This chapter considers the relationship between the rate of change of joint coordinates, the joint velocity, and the velocity of the end-effector. The 3-dimensional end-effector pose ξ ∈ SE(3) has a velocity which is represented by a 6-vector known as a spatial velocity. The joint velocity and the end-effector velocity are related by the manipulator Jacobian matrix which is a function of manipulator pose.

The chapter covers:

- the manipulator Jacobian matrix in world and tool coordinate frames
- transforming velocities and wrenches between coordinate frames
- the analytical Jacobian
- resolved-rate motion control
- Jacobian singularity and manipulability
- under- and over-actuated robots