Differential Kinematics

While forward kinematics maps joint angles to end-effector poses, differential kinematics studies how joint velocities map to end-effector velocities. This linearization of the kinematic map is captured by the Jacobian matrix, which is central to velocity control, singularity analysis, and numerical inverse kinematics.

Notes

• 5.1. Rigid Body Velocity — Angular velocity (spatial/body), full rigid body velocity ($V^s$, $V^b$), the adjoint map, velocity composition
• 5.2. Robot Jacobians — Spatial and body Jacobians, column construction via adjoint, force duality
• 5.3. Singularities and Manipulability — Rank deficiency, shoulder/elbow/wrist singularities, velocity ellipsoid, manipulability measures
• 5.4. Jacobian Inverse Kinematics — Pseudoinverse IK, damped least squares, redundant manipulators, null-space optimization