Robot Modeling and Control, 2nd Edition

CHAPTER 4VELOCITY KINEMATICS

In the previous chapter we derived the forward kinematic equations relating joint positions to end-effector positions and orientations. In this chapter we derive the velocity relationships, relating the linear and angular velocities of the end effector to the joint velocities.

Mathematically, the forward kinematic equations define a function from the configuration space of joint positions to the space of Cartesian positions and orientations. The velocity relationships are then determined by the Jacobian of this function. The Jacobian is a matrix that generalizes the notion of the ordinary derivative of a scalar function. The Jacobian is one of the most important quantities in the analysis and control of robot motion. It arises in virtually every aspect of robotic manipulation: in the planning and execution of smooth trajectories, in the determination of singular configurations, in the execution of coordinated anthropomorphic motion, in the derivation of the dynamic equations of motion, and in the transformation of forces and torques from the end effector to the manipulator joints.

We begin this chapter with an investigation of velocities and how to represent them. We first consider angular velocity about a fixed axis and then generalize this to rotation about an arbitrary, possibly moving axis with the aid of skew symmetric matrices. Equipped with this general representation of angular velocities, we are able to derive equations for both the angular ...