Comparison of cost functions for electrically driven running robots

In this work we apply optimal control to create running gaits for the model of an electrically driven one legged hopper, and compare the results obtained for five different objective functions. By using high compliant series elastic actuators, the motions of joint and motor are decoupled, which allows the exploitation of natural dynamics. Depending on the cost function, this exploitation varies. Energy is injected at different points of time, the amplitude of actuator action changes significantly, and the optimal gear ratios differ by a factor of two. Variations are, however, comparable over a wide range of hopping heights and running velocities. Purely force-based cost functions prove to be ill-suited for such non-conservative systems, and it is shown that thermal electrical losses, in contrast to common belief, do not dominate energy expenditure. The numerical results are corroborated by detailed analytical considerations which give general insights into optimal excitation with electric actuators.