Gregory S. Chirikjian
Department of Mechanical Engineering
Johns Hopkins University
Engineering Applications of Noncommutative Harmonic Analysis
Noncommutative harmonic analysis is an area of mathematics that is a generalization of classical Fourier analysis. A goal in noncommutative harmonic analysis is to expand functions on certain kinds of Lie groups in terms of irreducible unitary representations. Another goal is to examine how differential operators acting on functions on Lie groups are transformed into algebraic operations in generalized Fourier space. This is a form of operational calculus.
In this talk, techniques of noncommutative harmonic analysis applied to the group of rigid-body motions, SE(3), are used in the context of robotic manipulators and polymer chains. It is shown how probability density functions describing the relative position and orientation of one end of a manipulator (or polymer chain) can be generated using the techniques. Issues such as self-avoidance and applications of FFTs for groups will also be discussed.