Kinematics of Robots Alba Perez Gracia c Draft date October 2, 2007 Contents Contents i 1 Motion: An Introduction 3 1.1 Overview . 3 1.2 Introduction . 3 1.3 The finite displacement . 4 1.3.1 Translations . 5 1.3.2 Rotations . 6 1.3.3 General finite displacements . 10 1.3.4 The invariants of a general displacement . 11 1.4 A little bit of line geometry . 12 1.4.1 Dual vector algebra . 13 1.4.2 More line geometry . 15 1.4.3 Line motion . 15 1.4.4 Line Geometry and the Klein Quadric . 16 1.5 The screw axis of a general displacement and Chasles’ Theorem . 16 1.6 More on Matrix Representation . 19 1.6.1 How to create a rotation matrix . 19 1.6.2 Homogeneous matrix representation . 21 i ii CONTENTS 2 Robot Kinematics Using Matrix Algebra 25 2.1 Overview . 25 2.2 Robot Kinematics . 25 2.3 Basic joints . 27 2.4 Forward Kinematics Equations . 28 2.4.1 Denavit-Hartenberg parameters . 28 2.4.2 Kinematics Equations . 30 2.5 Forward Kinematics Example . 31 2.6 Inverse Kinematics . 32 2.7 Inverse Kinematics Example . 33 3 Group Theory and Motion 35 3.1 Overview . 35 3.2 Group Theory Review . 35 3.2.1 Subgroups . 36 3.2.2 Automorphisms . 37 3.2.3 Products of Groups . 38 3.2.4 Cosets and Actions . 39 3.3 The group of finite displacements . 40 3.3.1 Subgroups of the Special Euclidean Group . 41 4 Lie Groups and Lie Algebras 43 4.1 Introduction . 43 4.2 Lie groups . 43 4.3 One-parameter subgroups . 44 4.4 The Lie Algebra . 45 4.4.1 The Lie bracket . 46 CONTENTS iii 4.4.2 Chasles’ Theorem again . 48 4.5 The Lie algebra of the Special Euclidean Group SE(3) . 48 4.5.1 Relative Rotations . 48 4.5.2 Relative general motion . 49 4.5.3 Rigid body velocity . 50 4.6 Example . 51 5 The Product of Exponentials for Serial Manipulators 53 5.1 Introduction . 53 5.2 Forward Kinematics . 53 5.3 Example: Epson E2L Scara Robot . 54 5.4 Velocity and Jacobian . 56 5.4.1 Example . 57 5.5 Inverse and pseudoinverse Jacobian . 58 6 Grassmann Algebra 59 6.1 Overview . 59 6.2 Vector spaces and metric spaces . 59 6.3 Algebras . 61 6.4 Grassmann Algebra . 61 3 6.4.1 Grassmann algebra associated to R ....................... 62 2 6.4.2 Grassmann algebra associated to R ....................... 65 4 6.4.3 Grassmann algebra associated to R ....................... 65 6.4.4 The general Grassman algebra . 66 6.5 More on Grassmann algebra . 68 7 Quaternions 71 7.1 Overview . 71 iv CONTENTS 7.2 Quaternions . 71 7.3 Unit Quaternions and Rotations . 73 7.3.1 Rotation by a unit quaternion . 73 7.4 The Lie Algebra of Unit Quaternions . 75 7.4.1 Euler parameters revisited . 76 7.5 Applications . 76 7.5.1 Satellite navigation . 76 7.5.2 Singularities in the Euler angles matrix representation . 78 7.5.3 Trajectory planning and interpolation . 79 8 Clifford Algebras 81 8.1 Overview . 81 8.2 Clifford algebras and the Clifford product . 82 8.2.1 Properties of the geometric product . 83 8.2.2 Creating Clifford algebras . 83 8.2.3 Subalgebras . 84 8.3 Planar motion and planar geometry: Clifford algebras . 84 8.3.1 Clifford algebra of the Euclidean plane: C2,0,0 .................. 84 3 8.4 The Even Clifford algebra of the three-dimensional space R .............. 85 3 8.5 The even Clifford algebra of the projective space P .................. 86 8.6 The Clifford algebra of dual quaternions . 88 8.6.1 Spatial displacements using unit dual quaternions . 89 8.6.2 Examples . 91 + 4 8.7 The geometry of the Clifford algebra C (P , d)..................... 92 8.8 Lie algebra and Clifford algebra . 93 8.9 Clifford Analysis . 94 8.10 Other applications of Clifford algebras . 96 CONTENTS v 9 Kinematic Synthesis 95 9.1 Overview . 95 9.2 4R Synthesis . 95 Bibliography 73 vi CONTENTS List of Figures 1.1 Description of the motion of a rigid body . 4 1.2 (a) A pure translation and (b) a pure rotation of a rigid body . 5 1.3 A rigid displacement . 6 1.4 Composition of displacements . 11 1.5 A relative displacement . 12 1.6 A line defined using Pl¨ucker coordinates . 13 1.7 Dual vector geometry . 14 1.8 The screw axis, rotation angle and slide of a displacement . 18 1.9 Euler angles . 20 2.1 Reachable workspace, Adept Viper m650 robot (Adept Technologies, Inc.) . 27 2.2 Common types of joints . 28 2.3 Local transformations along the links of a robot. 29 2.4 Example: a three-jointed serial robot. 31 2.5 Welding Robot (Fanuc Robotics) . ..