NASSER SADATI is Professor in the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.GUY A. DUMONT is Professor in the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada.KAVEH AKBARI HAMED is Postdoctoral Research Fellow at the Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI, USA.WILLIAM A. GRUVER is Professor Emeritus in the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada.

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• Preface ix 1. Introduction 11.1 Objectives of Legged Locomotion and Challenges in Controlling Dynamic Walking and Running 11.2 Literature Overview 41.2.1 Tracking of Time Trajectories 41.2.2 Poincar´e Return Map and Hybrid Zero Dynamics 51.3 The Objective of the Book 71.3.1 Hybrid Zero Dynamics in Walking with Double Support Phase 71.3.2 Hybrid Zero Dynamics in Running with an Online Motion Planning Algorithm 81.3.3 Online Motion Planning Algorithms for Flight Phases of Running 91.3.4 Hybrid Zero Dynamics in 3D Running 101.3.5 Hybrid Zero Dynamics in Walking with Passive Knees 111.3.6 Hybrid Zero Dynamics with Continuous-Time Update Laws 122. Preliminaries in Hybrid Systems 132.1 Basic Definitions 132.2 Poincar´e Return Map for Hybrid Systems 162.3 Low-Dimensional Stability Analysis 232.4 Stabilization Problem 283. Asymptotic Stabilization of Periodic Orbits forWalking with Double Support Phase 353.1 Introduction 353.2 Mechanical Model of a Biped Walker 373.2.1 The Biped Robot 373.2.2 Dynamics of the Flight Phase 373.2.3 Dynamics of the Single Support Phase 393.2.4 Dynamics of the Double Support Phase 403.2.5 Impact Model 433.2.6 Transition from the Double Support Phase to the Single Support Phase 453.2.7 Hybrid Model of Walking 453.3 Control Laws for the Single and Double Support Phases 463.3.1 Single Support Phase Control Law 463.3.2 Double Support Phase Control Law 493.4 Hybrid Zero Dynamics (HZD) 543.4.1 Analysis of HZD in the Single Support Phase 553.4.2 Analysis of HZD in the Double Support Phase 573.4.3 Restricted Poincar´e Return Map 583.5 Design of an HZD Containing a Prespecified Periodic Solution 603.5.1 Design of the Output Functions 603.5.2 Design of u1d and u2d 623.6 Stabilization of the Periodic Orbit 673.7 Motion Planning Algorithm 713.7.1 Motion Planning Algorithm for the Single Support Phase 723.7.2 Motion Planning Algorithm for the Double Support Phase 733.7.3 Constructing a Period-One Orbit for the Open-Loop Hybrid Model of Walking 763.8 Numerical Example for the Motion Planning Algorithm 773.9 Simulation Results of the Closed-Loop Hybrid System 823.9.1 Effect of Double Support Phase on Angular Momentum Transfer and Stabilization 823.9.2 Effect of Event-Based Update Laws on Momentum Transfer and Stabilization 924. Asymptotic Stabilization of Periodic Orbits for Planar Monopedal Running 954.1 Introduction 954.2 Mechanical Model of a Monopedal Runner 974.2.1 The Monopedal Runner 974.2.2 Dynamics of the Flight Phase 974.2.3 Dynamics of the Stance Phase 984.2.4 Open-Loop Hybrid Model of Running 994.3 Reconfiguration Algorithm for the Flight Phase 994.3.1 Determination of the Reachable Set 1034.4 Control Laws for Stance and Flight Phases 1204.4.1 Stance Phase Control Law 1214.4.2 Flight Phase Control Law 1224.4.3 Event-Based Update Law 1244.5 Hybrid Zero Dynamics and Stabilization 1254.6 Numerical Results 1275. Online Generation of Joint Motions During Flight Phases of Planar Running 1375.1 Introduction 1375.2 Mechanical Model of a Planar Open Kinematic Chain 1385.3 Motion Planning Algorithm to Generate Continuous Joint Motions 1405.3.1 Determining the Reachable Set from the Origin 1435.3.2 Motion Planning Algorithm 1505.4 Motion Planning Algorithm to Generate Continuously Differentiable Joint Motions 1526. Stabilization of Periodic Orbits for 3D Monopedal Running 1596.1 Introduction 1596.2 Open-Loop Hybrid Model of a 3D Running 1606.2.1 Dynamics of the Flight Phase 1626.2.2 Dynamics of the Stance Phase 1636.2.3 Transition Maps 1646.2.4 Hybrid Model 1666.3 Design of a Period-One Solution for the Open-Loop Model of Running 1676.4 Numerical Example 1726.5 Within-Stride Controllers 1756.5.1 Stance Phase Control Law 1756.5.2 Flight Phase Control Law 1786.6 Event-Based Update Laws for Hybrid Invariance 1816.6.1 Takeoff Update Laws 1846.6.2 Impact Update Laws 1856.7 Stabilization Problem 1866.8 Simulation Results 1897. Stabilization of Periodic Orbits for Walking with Passive Knees 1937.1 Introduction 1937.2 Open-Loop Model of Walking 1947.2.1 Mechanical Model of the Planar Bipedal Robot 1947.2.2 Dynamics of the Single Support Phase 1957.2.3 Impact Map 1957.2.4 Open-Loop Impulsive Model of Walking 1967.3 Motion Planning Algorithm 1977.4 Numerical Example 2007.5 Continuous-Times Controllers 2027.6 Event-Based Controllers 2097.6.1 Hybrid Invariance 2097.6.2 Continuity of the Continuous-Time Controllers During the Within-Stride Transitions 2127.7 Stabilization Problem 2137.8 Simulation of the Closed-Loop Hybrid System 2178. Continuous-Time Update Laws During Continuous Phases of Locomotion 2218.1 Introduction 2218.2 Invariance of the Exponential Stability Behavior for a Class of Impulsive Systems 2228.3 Outline of the Proof of Theorem 8.1 2248.4 Application to Legged Locomotion 227A. Proofs Associated with Chapter 3 229A.1 Proof of Lemma 3.3 229A.2 Proof of Lemma 3.4 230A.3 Proof of Lemma 3.7 230B. Proofs Associated with Chapter 4 233B.1 Proof of Lemma 4.2 233B.2 Proof of Theorem 4.2 234C. Proofs Associated with Chapter 6 237C.1 Proof of Lemma 6.1 237C.2 Proof of Lemma 6.2 238C.3 Invertibility of the Stance Phase Decoupling Matrix on the Periodic Orbit 240Bibliography 241Index 249