| Motivated by a physical phenomenon, the diffusion process, this thesis develops diffusion-based algorithms to solve problems in several seemingly unrelated fields. These problems are workspace generation and inverse kinematics of highly articulated robotic manipulators, robot path planning, the effects of laser phase noise in coherent optical communications, and conformational statistics of stiff macromolecules in polymer science. First, a family of diffusion equations defined on motion groups is derived to model the above problems. Then, based on the techniques of noncommutative harmonic analysis, efficient and powerful computational methods are used to solve those equations. The concepts, techniques and methodologies used in this dissertation open up a wealth of tools for solving problems in other fields. |
