Exact isoholonomic motion of the planar Purcell's swimmer
This work addresses optimal control for low Reynolds number swimmers, but the approach is incremental, applying known principles to a specific system.
The authors solved the discrete-time isoholonomic problem for the planar Purcell's swimmer using the Discrete-time Pontryagin maximum principle, obtaining optimal motion that minimizes control effort. Numerical experiments demonstrate the algorithm's effectiveness.
In this article we present the discrete-time isoholonomic problem of the planar Purcell's swimmer and solve it using the Discrete-time Pontryagin maximum principle. The 3-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 state transfer while minimizing the control effort. 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.