Optimal control of a single leg hopper by Liouvillian system reduction
For legged robotics, this work provides a method to generate smooth trajectories for a single leg hopper, but it is incremental as it applies existing optimal control techniques to a specific system.
This paper applies optimal control to a single leg hopper by proving differential flatness and Liouvillian properties, then using a two-point BVP to minimize jerk. The method generates smooth, feasible trajectories that reach given waypoints.
The benefits of legged locomotion shown in nature overcome challenges such as obstacles or terrain smoothness typically encountered with wheeled vehicles. This paper evaluates the benefits of using optimal control on a single leg hopper during the entire hopping motion. Basic control without considering physical constraints is implemented through hand-tuned PD controllers following the Raibert control framework. The differential flatness of the first-order equations of motion and the Liouvillian property for the second-order equations for the hopper system are proved, enabling flat outputs for control. A two-point boundary value problem (BVP) is then used to minimize jerk in the flat system to gain implicit smoothness in the output controls. This smoothness ensures that the planned trajectories are feasible, allowing for given waypoints to be reached.