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E-grāmata: Feedback Control of Dynamic Bipedal Robot Locomotion

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, (The Ohio State University, Columbus, USA), , (University of Michigan, Ann Arbor, USA), (IRCCYN, Nantes Atlantic University, CNRS, France)
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Feedback Control of Dynamic Bipedal Robot LocomotionPalielināt
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Bipedal locomotion is among the most difficult challenges in control engineering. Most books treat the subject from a quasi-static perspective, overlooking the hybrid nature of bipedal mechanics. Feedback Control of Dynamic Bipedal Robot Locomotion is the first book to present a comprehensive and mathematically sound treatment of feedback design for achieving stable, agile, and efficient locomotion in bipedal robots.

In this unique and groundbreaking treatise, expert authors lead you systematically through every step of the process, including:

Mathematical modeling of walking and running gaits in planar robots

Analysis of periodic orbits in hybrid systems

Design and analysis of feedback systems for achieving stable periodic motions

Algorithms for synthesizing feedback controllers

Detailed simulation examples

Experimental implementations on two bipedal test beds

The elegance of the authors' approach is evident in the marriage of control theory and mechanics, uniting control-based presentation and mathematical custom with a mechanics-based approach to the problem and computational rendering. Concrete examples and numerous illustrations complement and clarify the mathematical discussion. A supporting Web site offers links to videos of several experiments along with MATLAB® code for several of the models. This one-of-a-kind book builds a solid understanding of the theoretical and practical aspects of truly dynamic locomotion in planar bipedal robots.
I. Preliminaries
1(42)
Introduction
3(26)
Why Study the Control of Bipedal Robots?
4(2)
Biped Basics
6(8)
Terminology
6(3)
Dynamics
9(2)
Challenges Inherent to Controlling Bipedal Locomotion
11(3)
Overview of the Literature
14(10)
Polypedal Robot Locomotion
15(2)
Bipedal Robot Locomotion
17(2)
Control of Bipedal Locomotion
19(5)
Feedback as a Mechanical Design Tool: The Notion of Virtual Constraints
24(5)
Time-Invariance, or, Self-Clocking of Periodic Motions
24(1)
Virtual Constraints
25(4)
Two Test Beds for Theory
29(14)
Rabbit
29(8)
Objectives of the Mechanism
29(1)
Structure of the Mechanism
30(1)
Lateral Stabilization
31(2)
Choice of Actuation
33(1)
Sizing the Mechanism
33(2)
Impacts
35(1)
Sensors
35(1)
Additional Details
36(1)
Ernie
37(6)
Objectives of the Mechanism
37(1)
Enabling Continuous Walking with Limited Lab Space
38(1)
Sizing the Mechanism
39(1)
Impacts
39(1)
Sensors
40(1)
Additional Details
40(3)
II. Modeling, Analysis, and Control of Robots with Passive Point Feet
43(256)
Modeling of Planar Bipedal Robots with Point Feet
45(36)
Why Point Feet?
46(1)
Robot, Gait, and Impact Hypotheses
47(5)
Some Remarks on Notation
52(1)
Dynamic Model of Walking
53(18)
Swing Phase Model
53(2)
Impact Model
55(2)
Hybrid Model of Walking
57(1)
Some Facts on Angular Momentum
58(2)
The MPFL-Normal Form
60(3)
Example Walker Models
63(8)
Dynamic Model of Running
71(10)
Flight Phase Model
72(1)
Stance Phase Model
73(1)
Impact Model
74(1)
Hybrid Model of Running
75(2)
Some Facts on Linear and Angular Momentum
77(4)
Periodic Orbits and Poincare Return Maps
81(30)
Autonomous Systems with Impulse Effects
82(5)
Hybrid System Hypotheses
83(1)
Definition of Solutions
84(2)
Periodic Orbits and Stability Notions
86(1)
Poincare's Method for Systems with Impulse Effects
87(4)
Formal Definitions and Basic Theorems
87(3)
The Poincare Return Map as a Partial Function
90(1)
Analyzing More General Hybrid Models
91(5)
Hybrid Model with Two Continuous Phases
92(1)
Basic Definitions
92(2)
Existence and Stability of Periodic Orbits
94(2)
A Low-Dimensional Stability Test Based on Finite-Time Convergence
96(3)
Preliminaries
96(1)
Invariance Hypotheses
96(1)
The Restricted Poincare Map
97(1)
Stability Analysis Based on the Restricted Poincare Map
97(2)
A Low-Dimensional Stability Test Based on Timescale Separation
99(3)
System Hypotheses
100(1)
Stability Analysis Based on the Restricted Poincare Map
101(1)
Including Event-Based Control
102(9)
Analyzing Event-Based Control with the Full-Order Model
103(4)
Analyzing Event-Based Actions with a Hybrid Restriction Dynamics Based on Finite-Time Attractivity
107(4)
Zero Dynamics of Bipedal Locomotion
111(26)
Introduction to Zero Dynamics and Virtual Constraints
111(6)
A Simple Zero Dynamics Example
112(2)
The Idea of Virtual Constraints
114(3)
Swing Phase Zero Dynamics
117(7)
Definitions and Preliminary Properties
117(5)
Interpreting the Swing Phase Zero Dynamics
122(2)
Hybrid Zero Dynamics
124(4)
Periodic Orbits of the Hybrid Zero Dynamics
128(4)
Poincare Analysis of the Hybrid Zero Dynamics
128(3)
Relating Modeling Hypotheses to the Properties of the Hybrid Zero Dynamics
131(1)
Creating Exponentially Stable, Periodic Orbits in the Full Hybrid Model
132(5)
Computed Torque with Finite-Time Feedback Control
133(1)
Computed Torque with Linear Feedback Control
134(3)
Systematic Design of Within-Stride Feedback Controllers for Walking
137(54)
A Special Class of Virtual Constraints
137(1)
Parameterization of hd by Bezier Polynomials
138(6)
Using Optimization of the HZD to Design Exponentially Stable Walking Motions
144(12)
Effects of Output Function Parameters on Gait Properties: An Example
145(2)
The Optimization Problem
147(5)
Cost
152(1)
Constraints
153(1)
The Optimization Problem in Mayer Form
154(2)
Further Properties of the Decoupling Matrix and the Zero Dynamics
156(6)
Decoupling Matrix Invertibility
156(3)
Computing Terms in the Hybrid Zero Dynamics
159(1)
Interpreting the Hybrid Zero Dynamics
160(2)
Designing Exponentially Stable Walking Motions on the Basis of a Prespecified Periodic Orbit
162(3)
Virtual Constraint Design
162(2)
Sample-Based Virtual Constraints and Augmentation Functions
164(1)
Example Controller Designs
165(26)
Designing Exponentially Stable Walking Motions without Invariance of the Impact Map
165(8)
Designs Based on Optimizing the HZD
173(5)
Designs Based on Sampled Virtual Constraints and Augmentation Functions
178(13)
Systematic Design of Event-Based Feedback Controllers for Walking
191(22)
Overview of Key Facts
192(3)
Transition Control
195(4)
Event-Based PI-Control of the Average Walking Rate
199(9)
Average Walking Rate
199(1)
Design and Analysis Based on the Hybrid Zero Dynamics
200(6)
Design and Analysis Based on the Full-Dimensional Model
206(2)
Examples
208(5)
Choice of δα
208(2)
Robustness to Disturbances
210(1)
Robustness to Parameter Mismatch
210(1)
Robustness to Structural Mismatch
210(3)
Experimental Results for Walking
213(36)
Implementation Issues
213(7)
Rabbit's Implementation Issues
213(5)
Ernie's Implementation Issues
218(2)
Control Algorithm Implementation: Imposing the Virtual Constraints
220(5)
Experiments
225(24)
Experimental Validation Using Rabbit
225(16)
Experimental Validation Using Ernie
241(8)
Running with Point Feet
249(50)
Related Work
250(1)
Qualitative Discussion of the Control Law Design
251(3)
Analytical Tractability through Invariance, Attractivity, and Configuration Determinism at Transitions
251(1)
Desired Geometry of the Closed-Loop System
252(2)
Control Law Development
254(4)
Stance Phase Control
255(1)
Flight Phase Control
256(2)
Closed-Loop Hybrid Model
258(1)
Existence and Stability of Periodic Orbits
258(8)
Definition of the Poincare Return Map
258(2)
Analysis of the Poincare Return Map
260(6)
Example: Illustration on Rabbit
266(11)
Stance Phase Controller Design
267(1)
Stability of the Periodic Orbits
268(1)
Flight Phase Controller Design
268(4)
Simulation without Modeling Error
272(5)
A Partial Robustness Evaluation
277(5)
Compliant Contact Model
278(1)
Simulation with Modeling Error
279(3)
Additional Event-Based Control for Running
282(5)
Deciding What to Control
283(1)
Implementing Stride-to-Stride Updates of Landing Configuration
283(1)
Simulation Results
284(3)
Alternative Control Law Design
287(9)
Controller Design
288(4)
Design of Running Motions with Optimization
292(4)
Experiment
296(3)
Hardware Modifications to Rabbit
296(1)
Result: Six Running Steps
296(2)
Discussion
298(1)
III. Walking with Feet
299(64)
Walking with Feet and Actuated Ankles
301(40)
Related Work
302(1)
Robot Model
302(13)
Robot and Gait Hypotheses
303(2)
Coordinates
305(1)
Underactuated Phase
305(1)
Fully Actuated phase
306(1)
Double-Support Phase
307(1)
Foot Rotation, or Transition from Full Actuation to Underactuation
308(1)
Overall Hybrid Model
309(1)
Comments on the FRI Point and Angular Momentum
309(6)
Creating the Hybrid Zero Dynamics
315(6)
Control Design for the Underactuated Phase
315(2)
Control Design for the Fully Actuated Phase
317(1)
Transition Map from the Fully Actuated Phase to the Underactuated Phase
318(1)
Transition Map from the Underactuated Phase to the Fully Actuated Phase
319(1)
Hybrid Zero Dynamics
320(1)
Ankle Control and Stability Analysis
321(5)
Analysis on the Hybrid Zero Dynamics for the Underactuated Phase
321(1)
Analysis on the Hybrid Zero Dynamics for the Fully Actuated Phase with Ankle Torque Used to Change Walking Speed
322(1)
Analysis on the Hybrid Zero Dynamics for the Fully Actuated Phase with Ankle Torque Used to Affect Convergence Rate
323(3)
Stability of the Robot in the Full-Dimensional Model
326(1)
Designing the Virtual Constraints
326(5)
Parametrization Using Bezier polynomials
326(2)
Achieving Impact Invariance of the Zero Dynamics Manifolds
328(2)
Specifying the Remaining Free Parameters
330(1)
Simulation
331(1)
Special Case of a Gait without Foot Rotation
332(2)
ZMP and Stability of an Orbit
334(7)
Directly Controlling the Foot Rotation Indicator Point
341(22)
Introduction
341(1)
Using Ankle Torque to Control FRI Position During the Fully Actuated Phase
342(5)
Ability to Track a Desired Profile of the FRI Point
343(1)
Analyzing the Zero Dynamics
344(3)
Special Case of a Gait without Foot Rotation
347(1)
Simulations
348(7)
Nominal Controller
348(2)
With Modeling Errors
350(1)
Effect of FRI Evolution on the Walking Gait
351(4)
A Variation on FRI Position Control
355(2)
Simulations
357(6)
Getting Started
363(12)
Graduate Student
363(5)
Professional Researcher
368(7)
Reader Already Has a Stabilizing Controller
368(4)
Controller Design Must Start from Scratch
372(1)
Walking with Feet
372(1)
3D Robot
373(2)
Essential Technical Background
375(64)
Smooth Surfaces and Associated Notions
376(11)
Manifolds and Embedded Submanifolds
376(2)
Local Coordinates and Smooth Functions
378(2)
Tangent Spaces and Vector Fields
380(3)
Invariant Submanifolds and Restriction Dynamics
383(2)
Lie Derivatives, Lie Brackets, and Involutive Distributions
385(2)
Elementary Notions in Geometric Nonlinear Control
387(12)
SISO Nonlinear Affine Control System
388(6)
MIMO Nonlinear Affine Control System
394(5)
Poincare's Method of Determining Limit Cycles
399(7)
Poincare Return Map
400(1)
Fixed Points and Periodic Orbits
401(2)
Utility of the Poincare Return Map
403(3)
Planar Lagrangian Dynamics
406(33)
Kinematic Chains
406(2)
Kinetic and Potential Energy of a Single Link
408(4)
Free Open Kinematic Chains
412(4)
Pinned Open Kinematic Chains
416(3)
The Lagrangian and Lagrange's Equations
419(1)
Generalized Forces and Torques
420(1)
Angular Momentum
420(1)
Further Remarks on Lagrange's Method
421(7)
Sign Convention on Measuring Angles
428(3)
Other Useful Facts
431(5)
Example: The Acrobot
436(3)
Proofs and Technical Details
439(18)
Proofs Associated with
Chapter 4
439(10)
Continuity of TI
439(1)
Distance of a Trajectory to a Periodic Orbit
439(1)
Proof of Theorem 4.1
440(1)
Proof of Proposition 4.1
441(1)
Proofs of Theorem 4.4 and Theorem 4.5
442(1)
Proof of Theorem 4.6
442(4)
Proof of Theorem 4.8
446(2)
Proof of Theorem 4.9
448(1)
Proofs Associated with
Chapter 5
449(2)
Proof of Theorem 5.4
449(1)
Proof of Theorem 5.5
450(1)
Proofs Associated with
Chapter 6
451(1)
Proof of Proposition 6.1
451(1)
Proof of Theorem 6.2
451(1)
Proof Associated with
Chapter 7
452(2)
Proof of Theorem 7.3
452(2)
Proofs Associated with
Chapter 9
454(3)
Proof of Theorem 9.2
454(1)
Proof of Theorem 9.3
455(1)
Proof of Theorem 9.4
455(2)
Derivation of the Equations of Motion for Three-Dimensional Mechanisms
457(8)
The Lagrangian
457(1)
The Kinetic Energy
458(4)
The Potential Energy
462(1)
Equations of Motion
462(2)
Invariance Properties of the Kinetic Energy
464(1)
Single Support Equations of Motion of Rabbit
465(6)
Nomenclature 471(2)
End Notes 473(6)
References 479(20)
Index 499(4)
Supplemental Indices 503
Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, Benjamin Morris
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