1.1 Introduction

The design and control of robots require certain mathematical models, such as:

• transformation models between the operational space (in which the position of the end-effector is defined) and the joint space (in which the configuration of the robot is defined). The following is distinguished:

• direct and inverse geometric models giving the location of the end-effector (or the tool) in terms of the joint coordinates of the mechanism and vice versa,

• direct and inverse kinematic models giving the velocity of the end-effector in terms of the joint velocities and vice versa,

• dynamic models giving the relations between the torques or forces of the actuators, and the positions, velocities and accelerations of the joints.

This chapter presents some methods to establish these models. It will also deal with identifying the parameters appearing in these models. We will limit the discussion to simple open structures. For complex structure robots, i.e. tree or closed structures, we refer the reader to [KHA 02].

Mathematical development is based on (4 4) homogenous transformation matrices. The homogenous matrix ⁱ T _j representing the transformation from frame R _ito frame R _j is defined as:

where ⁱ s _j, ⁱ n _j and ⁱ a _j of the orientation matrix ⁱ R _j indicate the unit vectors along the axes x _j, y _j and z _j of the frame R _j expressed in the frame R _i; and...