Using optimal control to obtain maximum displacement gait for Purcell's three-link swimmer

Purcell's swimmer is a classical model of a simple three-link swimmer moving in a highly viscous fluid, similar to the motion of microscopic organisms or robotic microswimmers. The two joint angles are commonly prescribed as periodic trajectories called gaits, so that the dynamics of Purcell's swimmer can be formulated as a driftless nonlinear control system. In a famous paper by Tam and Hosoi, they have found the optimal gait that maximizes net displacement over a cycle by representing the time-periodic joint angles as truncated Fourier series and numerically optimizing a finite set of their coefficients. In this work, the gait optimization is revisited and analytically formulated as an elegant problem of optimal control system with only two state variables and a single input, which can be solved using Pontryagin's maximum principle. Due to absence of any physical constraints on the control system's input, it turns out that the optimal solution must follow a 'singular arc'. Numerical solution of the boundary value problem is obtained, which exactly reproduces Tam and Hosoi's optimal gait.