Locomotion of Friction Coupled Systems
This note provides the general set-up and solution method for a friction-coupled system (the `body') moving in the landscape. This separates the shape (or configuration) space from the state space. The quotient space is the space where the motion of the body can be envisioned as a point mass. In particular we are interested in cases where the inertial forces are small in comparison to applied forces, and propose a simple perturbation expansion.
Next we consider a periodic regime for the motion in shape space and pose and solve the optimal control problem via Fourier techniques. This is illustrated on a simple toy system: the two-piece worm with differential friction. To add realism, some alternative friction models are proposed.