# Exact isoholonomic motion of the planar Purcell's swimmer

## TL;DR

This paper formulates and solves the exact isoholonomic motion planning problem for the planar Purcell's swimmer using a structure-preserving discrete model and the Discrete Pontryagin maximum principle, demonstrating effective numerical solutions.

## Contribution

It introduces a discrete-time, structure-preserving model for the Purcell's swimmer and applies the Discrete Pontryagin maximum principle to solve the isoholonomic motion planning problem.

## Key findings

- Successful formulation of the discrete isoholonomic problem.
- Derivation of necessary optimality conditions as a boundary value problem.
- Numerical experiments demonstrating the effectiveness of the approach.

## Abstract

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.