Rational Kinematics

1 Preliminary Notions.- 1.1 Introduction.- 1.2 On Notation and Basic Definitions.- 1.3 Further Definitions.- 1.4 Invariance.- 1.5 Generalized Inverses.- 2 Displacement of a Rigid Body.- 2.1 Introduction.- 2.2 Rotation of a Rigid Body About a Fixed Point.- 2.3 The Theorem of Euler.- 2.4 The Rotation Tensor.- 2.5 General Motion of a Rigid Body. The Pose and the Screw of a Rigid Body.- 2.6 Theorems About the General Motion of a Rigid Body.- 2.7 Compatibility Equations.- 3 Velocity Analysis of Rigid-Body Motions.- 3.1 Introduction.- 3.2 Motion of a Rigid Body About a Fixed Point.- 3.3 General Instantaneous Motion of a Rigid Body. The Twist of a Rigid Body.- 3.4 Further Results on the Velocity Distribution Throughout a Rigid Body.- 3.5 Compatibility Equations.- 4 Acceleration Analysis of Rigid-Body Motions.- 4.1 Introduction.- 4.2 Acceleration Field of a Body Moving About a Fixed Point.- 4.3 Acceleration Field of a Rigid Body Under General Motion.- 4.4 Change of Observer. Coriolis’ Theorem.- 4.5 Determination of Angular Acceleration. Compatibility Equations.- 5 Kinematic Chains.- 5.1 Introduction.- 5.2 Kinematic Pairs. Classification.- 5.3 Simple Kinematic Chains. Degree of Freedom by Groups of Motion.- 5.4 The Twist of a Link and the Jacobian of a Simple Kinematic Chain.- 5.4.1 Example 1.- 5.4.2 Example 2.- 5.5 The Condition Number of a Kinematic Chain. Isotropy.- 5.6 Complex Kinematic Chains. An Overview.- 5.7 The Jacobian of a Multiple-Closed-Loop Kinematic Chain. Degree of Freedom.- 5.8 Holonomic and Nonholonomic Constraints.- 5.9 Analysis of Kinematic Chains.- 5.10 Synthesis of Kinematic Chains.- 5.10.1 The Function-Generation Synthesis Problem.- 5.10.2 Kinematic Synthesis for Rigid-Body Guidance.- 5.10.3 Kinematic Synthesis for Path Generation.- References.