A further result on the optimal harmonic gait for locomotion of mechanical rectifier systems

This paper first formally defines a general class of three dimensional rectifier systems which capture the essential aspects of animal locomotion, then formulates an optimal gait problem, and finally solves an approximation of the problem to obtain a globally optimal solution. The approximation assumes small-amplitude harmonic oscillations of mechanical joints about a nominal posture. The problem is formulated as a minimization of a quadratic cost function subject to an average velocity constraint, which is solved with an additional amplitude constraint in a Pareto-optimal fashion to ensure that the solution to the approximate problem is valid for the original. The solution method is fast and numerically stable, using generalized eigenvalues and eigenvectors of a pair of Hermitian matrices, and is able to easily handle underactuated or hyper-redundant systems. We provide case studies of a chain of links representing a radially symmetric jellyfish-like animal, or two limbs pushing a central body forward. It is demonstrated that our method enables determination of various gaits through optimization of such cost functions as input power, rate of shape change, and torque derivative.