Optimal periodic locomotion for a two piece worm with an asymmetric dry friction model

This paper solves the optimization problem for a simplified one-dimensional worm model when the friction force depends on the direction of the motion. The motion of the worm is controlled by the actuator force $f(t)$ which is assumed to be piecewise continuous and always generates the same force in the opposite directions. The paper derives the necessary condition for the force which maximizes the average velocity or minimizes the power over a unit distance. The maximum excursion of the worm body and the force are bounded. A simulation is given at the end of the paper.