Feedback Control of Dynamic Bipedal Robot Locomotion

Transcription

1 Feedback Control of Dynamic Bipedal Robot Locomotion Eric R. Westervelt Jessy W. Grizzle Christine Chevaiiereau Jun Ho Choi Benjamin Morris CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor Sc Francis Group, an informa business

2 Contents I Preliminaries 1 1 Introduction Why Study the Control of Bipedal Robots? Biped Basics Terminology Dynamics Challenges Inherent to Controlling Bipedal Locomotion Overview of the Literature Polypedal Robot Locomotion Bipedal Robot Locomotion Control of Bipedal Locomotion Feedback as a Mechanical Design Tool: The Notion of Virtual Constraints Time-Invariance, or, Self-Clocking of Periodic Motions Virtual Constraints 25 2 Two Test Beds for Theory RABBIT Objectives of the Mechanism Structure of the Mechanism Lateral Stabilization Choice of Actuation Sizing the Mechanism Impacts Sensors Additional Details ERNIE Objectives of the Mechanism Enabling Continuous Walking with Limited Lab Space Sizing the Mechanism Impacts Sensors Additional Details 40

3 II Modeling, Analysis, and Control of Robots with Passive Point Feet 43 3 Modeling of Planar Bipedal Robots with Point Feet Why Point Feet? Robot, Gait, and Impact Hypotheses Some Remarks on Notation Dynamic Model of Walking Swing Phase Model Impact Model Hybrid Model of Walking Some Facts on Angular Momentum The MPFL-Normal Form Example Walker Models Dynamic Model of Running Flight Phase Model Stance Phase Model Impact Model Hybrid Model of Running Some Facts on Linear and Angular Momentum Periodic Orbits and Poincare Return Maps Autonomous Systems with Impulse Effects Hybrid System Hypotheses Definition of Solutions Periodic Orbits and Stability Notions Poincare's Method for Systems with Impulse Effects Formal Definitions and Basic Theorems The Poincare Return Map as a Partial Function Analyzing More General Hybrid Models Hybrid Model with Two Continuous Phases Basic Definitions Existence and Stability of Periodic Orbits A Low-Dimensional Stability Test Based on Finite-Time Convergence Preliminaries Invariance Hypotheses The Restricted Poincare Map Stability Analysis Based on the Restricted Poincare Map A Low-Dimensional Stability Test Based on Timescale Separation System Hypotheses Stability Analysis Based on the Restricted Poincare Map 101

4 4.6 Including Event-Based Control Analyzing Event-Based Control with the Full-Order Model Analyzing Event-Based Actions with a Hybrid Restriction Dynamics Based on Finite-Time Attractivity 107 Zero Dynamics of Bipedal Locomotion Introduction to Zero Dynamics and Virtual Constraints A Simple Zero Dynamics Example The Idea of Virtual Constraints Swing Phase Zero Dynamics Definitions and Preliminary Properties Interpreting the Swing Phase Zero Dynamics Hybrid Zero Dynamics Periodic Orbits of the Hybrid Zero Dynamics Poincare Analysis of the Hybrid Zero Dynamics Relating Modeling Hypotheses to the Properties of the Hybrid Zero Dynamics Creating Exponentially Stable, Periodic Orbits in the Füll Hybrid Model Computed Torque with Finite-Time Feedback Control Computed Torque with Linear Feedback Control Systematic Design of Within-Stride Feedback Controllers for Walking A Special Class of Virtual Constraints Parameterization of hd by Bezier Polynomials Using Optimization of the HZD to Design Exponentially Stable Walking Motions Effects of Output Function Parameters on Gait Properties: An Example The Optimization Problem Cost Constraints The Optimization Problem in Mayer Form Further Properties of the Decoupling Matrix and the Zero Dynamics Decoupling Matrix Invertibility Computing Terms in the Hybrid Zero Dynamics Interpreting the Hybrid Zero Dynamics Designing Exponentially Stable Walking Motions on the Basis of a Prespecified Periodic Orbit Virtual Constraint Design 162

5 6.5.2 Sample-Based Virtual Constraints and Augmentation Functions Example Controller Designs Designing Exponentially Stable Walking Motions without Invariance of the Impact Map Designs Based on Optimizing the HZD Designs Based on Sampled Virtual Constraints and Augmentation Functions 178 Systematic Design of Event-Based Feedback Controllers for Walking Overview of Key Facts Transition Control Event-Based Pl-Control of the Average Walking Rate Average Walking Rate Design and Analysis Based on the Hybrid Zero Dynamics Design and Analysis Based on the Full-Dimensional Model Examples Choice of Sa Robustness to Disturbances Robustness to Parameter Mismatch Robustness to Structural Mismatch 210 Experimental Results for Walking Implementation Issues RABBIT's Implementation Issues ERNIE's Implementation Issues Control Algorithm Implementation: Imposing the Virtual Constraints : Experiments Experimental Validation Using RABBIT Experimental Validation Using ERNIE 241 Running with Point Feet Related Work Qualitative Discussion of the Control Law Design Analytical Tractability through Invariance, Attractivity, and Configuration Determinism at Transitions Desired Geometry of the Closed-Loop System Control Law Development Stance Phase Control Flight Phase Control 256

6 9.3.3 Closed-Loop Hybrid Model Existence and Stability of Periodic Orbits Definition of the Poincare Return Map Analysis of the Poincare Return Map Example: Illustration on RABBIT Stance Phase Controller Design Stability of the Periodic Orbits Flight Phase Controller Design Simulation without Modeling Error A Partial Robustness Evaluation Compliant Contact Model Simulation with Modeling Error Additional Event-Based Control for Running Deciding What to Control Implementing Stride-to-Stride Updates of Landing Configuration Simulation Results Alternative Control Law Design Controller Design Design of Running Motions with Optimization Experiment Hardware Modifications to RABBIT Result: Six Running Steps Discussion 298 III Walking with Feet Walking with Feet and Actuated Ankles Related Work Robot Model Robot and Gait Hypotheses Coordinates Underactuated Phase FuUy Actuated phase Double-Support Phase Foot Rotation, or Transition from Füll Actuation to Underactuation Overall Hybrid Model Comments on the FRI Point and Angular Momentum Creating the Hybrid Zero Dynamics Control Design for the Underactuated Phase Control Design for the Fully Actuated Phase Transition Map from the Fully Actuated Phase to the Underactuated Phase 318

7 Transition Map from the Underactuated Phase to the Fully Actuated Phase Hybrid Zero Dynamics Ankle Control and Stability Analysis Analysis on the Hybrid Zero Dynamics for the Underactuated Phase Analysis on the Hybrid Zero Dynamics for the Fully Actuated Phase with Ankle Torque Used to Change Walking Speed Analysis on the Hybrid Zero Dynamics for the Fully Actuated Phase with Ankle Torque Used to Affect Convergence Rate Stability of the Robot in the Full-Dimensional Model Designing the Virtual Constraints Parametrization Using Bezier polynomials Achieving Impact Invariance of the Zero Dynamics Manifolds Specifying the Remaining Free Parameters Simulation Special Case of a Gait without Foot Rotation ZMP and Stability of an Orbit Directly Controlling the Foot Rotation Indicator Point Introduction Using Ankle Torque to Control FRI Position During the Fully Actuated Phase Ability to Track a Desired Profile of the FRI Point Analyzing the Zero Dynamics Special Case of a Gait without Foot Rotation Simulations Nominal Controller With Modeling Errors Effect of FRI Evolution on the Walking Gait A Variation on FRI Position Control Simulations 357 A Getting Started 363 A.l Graduate Student 363 A.2 Professional Researcher 368 A.2.1 Reader Already Has a Stabilizing Controller 368 A.2.2 Controller Design Must Start from Scratch 372 A.2.3 Walking with Feet 372 A.2.4 3D Robot 373

8 B Essential Technical Background 375 B.l Smooth Surfaces and Associated Notions 376 B.l.l Manifolds and Embedded Submanifolds 376 B.l.2 Local Coordinates and Smooth Functions 378 B.l.3 Tangent Spaces and Vector Fields 380 B.1.4 Invariant Submanifolds and Restriction Dynamics B.l.5 Lie Derivatives, Lie Brackets, and Involutive Distributions 385 B.2 Elementary Notions in Geometrie Nonlinear Control 387 B.2.1 SISO Nonlinear Affine Control System 388 B.2.2 MIMO Nonlinear Affine Control System 394 B.3 Poincare's Method of Determining Limit Cycles 399 B.3.1 Poincare Return Map 400 B.3.2 Fixed Points and Periodic Orbits 401 B.3.3 Utility of the Poincare Return Map 403 B.4 Planar Lagrangian Dynamics 406 B.4.1 Kinematic Chains 406 B.4.2 Kinetic and Potential Energy of a Single Link 408 B.4.3 Free Open Kinematic Chains 412 B.4.4 Pinned Open Kinematic Chains 416 B.4.5 The Lagrangian and Lagrange's Equations 419 B.4.6 Generalized Forces and Torques 420 B.4.7 Angular Momentum 420 B.4.8 Further Remarks on Lagrange's Method 421 B.4.9 Sign Convention on Measuring Angles 428 B.4.10 Other Useful Facts 431 B.4.11 Example: The Acrobot 436 C Proofs and Technical Details 439 Ol Proofs Associated with Chapter C.l.l Continuity of T, 439 C.1.2 Distance of a Trajectory to a Periodic Orbit 439 C.1.3 Proofof Theorem C.1.4 Proofof Proposition C.1.5 Proofs of Theorem 4.4 and Theorem C.1.6 Proof of Theorem Ol.7 Proof of Theorem C.1.8 Proof of Theorem C.2 Proofs Associated with Chapter C.2.1 Proof of Theorem C.2.2 Proof of Theorem C.3 Proofs Associated with Chapter C.3.1 Proofof Proposition C.3.2 Proofof Theorem C.4 Proof Associated with Chapter 7 452

9 C.4.1 Proof of Theorem C.5 Proofs Associated with Chapter C.5.1 Proof of Theorem C.5.2 Proof of Theorem C.5.3 Proof of Theorem D Derivation of the Equations of Motion for Three-Dimensional Mechanisms 457 D.l The Lagrangian 457 D.2 The Kinetic Energy 458 D.3 The Potential Energy 462 D.4 Equations of Motion 462 D.5 Invariance Properties of the Kinetic Energy 464 E Single Support Equations of Motion of RABBIT 465 Nomenclature 471 End Notes 473 References 479 Index 499 Supplemental Indices 503