A Symmetric Group Method for Controllability Characterization of Bilinear Systems on the Special Euclidean Group

Abstract

Abstract Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.