# Robot Manipulator Redundancy Resolution

Wiley-ASME Press Series

1. Edition November 2017

320 Pages, Hardcover*Practical Approach Book*

**978-1-119-38123-5**

Introduces a revolutionary, quadratic-programming based approach to solving long-standing problems in motion planning and control of redundant manipulators

This book describes a novel quadratic programming approach to solving redundancy resolutions problems with redundant manipulators. Known as ``QP-unified motion planning and control of redundant manipulators'' theory, it systematically solves difficult optimization problems of inequality-constrained motion planning and control of redundant manipulators that have plagued robotics engineers and systems designers for more than a quarter century.

An example of redundancy resolution could involve a robotic limb with six joints, or degrees of freedom (DOFs), with which to position an object. As only five numbers are required to specify the position and orientation of the object, the robot can move with one remaining DOF through practically infinite poses while performing a specified task. In this case redundancy resolution refers to the process of choosing an optimal pose from among that infinite set. A critical issue in robotic systems control, the redundancy resolution problem has been widely studied for decades, and numerous solutions have been proposed. This book investigates various approaches to motion planning and control of redundant robot manipulators and describes the most successful strategy thus far developed for resolving redundancy resolution problems.

* Provides a fully connected, systematic, methodological, consecutive, and easy approach to solving redundancy resolution problems

* Describes a new approach to the time-varying Jacobian matrix pseudoinversion, applied to the redundant-manipulator kinematic control

* Introduces The QP-based unification of robots' redundancy resolution

* Illustrates the effectiveness of the methods presented using a large number of computer simulation results based on PUMA560, PA10, and planar robot manipulators

* Provides technical details for all schemes and solvers presented, for readers to adopt and customize them for specific industrial applications

Robot Manipulator Redundancy Resolution is must-reading for advanced undergraduates and graduate students of robotics, mechatronics, mechanical engineering, tracking control, neural dynamics/neural networks, numerical algorithms, computation and optimization, simulation and modelling, analog, and digital circuits. It is also a valuable working resource for practicing robotics engineers and systems designers and industrial researchers.

List of Tables xxv

Preface xxvii

Acknowledgments xxxiii

Acronyms xxxv

Part I Pseudoinverse-Based ZD Approach 1

1 Redundancy Resolution via Pseudoinverse and ZD Models 3

1.1 Introduction 3

1.2 Problem Formulation and ZD Models 5

1.2.1 Problem Formulation 5

1.2.2 Continuous-Time ZD Model 6

1.2.3 Discrete-Time ZD Models 7

1.2.3.1 Euler-Type DTZD Model with J (t) Known 7

1.2.3.2 Euler-Type DTZD Model with J (t) Unknown 7

1.2.3.3 Taylor-Type DTZD Models 8

1.3 ZD Applications to Different-Type Robot Manipulators 9

1.3.1 Application to a Five-Link Planar Robot Manipulator 9

1.3.2 Application to a Three-Link Planar Robot Manipulator 12

1.4 Chapter Summary 14

Part II Inverse-Free Simple Approach 15

2 G1 Type Scheme to JVL Inverse Kinematics 17

2.1 Introduction 17

2.2 Preliminaries and RelatedWork 18

2.3 Scheme Formulation 18

2.4 Computer Simulations 19

2.4.1 Square-Path Tracking Task 19

2.4.2 "Z"-Shaped Path Tracking Task 22

2.5 Physical Experiments 25

2.6 Chapter Summary 26

3 D1G1 Type Scheme to JAL Inverse Kinematics 27

3.1 Introduction 27

3.2 Preliminaries and RelatedWork 28

3.3 Scheme Formulation 28

3.4 Computer Simulations 29

3.4.1 Rhombus-Path Tracking Task 29

3.4.1.1 Verifications 29

3.4.1.2 Comparisons 30

3.4.2 Triangle-Path Tracking Task 32

3.5 Chapter Summary 36

4 Z1G1 Type Scheme to JAL Inverse Kinematics 37

4.1 Introduction 37

4.2 Problem Formulation and Z1G1 Type Scheme 37

4.3 Computer Simulations 38

4.3.1 Desired Initial Position 38

4.3.1.1 Isosceles-Trapezoid Path Tracking 40

4.3.1.2 Isosceles-Triangle Path Tracking 41

4.3.1.3 Square Path Tracking 42

4.3.2 Nondesired Initial Position 44

4.4 Physical Experiments 45

4.5 Chapter Summary 45

Part III QP Approach and Unification 47

5 Redundancy Resolution via QP Approach and Unification 49

5.1 Introduction 49

5.2 Robotic Formulation 50

5.3 Handling Joint Physical Limits 52

5.3.1 Joint-Velocity Level 52

5.3.2 Joint-Acceleration Level 52

5.4 Avoiding Obstacles 53

5.5 Various Performance Indices 54

5.5.1 Resolved at Joint-Velocity Level 55

5.5.1.1 MVN scheme 55

5.5.1.2 RMP scheme 55

5.5.1.3 MKE scheme 55

5.5.2 Resolved at Joint-Acceleration Level 55

5.5.2.1 MAN scheme 55

5.5.2.2 MTN scheme 56

5.5.2.3 IIWT scheme 56

5.6 Unified QP Formulation 56

5.7 Online QP Solutions 57

5.7.1 Traditional QP Routines 57

5.7.2 Compact QP Method 57

5.7.3 Dual Neural Network 57

5.7.4 LVI-Aided Primal-Dual Neural Network 57

5.7.5 Numerical Algorithms E47 and 94LVI 59

5.7.5.1 Numerical Algorithm E47 59

5.7.5.2 Numerical Algorithm 94LVI 59

5.8 Computer Simulations 61

5.9 Chapter Summary 66

Part IV Illustrative JVL QP Schemes and Performances 67

6 Varying Joint-Velocity Limits Handled by QP 69

6.1 Introduction 69

6.2 Preliminaries and Problem Formulation 70

6.2.1 Six-DOF Planar Robot System 70

6.2.2 Varying Joint-Velocity Limits 73

6.3 9 4LVI Assisted QP Solution 76

6.4 Computer Simulations and Physical Experiments 77

6.4.1 Line-Segment Path-Tracking Task 77

6.4.2 Elliptical-Path Tracking Task 85

6.4.3 Simulations with Faster Tasks 87

6.4.3.1 Line-Segment-Path-Tracking Task 87

6.4.3.2 Elliptical-Path-Tracking Task 89

6.5 Chapter Summary 92

7 Feedback-AidedMinimum Joint Motion 95

7.1 Introduction 95

7.2 Preliminaries and Problem Formulation 97

7.2.1 Minimum Joint Motion Performance Index 97

7.2.2 Varying Joint-Velocity Limits 100

7.3 Computer Simulations and Physical Experiments 101

7.3.1 "M"-Shaped Path-Tracking Task 101

7.3.1.1 Simulation Comparisons with Different p 101

7.3.1.2 Simulation Comparisons with Different 103

7.3.1.3 Simulative and Experimental Verifications of FAMJM Scheme 105

7.3.2 "P"-Shaped Path Tracking Task 107

7.3.3 Comparisons with Pseudoinverse-Based Approach 108

7.3.3.1 Comparison with Tracking Task of Larger "M"-Shaped Path 110

7.3.3.2 Comparison with Tracking Task of Larger "P"-Shaped Path 112

7.4 Chapter Summary 119

8 QP Based Manipulator State Adjustment 121

8.1 Introduction 121

8.2 Preliminaries and Scheme Formulation 122

8.3 QP Solution and Control of Robot Manipulator 124

8.4 Computer Simulations and Comparisons 125

8.4.1 State Adjustment without ZIV Constraint 125

8.4.2 State Adjustment with ZIV Constraint 128

8.5 Physical Experiments 132

8.6 Chapter Summary 136

Part V Self-Motion Planning 137

9 QP-Based Self-Motion Planning 139

9.1 Introduction 139

9.2 Preliminaries and QP Formulation 140

9.2.1 Self-Motion Criterion 140

9.2.2 QP Formulation 141

9.3 LVIAPDNN Assisted QP Solution 141

9.4 PUMA560 Based Computer Simulations 142

9.4.1 From Initial Configuration A to Desired Configuration B 144

9.4.2 From Initial Configuration A to Desired Configuration C 146

9.4.3 From Initial Configuration E to Desired Configuration F 147

9.5 PA10 Based Computer Simulations 152

9.6 Chapter Summary 158

10 PseudoinverseMethod and Singularities Discussed 161

10.1 Introduction 161

10.2 Preliminaries and Scheme Formulation 162

10.2.1 Modified Performance Index for SMP 163

10.2.2 QP-Based SMP Scheme Formulation 163

10.3 LVIAPDNN Assisted QP Solution with Discussion 164

10.4 Computer Simulations 167

10.4.1 Three-Link Redundant PlanarManipulator 168

10.4.1.1 Verifications 168

10.4.1.2 Comparisons 171

10.4.2 PUMA560 Robot Manipulator 172

10.4.3 PA10 Robot Manipulator 176

10.5 Chapter Summary 180

Appendix 181

Equivalence Analysis in Limit Situation 181

11 Self-Motion Planning with ZIV Constraint 183

11.1 Introduction 183

11.2 Preliminaries and Scheme Formulation 184

11.2.1 Handling Joint Physical Limits 184

11.2.2 QP Reformulation 187

11.2.3 Design of ZIV Constraint 187

11.3 E47 Assisted QP Solution 188

11.4 Computer Simulations and Physical Experiments 189

11.5 Chapter Summary 197

Part VI Manipulability Maximization 199

12 Manipulability-Maximizing SMP Scheme 201

12.1 Introduction 201

12.2 Scheme Formulation 202

12.2.1 Derivation of Manipulability Index 202

12.2.2 Handling Physical Limits 203

12.2.3 QP Formulation 203

12.3 Computer Simulations and Physical Experiments 204

12.3.1 Computer Simulations 204

12.3.2 Physical Experiments 205

12.4 Chapter Summary 209

13 Time-Varying Coefficient AidedMMScheme 211

13.1 Introduction 211

13.2 Manipulability-Maximization with Time-Varying Coefficient 212

13.2.1 Nonzero Initial/Final Joint-Velocity Problem 212

13.2.2 Scheme Formulation 213

13.2.3 94LVI Assisted QP Solution 215

13.3 Computer Simulations and Physical Experiments 216

13.3.1 Computer Simulations 216

13.3.2 Physical Experiments 224

13.4 Chapter Summary 226

Part VII Encoder Feedback and Joystick Control 227

14 QP Based Encoder Feedback Control 229

14.1 Introduction 229

14.2 Preliminaries and Scheme Formulation 231

14.2.1 Joint Description 231

14.2.2 OMPFC Scheme 231

14.3 Computer Simulations 234

14.3.1 Petal-Shaped Path-Tracking Task 234

14.3.2 Comparative Simulations 238

14.3.2.1 Petal-Shaped Path Tracking Using Another Group of Joint-Angle Limits 238

14.3.2.2 Petal-Shaped Path Tracking via the Method 4 (M4) Algorithm 238

14.3.3 Hexagonal-Path-Tracking Task 239

14.4 Physical Experiments 240

14.5 Chapter Summary 248

15 QP Based Joystick Control 251

15.1 Introduction 251

15.2 Preliminaries and Hardware System 251

15.2.1 Velocity-Specified Inverse Kinematics Problem 252

15.2.2 Joystick-Controlled Manipulator Hardware System 252

15.3 Scheme Formulation 253

15.3.1 Cosine-Aided Position-to-VelocityMapping 253

15.3.2 Real-Time Joystick-Controlled Motion Planning 254

15.4 Computer Simulations and Physical Experiments 254

15.4.1 Movement Toward Four Directions 255

15.4.2 "MVN" LetterWriting 259

15.5 Chapter Summary 259

References 261

Index 277

Long Jin is pursuing his doctorate in Communication and Information Systems at the School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China. His main research interests include robotics, neural networks, and intelligent information processing.