# An optimal control approach to the design of periodic orbits for mechanical systems with impacts

## TL;DR

This paper presents an optimal control method for designing stable periodic orbits in hybrid mechanical systems with impacts, combining trajectory optimization and root finding for improved stability and performance.

## Contribution

It introduces a novel optimal control framework that efficiently designs periodic orbits for underactuated hybrid systems with impacts, ensuring numerical stability and boundary constraint satisfaction.

## Key findings

- Successfully designed periodic orbits for a compass biped model.
- Demonstrated numerical stability and effectiveness of the proposed method.
- Achieved optimized trajectories satisfying boundary constraints.

## Abstract

In this paper we study the problem of designing periodic orbits for a special class of hybrid systems, namely mechanical systems with underactuated continuous dynamics and impulse events. We approach the problem by means of optimal control. Specifically, we design an optimal control based strategy that combines trajectory optimization, dynamics embedding, optimal control relaxation and root finding techniques. The proposed strategy allows us to design, in a numerically stable manner, trajectories that optimize a desired cost and satisfy boundary state constraints consistent with a periodic orbit. To show the effectiveness of the proposed strategy, we perform numerical computations on a compass biped model with torso.