  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.
