Kinematics

Abstract

A manipulator can be schematically represented from a mechanical viewpoint as a kinematic chain of rigid bodies (links) connected by means of revolute or prismatic joints. One end of the chain is constrained to a base, while an end-effector is mounted to the other end. The resulting motion of the structure is obtained by composition of the elementary motions of each link with respect to the previous one. Therefore, in order to manipulate an object in space, it is necessary to describe the end-effector position and orientation. This chapter is dedicated to the derivation of the direct kinematics equation through a systematic, general approach based on linear algebra. This allows the end-effector position and orientation (pose) to be expressed as a function of the joint variables of the mechanical structure with respect to a reference frame. Both open-chain and closed-chain kinematic structures are considered. With reference to a minimal representation of orientation, the concept of operational space is introduced and its relationship with the joint space is established. Furthermore, a calibration technique of the manipulator kinematic parameters is presented. The chapter ends with the derivation of solutions to the inverse kinematics problem, which consists of the determination of the joint variables corresponding to a given end-effector pose.