# Optimal control of a single leg hopper by Liouvillian system reduction

arXiv: https://arxiv.org/abs/1710.02133 · HTML: https://mlsanity.com/paper/1710.02133/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Oct 2017.

## Built on by (most cited first, 1 of 1)

- [How Robot Dogs See the Unseeable: Improving Visual Interpretability via Peering for Exploratory Robots](https://mlsanity.com/paper/2511.16262/) (2025)

## Abstract (arXiv)

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.