Optimal control for principal kinematic systems on lie groups

Abstract

Abstract In this paper we present a simplified formulation of the necessary conditions for optimal controls of principal kinematic systems evolving on Lie groups. This class of systems is particularly meaningful because a number of different types of locomotion systems, such as kinematic snakes, paramecia, inchworms, mobile carts, and even the falling cat, can be represented in this form. Furthermore, it is shown that for systems on Abelian Lie groups, the equations describing the optimal control inputs take on an even simpler form, based purely on the curvature of the kinematic connection describing the locomotion. These ideas are presented with several examples, including the cylinder (paramecium) swimming at low Reynolds' number and a normal form for TV-trailer systems.