Humanoid Robots: Modelling and Control

__Humanoid Robots: Modeling and Control__provides systematic presentation of the models used in the analysis, design and control of humanoid robots. The book starts with a historical overview of the field, a summary of the current state of the art achievements and an outline of the related fields of research. It moves on to explain the theoretical foundations in terms of kinematic, kineto-static and dynamic relations. Further on, a detailed overview of biped balance control approaches is presented. Models and control algorithms for cooperative object manipulation with a multi-finger hand, a dual-arm and a multi-robot system are also discussed. One of the chapters is devoted to selected topics from the area of motion generation and control and their applications. The final chapter focuses on simulation environments, specifically on the step-by-step design of a simulator using the Matlab(R) environment and tools. This book will benefit readers with an advanced level of understanding of robotics, mechanics and control such as graduate students, academic and industrial researchers and professional engineers. Researchers in the related fields of multi-legged robots, biomechanics, physical therapy and physics-based computer animation of articulated figures can also benefit from the models and computational algorithms presented in the book. Provides a firm theoretical basis for modelling and control algorithm designGives a systematic presentation of models and control algorithmsContains numerous implementation examples demonstrated with 43 video clips Cover 1 HUMANOID ROBOTS: Modeling and Control 3 Copyright 4 Dedication 5 Preface 6 Acknowledgments 8 1 Introduction 9 1.1 Historical Development 9 1.2 Trends in Humanoid Robot Design 10 1.2.1 Human Likeness of a Humanoid Robot 10 1.2.2 Trade-Offs in Humanoid Robot Design 11 1.2.3 Human-Friendly Humanoid Robot Design 11 1.3 Characteristics of Humanoid Robots 12 1.4 Areas of Research Related to Humanoid Robots 13 1.4.1 Kinematic Redundancy, Task Constraints, and Optimal Inverse Kinematics Solutions 13 1.4.2 Constrained Multibody Systems and Contact Modeling 13 1.4.3 Multifingered Hands and Dual-Arm Object Manipulation 14 1.4.4 Underactuated Systems on a Floating Base 14 1.4.5 Other Related Areas of Research 14 Single-Leg, Multilegged, and Multilimb Robots 14 Physics-Based Animation of Articulated Figures 15 Studies on the Biomechanics of Human Movement 15 1.5 Prerequisite and Structure 15 References 16 2 Kinematics 23 2.1 Introduction 23 2.2 Kinematic Structure 23 2.3 Forward and Inverse Kinematic Problems 26 2.4 Differential Kinematics 27 2.4.1 Twist, Spatial Velocity, and Spatial Transform 27 2.4.2 Forward Differential Kinematic Relations 30 Jacobian Matrix 30 Multi-DoF Joint Models 30 Parametrization of Instantaneous Rotation 32 2.4.3 Inverse Differential Kinematic Relations 33 2.5 Differential Kinematics at Singular Configurations 34 2.6 Manipulability Ellipsoid 40 2.7 Kinematic Redundancy 40 2.7.1 Self-Motion 41 2.7.2 General Solution to the Inverse Kinematics Problem 43 2.7.3 Weighted Generalized Inverse 45 2.7.4 Redundancy Resolution via Gradient Projection 46 Joint-Limit Avoidance Subtask 46 Singularity Avoidance Subtask via the Manipulability Measure 46 2.7.5 Redundancy Resolution via the Extended Jacobian Technique 47 2.8 Inverse Kinematics Solution Under Multiple Task Constraints 48 2.8.1 Motion-Task Constraints 49 2.8.2 Redundancy Resolution Methods for Multiple Tasks 50 Restricted Generalized Inverse and Task Prioritization 50 Multiple Tasks With Fixed Priorities 51 Variable Task Priorities With Smooth Task Transitions 53 2.8.3 Iterative Optimization Methods 54 Introducing a Hierarchical Structure With Fixed Task Priorities 55 Variable Task Priorities With Smooth Task Transitions 56 Introducing Inequality Constraints 56 2.8.4 Summary and Discussion 57 2.9 Motion Constraints Through Contacts 58 2.9.1 Contact Joints 59 2.9.2 Contact Coordinate Frames 60 2.9.3 Kinematic Models of Frictionless Contact Joints 61 Examples 62 2.10 Differential Kinematics of Chains With Closed Loops 62 2.10.1 Instantaneous Motion Analysis of Chains With Closed Loops 63 Limb Velocities 65 Velocities Within the Closed Chain 66 2.10.2 Inverse Kinematics Solution 68 2.10.3 Forward Kinematics Solution 69 2.11 Differential Motion Relations of a Humanoid Robot 70 2.11.1 Quasivelocity, Holonomic and Nonholonomic Contact Constraints 70 2.11.2 First-Order Differential Motion Relations Expressed in Terms of Base Quasivelocity 71 Structural Changes 73 Constraint-Consistent Joint Velocity 73 Constraint-Consistent Generalized Velocity 74 2.11.3 Second-Order Differential Motion Constraints and Their Integrability 75 Example 78 2.11.4 First-Order Differential Motion Relations With Mixed Quasivelocity 78 Implementation Example 81 2.11.5 Summary and Discussion 83 References 83 3 Statics 91 3.1 Introduction 91 3.2 Wrench and Spatial Force 91 3.3 Contact Joints: Static Relations 92 3.3.1 Static Models of Frictionless Contact Joints 93 3.3.2 Models of Contact Joints With Friction 94 Point-Contact Model 94 Soft-Finger Contact Model 95 Polyhedral Convex Cone Model 96 Plane-Contact Model: the Contact Wrench Cone 97 Face/Span Polyhedral Convex Cone Double Representations 98 3.3.3 Motion/Force Duality Relations Across Contact Joints 98 Summary and Discussion 100 3.4 Kinetostatic Relations in Independent Closed-Loop Chains 101 3.4.1 Orthogonal Decomposition of the Contact Wrench 102 3.4.2 Orthogonal Decomposition of the Loop-Closure and Root Link Wrenches 102 3.4.3 Decomposition of the Limb Joint Torque 103 3.5 The Wrench Distribution Problem 104 3.5.1 General Solution to the Wrench Distribution Problem 105 3.5.2 Internal Force/Moments: the Virtual Linkage Model 106 Internal Forces 106 Internal Moments 107 Internal Wrench 108 3.5.3 Determining the Joint Torques in the Loop 110 3.5.4 Which Generalized Inverse? 110 3.5.5 Priorities Among the Joint Torque Components 111 3.6 Kinetostatic Relations of a Humanoid Robot 112 3.6.1 The Composite Rigid Body (CRB) and the CRB Wrench 112 3.6.2 Interdependent Closed Loops 114 3.6.3 Independent Closed Loops 115 3.6.4 Determining the Joint Torques 116 3.6.5 Illustrative Examples 117 Double Stance on Flat Floor in 2D (Lateral Plane) 117 Double Stance on Flat Floor With Friction 119 Double Stance on Frictionless Flat Floor 120 Double Stance With Noncoplanar Contacts 120 Double Stance on Flat Floor With High/Zero Friction at the Right/Left Foot 121 3.6.6 Summary and Discussion 121 3.7 Static Posture Stability and Optimization 122 3.7.1 Static Posture Stability 123 Simple Static Stability Test Based on Wrench Distribution 124 CRB-Wrench Cone and Contact-Consistent CRB Wrench 124 An Example 124 Contact Planning 125 Joint Torque Limit Test 125 3.7.2 Static Posture Optimization 126 3.8 Posture Characterization and Duality Relations 127 References 129 4 Dynamics 133 4.1 Introduction 133 4.2 Underactuated Robot Dynamics 134 4.3 Simple Underactuated Models on the Plane 136 4.3.1 The Linear Inverted Pendulum Model 136 4.3.2 Foot Modeling: CoM Dynamics Driven by the Center of Pressure 138 Linearized IP-on-Foot Model 138 IP-on-Cart Model 139 LIP-on-Cart Model 140 4.3.3 Linear Reaction Wheel Pendulum Model and Centroidal Moment Pivot 141 4.3.4 Reaction Mass Pendulum Model 144 4.3.5 Multilink Models on the Plane 145 4.4 Simple Underactuated Models in 3D 145 4.4.1 3D Inverted Pendulum With Variable Length 145 4.4.2 Spherical IP-on-Foot and Sphere-on-Plane Models 147 4.4.3 The 3D Reaction Wheel Pendulum Model 148 4.4.4 The 3D Reaction Mass Pendulum Model 149 4.4.5 Multilink Models in 3D 150 4.5 Dynamic Models of a Fixed-Base Manipulator 150 4.5.1 Dynamic Model in Joint-Space Coordinates 150 4.5.2 Dynamic Model in Spatial Coordinates 152 Operational Space Method [60] 153 Constrained Dynamics in Spatial Coordinates 154 Complete Dynamic Decoupling via the KD-JSD Method [116] 155 4.5.3 Null-Space Dynamics With Dynamically Decoupled Hierarchical Structure 156 4.6 Spatial Momentum of a Manipulator Floating Freely in Zero Gravity 157 4.6.1 Brief Historical Background 158 4.6.2 Spatial Momentum 158 4.6.3 Locked Joints: the Composite Rigid Body 159 4.6.4 Joints Unlocked: Multibody Notation 161 4.6.5 Instantaneous Motion of a Free-Floating Manipulator 164 4.7 Momentum-Based Redundancy Resolution 166 4.7.1 The Momentum Equilibrium Principle 166 4.7.2 Spatial Momentum-Based Redundancy Resolution 167 Coupling Spatial Momentum Conservation: the Reaction Null Space 167 System Spatial-Momentum Conservation 168 4.7.3 Angular Momentum-Based Redundancy Resolution 168 Coupling Angular Momentum Conservation 169 System Angular Momentum Conservation 169 Example: a Dual-Task Scenario With a Free-Floating Space Manipulator 170 4.7.4 Motion of a Free-Floating Humanoid Robot in Zero Gravity 170 4.8 Equation of Motion of a Free-Floating Manipulator in Zero Gravity 171 4.8.1 Representation in Terms of Base Quasivelocity 172 In the Case of Momentum Conservation (no External Forces) 173 In the Presence of External Forces 173 4.8.2 Representation in Terms of Mixed Quasivelocity 175 External Wrench Applied 176 4.8.3 Representation in Terms of Centroidal Quasivelocity 176 4.9 Reaction Null Space-Based Inverse Dynamics 178 4.10 Spatial Momentum of a Humanoid Robot 179 4.11 Equation of Motion of a Humanoid Robot 181 4.12 Constraint-Force Elimination Methods 183 4.12.1 Gauss' Principle of Least Constraint 184 4.12.2 Direct Elimination 186 4.12.3 Maggi's Equations (Null-Space Projection Method) 187 4.12.4 Range-Space Projection Method 190 4.12.5 Summary and Conclusions 191 4.13 Reduced-Form Representations of the Equation of Motion 192 4.13.1 Joint-Space Dynamics-Based Representation 192 4.13.2 Spatial Dynamics-Based Representation (Lagrange-d'Alembert Formulation) 193 Adjoining the Object Dynamics 194 4.13.3 Equation of Motion in End-Link Spatial Coordinates 195 Constrained Dynamics Projection Along the Constrained Motion Directions 196 System Dynamics Projection Along the Unconstrained Motion Directions 196 Projection Along the Constrained and Unconstrained Motion Directions 197 4.13.4 Summary and Discussion 198 4.14 Inverse Dynamics 200 4.14.1 Based on the Direct Elimination/Gauss/Maggi/Projection Methods 200 4.14.2 Based on Lagrange-d'Alembert's Formulation 202 4.14.3 Based on the Joint-Space Dynamics Elimination Approach 202 4.14.4 Summary and Conclusions 203 References 204 5 Balance Control 211 5.1 Overview 211 5.2 Dynamic Postural Stability 213 5.3 Inverted Pendulum-on-Foot Stability Analysis 215 5.3.1 The Extrapolated CoM and the Dynamic Stability Margin 215 5.3.2 Extrapolated CoM Dynamics 217 5.3.3 Discrete States With Transitions 218 5.3.4 Dynamic Stability Region in 2D 219 5.4 ZMP Manipulation-Type Stabilization on Flat Ground 220 5.4.1 The ZMP Manipulation-Type Stabilizer 222 5.4.2 Velocity-Based ZMP Manipulation-Type Stabilization in 3D 223 5.4.3 Regulator-Type ZMP Stabilizer 225 5.4.4 ZMP Stabilization in the Presence of GRF Estimation Time Lag 227 5.4.5 Torso Position Compliance Control (TPCC) 228 5.5 Capture Point-Based Analysis and Stabilization 230 5.5.1 Capture Point (CP) and Instantaneous Capture Point (ICP) 230 5.5.2 ICP-Based Stabilization 231 5.5.3 ICP Stabilization in the Presence of GRF Estimation Time Lag 232 5.5.4 ICP Dynamics and Stabilization in 2D 233 5.6 Stability Analysis and Stabilization With Angular Momentum Component 234 5.6.1 Stability Analysis Based on the LRWP Model 234 5.6.2 Stability Analysis in 3D: the Divergent Component of Motion 236 5.6.3 DCM Stabilizer 239 5.6.4 Summary and Conclusions 240 5.7 Maximum Output Admissible Set Based Stabilization 241 5.8 Balance Control Based on Spatial Momentum and Its Rate of Change 243 5.8.1 Fundamental Functional Dependencies in Balance Control 243 5.8.2 Resolved Momentum Control 245 5.8.3 Whole-Body Balance Control With Relative Angular Momentum/Velocity 245 Relative Angular Momentum/Velocity (RAM/V) Balance Control 247 Special Cases: Balance Control That Conserves the System or the Coupling Angular Momentum 249 5.8.4 RNS-Based Stabilization of Unstable Postures 250 Stabilization of Postures With Rolling Feet 251 Posture Stabilization on a Balance Board 251 Summary and Conclusions 251 5.8.5 An Approach to Contact Stabilization Within the Resolved Momentum Framework 252 5.8.6 Spatial Momentum Rate Stabilization Parametrized by the CMP/VRP 254 5.8.7 CRB Motion Trajectory Tracking With Asymptotic Stability 255 5.9 Task-Space Controller Design for Balance Control 256 5.9.1 Generic Task-Space Controller Structure 257 5.9.2 Optimization Task Formulation and Constraints 258 5.10 Noniterative Body Wrench Distribution Methods 261 5.10.1 Pseudoinverse-Based Body-Wrench Distribution 261 5.10.2 The ZMP Distributor 262 5.10.3 Proportional Distribution Approach 263 5.10.4 The DCM Generalized Inverse 264 Vertical GRF Force Distribution Policy 265 Friction Policy 266 CoP Allocation Policy 267 Final Result 268 Simple Balance Controller With DCM-GI-Based Body Wrench Distribution 269 Implementation Example 269 5.10.5 The VRP Generalized Inverse 270 5.10.6 Joint Torque-Based Contact Wrench Optimization 272 5.11 Noniterative Spatial Dynamics-Based Motion Optimization 274 5.11.1 Independent Motion Optimization With CRB Wrench-Consistent Input 274 5.11.2 Stabilization With Angular Momentum Damping 275 5.11.3 Motion Optimization With Task-Based Hand Motion Constraints 278 5.12 Noniterative Whole-Body Motion/Force Optimization 279 5.12.1 Multicontact Motion/Force Controller Based on the Closed-Chain Model 279 5.12.2 Motion/Force Optimization Based on the Operational-Space Formulation 281 Real-Time Implementation for Balance Control 283 Example 283 5.13 Reactive Balance Control in Response to Weak External Disturbances 286 5.13.1 Gravity Compensation-Based Whole-Body Compliance With Passivity 287 5.13.2 Whole-Body Compliance With Multiple Contacts and Passivity 288 5.13.3 Multicontact Motion/Force Control With Whole-Body Compliance 291 5.14 Iterative Optimization in Balance Control 292 5.14.1 A Brief Historical Overview 293 5.14.2 SOCP-Based Optimization 294 5.14.3 Iterative Contact Wrench Optimization 295 5.14.4 Iterative Spatial Dynamics Optimization 296 Sequential Approach 296 Nonsequential Approach 297 5.14.5 Complete Dynamics-Based Optimization 298 Hierarchical Multiobjective Optimization With Hard Constraints 298 Penalty-Based Multiobjective Optimization With Soft Constraints 299 Inverse Dynamics Plus Inverse Kinematics-Based Optimization Approach 300 5.14.6 Mixed Iterative/Noniterative Optimization Approaches 300 Using Generalized Acceleration Input Data Obtained From a Motion Capture System 301 Hierarchical Task Formulation With Decoupling 301 5.14.7 Computational Time Requirements 302 References 303 6 Cooperative Object Manipulation and Control 311 6.1 Introduction 311 6.2 Multifinger Grasping 312 6.2.1 Grasp Matrix and Hand Jacobian Matrix 312 6.2.2 Statics of Grasping 315 6.2.3 Constraint Types 315 6.2.4 Form Closure 316 Case Study 317 6.2.5 Force Closure 318 Definition of Force Closure 318 Case Study 319 6.3 Multiarm Object Manipulation Control Methods 321 6.3.1 Background of Multiarm Object Manipulation 321 6.3.2 Kinematics and Statics of Multiarm Cooperation 322 6.3.3 Force and Moment Applied to the Object 324 Case Study 325 6.3.4 Load Distribution 326 6.3.5 Control of the External and Internal Wrenches 326 Virtual Linkage [28] (see also Section 3.5.2) 327 Virtual Stick [25,27] 330 Cooperation Among Three Robot Arms 331 Cooperation Between Two Humanoid Robots 332 Cooperation Among Four Humanoid Robots 334 6.3.6 Hybrid Position/Force Control 336 Position Controller 337 Position-Based Force Controller 337 Hybrid Position/Force Controller 337 6.4 Cooperation Between Multiple Humanoids 338 6.4.1 On-Line Footstep Planning 338 6.4.2 Coordinated Movement of Hands and Feet 339 6.4.3 Leader-Follower- and Symmetry-Type Cooperation 341 6.4.4 Leader-Follower-Type Cooperative Object Manipulation 341 Concept of a Leader-Follower-Type Cooperative Object Manipulation 342 Experiment of Object Transportation 343 6.4.5 Symmetry-Type Cooperative Object Manipulation 344 Simulation of Symmetry-Type Cooperation 344 Simulation Results 345 6.4.6 Comparison Between Leader-Follower-Type and Symmetry-Type Cooperation 346 6.5 Dual-Arm Dynamic Object Manipulation Control 348 6.5.1 Equation of Motion of the Object 348 6.5.2 Controller 349 References 353 7 Motion Generation and Control: Selected Topics With Applications 355 7.1 Overview 355 7.2 ICP-Based Gait Generation and Walking Control 357 7.2.1 CP-Based Walking Control 357 7.2.2 CP-Based Gait Generation 358 7.2.3 ICP Controller 361 7.2.4 CP-Based Gait Generation and ZMP Control 362 7.3 Biped Walk on Sand 363 7.3.1 Landing Position Control for Walking on Sand 363 7.3.2 Experiments of Walking on Sand 364 7.3.3 Summary and Discussion 369 7.4 Gait Generation for Irregular Terrain and VRP-GI-Based Walking Control 370 7.4.1 Continuous Double-Support (CDS) Gait Generation 370 7.4.2 Heel-to-Toe (HT) Gait Generation 372 7.4.3 Simulation 373 7.5 Synergy-Based Motion Generation 374 7.5.1 Primitive Motion Synergies 376 7.5.2 Combinations of Primitive Synergies 376 7.5.3 Multiple Synergies Generated With a Single Command Input 378 7.6 Synergy-Based Reactive Balance Control With Planar Models 378 7.6.1 Motion Synergies for Balance Control Used by Humans 379 Sagittal Plane 379 Lateral Plane 379 7.6.2 RNS-Based Reactive Synergies 381 7.6.3 Sagittal-Plane Ankle/Hip Synergies 381 7.6.4 Lateral Plane Ankle, Load/Unload and Lift-Leg Synergies 385 7.6.5 Transverse-Plane Twist Synergy 387 7.6.6 Complex Reactive Synergies Obtained by Superposition of Simple Ones 388 7.6.7 Summary and Discussion 389 7.7 Reactive Synergies Obtained With a Whole-Body Model 390 7.7.1 Reactive Synergies Generated With a Simple Dynamic Torque Controller 390 7.7.2 The Load/Unload and Lift-Leg Strategies Revisited 391 7.7.3 Compliant-Body Response 392 7.7.4 Impact Accommodation With Angular Momentum Damping From the RNS 394 Anticipatory-Type Impact Accommodation 396 Nonanticipatory-Type Impact Accommodation 397 7.7.5 Reactive Stepping 398 Impact Phase 400 Stepping Phase 401 Recovery Phase 401 Simulation 401 7.7.6 Accommodating a Large Impact Without Stepping 404 7.8 Impact Motion Generation 407 7.8.1 Historical Background 407 7.8.2 Considering the Effects of the Reduction Gear Train 408 7.8.3 Ground Reaction Force and Moment 409 7.8.4 Dynamic Effects Caused by Impacts 410 7.8.5 Virtual Mass 412 7.8.6 CoP Displacement Induced by the Impulsive Force 414 7.8.7 Optimization Problems for Impact Motion Generation 414 7.8.8 A Case Study: Karate Chop Motion Generation 416 A Simplified Model of the Humanoid Robot HOAP-2 416 Performance Index for Impact Force Evaluation 417 Performance Index for Stability Margin Evaluation 417 Optimization of the Posture and Velocity at the Impact 418 Optimization of the Velocity Before/After the Impact 419 7.8.9 Experimental Verification of the Generated Impact Motion 421 References 423 8 Simulation 429 8.1 Overview 429 8.2 Robot Simulators 430 8.3 Structure of a Robot Simulator 432 8.4 Dynamics Simulation Using MATLAB/Simulink 437 8.4.1 Generating a Robot Tree Model for Simulink 437 Using CAD Files 437 Using the URDF File 442 8.4.2 Generating the Simulink Model 445 8.4.3 Joint Mode Configuration 448 8.4.4 Modeling of Contact Forces 458 8.4.5 Computing the ZMP 466 8.4.6 Motion Design 471 8.4.7 Simulation 473 References 477 A Appendix 480 A.1 Model Parameters for a Small-Size Humanoid Robot With 4-DoF Arms 480 A.2 Model Parameters for a Small-Size Humanoid Robot With 7-DoF Arms 480 References 486 Index 487 Back Cover 498 Cover......Page 1 HUMANOID ROBOTS: Modeling and Control ......Page 3 Copyright ......Page 4 Dedication......Page 5 Preface......Page 6 Acknowledgments......Page 8 1.1 Historical Development......Page 9 1.2.1 Human Likeness of a Humanoid Robot......Page 10 1.2.3 Human-Friendly Humanoid Robot Design......Page 11 1.3 Characteristics of Humanoid Robots......Page 12 1.4.2 Constrained Multibody Systems and Contact Modeling......Page 13 Single-Leg, Multilegged, and Multilimb Robots......Page 14 1.5 Prerequisite and Structure......Page 15 References......Page 16 2.2 Kinematic Structure......Page 23 2.3 Forward and Inverse Kinematic Problems......Page 26 2.4.1 Twist, Spatial Velocity, and Spatial Transform......Page 27 Multi-DoF Joint Models......Page 30 Parametrization of Instantaneous Rotation......Page 32 2.4.3 Inverse Differential Kinematic Relations......Page 33 2.5 Differential Kinematics at Singular Configurations......Page 34 2.7 Kinematic Redundancy......Page 40 2.7.1 Self-Motion......Page 41 2.7.2 General Solution to the Inverse Kinematics Problem......Page 43 2.7.3 Weighted Generalized Inverse......Page 45 Singularity Avoidance Subtask via the Manipulability Measure......Page 46 2.7.5 Redundancy Resolution via the Extended Jacobian Technique......Page 47 2.8 Inverse Kinematics Solution Under Multiple Task Constraints......Page 48 2.8.1 Motion-Task Constraints......Page 49 Restricted Generalized Inverse and Task Prioritization......Page 50 Multiple Tasks With Fixed Priorities......Page 51 Variable Task Priorities With Smooth Task Transitions......Page 53 2.8.3 Iterative Optimization Methods......Page 54 Introducing a Hierarchical Structure With Fixed Task Priorities......Page 55 Introducing Inequality Constraints......Page 56 2.8.4 Summary and Discussion......Page 57 2.9 Motion Constraints Through Contacts......Page 58 2.9.1 Contact Joints......Page 59 2.9.2 Contact Coordinate Frames......Page 60 2.9.3 Kinematic Models of Frictionless Contact Joints......Page 61 2.10 Differential Kinematics of Chains With Closed Loops......Page 62 2.10.1 Instantaneous Motion Analysis of Chains With Closed Loops......Page 63 Limb Velocities......Page 65 Velocities Within the Closed Chain......Page 66 2.10.2 Inverse Kinematics Solution......Page 68 2.10.3 Forward Kinematics Solution......Page 69 2.11.1 Quasivelocity, Holonomic and Nonholonomic Contact Constraints......Page 70 2.11.2 First-Order Differential Motion Relations Expressed in Terms of Base Quasivelocity......Page 71 Constraint-Consistent Joint Velocity......Page 73 Constraint-Consistent Generalized Velocity......Page 74 2.11.3 Second-Order Differential Motion Constraints and Their Integrability......Page 75 2.11.4 First-Order Differential Motion Relations With Mixed Quasivelocity......Page 78 Implementation Example......Page 81 References......Page 83 3.2 Wrench and Spatial Force......Page 91 3.3 Contact Joints: Static Relations......Page 92 3.3.1 Static Models of Frictionless Contact Joints......Page 93 Point-Contact Model......Page 94 Soft-Finger Contact Model......Page 95 Polyhedral Convex Cone Model......Page 96 Plane-Contact Model: the Contact Wrench Cone......Page 97 3.3.3 Motion/Force Duality Relations Across Contact Joints......Page 98 Summary and Discussion......Page 100 3.4 Kinetostatic Relations in Independent Closed-Loop Chains......Page 101 3.4.2 Orthogonal Decomposition of the Loop-Closure and Root Link Wrenches......Page 102 3.4.3 Decomposition of the Limb Joint Torque......Page 103 3.5 The Wrench Distribution Problem......Page 104 3.5.1 General Solution to the Wrench Distribution Problem......Page 105 Internal Forces......Page 106 Internal Moments......Page 107 Internal Wrench......Page 108 3.5.4 Which Generalized Inverse?......Page 110 3.5.5 Priorities Among the Joint Torque Components......Page 111 3.6.1 The Composite Rigid Body (CRB) and the CRB Wrench......Page 112 3.6.2 Interdependent Closed Loops......Page 114 3.6.3 Independent Closed Loops......Page 115 3.6.4 Determining the Joint Torques......Page 116 Double Stance on Flat Floor in 2D (Lateral Plane)......Page 117 Double Stance on Flat Floor With Friction......Page 119 Double Stance With Noncoplanar Contacts......Page 120 3.6.6 Summary and Discussion......Page 121 3.7 Static Posture Stability and Optimization......Page 122 3.7.1 Static Posture Stability......Page 123 An Example......Page 124 Joint Torque Limit Test......Page 125 3.7.2 Static Posture Optimization......Page 126 3.8 Posture Characterization and Duality Relations......Page 127 References......Page 129 4.1 Introduction......Page 133 4.2 Underactuated Robot Dynamics......Page 134 4.3.1 The Linear Inverted Pendulum Model......Page 136 Linearized IP-on-Foot Model......Page 138 IP-on-Cart Model......Page 139 LIP-on-Cart Model......Page 140 4.3.3 Linear Reaction Wheel Pendulum Model and Centroidal Moment Pivot......Page 141 4.3.4 Reaction Mass Pendulum Model......Page 144 4.4.1 3D Inverted Pendulum With Variable Length......Page 145 4.4.2 Spherical IP-on-Foot and Sphere-on-Plane Models......Page 147 4.4.3 The 3D Reaction Wheel Pendulum Model......Page 148 4.4.4 The 3D Reaction Mass Pendulum Model......Page 149 4.5.1 Dynamic Model in Joint-Space Coordinates......Page 150 4.5.2 Dynamic Model in Spatial Coordinates......Page 152 Operational Space Method [60]......Page 153 Constrained Dynamics in Spatial Coordinates......Page 154 Complete Dynamic Decoupling via the KD-JSD Method [116]......Page 155 4.5.3 Null-Space Dynamics With Dynamically Decoupled Hierarchical Structure......Page 156 4.6 Spatial Momentum of a Manipulator Floating Freely in Zero Gravity......Page 157 4.6.2 Spatial Momentum......Page 158 4.6.3 Locked Joints: the Composite Rigid Body......Page 159 4.6.4 Joints Unlocked: Multibody Notation......Page 161 4.6.5 Instantaneous Motion of a Free-Floating Manipulator......Page 164 4.7.1 The Momentum Equilibrium Principle......Page 166 Coupling Spatial Momentum Conservation: the Reaction Null Space......Page 167 4.7.3 Angular Momentum-Based Redundancy Resolution......Page 168 System Angular Momentum Conservation......Page 169 4.7.4 Motion of a Free-Floating Humanoid Robot in Zero Gravity......Page 170 4.8 Equation of Motion of a Free-Floating Manipulator in Zero Gravity......Page 171 4.8.1 Representation in Terms of Base Quasivelocity......Page 172 In the Presence of External Forces......Page 173 4.8.2 Representation in Terms of Mixed Quasivelocity......Page 175 4.8.3 Representation in Terms of Centroidal Quasivelocity......Page 176 4.9 Reaction Null Space-Based Inverse Dynamics......Page 178 4.10 Spatial Momentum of a Humanoid Robot......Page 179 4.11 Equation of Motion of a Humanoid Robot......Page 181 4.12 Constraint-Force Elimination Methods......Page 183 4.12.1 Gauss' Principle of Least Constraint......Page 184 4.12.2 Direct Elimination......Page 186 4.12.3 Maggi's Equations (Null-Space Projection Method)......Page 187 4.12.4 Range-Space Projection Method......Page 190 4.12.5 Summary and Conclusions......Page 191 4.13.1 Joint-Space Dynamics-Based Representation......Page 192 4.13.2 Spatial Dynamics-Based Representation (Lagrange-d'Alembert Formulation)......Page 193 Adjoining the Object Dynamics......Page 194 4.13.3 Equation of Motion in End-Link Spatial Coordinates......Page 195 System Dynamics Projection Along the Unconstrained Motion Directions......Page 196 Projection Along the Constrained and Unconstrained Motion Directions......Page 197 4.13.4 Summary and Discussion......Page 198 4.14.1 Based on the Direct Elimination/Gauss/Maggi/Projection Methods......Page 200 4.14.3 Based on the Joint-Space Dynamics Elimination Approach......Page 202 4.14.4 Summary and Conclusions......Page 203 References......Page 204 5.1 Overview......Page 211 5.2 Dynamic Postural Stability......Page 213 5.3.1 The Extrapolated CoM and the Dynamic Stability Margin......Page 215 5.3.2 Extrapolated CoM Dynamics......Page 217 5.3.3 Discrete States With Transitions......Page 218 5.3.4 Dynamic Stability Region in 2D......Page 219 5.4 ZMP Manipulation-Type Stabilization on Flat Ground......Page 220 5.4.1 The ZMP Manipulation-Type Stabilizer......Page 222 5.4.2 Velocity-Based ZMP Manipulation-Type Stabilization in 3D......Page 223 5.4.3 Regulator-Type ZMP Stabilizer......Page 225 5.4.4 ZMP Stabilization in the Presence of GRF Estimation Time Lag......Page 227 5.4.5 Torso Position Compliance Control (TPCC)......Page 228 5.5.1 Capture Point (CP) and Instantaneous Capture Point (ICP)......Page 230 5.5.2 ICP-Based Stabilization......Page 231 5.5.3 ICP Stabilization in the Presence of GRF Estimation Time Lag......Page 232 5.5.4 ICP Dynamics and Stabilization in 2D......Page 233 5.6.1 Stability Analysis Based on the LRWP Model......Page 234 5.6.2 Stability Analysis in 3D: the Divergent Component of Motion......Page 236 5.6.3 DCM Stabilizer......Page 239 5.6.4 Summary and Conclusions......Page 240 5.7 Maximum Output Admissible Set Based Stabilization......Page 241 5.8.1 Fundamental Functional Dependencies in Balance Control......Page 243 5.8.3 Whole-Body Balance Control With Relative Angular Momentum/Velocity......Page 245 Relative Angular Momentum/Velocity (RAM/V) Balance Control......Page 247 Special Cases: Balance Control That Conserves the System or the Coupling Angular Momentum......Page 249 5.8.4 RNS-Based Stabilization of Unstable Postures......Page 250 Summary and Conclusions......Page 251 5.8.5 An Approach to Contact Stabilization Within the Resolved Momentum Framework......Page 252 5.8.6 Spatial Momentum Rate Stabilization Parametrized by the CMP/VRP......Page 254 5.8.7 CRB Motion Trajectory Tracking With Asymptotic Stability......Page 255 5.9 Task-Space Controller Design for Balance Control......Page 256 5.9.1 Generic Task-Space Controller Structure......Page 257 5.9.2 Optimization Task Formulation and Constraints......Page 258 5.10.1 Pseudoinverse-Based Body-Wrench Distribution......Page 261 5.10.2 The ZMP Distributor......Page 262 5.10.3 Proportional Distribution Approach......Page 263 5.10.4 The DCM Generalized Inverse......Page 264 Vertical GRF Force Distribution Policy......Page 265 Friction Policy......Page 266 CoP Allocation Policy......Page 267 Final Result......Page 268 Implementation Example......Page 269 5.10.5 The VRP Generalized Inverse......Page 270 5.10.6 Joint Torque-Based Contact Wrench Optimization......Page 272 5.11.1 Independent Motion Optimization With CRB Wrench-Consistent Input......Page 274 5.11.2 Stabilization With Angular Momentum Damping......Page 275 5.11.3 Motion Optimization With Task-Based Hand Motion Constraints......Page 278 5.12.1 Multicontact Motion/Force Controller Based on the Closed-Chain Model......Page 279 5.12.2 Motion/Force Optimization Based on the Operational-Space Formulation......Page 281 Example......Page 283 5.13 Reactive Balance Control in Response to Weak External Disturbances......Page 286 5.13.1 Gravity Compensation-Based Whole-Body Compliance With Passivity......Page 287 5.13.2 Whole-Body Compliance With Multiple Contacts and Passivity......Page 288 5.13.3 Multicontact Motion/Force Control With Whole-Body Compliance......Page 291 5.14 Iterative Optimization in Balance Control......Page 292 5.14.1 A Brief Historical Overview......Page 293 5.14.2 SOCP-Based Optimization......Page 294 5.14.3 Iterative Contact Wrench Optimization......Page 295 Sequential Approach......Page 296 Nonsequential Approach......Page 297 Hierarchical Multiobjective Optimization With Hard Constraints......Page 298 Penalty-Based Multiobjective Optimization With Soft Constraints......Page 299 5.14.6 Mixed Iterative/Noniterative Optimization Approaches......Page 300 Hierarchical Task Formulation With Decoupling......Page 301 5.14.7 Computational Time Requirements......Page 302 References......Page 303 6.1 Introduction......Page 311 6.2.1 Grasp Matrix and Hand Jacobian Matrix......Page 312 6.2.3 Constraint Types......Page 315 6.2.4 Form Closure......Page 316 Case Study......Page 317 Definition of Force Closure......Page 318 Case Study......Page 319 6.3.1 Background of Multiarm Object Manipulation......Page 321 6.3.2 Kinematics and Statics of Multiarm Cooperation......Page 322 6.3.3 Force and Moment Applied to the Object......Page 324 Case Study......Page 325 6.3.5 Control of the External and Internal Wrenches......Page 326 Virtual Linkage [28] (see also Section 3.5.2)......Page 327 Virtual Stick [25,27]......Page 330 Cooperation Among Three Robot Arms......Page 331 Cooperation Between Two Humanoid Robots......Page 332 Cooperation Among Four Humanoid Robots......Page 334 6.3.6 Hybrid Position/Force Control......Page 336 Hybrid Position/Force Controller......Page 337 6.4.1 On-Line Footstep Planning......Page 338 6.4.2 Coordinated Movement of Hands and Feet......Page 339 6.4.4 Leader-Follower-Type Cooperative Object Manipulation......Page 341 Concept of a Leader-Follower-Type Cooperative Object Manipulation......Page 342 Experiment of Object Transportation......Page 343 Simulation of Symmetry-Type Cooperation......Page 344 Simulation Results......Page 345 6.4.6 Comparison Between Leader-Follower-Type and Symmetry-Type Cooperation......Page 346 6.5.1 Equation of Motion of the Object......Page 348 6.5.2 Controller......Page 349 References......Page 353 7.1 Overview......Page 355 7.2.1 CP-Based Walking Control......Page 357 7.2.2 CP-Based Gait Generation......Page 358 7.2.3 ICP Controller......Page 361 7.2.4 CP-Based Gait Generation and ZMP Control......Page 362 7.3.1 Landing Position Control for Walking on Sand......Page 363 7.3.2 Experiments of Walking on Sand......Page 364 7.3.3 Summary and Discussion......Page 369 7.4.1 Continuous Double-Support (CDS) Gait Generation......Page 370 7.4.2 Heel-to-Toe (HT) Gait Generation......Page 372 7.4.3 Simulation......Page 373 7.5 Synergy-Based Motion Generation......Page 374 7.5.2 Combinations of Primitive Synergies......Page 376 7.6 Synergy-Based Reactive Balance Control With Planar Models......Page 378 Lateral Plane......Page 379 7.6.3 Sagittal-Plane Ankle/Hip Synergies......Page 381 7.6.4 Lateral Plane Ankle, Load/Unload and Lift-Leg Synergies......Page 385 7.6.5 Transverse-Plane Twist Synergy......Page 387 7.6.6 Complex Reactive Synergies Obtained by Superposition of Simple Ones......Page 388 7.6.7 Summary and Discussion......Page 389 7.7.1 Reactive Synergies Generated With a Simple Dynamic Torque Controller......Page 390 7.7.2 The Load/Unload and Lift-Leg Strategies Revisited......Page 391 7.7.3 Compliant-Body Response......Page 392 7.7.4 Impact Accommodation With Angular Momentum Damping From the RNS......Page 394 Anticipatory-Type Impact Accommodation......Page 396 Nonanticipatory-Type Impact Accommodation......Page 397 7.7.5 Reactive Stepping......Page 398 Impact Phase......Page 400 Simulation......Page 401 7.7.6 Accommodating a Large Impact Without Stepping......Page 404 7.8.1 Historical Background......Page 407 7.8.2 Considering the Effects of the Reduction Gear Train......Page 408 7.8.3 Ground Reaction Force and Moment......Page 409 7.8.4 Dynamic Effects Caused by Impacts......Page 410 7.8.5 Virtual Mass......Page 412 7.8.7 Optimization Problems for Impact Motion Generation......Page 414 A Simplified Model of the Humanoid Robot HOAP-2......Page 416 Performance Index for Stability Margin Evaluation......Page 417 Optimization of the Posture and Velocity at the Impact......Page 418 Optimization of the Velocity Before/After the Impact......Page 419 7.8.9 Experimental Verification of the Generated Impact Motion......Page 421 References......Page 423 8.1 Overview......Page 429 8.2 Robot Simulators......Page 430 8.3 Structure of a Robot Simulator......Page 432 Using CAD Files......Page 437 Using the URDF File......Page 442 8.4.2 Generating the Simulink Model......Page 445 8.4.3 Joint Mode Configuration......Page 448 8.4.4 Modeling of Contact Forces......Page 458 8.4.5 Computing the ZMP......Page 466 8.4.6 Motion Design......Page 471 8.4.7 Simulation......Page 473 References......Page 477 A.2 Model Parameters for a Small-Size Humanoid Robot With 7-DoF Arms......Page 480 References......Page 486 Index......Page 487 Back Cover......Page 498 Humanoid Robots: Modeling and Control provides systematic presentation of the models used in the analysis, design and control of humanoid robots. The book starts with a historical overview of the field, a summary of the current state of the art achievements and an outline of the related fields of research. It moves on to explain the theoretical foundations in terms of kinematic, kineto-static and dynamic relations. Further on, a detailed overview of biped balance control approaches is presented. Models and control algorithms for cooperative object manipulation with a multi-finger hand, a dual-arm and a multi-robot system are also discussed. One of the chapters is devoted to selected topics from the area of motion generation and control and their applications. The final chapter focuses on simulation environments, specifically on the step-by-step design of a simulator using the Matlab(R) environment and tools. This book will benefit readers with an advanced level of understanding of robotics, mechanics and control such as graduate students, academic and industrial researchers and professional engineers. Researchers in the related fields of multi-legged robots, biomechanics, physical therapy and physics-based computer animation of articulated figures can also benefit from the models and computational algorithms presented in the book. Provides a firm theoretical basis for modelling and control algorithm design Gives a systematic presentation of models and control algorithms Contains numerous implementation examples demonstrated with 43 video clips "Humanoid Robots: Modeling and Control provides systematic presentation of the models used in the analysis, design and control of humanoid robots. The book starts with a historical overview of the field, a summary of the current state of the art achievements and an outline of the related fields of research. It moves on to explain the theoretical foundations in terms of kinematic, kineto-static and dynamic relations. Further on, a detailed overview of biped balance control approaches is presented. Models and control algorithms for cooperative object manipulation with a multi-finger hand, a dual-arm and a multi-robot system are also discussed. One of the chapters is devoted to selected topics from the area of motion generation and control and their applications. The final chapter focuses on simulation environments, specifically on the step-by-step design of a simulator using the Matlab® environment and tools. This book will benefit readers with an advanced level of understanding of robotics, mechanics and control such as graduate students, academic and industrial researchers and professional engineers. Researchers in the related fields of multi-legged robots, biomechanics, physical therapy and physics-based computer animation of articulated figures can also benefit from the models and computational algorithms presented in the book"--Page 4 of cover