# Control Contraction Metrics on Finsler Manifolds

## TL;DR

This paper extends Control Contraction Metrics to Finsler geometry, enabling nonlinear control design with simplified computation by avoiding real-time shortest path calculations.

## Contribution

It introduces a generalization of CCMs to Finsler manifolds and provides practical open loop and sampled data controllers that reduce computational complexity.

## Key findings

- Finsler-based CCMs enable more flexible metric choices.
- Sampled data controllers do not require real-time shortest path computation.
- The approach simplifies nonlinear control design on complex manifolds.

## Abstract

Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of non-Riemannian metrics. We provide open loop and sampled data controllers. The sampled data control construction presented here does not require real time computation of globally shortest paths, simplifying computation.