Optimal Control of a Two-body: Vibration-driven Locomotion System in a Resistive Environment*

A two-body system moving along a horizontal line in a resistive medium with a nonlinear resistance law is considered. One of the bodies (main body) interacts with the environment and with the other body (internal body), which interacts with the main body but does not interact with the environment. A periodic motion of the internal body relative to the main body sustains the progressive motion of the main body with periodically changing velocity. By solving an optimal control problem, the optimal motion of the internal body that maximizes the average velocity of the system is found.