Optimal control of a swimming robot based on Purcell's microswimmer model

Abstract

Purcell's swimmer is a well-known planar model of a swimming microorganism, governed by low Reynolds number hydrodynamics, which is comprised of three rigid links connected by actuated rotary joints. This model has been analyzed as a robotic locomotion system governed by first-order nonlinear dynamics with a periodic input (gait) of the two joint angles. In this work, we present a robotic macro-scale realization of this three-link swimmer moving in a highly viscous fluid. We propose a simple variant of Purcell's theoretical model with non-slender links and a central rigid sphere which represents the added drag of the robot's central flotation block, and calibrate the model's parameters to fit experimental measurements. Next, we apply optimal control formulation based on Pontryagin's Maximum Principle (PMP) in order to find optimal gaits that maximize the displacement per cycle under bounds on the joint angles. Employing a differential geometric method that transforms the problem to area integral enclosed by the gait trajectory in the plane of joint angles, enables visual interpretation which explains topological changes in displacement-optimal gaits upon varying the bound on the joint angles. We then apply PMP formulation to the problem of maximizing Lighthill's energy efficiency in order to obtain a boundary value problem (BVP) whose solution gives efficiency-optimal gaits for Purcell's swimmer model, as well as its variant with a central sphere. Finally, we utilize numerical methods such as parameterizing the input gait as a truncated Fourier series, as well as GPOPS-II solver, to produce sufficient initial guess values for solving the BVPs and obtaining efficiency-optimal gaits.