# Symmetry and Motion Primitives in Model Predictive Control

## TL;DR

This paper explores how symmetries in mechanical systems can be leveraged within Model Predictive Control to improve stability and control strategies, demonstrated through a mobile robot example and motion primitives.

## Contribution

It establishes the foundation for integrating symmetries into MPC, proving stability in a mobile robot scenario and providing guidelines for stability when symmetry is not aligned with the cost function.

## Key findings

- Asymptotic stability of a set point in MPC with symmetries is proven.
- Numerical validation using motion primitives in a parallel parking scenario.
- Guidelines for stability guarantees when the optimization criterion conflicts with symmetry actions.

## Abstract

Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is well-established in both motion planning and control. However, the linkage between the respective techniques to optimal control is not yet fully explored. In this manuscript, we want to lay the foundation for the usage of symmetries in Model Predictive Control (MPC). To this end, we investigate a mobile robot example in detail where our contribution is twofold: Firstly, we establish asymptotic stability of a desired set point w.r.t. the MPC closed loop, which is also demonstrated numerically by using motion primitives applied to the parallel parking scenario. Secondly, if the optimization criterion is not consistent with the symmetry action, we provide guidelines to rigorously derive stability guarantees based on symmetry exploitation.