Transferring solutions found by trajectory optimization to robotic hardware
remains a challenging task. When the optimization fully exploits the provided
model to perform dynamic tasks, the presence of unmodeled dynamics renders the
motion infeasible on the real system. Model errors can be a result of model
simplifications, but also naturally arise when deploying the robot in
unstructured and nondeterministic environments. Predominantly, compliant
contacts and actuator dynamics lead to bandwidth limitations. While classical
control methods provide tools to synthesize controllers that are robust to a
class of model errors, such a notion is missing in modern trajectory
optimization, which is solved in the time domain. We propose frequency-shaped
cost functions to achieve robust solutions in the context of optimal control
for legged robots. Through simulation and hardware experiments we show that
motion plans can be made compatible with bandwidth limits set by actuators and
contact dynamics. The smoothness of the model predictive solutions can be
continuously tuned without compromising the feasibility of the problem.
Experiments with the quadrupedal robot ANYmal, which is driven by
highly-compliant series elastic actuators, showed significantly improved
tracking performance of the planned motion, torque, and force trajectories and
enabled the machine to walk robustly on terrain with unmodeled compliance
