Yasuhiko Kamiyama

By Yasuhiko Kamiyama

As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.

Part of the book: Advanced Topics of Topology

By Yasuhiko Kamiyama

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.

Part of the book: Motion Planning for Dynamic Agents