A Differentiable Distance Metric for Robotics Through Generalized Alternating Projection

TL;DR

This paper introduces a new differentiable distance metric for robotics that improves upon previous methods by providing simpler expressions, ensuring the metric vanishes upon object overlap, and demonstrating practical effectiveness in experiments.

Contribution

The paper presents a simplified, practical differentiable distance metric for convex polytopes that addresses limitations of prior approaches and is validated through experiments.

Findings

Effective in robotics control scenarios

Simpler expressions for the projection function

Ensures distance vanishes when objects overlap

Abstract

In many robotics applications, it is necessary to compute not only the distance between the robot and the environment, but also its derivative - for example, when using control barrier functions. However, since the traditional Euclidean distance is not differentiable, there is a need for alternative distance metrics that possess this property. Recently, a metric with guaranteed differentiability was proposed [1]. This approach has some important drawbacks, which we address in this paper. We provide much simpler and practical expressions for the smooth projection for general convex polytopes. Additionally, as opposed to [1], we ensure that the distance vanishes as the objects overlap. We show the efficacy of the approach in experimental results. Our proposed distance metric is publicly available through the Python-based simulation package UAIBot.