Perspective Chapter: On the Morse Property for the Distance Function of a Robot Arm

Abstract

Morse theory plays a central role when we study the configuration space of various mechanical linkages. As an important linkage, we consider the planar robot arm. It is known that the distance function on its configuration space is a Morse function. On the other hand, for a fixed angle θ, we consider the spatial robot arm whose adjacent bond angles are θ. In chemistry, such an arm is used as a model for protein backbones and has been studied extensively. We consider the distance function on its configuration space. The purpose of this chapter is threefold: First, we study whether the distance function is a Morse function. Second, we determine the minimum and maximum values of the function. Consider the case that the arm consists of four bars. Then our third purpose is to study how the distance function is different from the usual Morse function on the torus.