Exact Isoholonomic Motion of the Planar Purcell's Swimmer

In this article, we present the discrete-time isoholonomic problem of the planar Purcell's swimmer and solve it using the discrete-time Pontryagin's maximum principle. The three-link Purcell's swimmer is a locomotion system moving in a low Reynolds number environment. The kinematics of the system evolves on a principal fiber bundle. A structure-preserving discrete-time kinematic model of the system is obtained in terms of the local form of a discrete connection. An adapted version of the discrete maximum principle on matrix Lie groups is then employed to come up with the necessary optimality conditions for an optimal transfer from a given initial state while minimizing the mechanical energy expended in the presence of constraints on the controls. These necessary conditions appear as a two-point boundary value problem and are solved using a numerical technique. Results from numerical experiments are presented to illustrate the algorithm and compared with the existing results for a similar case in the literature.