Front Cover

Bioinspired Legged Locomotion

Copyright

Contents

List of Contributors

About the Editors

1 Introduction

1.1 What Is Bio-Inspired Legged Locomotion?

1.2 Organization of the Book

References

Part I Concepts

2 Fundamental Subfunctions of Locomotion

Preamble-Things to Consider "Before Walking"

2.1 Stance

2.1.1 Effects of Gait

2.1.2 Effects of Size

2.1.3 Summary

References

2.2 Leg Swinging

2.2.1 Characterizing Features of Leg Swinging

2.2.2 Leg Swinging Effects in Locomotion

2.2.2.1 Contributing to Stance Phase Dynamics

2.2.2.2 Trade-off Between Versatility, Robustness, and Efficiency

2.2.2.3 Distribution of Energies in Forward, Lateral, and Vertical Directions

2.2.2.4 Recovery from Perturbations

2.2.3 Swing Leg Modeling and Control

2.2.3.1 Massless Swing Leg

2.2.3.2 Mass in the Swing Leg

References

2.3 Balancing

2.3.1 The Neural Control of Balance: Standing vs. Walking

Sensory feedback control

References

3 Conceptual Models of Legged Locomotion

A Role for Simple Conceptual Models

3.1 Conceptual Models Based on Empirical Observations

3.1.1 Observing, Imagining, and Gaining Insights into Locomotion

3.1.2 Locomotion as a Complex System Behavior

3.1.3 Some Characteristics of Whole-Body Locomotion

3.1.3.1 The Trunk: Bouncing Along

3.1.3.2 The Stance Leg: Acting Like a Spring

3.1.3.3 The Swing Leg: Recirculating for Touchdown

3.1.4 Whole-Body Conceptual Models as an Integration of Parts or Subfunctions

References

3.2 Templates and Anchors

3.2.1 A Mathematical Framework for Legged Locomotion

3.2.2 Templates and Anchors: Hierarchies of Models

3.2.3 Templates in Dynamics, Control, and Modeling

3.2.4 Sources of Templates; Notions of Templates

3.2.4.1 Dimensionality Reduction in Dynamical Systems

3.2.4.2 Templates Based on Mechanical Intuition

3.2.4.3 Data-Driven Model Reduction

3.2.5 Conclusion

References

3.3 A Simple Model of Running

3.3.1 Running Like a Spring-Loaded Inverted Pendulum (SLIP)

3.3.1.1 Physical Mechanisms and Robots Related to the SLIP Model

3.3.2 Mathematical and Physics-Based SLIP Model

3.3.2.1 Ground Reaction Forces During Stance

3.3.2.2 Stride Maps: Behavior Investigated Step-by-Step

3.3.2.3 Stability of Locomotion

3.3.3 Some Insights into Running Aided by SLIP-based Models

3.3.3.1 Adaptive, Resilient Locomotion Based on Open-Loop Stability

3.3.3.2 Reducing Energetic Costs through Compliant Interaction

3.3.3.3 Momentum Trading to Benefit Stability

3.3.3.4 Useful Inefficiency: Inefficiency can Benefit Robustness

References

3.4 Simple Models of Walking

3.4.1 Walking Like an Inverted Pendulum

3.4.2 Passive Walking Mechanisms: Physical Models and Physics-Based Math Models

3.4.3 Mathematical Equations Governing a Bipedal Inverted Pendulum (IP) Model

3.4.3.1 Behavior Within a Single Stance Phase

3.4.3.2 Stance Leg Liftoff and Swing Leg Touchdown

3.4.3.3 The Mechanics of Switching from One Stance Leg to the Next

3.4.3.4 Stride Maps: Behavior Investigated Step-by-Step

3.4.3.5 Stability of Locomotion

3.4.4 Some Insights into Walking Aided by Inverted Pendulum Models

3.4.4.1 Walking Includes a Pendular Flow of Energy

3.4.4.2 Walking Includes the Catching of Repeated Falls

3.4.4.3 Momentum is Exchanged During Double Stance

3.4.5 Integration of Walking and Running Models

References

3.5 Locomotion as an Oscillator

3.5.1 Locomotion as an Oscillator

3.5.2 Stride Registration as Phase Estimation

3.5.3 Recovery from Perturbations

3.5.4 Subsystems as Coupled Oscillators

3.5.5 Legged Locomotion Oscillators are Hybrid Dynamical Systems

3.5.6 Advanced Application: Data Driven Floquet Models

3.5.7 Summary

References

3.6 Model Zoo: Extended Conceptual Models

3.6.1 More Detailed Representations of the Leg

3.6.1.1 Extending the Number of Limbs (B-SLIP, Q-SLIP)

3.6.1.2 Rimless Wheel

3.6.1.3 Stance Leg Adaptation (VLS and E-SLIP)

3.6.1.4 Clock-Torque SLIP (CT-SLIP)

3.6.1.5 Linear Inverted Pendulum Mode (LIPM)

3.6.1.6 Addition of Leg Mass to IP (Acrobot, Simplest Walking Model)

3.6.1.7 Addition of Mass to SLIP Leg (M-SLIP)

3.6.1.8 Extending SLIP with Leg Segments (F-SLIP, 2-SEG, 3-SEG)

3.6.1.9 Ankle Actuated IP

3.6.1.10 Curved Feet Model

3.6.2 Upper Body Modeling

3.6.2.1 Virtual Pivot Point (VPP)

3.6.2.2 Force Modulated Compliant Hip (FMCH)

3.6.3 Extension to 3D

3.6.3.1 3D SLIP

3.6.3.2 3D IP

3.6.4 Extension with Muscle Models

References

Part II Control

4 Control of Motion and Compliance

Introduction

4.1 Stability and Robustness of Bipedal Walking

4.1.1 Introduction

4.1.2 Stability Criteria Related to Instantaneous Properties of the Walking System

4.1.2.1 Projected Center of Mass

4.1.2.2 Zero Moment Point

4.1.2.3 Capture Point or Extrapolated Center of Mass

4.1.2.4 Virtual Pivot Point

4.1.2.5 Angular Momentum

4.1.2.6 Zero Rate of Angular Momentum Point

4.1.3 Stability Criteria for Limit Cycles

4.1.3.1 Definition of Stability and Orbital Stability in the Sense of Lyapunov

4.1.3.2 Stability Analysis of Walking Using Lyapunov's First Method

4.1.3.3 Applicability of Limit Cycle Stability Concepts to Feedback-Controlled Robots and Humans

4.1.4 Robustness Measures of Walking

4.1.4.1 Robustness Analysis via the Basin of Attraction

4.1.4.2 Robustness Analysis via the Gait Sensitivity Norm

4.1.4.3 Robustness Analysis Based on Lyapunov's Second Method

4.1.4.4 Pseudospectra for Robustness Analysis of the Matrix Spectrum

4.1.5 Recovery from Large Perturbations and Pushes

4.1.6 Discussion & Outlook

References

4.2 Optimization as Guiding Principle of Locomotion

4.2.1 Introduction

4.2.2 Forward and Inverse Dynamics Models of Locomotion

4.2.3 Formulating Legged Locomotion as Optimal Control Problem

4.2.4 Application of Optimal Control to Generate Locomotion in Humans and Robots

4.2.5 What Is the Cost Function of Human Locomotion? The Inverse Optimal Control Problem

4.2.6 Application of Inverse Optimal Control to Analyze Optimality in Human Locomotion

4.2.7 Discussion & Outlook

Acknowledgements

References

4.3 Efficiency and Compliance in Bipedal Walking

4.3.1 Introduction

4.3.2 Different Models of Compliance

4.3.2.1 Constant Compliance

4.3.2.2 Variable Compliance

4.3.2.3 Extension of Compliance Models to Coupled Joints

4.3.3 Using Optimal Control for Compliance Studies

4.3.4 Optimization-Based Compliance Studies in Humans

4.3.4.1 Constant Parallel Compliance Models for Running and Walking

4.3.4.2 Compliance Modulation in Human Walking in Different Situations

4.3.5 Optimization-Based Compliance Studies in Robots

4.3.5.1 Constant Serial Compliance in Robots

4.3.5.2 Variable Rest Length Results in Robots

4.3.5.3 Variable Compliance in Robots

4.3.6 Discussion & Outlook

References

4.4 Impedance Control for Bio-inspired Robots

4.4.1 Rigid Body Dynamics

4.4.2 Task/Operational Spaces

4.4.3 Impedance & Admittance

4.4.4 Impedance of a Robot

4.4.5 Impedance Control

4.4.5.1 Impedance Control Through Joint Control

4.4.5.2 Impedance Control Through Kinematic Configuration Control

4.4.5.3 Impedance Control Through Contact Control

4.4.6 Emulation of Muscle Models

References

4.5 Template Models for Control

4.5.1 Introduction

4.5.1.1 A Design Process for Template-Based Control

4.5.2 Template Model Selection

4.5.2.1 Linear CoM Models for Walking

4.5.2.2 SLIP Models for Running

4.5.2.3 Perspectives of Template Model Selection

4.5.3 Template Model Control

4.5.3.1 Control of Linear CoM Models for Walking

4.5.3.2 Control for SLIP-Based Models

4.5.3.3 Beyond Tracking Control for Pendular Models

4.5.4 Establishing a Template/Anchor Relationship

4.5.4.1 Realizing Template Dynamics Through Task-Space Control

Replicating the Dynamics of CoM Templates

Exploiting Redundancy

4.5.4.2 Lifting Other Properties of Template Control

4.5.4.3 Anchoring the Template Through Less Model-Intensive Methods

4.5.4.4 Template-Inspired Mechanical Design

4.5.5 Conclusions

References

4.6 Control Based on Passive Dynamic Walking

4.6.1 Introduction

4.6.2 Passive Dynamic Walking on a Slope

4.6.2.1 Model Description and Equations of Motion

Single Stance Phase (Continuous Dynamics)

Foot-Ground Contact Event

Foot-Strike Phase (Discontinuous Dynamics)

4.6.2.2 Analysis Using Poincaré Return Map

4.6.2.3 Passive Dynamic Walking in 3-Dimensions

4.6.3 Powered Bipedal Robots Inspired from Passive Dynamics

4.6.3.1 Collisionless Walking

4.6.3.2 Actuating Passive Dynamic Walking Robots

4.6.3.3 Discrete-Decision Continuous Action Control

Control Problem

Schematic Example

Discrete Linear Quadratic Regulator (DLQR)

Other Goals

Factors to Consider While Designing the Controller

Example: Controlling a Bipedal Walking Robot

Computing the Linearization

4.6.4 Discussion and Challenges

4.6.4.1 Energy Efficiency and Dynamic Walking

4.6.4.2 Stability and Robustness

4.6.4.3 Versatility, Maneuverability, and Agility

4.6.4.4 Mechanical Design

4.6.4.5 Estimation

4.6.4.6 Higher Dimensional Systems

4.6.5 Conclusion

Acknowledgement

Appendix 4.6.6

4.6.6.1 Derivation of Equations of Motion for the Simplest Walker

Single Stance Phase

Foot-Strike Phase

References

4.7 Hybrid Zero Dynamics Control of Legged Robots

4.7.1 Bipedal Robots with HZD Controllers

4.7.2 Modeling Legged Robots as Hybrid Dynamical Systems

4.7.2.1 Continuous Dynamics

4.7.2.2 Discrete Dynamics

4.7.2.3 Hybrid Control System

4.7.2.4 Advanced Models of Locomotion

4.7.3 Virtual Constraints for Locomotion

4.7.3.1 Virtual Constraints

4.7.3.2 Designing Virtual Constraints for Locomotion Tasks

4.7.4 Using Feedback Control to Impose Virtual Constraints

4.7.4.1 Feedback Linearization

4.7.4.2 Zero Dynamics

4.7.4.3 Partial Zero Dynamics

4.7.5 Generating Periodic Motions

4.7.5.1 Hybrid Zero Dynamics

4.7.5.2 Partial Hybrid Zero Dynamics

4.7.5.3 Control Lyapunov Functions

4.7.6 Extensions of Hybrid Zero Dynamics

4.7.6.1 CLF-Based QPs

4.7.6.2 Multidomain Hybrid Zero Dynamics

4.7.6.3 Application to Prostheses

4.7.6.4 Compliant Hybrid Zero Dynamics

The Asymmetric Spring-Loaded Inverted Pendulum

Embedding the SLIP in the Dynamics of the ASLIP

Implications to the Control of Robots

4.7.7 Summary

References

4.8 Robot Locomotion Control Based on Central Pattern Generators

4.8.1 Introduction

4.8.2 Central Pattern Generators in Animals

4.8.3 CPGs as Robot Controllers

4.8.3.1 Different Types of Implementation

4.8.3.2 Examples of CPG Controllers

Taga's Neuromechanical Simulation of Biped Locomotion

CPG Models for Quadruped Locomotion

CPG Models for Biped Locomotion

CPGs for Amphibious Locomotion

4.8.3.3 Design Methods for CPG Controllers

4.8.4 Discussion

4.8.5 Conclusion

References

5 Torque Control in Legged Locomotion

5.1 Introduction

5.2 System Overview

5.2.1 System Modeling

5.2.2 Potential Control Issues

5.3 A Case Study with an Ankle Exoskeleton

5.3.1 Exoskeleton System

5.3.2 Low-Level Torque Controllers

5.3.2.1 Motor Velocity Control

5.3.2.2 Model-Free Feedback Control

5.3.2.3 Model-Based Feed-Forward Control

5.3.2.4 Model-Based Feedback Control

5.3.2.5 Model-Free Feed-Forward Control

5.3.2.6 Additional Feedback Control Terms Piloted

5.3.3 High-Level Assistance Controllers

5.3.3.1 Stance Torque Control

5.3.3.2 Swing Control

5.3.4 Experimental Methods

5.3.5 Results

5.4 Discussion

5.4.1 Proportional-Learning-Damping Control

5.4.2 Benefits of Additional Control Elements

5.4.2.1 Continuous-Time Integration

5.4.2.2 Model-Based Control Elements

5.4.2.3 Gain Scheduling, Optimal Control, and Learning

5.4.3 Factors Limiting Interpretation

5.4.3.1 High-Level Controllers

5.4.3.2 Interactions with Human Response

5.4.3.3 Hardware Dependence

5.4.4 Implications for Control of Future Systems

5.5 Conclusions

Acknowledgements

Appendix 5.A Stability and Convergence of the Passivity Based Controller

5.A.1 Passivity

5.A.2 Convergence

Appendix 5.B PD*+ΔLRN Versus LRN+PD*

Appendix 5.C Neuromuscular Reflex Model

References

6 Neuromuscular Models for Locomotion

6.1 Introduction: Feedforward vs Feedback in Neural Control: Central Pattern Generators (CPGs) Versus Reflexive Control

References

6.2 Locomotor Central Pattern Generators

6.2.1 Neuronal Networks that Make up the Locomotor CPG

6.2.2 In Vivo Preparations Used to Study the Locomotor CPG

6.2.3 In Vitro Preparations Used to Study the Locomotor CPG

6.2.4 Implementation of Molecular Genetic Techniques to Study the Locomotor CPG

6.2.5 Network Models of the Locomotor CPG

6.2.6 CPG Control of Locomotor Phase Durations

References

6.3 Corticospinal Control of Human Walking

6.3.1 Forward Walking

6.3.2 Backward Walking

6.3.3 Comments on the Role of Motor Cortex in Forward and Backward Walking

6.3.4 Conclusions on Corticospinal Control

References

6.4 Feedback Control: Interaction Between Centrally Generated Commands and Sensory Input

6.4.1 Locomotor Control

6.4.2 Effect on Locomotion of Sensory Loss

6.4.3 Centrally-Generated Commands Versus Sensory-Dominated Control

6.4.4 Sensory Inputs

6.4.5 Stretch Reflexes and "Preflexes": Displacement and Force Feedback

6.4.6 Role of Sensory Input in Phase-Switching

References

6.5 Neuromechanical Control Models

6.5.1 Are Extensor-Dominated Phase Durations Obligatory for Biomechanical Reasons?

6.5.2 Neuromechanical Entrainment in Human Models of Locomotion

6.5.3 Alternative Roles of CPGs in the Limb Controller

6.5.4 Inspiration for Control in Robotics

6.5.5 Modeling the Mammalian Locomotor System

6.5.6 Explicit Example of Neuromechanical Model of Human Locomotion

6.5.7 Concluding Remarks

References

6.6 Appendix

6.6.1 Muscle Activation Function

6.6.2 Force-Velocity Function

6.6.3 Force-Length (Length-Tension) Function

6.6.4 Passive Stiffness

6.6.5 Tendon Compliance

6.6.6 Muscle Spindle Length Response Function

6.6.7 Fusimotor Offset and Gain Function

6.6.8 γ-Fusimotor Drive

6.6.9 Golgi Tendon Organ Model

Acknowledgments

References

Part III Implementation

7 Legged Robots with Bioinspired Morphology

Preface

7.1 Biological Feet: Evolution, Mechanics and Applications

7.1.1 Overview

7.1.2 The Human Foot

7.1.2.1 Anatomy

7.1.2.2 Evolution

7.1.3 Cost-Benefit Analysis of the Human Foot

7.1.3.1 Costs

7.1.3.2 Benefits

7.1.4 Temporal Filtering

7.1.4.1 Point-Like Foot

7.1.4.2 Spatially Extended and Rigid Foot

7.1.4.3 Conclusion: Temporal Filtering

7.1.5 Spatial Filtering

7.1.5.1 Smoothing Over Rough Terrains

7.1.5.2 Effect of the Foot Arches on Stiffness

7.1.6 Conclusion

References

7.2 Bioinspired Leg Design

7.2.1 Functions of a Leg in a Robot

7.2.1.1 Four Basic Functions

7.2.1.2 Obstacle Clearance and Foot Scuffing

7.2.1.3 Material and Manufacturing Considerations

7.2.2 Actuation Strategies

7.2.2.1 Pneumatics, Hydraulics, and DC Motors

7.2.2.2 Active Materials

7.2.2.3 Transmission Strategies

7.2.2.4 Variable Stiffness Mechanisms

7.2.3 Bio-inspiration: Morphology

7.2.4 Bio-inspiration: Dynamics

7.2.4.1 Single Active DOF Legs

7.2.4.2 2+ Active DOF Legs

7.2.4.3 Climbing and Other Uses of Legs

Dynamic Climbing

7.2.4.4 Multiuse Leg Designs

Control vs. Design-Based Approaches

7.2.5 Summary and Future Directions

References

7.3 Human-Inspired Bipeds

7.3.1 Mimicking the Human Figure

7.3.1.1 Early Control-Based Approaches

7.3.1.2 Morphologically Inspired Bipeds and Quasistatic Balancing

7.3.1.3 Passive Walking and Dynamic Balancing

7.3.2 Human-Inspired Musculoskeletal Bipeds

7.3.2.1 Biarticular Muscles: Biomechanics and Inspiration

Coupling Joint Movements

Low Contraction Velocity

Transport of Energy

7.3.2.2 Applications to Robotics

7.3.3 Conclusions

References

7.4 Bioinspired Robotic Quadrupeds

7.4.1 Preliminaries on Gaits

7.4.2 The Role of the Torso: Observations from Biology

7.4.3 Modeling: Template Candidates for Quadrupedal Locomotion

7.4.3.1 Spring-Mass Models for Quadrupedal Locomotion

7.4.3.2 A Passive Template Candidate for Bounding With a Flexible Torso

7.4.4 Quadrupedal Robot Design: Rigid or Flexible Torsos?

7.4.4.1 Robots With Rigid Torso

7.4.4.2 Robots With Segmented Torsos

7.4.5 Conclusions

Acknowledgements

References

8 Actuation in Legged Locomotion

8.1 Muscle-Like Actuation for Locomotion

8.1.1 Fundamental Phenomenological Muscle Mechanics

8.1.2 Active and Semiactive Mechanisms of Force-Production

8.1.3 How Human Muscles Work as Actuators in Locomotion

8.1.4 Redundancy of the Actuation System - Functionally Resolved?

References

8.2 From Stiff to Compliant Actuation

8.2.1 Stiff Servomotor

8.2.2 Stiff Servomotor with Active Compliance

8.2.3 Series Elastic Actuator (SEA)

8.2.3.1 Example: SEA in an Ankle Prosthesis

8.2.4 Variable Stiffness Actuator (VSA)

8.2.4.1 Spring Preload

8.2.4.2 Changing Transmission Between Load and Spring

8.2.4.3 Changing the Physical Properties of the Spring

8.2.5 Parallel Stiffness

8.2.6 Locking Mechanisms

8.2.7 Multi-DoF Joints

References

8.3 Actuators in Robotics as Artificial Muscles

8.3.1 Muscle-Like Actuators Driven by Electric Rotational Motors

8.3.2 Linear Actuators without Slack

8.3.3 Pneumatic Artificial Muscles

8.3.4 Artificial Muscle Emulating Dynamics of Biological Muscle

References

9 Conclusion

9.1 Versatility, Robustness and Economy

References

9.2 Application in Daily Life (Assistive Systems)

9.2.1 Rehabilitation

9.2.2 Spinal Cord Injury

9.2.3 Passive Ankle Foot Orthoses

9.2.4 Powered Ankle Foot Orthoses

9.2.5 Passive Prosthetic Ankles

9.2.6 Powered Prosthetic Ankles

9.2.7 Wearable Robots for Manufacturing

9.2.8 Wearable Robots for Recreation

References

9.3 Related Research Projects and Future Directions

9.3.1 European Projects

9.3.2 Research Projects in North America

9.3.2.1 National Science Foundation (NSF)

9.3.2.2 Defense Advanced Research Projects Agency (DARPA)

9.3.2.3 CAREER: Robust Bipedal Locomotion in Real-World Environments

9.3.2.4 MIT Cheetah Robot

9.3.3 Research in Asia

9.3.3.1 Humanoid Robotics Institute, Waseda University

9.3.3.2 JST Laboratory, University of Tokyo

9.3.3.3 Honda Robotics

9.3.3.4 Humanoid Robotics Project (HRP)

9.3.3.5 HUBO Project, KAIST

9.3.3.6 Adaptive Robotics Laboratory, Osaka University (Hosoda Laboratory)

9.3.3.7 Surena Bipedal Robot Series

9.3.4 Novel Techonologies

9.3.4.1 Soft Exosuit

9.3.4.2 Superflex

9.3.4.3 JTAR from SpringActive

9.3.4.4 Bionics at MIT

9.3.4.5 Quadrupedal Robots of Boston Dynamics

9.3.4.6 SRI PROXI Humanoid Robot

9.3.4.7 SCHAFT Biped Robot

9.3.4.8 "Spring-Mass" Technology in the Future of Walking Robots

9.3.4.9 Summary

References

Index

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