Optimal Gaits for Mechanical Rectifier Systems

The essential mechanism underlying animal locomotion can be viewed as mechanical rectification that converts periodic body movements to thrust force through interactions with the environment. This paper defines a general class of mechanical rectifiers as multi-body systems equipped with such thrust generation mechanisms. A simple model is developed from the Euler-Lagrange equation by assuming small body oscillations around a given nominal posture. The model reveals that the rectifying dynamics can be captured by a bilinear, but not linear, term of body shape variables. An optimal gait problem is formulated for the bilinear rectifier model as a minimization of a quadratic cost function over the set of periodic functions subject to a constraint on the average locomotion velocity. We prove that a globally optimal solution is given by a harmonic gait that can be found by generalized eigenvalue computation with a line search over cycle frequencies. We provide case studies of a chain of links for which snake-like undulations and jellyfish-like flapping gaits are found to be optimal.