Direct kinematics

Abstract

This chapter will address the problem of determining the pose of the endeffector of a parallel robot from its actuated joint coordinates. This relation has a clear practical interest for the control of the pose of the manipulator, but also for the velocity control of the end-effector.

Determining the pose of the end-effector from measurements of the minimal set of joint coordinates that are necessary for control purposes, is equivalent to solving the system of inverse kinematics equations. We will show that, in general, the solution for this problem is not unique i.e. there are several ways of assembling a parallel manipulator with given actuated joint coordinates, and that generally we cannot express in an analytical manner the generalized coordinates as functions of the actuated joint coordinates. We will present methods for finding all the solutions for this problem. We will see that their computation times, although decreasing, are still too large for use in a real time context. Furthermore, there is no known algorithm that allows the determination of the current pose of the platform among the set of solutions. We will then present numerical methods using a-priori information on the current pose, that are more compatible with a real-time context, and emphasize that their convergence and robustness is an important issue. Finally we will investigate the influence of additional sensory information on the direct kinematics.