Some Notes on Vector Differentiation . . . . . . . . . . . . . 29
2.8 Kinematics of Systems of Bodies . . . . . . . . . . . . . . . . . . . . 30
2.8.1 Generalized Coordinates and Joint Configuration . . . . . . . 30
2.8.2 Task-Space Coordinates . . . . . . . . . . . . . . . . . . . . 31
End-Effector Configuration Parameters . . . . . . . . . . . . 31
Operational Space Coordinates . . . . . . . . . . . . . . . . . 32
2.8.3 Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . 33
2.8.4 Differential Kinematics and Analytical Jacobian . . . . . . . 34
Position and Rotation Jacobian . . . . . . . . . . . . . . . . . 35
Dependency on Parameterization . . . . . . . . . . . . . . . . 35
2.8.5 Geometric or Basic Jacobian . . . . . . . . . . . . . . . . . . 37
Addition and Subtraction of Geometric Jacobians . . . . . . . 37
Calculation of geometric Jacobian using Rigid Body Formulation 38
2.8.6 Relation between Geometric and Analytic Jacobian Matrix . . 41
2.9 Kinematic Control Methods . . . . . . . . . . . . . . . . . . . . . . 42
2.9.1 Inverse Differential Kinematics . . . . . . . . . . . . . . . . 42
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9.2 Multi-task Inverse Differential Kinematics Control . . . . . . 43
Multi-task with Equal Priority . . . . . . . . . . . . . . . . . 43
Multi-task with Prioritization . . . . . . . . . . . . . . . . . . 44
2.9.3 Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . . 47
Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . 47
Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . 47
Appropriate Rotation Error . . . . . . . . . . . . . . . . . . . 50
2.9.4 Trajectory Control . . . . . . . . . . . . . . . . . . . . . . . 53
Position Trajectory Control . . . . . . . . . . . . . . . . . . . 53
Orientation Trajectory Control . . . . . . . . . . . . . . . . . 53
2.10 Floating Base Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 54
2.10.1 Generalized Velocity and Acceleration . . . . . . . . . . . . . 54
2.10.2 Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . 55
2.10.3 Differential Kinematics of Floating Base Systems . . . . . . . 55
2.10.4 Contacts and Constraints . . . . . . . . . . . . . . . . . . . . 56
Point Contacts - Quadruped . . . . . . . . . . . . . . . . . . 57
Extended Contacts - Humanoid . . . . . . . . . . . . . . . . 57
2.10.5 Support Consistent Inverse Kinematics . . . . . . . . . . . . 57
3 Dynamics 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Foundations from Classical Mechanics . . . . . . . . . . . . . . . . . 60
3.2.1 Newton’s Law for Particles . . . . . . . . . . . . . . . . . . . 60
3.2.2 Virtual Displacements . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 Virtual Displacement of Single Rigid Bodies . . . . . . . . . 61
3.2.4 Virtual Displacement of Multi-Body Systems . . . . . . . . . 62
3.2.5 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . 62
3.3 Newton-Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Newton-Euler for Single Bodies . . . . . . . . . . . . . . . . 63
3.3.2 Newton-Euler for Multi-Body Systems . . . . . . . . . . . . 64
3.4 Lagrange Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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