Kinematics of General Spatial Mechanical Systems
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Kinematics of General Spatial Mechanical Systems
John Wiley and Sons Ltd
02/2020
650
Dura
Inglês
9781119195733
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Chapter 1. Vectors and Their Matrix Representations in Selected Reference Frames 1.1. General Features of Notation 1.2. Vectors 1.2.1. Definition and Description of a Vector 1.2.2. Equality of Vectors 1.2.3. Opposite Vectors 1.3. Vector Products 1.3.1. Dot Product 1.3.2. Cross Product 1.4. Reference Frames 1.5. Representation of a Vector in a Selected Reference Frame 1.6. Matrix Operations Corresponding to Vector Operations 1.6.1. Dot Product 1.6.2. Cross Product and Skew Symmetric Cross Product Matrices 1.7. Mathematical Properties of the Skew Symmetric Matrices 1.8. Examples Involving Skew Symmetric Matrices Example 1.8.1 Example 1.8.2 Example 1.8.3 Chapter 2. Rotation of Vectors and Rotation Matrices 2.1. Vector Equation of Rotation and the Rodrigues Formula 2.2. Matrix Equation of Rotation and the Rotation Matrix 2.3. Exponentially Expressed Rotation Matrix 2.4. Basic Rotation Matrices 2.5. Successive Rotations 2.6. Orthonormality of the Rotation Matrices 2.7. Mathematical Properties of the Rotation Matrices 2.7.1. Mathematical Properties of General Rotation Matrices 2.7.2. Mathematical Properties of the Basic Rotation Matrices 2.8. Examples Involving Rotation Matrices Example 2.8.1 Example 2.8.2 Example 2.8.3 Example 2.8.4 2.9. Determination of the Angle and Axis of a Specified Rotation Matrix 2.9.1. Scalar Equations of Rotation 2.9.2. Determination of the Angle of Rotation 2.9.3. Determination of the Axis of Rotation a) General Case with b) Special Cases with (No Rotation, Full Rotation, Half Rotation) 2.9.4. Discussion about the Optional Sign Variables a) General Case with b) Special Case with a Half Rotation 2.10. Definition and Properties of the Double Argument Arctangent Function Chapter 3. Matrix Representations of Vectors in Different Reference Frames and the Component Transformation Matrices 3.1. Matrix Representations of a Vector in Different Reference Frames 3.2. Transformation Matrices between Reference Frames 3.2.1. Definition and Usage of a Transformation Matrix 3.2.2. Basic Properties of a Transformation Matrix a) Inversion Property b) Orthonormality Property c) Combination Property 3.3. Expression of a Transformation Matrix in terms of Basis Vectors 3.3.1. Column-by-Column Expression 3.3.2. Row-by-Row Expression 3.3.3. Example 3.4. Expression of a Transformation Matrix as a Direction Cosine Matrix 3.4.1. Definitions of Direction Angles and Direction Cosines 3.4.2. Transformation Matrix Formed as a Direction Cosine Matrix 3.5. Expression of a Transformation Matrix as a Rotation Matrix 3.5.1. Correlation between the Rotation and Transformation Matrices 3.5.2. Distinction between the Rotation and Transformation Matrices 3.6. Relationship between the Matrix Representations of a Rotation Operator in Different Reference Frames 3.7. Expression of a Transformation Matrix in a Case of Several Successive Rotations 3.7.1. Rotated Frame Based (RFB) Formulation 3.7.2. Initial Frame Based (IFB) Formulation 3.8. Expression of a Transformation Matrix in terms of Euler Angles 3.8.1. General Definition of Euler Angles 3.8.2. IFB (Initial Frame Based) Euler Angle Sequences 3.8.3. RFB (Rotated Frame Based) Euler Angle Sequences 3.8.4. Commonly Used Euler Angle Sequences a) RFB 1-2-3 Sequence b) RFB 3-2-1 Sequence c) RFB 3-1-3 Sequence d) RFB 3-2-3 Sequence 3.8.5. Extraction of Euler Angles from a Given Transformation Matrix a) Extraction of the 3-2-3 Euler Angles b) Extraction of the 1-2-3 Euler Angles 3.9. Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices 3.9.1. Position of a Point Expressed in Different Reference Frames 3.9.2. Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships a) Homogeneous versus Nonhomogeneous Relationships b) Linear versus Nonlinear Relationships c) Affine Relationships 3.9.3. Affine Coordinate Transformation between Two Reference Frames 3.9.4. Homogeneous Coordinate Transformation between Two Reference Frames 3.9.5. Mathematical Properties of the Homogeneous Transformation Matrices (HTMs) a) Determinant of an HTM b) Inverse of an HTM c) Decomposition of an HTM d) HTM of a Pure Rotation e) HTM of a Pure Translation f) Observation in a Third Different Reference Frame 9.3.6. Example Chapter 4. Vector Differentiation Accompanied by Velocity and Acceleration Expressions 4.1. Derivatives of a Vector with respect to Different Refrence Frames 4.1.1. Differentiation and Resolution Frames 4.1.2. Components in Different Differentiation and Resolution Frames 4.1.3. Example 4.2. Vector Derivatives with respect to Different Reference Frames and the Coriolis Transport Theorem 4.2.1. First Derivatives and the Relative Angular Velocity 4.2.2. Second Derivatives and the Relative Angular Acceleration 4.3. Combination of Relative Angular Velocities and Accelerations 4.3.1. Combination of Relative Angular Velocities 4.3.2. Combination of Relative Angular Accelerations 4.4. Angular Velocities and Accelerations Associated with Rotation Sequences 4.4.1. Relative Angular Velocities and Accelerations about Relatively Fixed Axes 4.4.2. Example 4.4.3. Angular Velocities Associated with the Euler Angle Sequences a) Rotated Frame Based (RFB) Euler Angle Sequences b) Initial Frame Based (IFB) Euler Angle Sequences 4.5. Velocity and Acceleration of a Point with respect to Different References Frames 4.5.1. Velocity of a Point with respect to Different References Frames 4.5.2. Acceleration of a Point with respect to Different References Frames 4.5.3. Velocity and Acceleration Expressions with Simplified Notations Chapter 5. Kinematics of Rigid Body Systems 5.1. Kinematic Description of a Rigid Body System 5.1.1. Body Frames and Joint Frames 5.1.2. Kinematic Chains, Kinematic Branches, and Kinematic Loops 5.1.3. Joints or Kinematic Pairs 5.2. Position Equations for a Kinematic Chain of Rigid Bodies 5.2.1. Relative Orientation Equation between Successive Bodies 5.2.2. Relative Location Equation between Successive Bodies 5.2.3. Orientation of a Body with respect to the Base of the Kinematic Chain 5.2.4. Location of a Body with respect to the Base of the Kinematic Chain 5.2.5. Loop Closure Equations for a Kinematic Loop a) Orientation Equation for the Closure at Joint Joz b) Location Equation for the Closure at Joint Joz c) Orientation Equation for the Closure at Joint Jpq d) Location Equation for the Closure at Joint Jpq 5.3. Velocity Equations for a Kinematic Chain of Rigid Bodies 5.3.1. Relative Angular Velocity between Successive Bodies 5.3.2. Relative Translational Velocity between Successive Bodies 5.3.3. Angular Velocity of a Body with respect to the Base 5.3.4. Translational Velocity of a Body with respect to the Base 5.3.5. Velocity Equations for a Kinematic Loop a) Angular Velocity Equation for the Kinematic Loop b) Translational Velocity Equation for the Kinematic Loop 5.4. Acceleration Equations for a Kinematic Chain of Rigid Bodies 5.4.1. Relative Angular Acceleration between Successive Bodies 5.4.2. Relative Translational Acceleration between Successive Bodies 5.4.3. Angular Acceleration of a Body with respect to the Base 5.4.4. Translational Acceleration of a Body with respect to the Base 5.4.5. Acceleration Equations for a Kinematic Loop a) Angular Acceleration Equation for the Kinematic Loop b) Translational Acceleration Equation for the Kinematic Loop 5.5. Example 1: A Serial Manipulator with an RRP Arm 5.5.1. Kinematic Description of the System 5.5.2. Position Analysis a) Orientations of the Links b) Locations of the Link Frame Origins c) Inverse Kinematic Solution d) Multiple Solutions e) Position Singularity 5.5.3. Velocity Analysis a) Angular Velocities of the Links b) Velocities of the Link Frame Origins c) Wrist Point Jacobian Matrix d) Inverse Velocity Solution e) Motion Singularities Involving Velocities f) Consequences of the Motion Singularities 5.5.4. Acceleration Analysis a) Angular Accelerations of the Links b) Accelerations of the Link Frame Origins c) Wrist Point Acceleration by Using the Wrist Point Jacobian Matrix d) Inverse Acceleration Solution e) Motion Singularities Involving Accelerations 5.6. Example 2: A Spatial Slider-Crank Mechanism 5.6.1. Kinematic Description of the Mechanism 5.6.2. Loop Closure Equations a) Orientational Loop Closure Equation b) Locational Loop Closure Equation 5.6.3. Degree of Freedom (DoF) or Mobility () 5.6.4. Position Analysis a) Solution of the Loop Closure Equations for a Specified Crank Angle b) Existence of Solutions with Continuous Crank Rotations c) Multiple Solutions for the Same Crank Angle d) Position Singularity Associated with the Crank Angle e) Solution of the Loop Closure Equations for a Specified Slider Position f) Existence of Solutions g) Multiple Solutions for the Same Slider Position h) Position Singularities Associated with the Slider Position 5.6.5. Velocity Analysis a) Solution of the Velocity Equations for a Specified Crank Motion b) Motion Singularities Associated with the Crank Motion c) Solution of the Velocity Equations for a Specified Slider Motion h) Motion Singularities Associated with the Slider Motion 5.6.6. Acceleration Analysis Chapter 6. Joints and Their Kinematic Characteristics 6.1. Kinematic Details of the Joints 6.1.1. Description of a Joint as a Kinematic Pair 6.1.2. Degree of Freedom (DoF) or Mobility of a Joint 6.1.3. Number of Distinct Joints between Two Rigid Bodies 6.1.4. Classification of the Joints 6.2. Typical Lower Order Joints 6.2.1. Single-Axis Joints a) Cylindrical Joint b) Revolute Joint c) Prismatic Joint d) Screw Joint 6.2.2. Universal Joint 6.2.3. Spherical Joint 6.2.4. Plane-on-Plane Joint 6.3. Higher Order Joints with Simple Contacts 6.3.1. Line-on-Plane Joint 6.3.2. Point-on-Plane Joint 6.3.3. Point-on-Surface Joint 6.4. Typical Multi-Joint Connections 6.4.1. Fork-on-Surface Joint 6.4.2. Triangle-on-Surface Joint 6.5. Rolling Contact Joints with Point Contacts 6.5.1. Surface-on-Surface Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.5.2. Curve-on-Surface Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.5.3. Curve-on-Curve Joint a) Relative Position Equations b) Relative Velocity Equations 6.6. Rolling Contact Joints with Line Contacts 6.6.1. Cone-on-Cone Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.6.2. Cone-on-Cylinder Joint a) Relative Position Equations b) Relative Velocity Equations 6.6.3. Cone-on-Plane Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.6.4. Cylinder-on-Cylinder Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.6.5. Cylinder-on-Plane Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.7. Examples 6.7.1. Example 1: An RRRSP Mechanism a) Kinematic Description of the System b) Loop Closure Equations c) Determination of the Unspecified Variables for Specified d) Determination of the Unspecified Variables for Specified 6.7.2. Example 2: A Two-Link Mechanism with Three Point-on-Plane Joints a) Kinematic Description of the System b) Determination of the Unspecified Variables c) Appendix about Closed-Form Expressions and Analytical Solutions 6.7.3. Example 3: A Spatial Cam Mechanism a) Kinematic Description of the System b) Loop Closure Equations c) Determination of the Unspecified Variables for Specified 6.7.4. Example 4: A Spatial Cam Mechanism That Allows Rolling Without Slipping a) Kinematic Description of the System b) Loop Closure Equations c) Determination of the Unspecified Variables for Specified , , and d) Velocity Equations and their Solution in the Case of Rolling with Slipping e) Velocity Equations and their Solution in the Case of Rolling without Slipping f) Finding the Pose of the Mechanism during a Rolling-Without-Slipping Motion Chapter 7. Kinematic Features of Serial Manipulators 7.1. Kinematic Description of a General Serial Manipulator 7.2. Denavit-Hartenberg (D-H) Convention 7.3. D-H Convention for Successive Intermediate Links and Joints 7.3.1. Assignment and Description of the Link Frames 7.3.2. D-H Parameters 7.3.3. Relative Position Equation between Successive Links 7.3.4. Alternative Multi-Index Notation for the D-H Convention 7.4. D-H Convention for the First Joint 7.5. D-H Convention for the Last Joint 7.6. D-H Convention for Successive Joints with Perpendicularly Intersecting Axes 7.7. D-H Convention for Successive Joints with Parallel Axes 7.8. D-H Convention for Successive Joints with Coincident Axes Chapter 8. Position and Motion Analyses of Generic Serial Manipulators 8.1. Forward Kinematics 8.2. Compact Formulation of Forward Kinematics 8.3. Detailed Formulation of Forward Kinematics 8.4. Manipulators with or without Spherical Wrists 8.5. Inverse Kinematics 8.6. Inverse Kinematic Solution for a Regular Manipulator 8.6.1. Regular Manipulator with a Spherical Wrist 8.6.2. Regular Manipulator with a Nonspherical Wrist 8.7. Inverse Kinematic Solution for a Redundant Manipulator 8.7.1. Solution by Specifying the Variables of Certain Joints 8.7.2. Solution by Optimization 8.8. Inverse Kinematic Solution for a Deficient Manipulator 8.8.1. Compromise in Orientation in Favor of a Completely Specified Location 8.8.2. Compromise in Location in Favor of a Completely Specified Orientation 8.9. Forward Kinematics of Motion 8.9.1. Forward Kinematics of Velocity Relationships 8.9.2. Forward Kinematics of Acceleration Relationships 8.10. Jacobian Matrices Associated with the Wrist and Tip Points 8.11. Recursive Position, Velocity, and Acceleration Relationships 8.11.1. Orientations of the Links 8.11.2. Locations of the Link Frame Origins 8.11.3. Locations of the Mass Centers of the Links 8.11.4. Angular Velocities of the Links 8.11.5. Velocities of the Link Frame Origins 8.11.6. Velocities of the Mass Centers of the Links 8.11.7. Angular Accelerations of the Links 8.11.8. Accelerations of the Link Frame Origins 8.11.9. Accelerations of the Mass Centers of the Links 8.12. Inverse Motion Analysis of a Manipulator based on the Jacobian Matrix 8.12.1. Inverse Velocity Analysis of a Regular Manipulator 8.12.2. Inverse Acceleration Analysis of a Regular Manipulator 8.13. Inverse Motion Analysis of a Redundant Manipulator 8.13.1. Inverse Velocity Analysis 8.13.2. Inverse Acceleration Analysis 8.14. Inverse Motion Analysis of a Deficient Manipulator 8.15. Inverse Motion Analysis of a Regular Manipulator Using The Detailed Formulation 8.15.1. Inverse Velocity Solution 8.15.2. Inverse Acceleration Solution Chapter 9. Kinematic Analyses of Typical Serial Manipulators 9.1. Puma Manipulator 9.1.1. Kinematic Description According to the D-H Convention 9.1.2. Forward Kinematics in the Position Domain 9.1.3. Inverse Kinematics in the Position Domain 9.1.4. Multiplicity Analysis 9.1.5. Singularity Analysis in the Position Domain 9.1.6. Forward Kinematics in the Velocity Domain 9.1.7. Inverse Kinematics in the Velocity Domain 9.1.8. Singularity Analysis in the Velocity Domain 9.2. Stanford Manipulator 9.2.1. Kinematic Description According to the D-H Convention 9.2.2. Forward Kinematics in the Position Domain 9.2.3. Inverse Kinematics in the Position Domain 9.2.4. Multiplicity Analysis 9.2.5. Singularity Analysis in the Position Domain 9.2.6. Forward Kinematics in the Velocity Domain 9.2.7. Inverse Kinematics in the Velocity Domain 9.2.8. Singularity Analysis in the Velocity Domain 9.3. Elbow Manipulator 9.3.1. Kinematic Description According to the D-H Convention 9.3.2. Forward Kinematics in the Position Domain 9.3.3. Inverse Kinematics in the Position Domain 9.3.4. Multiplicity Analysis 9.3.5. Singularity Analysis in the Position Domain 9.3.6. Forward Kinematics in the Velocity Domain 9.3.7. Inverse Kinematics in the Velocity Domain 9.3.8. Singularity Analysis in the Velocity Domain 9.4. Scara Manipulator 9.4.1. Kinematic Description According to the D-H Convention 9.4.2. Forward Kinematics in the Position Domain 9.4.3. Inverse Kinematics in the Position Domain 9.4.4. Multiplicity Analysis 9.4.5. Singularity Analysis in the Position Domain 9.4.6. Forward Kinematics in the Velocity Domain 9.4.7. Inverse Kinematics in the Velocity Domain 9.4.8. Singularity Analysis in the Velocity Domain 9.5. An RP2R3 Manipulator without an Analytical Solution 9.5.1. Kinematic Description According to the D-H Convention 9.5.2. Forward Kinematics in the Position Domain 9.5.3. Inverse Kinematics in the Position Domain 9.5.4. Multiplicity Analysis 9.5.5. Singularity Analysis in the Position Domain 9.5.6. Forward Kinematics in the Velocity Domain 9.5.7. Inverse Kinematics in the Velocity Domain 9.5.8. Singularity Analysis in the Velocity Domain 9.6. An RPRPR2 Manipulator with an Uncustomary Analytical Solution 9.6.1. Kinematic Description According to the D-H Convention 9.6.2. Forward Kinematics in the Position Domain 9.6.3. Inverse Kinematics in the Position Domain 9.6.4. Multiplicity Analysis 9.6.5. Singularity Analysis in the Position Domain 9.6.6. Forward Kinematics in the Velocity Domain 9.6.7. Inverse Kinematics in the Velocity Domain 9.6.8. Singularity Analysis in the Velocity Domain 9.7. A Deficient Puma Manipulator with Five Active Joints 9.7.1. Kinematic Description According to the D-H Convention 9.7.2. Forward Kinematics in the Position Domain 9.7.3. Inverse Kinematics in the Position Domain 9.7.3.1. Solution in the Case of Fully Specified Tip Point Location 9.7.3.2. Solution in the Case of Fully Specified End-Effector Orientation 9.7.4. Multiplicity Analysis in the Position Domain 9.7.4.1. Analysis in the Case of Fully Specified Tip Point Location 9.7.4.2. Analysis in the Case of Fully Specified End-Effector Orientation 9.7.5. Singularity Analysis in the Position Domain 9.7.5.1. Analysis in the Case of Fully Specified Tip Point Location 9.7.5.2. Analysis in the Case of Fully Specified End-Effector Orientation 9.7.6. Forward Kinematics in the Velocity Domain 9.7.7. Inverse Kinematics in the Velocity Domain 9.7.7.1. Solution in the Case of Fully Specified Tip Point Velocity 9.7.7.2. Solution in the Case of Fully Specified End-Effector Angular Velocity 9.7.8. Singularity Analysis in the Velocity Domain 9.7.8.1. Analysis in the Case of Fully Specified Tip Point Velocity 9.7.8.2. Analysis in the Case of Fully Specified End-Effector Angular Velocity 9.8. A Redundant Humanoid Manipulator with Eight Joints 9.8.1. Kinematic Description According to the D-H Convention 9.8.2. Forward Kinematics in the Position Domain 9.8.3. Inverse Kinematics in the Position Domain 9.8.4. Multiplicity Analysis 9.8.5. Singularity Analysis in the Position Domain 9.8.6. Forward Kinematics in the Velocity Domain 9.8.7. Inverse Kinematics in the Velocity Domain 9.8.8. Singularity Analysis in the Velocity Domain 9.8.9. Consistency of the Inverse Kinematics in the Position and Velocity Domains Chapter 10. Position and Velocity Analyses of Parallel Manipulators 10.1. General Kinematic Features of Parallel Manipulators Example 10.1.1: 3RPR Planar Parallel Manipulator Example 10.1.2: 3PRR+3RPR Planar Parallel Manipulator Example 10.1.3: Parallel Manipulator Formed by Two Serial Manipulators 10.2. Position Equations of a Parallel Manipulator Example 10.2.1: Position Equations of a 3RRR Planar Parallel Manipulator 10.3. Forward Kinematics in the Position Domain Example 10.3.1: Forward Kinematics of the 3RRR Planar Parallel Manipulator a) Forward Kinematic Solution b) Posture Modes of Forward Kinematics and Posture Mode Changing Poses c) Position Singularities of Forward Kinematics 10.4. Inverse Kinematics in the Position Domain Example 10.4.1: Inverse Kinematics of the 3RRR Planar Parallel Manipulator a) Inverse Kinematic Solutions for the Legs b) Posture Modes of Inverse Kinematics and Posture Mode Changing Poses c) Position Singularities of Inverse Kinematics 10.5. Velocity Equations of a Parallel Manipulator Example 10.5.1: Velocity Equations of a 3RRR Planar Parallel Manipulator 10.6. Forward Kinematics in the Velocity Domain Example 10.6.1: Forward Velocity Analysis of the 3RRR Planar Parallel Manipulator a) Forward Velocity Solution b) Motion Singularities of Forward Kinematics c) Comparison of the Position and Motion Singularities 10.7. Inverse Kinematics in the Velocity Domain Example 10.7.1: Inverse Velocity Analysis of the 3RRR Planar Parallel Manipulator a) Inverse Velocity Solutions for the Legs b) Motion Singularities of Inverse Kinematics 10.8. Stewart-Gough Platform as a 6UPS Spatial Parallel Manipulator 10.8.1. Kinematic Description 10.8.2. Position Equations a) End-Effector Orientation Equations through the Legs b) Tip Point Location Equations through the Legs 10.8.3. Inverse Kinematics in the Position Domain a) Inverse Kinematic Solution b) Position Singularities of Inverse Kinematics (PSIKs) 10.8.4. Forward Kinematics in the Position Domain a) Independent Kinematic Loops (IKLs) b) Loop Equations c) Solution of the Loop Equations and Finding the Position of the End-Effector d) Position Singularities of Forward Kinematics (PSFKs) 10.8.5. Velocity Equations a) Angular Velocity Equations for the End-Effector through the Legs b) Tip Point Velocity Equations through the Legs 10.8.6. Inverse Kinematics in the Velocity Domain a) Inverse Velocity Solution b) Motion Singularities of Inverse Kinematics (MSIKs) 10.8.7. Forward Kinematics in the Velocity Domain a) Loop Equations in the Velocity Domain b) Forward Velocity Solution c) Motion Singularities of Forward Kinematics (MSFKs) 10.9. Delta Robot: A 3RS2S2 Spatial Parallel Manipulator 10.9.1. Kinematic Description 10.9.2. Position Equations a) End-Effector Orientation Equations through the Legs b) Tip Point Location Equations through the Legs 10.9.3. Independent Kinematic Loops and the Associated Equations a) Independent Kinematic Loops (IKLs) b) Orientation Equations for the IKLs c) Characteristic Point Location Equations for the IKLs 10.9.4. Inverse Kinematics in the Position Domain a) Inverse Kinematic Solution b) Multiplicity Analysis of Inverse Kinematics c) Position Singularities of Inverse Kinematics (PSIKs) 10.9.5. Forward Kinematics in the Position Domain a) Forward Kinematic Solution b) Multiplicity Analysis of Forward Kinematics c) Position Singularities of Forward Kinematics (PSFKs) 10.9.6. Velocity Equations 10.9.7. Inverse Kinematics in the Velocity Domain a) Inverse Velocity Solution b) Motion Singularities of Inverse Kinematics (MSIKs) 10.9.8. Forward Kinematics in the Velocity Domain a) Forward Velocity Solution b) Motion Singularities of Forward Kinematics (MSFKs)
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Chapter 1. Vectors and Their Matrix Representations in Selected Reference Frames 1.1. General Features of Notation 1.2. Vectors 1.2.1. Definition and Description of a Vector 1.2.2. Equality of Vectors 1.2.3. Opposite Vectors 1.3. Vector Products 1.3.1. Dot Product 1.3.2. Cross Product 1.4. Reference Frames 1.5. Representation of a Vector in a Selected Reference Frame 1.6. Matrix Operations Corresponding to Vector Operations 1.6.1. Dot Product 1.6.2. Cross Product and Skew Symmetric Cross Product Matrices 1.7. Mathematical Properties of the Skew Symmetric Matrices 1.8. Examples Involving Skew Symmetric Matrices Example 1.8.1 Example 1.8.2 Example 1.8.3 Chapter 2. Rotation of Vectors and Rotation Matrices 2.1. Vector Equation of Rotation and the Rodrigues Formula 2.2. Matrix Equation of Rotation and the Rotation Matrix 2.3. Exponentially Expressed Rotation Matrix 2.4. Basic Rotation Matrices 2.5. Successive Rotations 2.6. Orthonormality of the Rotation Matrices 2.7. Mathematical Properties of the Rotation Matrices 2.7.1. Mathematical Properties of General Rotation Matrices 2.7.2. Mathematical Properties of the Basic Rotation Matrices 2.8. Examples Involving Rotation Matrices Example 2.8.1 Example 2.8.2 Example 2.8.3 Example 2.8.4 2.9. Determination of the Angle and Axis of a Specified Rotation Matrix 2.9.1. Scalar Equations of Rotation 2.9.2. Determination of the Angle of Rotation 2.9.3. Determination of the Axis of Rotation a) General Case with b) Special Cases with (No Rotation, Full Rotation, Half Rotation) 2.9.4. Discussion about the Optional Sign Variables a) General Case with b) Special Case with a Half Rotation 2.10. Definition and Properties of the Double Argument Arctangent Function Chapter 3. Matrix Representations of Vectors in Different Reference Frames and the Component Transformation Matrices 3.1. Matrix Representations of a Vector in Different Reference Frames 3.2. Transformation Matrices between Reference Frames 3.2.1. Definition and Usage of a Transformation Matrix 3.2.2. Basic Properties of a Transformation Matrix a) Inversion Property b) Orthonormality Property c) Combination Property 3.3. Expression of a Transformation Matrix in terms of Basis Vectors 3.3.1. Column-by-Column Expression 3.3.2. Row-by-Row Expression 3.3.3. Example 3.4. Expression of a Transformation Matrix as a Direction Cosine Matrix 3.4.1. Definitions of Direction Angles and Direction Cosines 3.4.2. Transformation Matrix Formed as a Direction Cosine Matrix 3.5. Expression of a Transformation Matrix as a Rotation Matrix 3.5.1. Correlation between the Rotation and Transformation Matrices 3.5.2. Distinction between the Rotation and Transformation Matrices 3.6. Relationship between the Matrix Representations of a Rotation Operator in Different Reference Frames 3.7. Expression of a Transformation Matrix in a Case of Several Successive Rotations 3.7.1. Rotated Frame Based (RFB) Formulation 3.7.2. Initial Frame Based (IFB) Formulation 3.8. Expression of a Transformation Matrix in terms of Euler Angles 3.8.1. General Definition of Euler Angles 3.8.2. IFB (Initial Frame Based) Euler Angle Sequences 3.8.3. RFB (Rotated Frame Based) Euler Angle Sequences 3.8.4. Commonly Used Euler Angle Sequences a) RFB 1-2-3 Sequence b) RFB 3-2-1 Sequence c) RFB 3-1-3 Sequence d) RFB 3-2-3 Sequence 3.8.5. Extraction of Euler Angles from a Given Transformation Matrix a) Extraction of the 3-2-3 Euler Angles b) Extraction of the 1-2-3 Euler Angles 3.9. Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices 3.9.1. Position of a Point Expressed in Different Reference Frames 3.9.2. Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships a) Homogeneous versus Nonhomogeneous Relationships b) Linear versus Nonlinear Relationships c) Affine Relationships 3.9.3. Affine Coordinate Transformation between Two Reference Frames 3.9.4. Homogeneous Coordinate Transformation between Two Reference Frames 3.9.5. Mathematical Properties of the Homogeneous Transformation Matrices (HTMs) a) Determinant of an HTM b) Inverse of an HTM c) Decomposition of an HTM d) HTM of a Pure Rotation e) HTM of a Pure Translation f) Observation in a Third Different Reference Frame 9.3.6. Example Chapter 4. Vector Differentiation Accompanied by Velocity and Acceleration Expressions 4.1. Derivatives of a Vector with respect to Different Refrence Frames 4.1.1. Differentiation and Resolution Frames 4.1.2. Components in Different Differentiation and Resolution Frames 4.1.3. Example 4.2. Vector Derivatives with respect to Different Reference Frames and the Coriolis Transport Theorem 4.2.1. First Derivatives and the Relative Angular Velocity 4.2.2. Second Derivatives and the Relative Angular Acceleration 4.3. Combination of Relative Angular Velocities and Accelerations 4.3.1. Combination of Relative Angular Velocities 4.3.2. Combination of Relative Angular Accelerations 4.4. Angular Velocities and Accelerations Associated with Rotation Sequences 4.4.1. Relative Angular Velocities and Accelerations about Relatively Fixed Axes 4.4.2. Example 4.4.3. Angular Velocities Associated with the Euler Angle Sequences a) Rotated Frame Based (RFB) Euler Angle Sequences b) Initial Frame Based (IFB) Euler Angle Sequences 4.5. Velocity and Acceleration of a Point with respect to Different References Frames 4.5.1. Velocity of a Point with respect to Different References Frames 4.5.2. Acceleration of a Point with respect to Different References Frames 4.5.3. Velocity and Acceleration Expressions with Simplified Notations Chapter 5. Kinematics of Rigid Body Systems 5.1. Kinematic Description of a Rigid Body System 5.1.1. Body Frames and Joint Frames 5.1.2. Kinematic Chains, Kinematic Branches, and Kinematic Loops 5.1.3. Joints or Kinematic Pairs 5.2. Position Equations for a Kinematic Chain of Rigid Bodies 5.2.1. Relative Orientation Equation between Successive Bodies 5.2.2. Relative Location Equation between Successive Bodies 5.2.3. Orientation of a Body with respect to the Base of the Kinematic Chain 5.2.4. Location of a Body with respect to the Base of the Kinematic Chain 5.2.5. Loop Closure Equations for a Kinematic Loop a) Orientation Equation for the Closure at Joint Joz b) Location Equation for the Closure at Joint Joz c) Orientation Equation for the Closure at Joint Jpq d) Location Equation for the Closure at Joint Jpq 5.3. Velocity Equations for a Kinematic Chain of Rigid Bodies 5.3.1. Relative Angular Velocity between Successive Bodies 5.3.2. Relative Translational Velocity between Successive Bodies 5.3.3. Angular Velocity of a Body with respect to the Base 5.3.4. Translational Velocity of a Body with respect to the Base 5.3.5. Velocity Equations for a Kinematic Loop a) Angular Velocity Equation for the Kinematic Loop b) Translational Velocity Equation for the Kinematic Loop 5.4. Acceleration Equations for a Kinematic Chain of Rigid Bodies 5.4.1. Relative Angular Acceleration between Successive Bodies 5.4.2. Relative Translational Acceleration between Successive Bodies 5.4.3. Angular Acceleration of a Body with respect to the Base 5.4.4. Translational Acceleration of a Body with respect to the Base 5.4.5. Acceleration Equations for a Kinematic Loop a) Angular Acceleration Equation for the Kinematic Loop b) Translational Acceleration Equation for the Kinematic Loop 5.5. Example 1: A Serial Manipulator with an RRP Arm 5.5.1. Kinematic Description of the System 5.5.2. Position Analysis a) Orientations of the Links b) Locations of the Link Frame Origins c) Inverse Kinematic Solution d) Multiple Solutions e) Position Singularity 5.5.3. Velocity Analysis a) Angular Velocities of the Links b) Velocities of the Link Frame Origins c) Wrist Point Jacobian Matrix d) Inverse Velocity Solution e) Motion Singularities Involving Velocities f) Consequences of the Motion Singularities 5.5.4. Acceleration Analysis a) Angular Accelerations of the Links b) Accelerations of the Link Frame Origins c) Wrist Point Acceleration by Using the Wrist Point Jacobian Matrix d) Inverse Acceleration Solution e) Motion Singularities Involving Accelerations 5.6. Example 2: A Spatial Slider-Crank Mechanism 5.6.1. Kinematic Description of the Mechanism 5.6.2. Loop Closure Equations a) Orientational Loop Closure Equation b) Locational Loop Closure Equation 5.6.3. Degree of Freedom (DoF) or Mobility () 5.6.4. Position Analysis a) Solution of the Loop Closure Equations for a Specified Crank Angle b) Existence of Solutions with Continuous Crank Rotations c) Multiple Solutions for the Same Crank Angle d) Position Singularity Associated with the Crank Angle e) Solution of the Loop Closure Equations for a Specified Slider Position f) Existence of Solutions g) Multiple Solutions for the Same Slider Position h) Position Singularities Associated with the Slider Position 5.6.5. Velocity Analysis a) Solution of the Velocity Equations for a Specified Crank Motion b) Motion Singularities Associated with the Crank Motion c) Solution of the Velocity Equations for a Specified Slider Motion h) Motion Singularities Associated with the Slider Motion 5.6.6. Acceleration Analysis Chapter 6. Joints and Their Kinematic Characteristics 6.1. Kinematic Details of the Joints 6.1.1. Description of a Joint as a Kinematic Pair 6.1.2. Degree of Freedom (DoF) or Mobility of a Joint 6.1.3. Number of Distinct Joints between Two Rigid Bodies 6.1.4. Classification of the Joints 6.2. Typical Lower Order Joints 6.2.1. Single-Axis Joints a) Cylindrical Joint b) Revolute Joint c) Prismatic Joint d) Screw Joint 6.2.2. Universal Joint 6.2.3. Spherical Joint 6.2.4. Plane-on-Plane Joint 6.3. Higher Order Joints with Simple Contacts 6.3.1. Line-on-Plane Joint 6.3.2. Point-on-Plane Joint 6.3.3. Point-on-Surface Joint 6.4. Typical Multi-Joint Connections 6.4.1. Fork-on-Surface Joint 6.4.2. Triangle-on-Surface Joint 6.5. Rolling Contact Joints with Point Contacts 6.5.1. Surface-on-Surface Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.5.2. Curve-on-Surface Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.5.3. Curve-on-Curve Joint a) Relative Position Equations b) Relative Velocity Equations 6.6. Rolling Contact Joints with Line Contacts 6.6.1. Cone-on-Cone Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.6.2. Cone-on-Cylinder Joint a) Relative Position Equations b) Relative Velocity Equations 6.6.3. Cone-on-Plane Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.6.4. Cylinder-on-Cylinder Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.6.5. Cylinder-on-Plane Joint a) Relative Position Equations b) Relative Velocity Equations c) Relative Velocity Equations in Case of Rolling without Slipping 6.7. Examples 6.7.1. Example 1: An RRRSP Mechanism a) Kinematic Description of the System b) Loop Closure Equations c) Determination of the Unspecified Variables for Specified d) Determination of the Unspecified Variables for Specified 6.7.2. Example 2: A Two-Link Mechanism with Three Point-on-Plane Joints a) Kinematic Description of the System b) Determination of the Unspecified Variables c) Appendix about Closed-Form Expressions and Analytical Solutions 6.7.3. Example 3: A Spatial Cam Mechanism a) Kinematic Description of the System b) Loop Closure Equations c) Determination of the Unspecified Variables for Specified 6.7.4. Example 4: A Spatial Cam Mechanism That Allows Rolling Without Slipping a) Kinematic Description of the System b) Loop Closure Equations c) Determination of the Unspecified Variables for Specified , , and d) Velocity Equations and their Solution in the Case of Rolling with Slipping e) Velocity Equations and their Solution in the Case of Rolling without Slipping f) Finding the Pose of the Mechanism during a Rolling-Without-Slipping Motion Chapter 7. Kinematic Features of Serial Manipulators 7.1. Kinematic Description of a General Serial Manipulator 7.2. Denavit-Hartenberg (D-H) Convention 7.3. D-H Convention for Successive Intermediate Links and Joints 7.3.1. Assignment and Description of the Link Frames 7.3.2. D-H Parameters 7.3.3. Relative Position Equation between Successive Links 7.3.4. Alternative Multi-Index Notation for the D-H Convention 7.4. D-H Convention for the First Joint 7.5. D-H Convention for the Last Joint 7.6. D-H Convention for Successive Joints with Perpendicularly Intersecting Axes 7.7. D-H Convention for Successive Joints with Parallel Axes 7.8. D-H Convention for Successive Joints with Coincident Axes Chapter 8. Position and Motion Analyses of Generic Serial Manipulators 8.1. Forward Kinematics 8.2. Compact Formulation of Forward Kinematics 8.3. Detailed Formulation of Forward Kinematics 8.4. Manipulators with or without Spherical Wrists 8.5. Inverse Kinematics 8.6. Inverse Kinematic Solution for a Regular Manipulator 8.6.1. Regular Manipulator with a Spherical Wrist 8.6.2. Regular Manipulator with a Nonspherical Wrist 8.7. Inverse Kinematic Solution for a Redundant Manipulator 8.7.1. Solution by Specifying the Variables of Certain Joints 8.7.2. Solution by Optimization 8.8. Inverse Kinematic Solution for a Deficient Manipulator 8.8.1. Compromise in Orientation in Favor of a Completely Specified Location 8.8.2. Compromise in Location in Favor of a Completely Specified Orientation 8.9. Forward Kinematics of Motion 8.9.1. Forward Kinematics of Velocity Relationships 8.9.2. Forward Kinematics of Acceleration Relationships 8.10. Jacobian Matrices Associated with the Wrist and Tip Points 8.11. Recursive Position, Velocity, and Acceleration Relationships 8.11.1. Orientations of the Links 8.11.2. Locations of the Link Frame Origins 8.11.3. Locations of the Mass Centers of the Links 8.11.4. Angular Velocities of the Links 8.11.5. Velocities of the Link Frame Origins 8.11.6. Velocities of the Mass Centers of the Links 8.11.7. Angular Accelerations of the Links 8.11.8. Accelerations of the Link Frame Origins 8.11.9. Accelerations of the Mass Centers of the Links 8.12. Inverse Motion Analysis of a Manipulator based on the Jacobian Matrix 8.12.1. Inverse Velocity Analysis of a Regular Manipulator 8.12.2. Inverse Acceleration Analysis of a Regular Manipulator 8.13. Inverse Motion Analysis of a Redundant Manipulator 8.13.1. Inverse Velocity Analysis 8.13.2. Inverse Acceleration Analysis 8.14. Inverse Motion Analysis of a Deficient Manipulator 8.15. Inverse Motion Analysis of a Regular Manipulator Using The Detailed Formulation 8.15.1. Inverse Velocity Solution 8.15.2. Inverse Acceleration Solution Chapter 9. Kinematic Analyses of Typical Serial Manipulators 9.1. Puma Manipulator 9.1.1. Kinematic Description According to the D-H Convention 9.1.2. Forward Kinematics in the Position Domain 9.1.3. Inverse Kinematics in the Position Domain 9.1.4. Multiplicity Analysis 9.1.5. Singularity Analysis in the Position Domain 9.1.6. Forward Kinematics in the Velocity Domain 9.1.7. Inverse Kinematics in the Velocity Domain 9.1.8. Singularity Analysis in the Velocity Domain 9.2. Stanford Manipulator 9.2.1. Kinematic Description According to the D-H Convention 9.2.2. Forward Kinematics in the Position Domain 9.2.3. Inverse Kinematics in the Position Domain 9.2.4. Multiplicity Analysis 9.2.5. Singularity Analysis in the Position Domain 9.2.6. Forward Kinematics in the Velocity Domain 9.2.7. Inverse Kinematics in the Velocity Domain 9.2.8. Singularity Analysis in the Velocity Domain 9.3. Elbow Manipulator 9.3.1. Kinematic Description According to the D-H Convention 9.3.2. Forward Kinematics in the Position Domain 9.3.3. Inverse Kinematics in the Position Domain 9.3.4. Multiplicity Analysis 9.3.5. Singularity Analysis in the Position Domain 9.3.6. Forward Kinematics in the Velocity Domain 9.3.7. Inverse Kinematics in the Velocity Domain 9.3.8. Singularity Analysis in the Velocity Domain 9.4. Scara Manipulator 9.4.1. Kinematic Description According to the D-H Convention 9.4.2. Forward Kinematics in the Position Domain 9.4.3. Inverse Kinematics in the Position Domain 9.4.4. Multiplicity Analysis 9.4.5. Singularity Analysis in the Position Domain 9.4.6. Forward Kinematics in the Velocity Domain 9.4.7. Inverse Kinematics in the Velocity Domain 9.4.8. Singularity Analysis in the Velocity Domain 9.5. An RP2R3 Manipulator without an Analytical Solution 9.5.1. Kinematic Description According to the D-H Convention 9.5.2. Forward Kinematics in the Position Domain 9.5.3. Inverse Kinematics in the Position Domain 9.5.4. Multiplicity Analysis 9.5.5. Singularity Analysis in the Position Domain 9.5.6. Forward Kinematics in the Velocity Domain 9.5.7. Inverse Kinematics in the Velocity Domain 9.5.8. Singularity Analysis in the Velocity Domain 9.6. An RPRPR2 Manipulator with an Uncustomary Analytical Solution 9.6.1. Kinematic Description According to the D-H Convention 9.6.2. Forward Kinematics in the Position Domain 9.6.3. Inverse Kinematics in the Position Domain 9.6.4. Multiplicity Analysis 9.6.5. Singularity Analysis in the Position Domain 9.6.6. Forward Kinematics in the Velocity Domain 9.6.7. Inverse Kinematics in the Velocity Domain 9.6.8. Singularity Analysis in the Velocity Domain 9.7. A Deficient Puma Manipulator with Five Active Joints 9.7.1. Kinematic Description According to the D-H Convention 9.7.2. Forward Kinematics in the Position Domain 9.7.3. Inverse Kinematics in the Position Domain 9.7.3.1. Solution in the Case of Fully Specified Tip Point Location 9.7.3.2. Solution in the Case of Fully Specified End-Effector Orientation 9.7.4. Multiplicity Analysis in the Position Domain 9.7.4.1. Analysis in the Case of Fully Specified Tip Point Location 9.7.4.2. Analysis in the Case of Fully Specified End-Effector Orientation 9.7.5. Singularity Analysis in the Position Domain 9.7.5.1. Analysis in the Case of Fully Specified Tip Point Location 9.7.5.2. Analysis in the Case of Fully Specified End-Effector Orientation 9.7.6. Forward Kinematics in the Velocity Domain 9.7.7. Inverse Kinematics in the Velocity Domain 9.7.7.1. Solution in the Case of Fully Specified Tip Point Velocity 9.7.7.2. Solution in the Case of Fully Specified End-Effector Angular Velocity 9.7.8. Singularity Analysis in the Velocity Domain 9.7.8.1. Analysis in the Case of Fully Specified Tip Point Velocity 9.7.8.2. Analysis in the Case of Fully Specified End-Effector Angular Velocity 9.8. A Redundant Humanoid Manipulator with Eight Joints 9.8.1. Kinematic Description According to the D-H Convention 9.8.2. Forward Kinematics in the Position Domain 9.8.3. Inverse Kinematics in the Position Domain 9.8.4. Multiplicity Analysis 9.8.5. Singularity Analysis in the Position Domain 9.8.6. Forward Kinematics in the Velocity Domain 9.8.7. Inverse Kinematics in the Velocity Domain 9.8.8. Singularity Analysis in the Velocity Domain 9.8.9. Consistency of the Inverse Kinematics in the Position and Velocity Domains Chapter 10. Position and Velocity Analyses of Parallel Manipulators 10.1. General Kinematic Features of Parallel Manipulators Example 10.1.1: 3RPR Planar Parallel Manipulator Example 10.1.2: 3PRR+3RPR Planar Parallel Manipulator Example 10.1.3: Parallel Manipulator Formed by Two Serial Manipulators 10.2. Position Equations of a Parallel Manipulator Example 10.2.1: Position Equations of a 3RRR Planar Parallel Manipulator 10.3. Forward Kinematics in the Position Domain Example 10.3.1: Forward Kinematics of the 3RRR Planar Parallel Manipulator a) Forward Kinematic Solution b) Posture Modes of Forward Kinematics and Posture Mode Changing Poses c) Position Singularities of Forward Kinematics 10.4. Inverse Kinematics in the Position Domain Example 10.4.1: Inverse Kinematics of the 3RRR Planar Parallel Manipulator a) Inverse Kinematic Solutions for the Legs b) Posture Modes of Inverse Kinematics and Posture Mode Changing Poses c) Position Singularities of Inverse Kinematics 10.5. Velocity Equations of a Parallel Manipulator Example 10.5.1: Velocity Equations of a 3RRR Planar Parallel Manipulator 10.6. Forward Kinematics in the Velocity Domain Example 10.6.1: Forward Velocity Analysis of the 3RRR Planar Parallel Manipulator a) Forward Velocity Solution b) Motion Singularities of Forward Kinematics c) Comparison of the Position and Motion Singularities 10.7. Inverse Kinematics in the Velocity Domain Example 10.7.1: Inverse Velocity Analysis of the 3RRR Planar Parallel Manipulator a) Inverse Velocity Solutions for the Legs b) Motion Singularities of Inverse Kinematics 10.8. Stewart-Gough Platform as a 6UPS Spatial Parallel Manipulator 10.8.1. Kinematic Description 10.8.2. Position Equations a) End-Effector Orientation Equations through the Legs b) Tip Point Location Equations through the Legs 10.8.3. Inverse Kinematics in the Position Domain a) Inverse Kinematic Solution b) Position Singularities of Inverse Kinematics (PSIKs) 10.8.4. Forward Kinematics in the Position Domain a) Independent Kinematic Loops (IKLs) b) Loop Equations c) Solution of the Loop Equations and Finding the Position of the End-Effector d) Position Singularities of Forward Kinematics (PSFKs) 10.8.5. Velocity Equations a) Angular Velocity Equations for the End-Effector through the Legs b) Tip Point Velocity Equations through the Legs 10.8.6. Inverse Kinematics in the Velocity Domain a) Inverse Velocity Solution b) Motion Singularities of Inverse Kinematics (MSIKs) 10.8.7. Forward Kinematics in the Velocity Domain a) Loop Equations in the Velocity Domain b) Forward Velocity Solution c) Motion Singularities of Forward Kinematics (MSFKs) 10.9. Delta Robot: A 3RS2S2 Spatial Parallel Manipulator 10.9.1. Kinematic Description 10.9.2. Position Equations a) End-Effector Orientation Equations through the Legs b) Tip Point Location Equations through the Legs 10.9.3. Independent Kinematic Loops and the Associated Equations a) Independent Kinematic Loops (IKLs) b) Orientation Equations for the IKLs c) Characteristic Point Location Equations for the IKLs 10.9.4. Inverse Kinematics in the Position Domain a) Inverse Kinematic Solution b) Multiplicity Analysis of Inverse Kinematics c) Position Singularities of Inverse Kinematics (PSIKs) 10.9.5. Forward Kinematics in the Position Domain a) Forward Kinematic Solution b) Multiplicity Analysis of Forward Kinematics c) Position Singularities of Forward Kinematics (PSFKs) 10.9.6. Velocity Equations 10.9.7. Inverse Kinematics in the Velocity Domain a) Inverse Velocity Solution b) Motion Singularities of Inverse Kinematics (MSIKs) 10.9.8. Forward Kinematics in the Velocity Domain a) Forward Velocity Solution b) Motion Singularities of Forward Kinematics (MSFKs)
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