Optimal control of a single leg hopper by Liouvillian system reduction

TL;DR

This paper explores optimal control of a single leg hopper using Liouvillian system reduction, demonstrating how differential flatness and smooth trajectory planning improve hopping motion feasibility.

Contribution

It introduces a novel application of Liouvillian system reduction and differential flatness to optimize control trajectories for a leg hopper.

Findings

Liouvillian property enables flat outputs for control.

Trajectory planning minimizes jerk for smoothness.

Feasible hopping trajectories reach specified waypoints.

Abstract

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.