Chapter 18 - Equation solving in robotics and mechanisms

Abstract

This chapter discusses equation solving in robotics and mechanisms. The problems include the inverse kinematics of several kinds of manipulators, the forward kinematics of the 3-TPS parallel manipulator, the forward kinematics of the general 6-6 Stewart platform, the five position synthesis of the planar four-bar linkage, and the reverse force analysis of two kinds of three-spring systems. The Stewart platform is the foundation for the so-called NC machine tools of the 21st century. The inverse kinematics of PUMA type and PUMA560 manipulators is explained. The inverse kinematics of 6R manipulators with simple geometry is elaborated. Synthesis of mechanisms is the first step in the design of a machine. The synthesis is called five positions rigid body guidance, and the five positions and orientations of a plane are given. The problem is called Burmester's problem, and was solved once by drawing a series of graphs. It is found that the mathematical model of the five position plane rigid body guidance consists of 12 equations in 12 unknowns as desired. The reverse force analysis of a spatial three-spring system is analyzed. The chapter also discusses geometric constraint equations for the spring system.