Kinematics of General Spatial Mechanical Systems

M. KEMAL OZGOREN, PhD, is currently an Emeritus Professor at METU (Middle East Technical University), Ankara, Turkey. He was an active faculty member at the same university between 1976 and 2015. He received the BSc and MSc degrees from the Mechanical Engineering Department of METU in 1971 and 1972. He received the doctoral degree from the Mechanical Engineering Department of the Columbia University, USA, in 1976. His specific interest lies in the area of Spatial Kinematics but he is generally interested in the field of Kinematics, Dynamics, and Control of Mechanical Systems. In this field, his academic activities include teaching several courses, supervising numerous MSc and PhD theses, presenting papers in various national and international conferences, and publishing several archive journal papers. Additionally, he has acted as a Research Adviser for various National Defense Institutions and previously served as a Panel Member on the NATO-AGARD, Flight Mechanics Panel.

• Preface xvAcknowledgments xixList of Commonly Used Symbols, Abbreviations, and Acronyms xxiAbout the Companion Website xxvii1 Vectors and Their Matrix Representations in Selected Reference Frames 11.1 General Features of Notation 11.2 Vectors 21.2.1 Definition and Description of a Vector 21.2.2 Equality of Vectors 21.2.3 Opposite Vectors 31.3 Vector Products 31.3.1 Dot Product 31.3.2 Cross Product 31.4 Reference Frames 41.5 Representation of a Vector in a Selected Reference Frame 61.6 Matrix Operations Corresponding to Vector Operations 71.6.1 Dot Product 71.6.2 Cross Product and Skew Symmetric Cross Product Matrices 81.7 Mathematical Properties of the Skew Symmetric Matrices 91.8 Examples Involving Skew Symmetric Matrices 101.8.1 Example 1.1 101.8.2 Example 1.2 111.8.3 Example 1.3 112 Rotation of Vectors and Rotation Matrices 132.1 Vector Equation of Rotation and the Rodrigues Formula 132.2 Matrix Equation of Rotation and the Rotation Matrix 152.3 Exponentially Expressed Rotation Matrix 162.4 Basic Rotation Matrices 162.5 Successive Rotations 172.6 Orthonormality of the Rotation Matrices 182.7 Mathematical Properties of the Rotation Matrices 202.7.1 Mathematical Properties of General Rotation Matrices 202.7.2 Mathematical Properties of the Basic Rotation Matrices 222.8 Examples Involving Rotation Matrices 222.8.1 Example 2.1 222.8.2 Example 2.2 232.8.3 Example 2.3 242.8.4 Example 2.4 242.9 Determination of the Angle and Axis of a Specified Rotation Matrix 252.9.1 Scalar Equations of Rotation 252.9.2 Determination of the Angle of Rotation 262.9.3 Determination of the Axis of Rotation 262.9.4 Discussion About the Optional Sign Variables 292.10 Definition and Properties of the Double Argument Arctangent Function 293 Matrix Representations of Vectors in Different Reference Frames and the Component Transformation Matrices 313.1 Matrix Representations of a Vector in Different Reference Frames 313.2 Transformation Matrices Between Reference Frames 323.2.1 Definition and Usage of a Transformation Matrix 323.2.2 Basic Properties of a Transformation Matrix 333.3 Expression of a Transformation Matrix in Terms of Basis Vectors 343.3.1 Column-by-Column Expression 343.3.2 Row-by-Row Expression 343.3.3 Remark 3.1 353.3.4 Remark 3.2 353.3.5 Remark 3.3 363.3.6 Example 3.1 363.4 Expression of a Transformation Matrix as a Direction Cosine Matrix 373.4.1 Definitions of Direction Angles and Direction Cosines 373.4.2 Transformation Matrix Formed as a Direction Cosine Matrix 383.5 Expression of a Transformation Matrix as a Rotation Matrix 383.5.1 Correlation Between the Rotation and Transformation Matrices 383.5.2 Distinction Between the Rotation and Transformation Matrices 393.6 Relationship Between the Matrix Representations of a Rotation Operator in Different Reference Frames 403.7 Expression of a Transformation Matrix in a Case of Several Successive Rotations 403.7.1 Rotated Frame Based (RFB) Formulation 413.7.2 Initial Frame Based (IFB) Formulation 413.8 Expression of a Transformation Matrix in Terms of Euler Angles 423.8.1 General Definition of Euler Angles 423.8.2 IFB (Initial Frame Based) Euler Angle Sequences 423.8.3 RFB (Rotated Frame Based) Euler Angle Sequences 433.8.4 Remark 3.4 443.8.5 Remark 3.5 443.8.6 Remark 3.6: Preference Between IFB and RFB Sequences 453.8.7 Commonly Used Euler Angle Sequences 453.8.8 Extraction of Euler Angles from a Given Transformation Matrix 463.9 Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices 513.9.1 Position of a Point Expressed in Different Reference Frames 513.9.2 Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships 523.9.3 Affine Coordinate Transformation Between Two Reference Frames 533.9.4 Homogeneous Coordinate Transformation Between Two Reference Frames 543.9.5 Mathematical Properties of the Homogeneous Transformation Matrices 553.9.6 Example 3.2 584 Vector Differentiation Accompanied by Velocity and Acceleration Expressions 634.1 Derivatives of a Vector with Respect to Different Reference Frames 634.1.1 Differentiation and Resolution Frames 634.1.2 Components in Different Differentiation and Resolution Frames 644.1.3 Example 654.2 Vector Derivatives with Respect to Different Reference Frames and the Coriolis Transport Theorem 664.2.1 First Derivatives and the Relative Angular Velocity 664.2.2 Second Derivatives and the Relative Angular Acceleration 684.3 Combination of Relative Angular Velocities and Accelerations 704.3.1 Combination of Relative Angular Velocities 704.3.2 Combination of Relative Angular Accelerations 714.4 Angular Velocities and Accelerations Associated with Rotation Sequences 714.4.1 Relative Angular Velocities and Accelerations about Relatively Fixed Axes 714.4.2 Example 724.4.3 Angular Velocities Associated with the Euler Angle Sequences 744.5 Velocity and Acceleration of a Point with Respect to Different Reference Frames 774.5.1 Velocity of a Point with Respect to Different Reference Frames 774.5.2 Acceleration of a Point with Respect to Different Reference Frames 784.5.3 Velocity and Acceleration Expressions with Simplified Notations 795 Kinematics of Rigid Body Systems 815.1 Kinematic Description of a Rigid Body System 825.1.1 Body Frames and Joint Frames 825.1.2 Kinematic Chains, Kinematic Branches, and Kinematic Loops 835.1.3 Joints or Kinematic Pairs 835.2 Position Equations for a Kinematic Chain of Rigid Bodies 845.2.1 Relative Orientation Equation Between Successive Bodies 855.2.2 Relative Location Equation Between Successive Bodies 855.2.3 Orientation of a Body with Respect to the Base of the Kinematic Chain 855.2.4 Location of a Body with Respect to the Base of the Kinematic Chain 865.2.5 Loop Closure Equations for a Kinematic Loop 865.3 Velocity Equations for a Kinematic Chain of Rigid Bodies 875.3.1 Relative Angular Velocity between Successive Bodies 875.3.2 Relative Translational Velocity Between Successive Bodies 885.3.3 Angular Velocity of a Body with Respect to the Base 895.3.4 Translational Velocity of a Body with Respect to the Base 895.3.5 Velocity Equations for a Kinematic Loop 905.4 Acceleration Equations for a Kinematic Chain of Rigid Bodies 905.4.1 Relative Angular Acceleration Between Successive Bodies 915.4.2 Relative Translational Acceleration Between Successive Bodies 925.4.3 Angular Acceleration of a Body with Respect to the Base 925.4.4 Translational Acceleration of a Body with Respect to the Base 935.4.5 Acceleration Equations for a Kinematic Loop 935.5 Example 5.1 :A Serial Manipulator with an RRP Arm 945.5.1 Kinematic Description of the System 945.5.2 Position Analysis 955.5.3 Velocity Analysis 1005.5.4 Acceleration Analysis 1035.6 Example 5.2 :A Spatial Slider-Crank (RSSP) Mechanism 1065.6.1 Kinematic Description of the Mechanism 1065.6.2 Loop Closure Equations 1085.6.3 Degree of Freedom or Mobility 1095.6.4 Position Analysis 1105.6.5 Velocity Analysis 1195.6.6 Acceleration Analysis 1226 Joints and Their Kinematic Characteristics 1256.1 Kinematic Details of the Joints 1256.1.1 Description of a Joint as a Kinematic Pair 1256.1.2 Degree of Freedom or Mobility of a Joint 1266.1.3 Number of Distinct Joints Between Two Rigid Bodies 1266.1.4 Classification of the Joints 1276.2 Typical Lower Order Joints 1286.2.1 Single-Axis Joints 1286.2.2 Universal Joint 1306.2.3 Spherical Joint 1316.2.4 Plane-on-Plane Joint 1326.3 Higher Order Joints with Simple Contacts 1326.3.1 Line-on-Plane Joint 1326.3.2 Point-on-Plane Joint 1336.3.3 Point-on-Surface Joint 1336.4 Typical Multi-Joint Connections 1346.4.1 Fork-on-Surface Joint 1346.4.2 Triangle-on-Surface Joint 1366.5 Rolling Contact Joints with Point Contacts 1386.5.1 Surface-on-Surface Joint 1386.5.2 Curve-on-Surface Joint 1446.5.3 Curve-on-Curve Joint 1476.6 Rolling Contact Joints with Line Contacts 1486.6.1 Cone-on-Cone Joint 1486.6.2 Cone-on-Cylinder Joint 1556.6.3 Cone-on-Plane Joint 1576.6.4 Cylinder-on-Cylinder Joint 1616.6.5 Cylinder-on-Plane Joint 1646.7 Examples 1676.7.1 Example 6.1: An RRRSP Mechanism 1676.7.2 Example 6.2: A Two-Link Mechanism with Three Point-on-Plane Joints 1716.7.3 Example 6.3: A Spatial Cam Mechanism 1746.7.4 Example 6.4: A Spatial Cam Mechanism That Allows Rolling Without Slipping 1777 Kinematic Features of Serial Manipulators 1857.1 Kinematic Description of a General Serial Manipulator 1857.2 Denavit–Hartenberg Convention 1867.3 D–H Convention for Successive Intermediate Links and Joints 1877.3.1 Assignment and Description of the Link Frames 1877.3.2 D–H Parameters 1887.3.3 Relative Position Formulas Between Successive Links 1897.3.4 Alternative Multi-Index Notation for the D–H Convention 1897.4 D–H Convention for the First Joint 1907.5 D–H Convention for the Last Joint 1937.6 D–H Convention for Successive Joints with Perpendicularly Intersecting Axes 1957.7 D–H Convention for Successive Joints with Parallel Axes 1957.8 D–H Convention for Successive Joints with Coincident Axes 1978 Position and Motion Analyses of Generic Serial Manipulators 1998.1 Forward Kinematics 2018.2 Compact Formulation of Forward Kinematics 2028.3 Detailed Formulation of Forward Kinematics 2038.4 Manipulators with or without Spherical Wrists 2058.5 Inverse Kinematics 2078.6 Inverse Kinematic Solution for a Regular Manipulator 2088.6.1 Regular Manipulator with a Spherical Wrist 2088.6.2 Regular Manipulator with a Nonspherical Wrist 2118.7 Inverse Kinematic Solution for a Redundant Manipulator 2128.7.1 Solution by Specifying the Variables of Certain Joints 2128.7.2 Solution by Optimization 2138.8 Inverse Kinematic Solution for a Deficient Manipulator 2148.8.1 Compromise in Orientation in Favor of a Completely Specified Location 2148.8.2 Compromise in Location in Favor of a Completely Specified Orientation 2158.9 Forward Kinematics of Motion 2158.9.1 Forward Kinematics of Velocity Relationships 2158.9.2 Forward Kinematics of Acceleration Relationships 2168.10 Jacobian Matrices Associated with the Wrist and Tip Points 2188.11 Recursive Position, Velocity, and Acceleration Formulations 2208.11.1 Orientations of the Links 2208.11.2 Locations of the Link Frame Origins 2218.11.3 Locations of the Mass Centers of the Links 2218.11.4 Angular Velocities of the Links 2218.11.5 Velocities of the Link Frame Origins 2228.11.6 Velocities of the Mass Centers of the Links 2228.11.7 Angular Accelerations of the Links 2228.11.8 Accelerations of the Link Frame Origins 2228.11.9 Accelerations of the Mass Centers of the Links 2238.12 Inverse Motion Analysis of a Manipulator Based on the Jacobian Matrix 2238.12.1 Inverse Velocity Analysis of a Regular Manipulator 2248.12.2 Inverse Acceleration Analysis of a Regular Manipulator 2258.13 Inverse Motion Analysis of a Redundant Manipulator 2258.13.1 Inverse Velocity Analysis 2258.13.2 Inverse Acceleration Analysis 2288.14 Inverse Motion Analysis of a Deficient Manipulator 2298.15 Inverse Motion Analysis of a Regular Manipulator Using the Detailed Formulation 2308.15.1 Inverse Velocity Solution 2308.15.2 Inverse Acceleration Solution 2319 Kinematic Analyses of Typical Serial Manipulators 2339.1 Puma Manipulator 2339.1.1 Kinematic Description According to the D–H Convention 2349.1.2 Forward Kinematics in the Position Domain 2359.1.3 Inverse Kinematics in the Position Domain 2379.1.4 Multiplicity Analysis 2409.1.5 Singularity Analysis in the Position Domain 2429.1.6 Forward Kinematics in the Velocity Domain 2449.1.7 Inverse Kinematics in the Velocity Domain 2459.1.8 Singularity Analysis in the Velocity Domain 2479.2 Stanford Manipulator 2509.2.1 Kinematic Description According to the D–H Convention 2509.2.2 Forward Kinematics in the Position Domain 2519.2.3 Inverse Kinematics in the Position Domain 2539.2.4 Multiplicity Analysis 2549.2.5 Singularity Analysis in the Position Domain 2559.2.6 Forward Kinematics in the Velocity Domain 2559.2.7 Inverse Kinematics in the Velocity Domain 2569.2.8 Singularity Analysis in the Velocity Domain 2579.3 Elbow Manipulator 2589.3.1 Kinematic Description According to the D–H Convention 2599.3.2 Forward Kinematics in the Position Domain 2609.3.3 Inverse Kinematics in the Position Domain 2629.3.4 Multiplicity Analysis 2649.3.5 Singularity Analysis in the Position Domain 2669.3.6 Forward Kinematics in the Velocity Domain 2699.3.7 Inverse Kinematics in the Velocity Domain 2699.3.8 Singularity Analysis in the Velocity Domain 2719.4 Scara Manipulator 2739.4.1 Kinematic Description According to the D–H Convention 2739.4.2 Forward Kinematics in the Position Domain 2749.4.3 Inverse Kinematics in the Position Domain 2759.4.4 Multiplicity Analysis 2779.4.5 Singularity Analysis in the Position Domain 2789.4.6 Forward Kinematics in the Velocity Domain 2799.4.7 Inverse Kinematics in the Velocity Domain 2799.4.8 Singularity Analysis in the Velocity Domain 2809.5 An RP2R3 Manipulator without an Analytical Solution 2819.5.1 Kinematic Description According to the D–H Convention 2829.5.2 Forward Kinematics in the Position Domain 2829.5.3 Inverse Kinematics in the Position Domain 2839.5.4 Multiplicity Analysis 2859.5.5 Singularity Analysis in the Position Domain 2879.5.6 Forward Kinematics in the Velocity Domain 2879.5.7 Inverse Kinematics in the Velocity Domain 2879.5.8 Singularity Analysis in the Velocity Domain 2899.6 An RPRPR2 Manipulator with an Uncustomary Analytical Solution 2909.6.1 Kinematic Description According to the D–H Convention 2909.6.2 Forward Kinematics in the Position Domain 2919.6.3 Inverse Kinematics in the Position Domain 2939.6.4 Multiplicity Analysis 2979.6.5 Singularity Analysis in the Position Domain 2989.6.6 Forward Kinematics in the Velocity Domain 2989.6.7 Inverse Kinematics in the Velocity Domain 2999.6.8 Singularity Analysis in the Velocity Domain 3019.7 A Deficient Puma Manipulator with Five Active Joints 3039.7.1 Kinematic Description According to the D–H Convention 3039.7.2 Forward Kinematics in the Position Domain 3049.7.3 Inverse Kinematics in the Position Domain 3059.7.3.1 Solution in the Case of Fully Specified Tip Point Location 3059.7.3.2 Solution in the Case of Fully Specified End-Effector Orientation 3079.7.4 Multiplicity Analysis in the Position Domain 3079.7.4.1 Analysis in the Case of Fully Specified Tip Point Location 3079.7.4.2 Analysis in the Case of Fully Specified End-Effector Orientation 3089.7.5 Singularity Analysis in the Position Domain 3089.7.5.1 Analysis in the Case of Fully Specified Tip Point Location 3089.7.5.2 Analysis in the Case of Fully Specified End-Effector Orientation 3099.7.6 Forward Kinematics in the Velocity Domain 3109.7.7 Inverse Kinematics in the Velocity Domain 3109.7.7.1 Solution in the Case of Fully Specified Tip Point Velocity 3109.7.7.2 Solution in the Case of Fully Specified End-Effector Angular Velocity 3119.7.8 Singularity Analysis in the Velocity Domain 3129.7.8.1 Analysis in the Case of Fully Specified Tip Point Velocity 3129.7.8.2 Analysis in the Case of Fully Specified End-Effector Angular Velocity 3139.8 A Redundant Humanoid Manipulator with Eight Joints 3139.8.1 Kinematic Description According to the D–H Convention 3139.8.2 Forward Kinematics in the Position Domain 3159.8.3 Inverse Kinematics in the Position Domain 3169.8.4 Multiplicity Analysis 3239.8.5 Singularity Analysis in the Position Domain 3269.8.6 Forward Kinematics in the Velocity Domain 3289.8.7 Inverse Kinematics in the Velocity Domain 3289.8.8 Singularity Analysis in the Velocity Domain 3339.8.9 Consistency of the Inverse Kinematics in the Position and Velocity Domains 33510 Position and Velocity Analyses of Parallel Manipulators 34110.1 General Kinematic Features of Parallel Manipulators 34310.2 Position Equations of a Parallel Manipulator 34710.3 Forward Kinematics in the Position Domain 35110.4 Inverse Kinematics in the Position Domain 35910.5 Velocity Equations of a Parallel Manipulator 36810.6 Forward Kinematics in the Velocity Domain 37110.7 Inverse Kinematics in the Velocity Domain 37710.8 Stewart–Gough Platform as a 6UPS Spatial Parallel Manipulator 38410.8.1 Kinematic Description 38410.8.2 Position Equations 38610.8.3 Inverse Kinematics in the Position Domain 38710.8.4 Forward Kinematics in the Position Domain 38910.8.5 Velocity Equations 39610.8.6 Inverse Kinematics in the Velocity Domain 39710.8.7 Forward Kinematics in the Velocity Domain 39810.9 Delta Robot: A 3RS2S2 Spatial Parallel Manipulator 40210.9.1 Kinematic Description 40210.9.2 Position Equations 40410.9.3 Independent Kinematic Loops and the Associated Equations 40710.9.4 Inverse Kinematics in the Position Domain 41010.9.5 Forward Kinematics in the Position Domain 41210.9.6 Velocity Equations 41710.9.7 Inverse Kinematics in the Velocity Domain 41810.9.8 Forward Kinematics in the Velocity Domain 420Bibliography 423Index 425