Summary
Purcell’s swimmer is a well-known planar model of a swimming microorganism, 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 two periodic inputs. In this work we present a macro-scale realization of a three-link robotic swimmer moving in a highly viscous fluid. We propose a simple variant of Purcell’s model with unequal 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 periodic gaits for maximizing the displacement per cycle under bounds on the of joint angles. Employing differential geometric method that transforms the problem to area integral enclosed by the gait trajectory enables visual interpretation which explains topological changes in optimal gaits upon varying the joint angle bounds. Finally, we apply PMP formulation to the problem of maximizing Lighthill’s energy efficiency in order to obtain a boundary value problem whose solution gives energy-optimal gaits for Purcell’s swimmer as well as its variant with a central sphere.
