# Residual Force Polytope: Admissible Task-Space Forces of Dynamic Trajectories

## TL;DR

This paper introduces the residual force polytope, a new representation of the forces a robot can counteract during dynamic motion, enabling better analysis and optimization of trajectories for robustness.

## Contribution

The paper presents the residual force polytope as a novel tool for analyzing and optimizing robot trajectories based on admissible forces, improving robustness to disturbances.

## Key findings

- Trajectories optimized with residual force polytope metrics are more disturbance-resistant.
- The residual force polytope allows for meaningful comparison of different trajectories.
- Computational cost is high, suitable for offline planning but not real-time applications.

## Abstract

We propose a representation for the set of forces a robot can counteract using full system dynamics: the residual force polytope. Given the nominal torques required by a dynamic motion, this representation models the forces which can be sustained without interfering with that motion. The residual force polytope can be used to analyze and compare the set of admissible forces of different trajectories, but it can also be used to define metrics for solving optimization problems, such as in trajectory optimization or system design. We demonstrate how such a metric can be applied to trajectory optimization and compare it against other objective functions typically used. Our results show that the trajectories computed by optimizing objectives defined as functions of the residual force polytope are more robust to unknown external disturbances. The computational cost of these metrics is relatively high and not compatible with the short planning times required by online methods, but they are acceptable for planning motions offline.