters of stabilizing controllers during continuous
phases is proposed such that (i) a general cost function, such as the energy of the
control input over single support, can be minimized online, and (ii) the exponential
stability behavior of the limit cycle for the closed-loop system is not affected.
Book Webpage: Supplemental materials are available at the following URL:
http://booksupport.wiley.com.
This webpage includes MATLAB codes for motion planning algorithms and hybrid
control schemes of several legged robots studied in this book, an erratum, and a link
to submit errors found in this book.
Nasser Sadati
Guy A. Dumont
Kaveh Akbari Hamed
William A. Gruver
November 14, 2011