Iterative Inverse Kinematics with Manipulator Configuration Control and Proof of Convergence Gregory Z. Grudid B.A.Sc. University of British Columbia, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPUED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1990 © Gregory Z. Grudid, 1990 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of The University of British Columbia Vancouver, Canada Date Aua. 20, 1110. DE-6 (2/88) Abstract A complete solution to the inverse kinematics problem for a large class of practical manipulators, which includes manipulators with no closed form inverse kinematics equations, is presented in this thesis. A complete solution to the inverse kinematics problem of a manipulator is defined as a method for obtaining the required joint variable values to establish the desired endpoint position, endpoint orientation, and manipulator configuration; the only requirement being that the desired solution exists. For all manipulator geometries that satisfy a set of conditions (THEOREM I), an algorithm is presented that is theoretically guaranteed to always converge to the desired solution (if it exists). The algorithm is extensively tested on two complex 6 degree of freedom manipulators which have no known closed form inverse kinematics equations. It is shown that the algorithm can be used in real time manipulator control. Applications of the method to other 6 DOF manipulator geometries and to redundant manipulators are discussed. ii Table of Contents Abstract 11 List of Tables v List of Figures vi Acknowledgments vii 1 Introduction 1 1.1 Current Approaches to the Inverse Kinematics Problem 3 1.2 Thesis Organization 8 2 The Theoretical Development 9 2.1 Summary of the Approach 9 2.2 Forward Kinematics Representation : 9 2.3 The Theoretical Framework 11 2.4 Application to 6 DOF Manipulators with Non-Intersecting, Non-Parallel Rotational Wrist Joints 14 3 The Nonlinear Equation Solver 16 3.1 The Main Algorithm 17 3.2 The Fixed Point Algorithm 18 3.3 The Search Algorithm 20 3.4 The Modified Powell's Algorithm 24 3.5 Numerical Approximation of the Jacobian 28 4 Numerical Examples " 31 4.1 The Spherical Manipulator 31 4.1.1 Evaluation of the Algorithm Over 100,000 Uniformly Generated Random Points . 35 4.1.2 Rate of Convergence Comparison with an Existing Algorithm 37 iii 4.2 The Modified PUMA Manipulator 39 4.2.1 Evaluation of the Algorithm Over 1,000,000 Uniformly Generated Random Points 4.2.2 Evaluation of the Algorithm On Various Wrist Offset Magnitudes 44 47 5 Discussion 5.1 Analysis of Results 47 5.2 Improving Convergence Near Singularity Regions and Workspace Boundaries 48 5.3 Application to Other Manipulator Geometries 48 5.4 Application to Redundant Manipulators 49 6 Conclusions 51 6.1 Contributions 51 6.2 Further Research References Appendix A Proof of Theorems 56 Appendix B Pseudo-code Descriptions of the Procedure and Algorithms 65 Appendix C The Inverse Kinematics Equations of the Spherical Model Manipulator 76 Appendix D The Inverse Kinematics Equations of the Modified PUMA Model Manipulator 78 iv List of Tables 1.1 Summary of Current Inverse Kinematics Methods 5 4.1 The Denavit-Hartenberg Parameters of the "Real" Spherical Manipulator 33 4.2 The Denavit-Hartenberg Parameters of the "Model" Spherical Manipulator 33 4.3 Computational Requirements of The Spherical Manipulator 37 4.4 The Denavit-Hartenberg Parameters of the "Real" Modified PUMA Manipulator 40 4.5 The Denavit-Hartenberg Parameters of the "Model" Modified PUMA Manipulator .... 40 4.6 Simulation Results of the Modified PUMA Manipulator 43 4.7 Computational Requirements for the Modified PUMA Manipulator 44 4.8 Simulation Results on Various Wrist Offset Magnitudes: Analytical Jacobian 45 4.9 Simulation Results on Various Wrist Offset Magnitudes: Numerically Approximated Jacobian ^ V List of Figures 1.1 The Modified PUMA Manipulator 1.2 Manipulator Configurations 3.1 Hierarchy of Algorithms 3.2 Search Grid 4.1 The Spherical Manipulator A.l Relative Positioning of Manipulator Endpoint Positions vi Acknowledgments Foremost, I would like to thank my supervisor, Dr. Lawrence, for his continued support throughout the course of my thesis. Many of the ideas presented in this thesis have resulted directiy from discussions with him. I would also like to thank Jane Mulligan, Kevin O'Donnell, and The Iceman for proof reading earlier versions of thesis. vii Chapter 1: Introduction Chapter 1 Introduction A complete solution to the inverse kinematics problem for a robot manipulator, as defined in this thesis, is a method of obtaining the required manipulator joint variable values for any desired endpoint position, endpoint orientation (see Figure 1.1), and manipulator configuration (i.e. elbow up/elbow down, right arm/left arm, etc.; see Figure 1.2); the only requirement being that the desired solution exists. For 6 degree of freedom (DOF) manipulators, a complete solution to the inverse kinematics problem is known when any three adjacent joints of the manipulator are either intersecting [1] or parallel to one another [2]. Because of this, most industrial manipulators are built with the last three joints having intersecting axes. Most 6 DOF manipulators which fall into this restricted class have closed form inverse kinematics equations. Many iterative algorithms have been developed which attempt to solve the complete inverse kinematics problem for manipulators with no closed form inverse kinematics equations. The author is not aware of any existing algorithms which can theoretically guarantee a complete solution to this problem. In this thesis, an iterative method is presented which gives a complete solution to the inverse kinematics problem for a large class of manipulators that may or may not have closed form inverse kinematics equations. The method is applicable to any manipulator geometry that satisfies a specific set of conditions (the conditions of THEOREM I), and it is shown that these conditions include a class of 6 DOF manipulators which do not have three adjacent joints that are either intersecting or parallel. It is advantageous to have a complete solution to the inverse kinematics problem for such manipulators because, in general, these manipulators are easier to build and maintain, and they can be made more dextrous [3] than those manipulators which fall into more restrictive classes. Subsea and heavy equipment manipulators are often built which do not have three adjacent joints that are 1 Chapter 1: Introduction The inverse kinematics problem is one of calculating the required joint variable values (?i, «•>, • • •, ?o) in order to achieve the desired endpoint position (pd) and orienting (defined by the 3 unit vectors nd, sd, &d). Note that the desired endpoint position and orientation can usually be achieved at more than one manipulator configuration (see Figure 1.2). Figure 1.1: The Modified PUMA Manipulator either intersecting or parallel. Applications of the method to other 6 DOF manipulator geometries and to redundant manipulators (manipulators with more than 6 DOF) are also discussed in this thesis. The method is thoroughly tested on two complex 6 DOF manipulators which have no known closed form inverse kinematics equations. It is shown that the method never fails and is applicable to real time manipulator control (defined below). 2 Chapter 1: Introduction b) Left Arm, Elbow Down. a) Left Arm, Elbow UP. d) Right Arm, Elbow Down. c) Right Arm, Elbow UP. Figure 1.2: Manipulator Configurations 1.1 Current Approaches to the Inverse Kinematics Problem A summary of the characteristics of some current inverse kinematics methods is given Table 1.1. A brief description of each of these methods is presented here. The characteristics of the inverse kinematics methods are compared in the following categories: 1. Convergence conditions given. If convergence conditions for a particular inverse kinematics method are not given, then one can never be sure that the algorithm being used will converge to the desired solution. 2. Guaranteed convergence if solution exists. Ideally, the inverse kinematics algorithm should always be able to converge to the desired solution if it exists. Also, the algorithm should be able to detect when a solution does not exist 3 Chapter 1: Introduction 3. Allows real time manipulator control. It is desirable to have an algorithm which is fast enough to be used in a real time manipulator control scheme. For the puiposes of this thesis, a loose definition for an inverse kinematics algorithm that is suitable to a real time control scheme, is one for which the convergence time in most of the manipulator's workspace (independent of the initial position of the manipulator) is less than 1 second. The time of 1 second was chosen in order to meet the real time claims of [5-17], Convergence time is defined as the required time for the algorithm to converge to the desired solution, to within an accuracy that is equal to the positioning repeatability accuracy of the manipulator. 4. Allows direct control over manipulator configuration. In order to avoid obstacles in the manip- ulator's workspace, it is often essential to have control over the manipulator's configuration. 5. Extendable to all 6 DOF manipulators. The algorithm should ideally be applicable to all practical 6 DOF geometries.