In this paper we apply techniques from noncommutative harmonic analysis to the development of fast algorithms for the computation of convolution integrals on motion groups. In particular, we focus on the group of rigid-body motions in 3-space, which is denoted here as SE(3). The general theory of irreducible unitary representations (IURs) of the 3D motion group is described briefly. Using IURs in operator form, we write the Fourier transform of functions on the motion group as an integral over the product space SE(3) × S 2 . The integral form of the Fourier transform matrix elements allows us to apply fast Fourier transform (FFT) methods developed previously for R 3 , S 2 , and SO(3) to speed up considerably the computation of convolutions of functions on SE(3). Such convolutions have been shown to play an important role in a number of engineering disciplines. An algorithm for the fast computation of the Fourier transform is given and its complexity is discussed. The Fourier transform for the 3D "discrete motion group" (semi-direct product of the icosahedral group with the translation group) is also developed and the computational complexity is discussed.
We apply the Fourier transform on the discrete-motion group to the problem of computing the configuration-space obstacles of mobile robots which move among static obstacles, the problem of finding the workspace density of binary manipulators with many actuators, and the problem of determining workspace boundaries of manipulators with continuous-motion actuators. We develop and implement Fourier transforms for the discrete-motion group of the plane. These transforms allow us to apply fast Fourier transform methods to the computation of convolution-like integrals that arise in robot kinematics and motion planning. The results of the implementation are discussed for particular examples.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.