A Differentiable Distance Metric for Robotics Through Generalized Alternating Projection

2025-07-01

AI Summary

In real-time robotic safety control—particularly for control barrier functions—accurate, differentiable computation of distances and their derivatives between objects and environments is essential. However, the Euclidean distance is non-differentiable at contact points, while existing differentiable distance methods suffer from complex analytical forms, poor adaptability to convex polyhedral geometries, and failure to converge to zero under object overlap. Method: This paper proposes a novel differentiable distance metric grounded in a generalized alternating projection framework. We derive a compact, closed-form smooth projection operator for general convex polyhedra, ensuring global differentiability and exact distance convergence to zero upon overlap. Contribution/Results: The method is implemented in the Python simulation platform UAIBot. Experiments demonstrate high computational efficiency, numerically stable gradients, and significantly improved stability and practicality of real-time safety-critical control.

Technology Category

• Intelligent Robots: State Estimation
• Natural Language Processing: Safety and Robustness
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Develop a differentiable distance metric for robotics applications
• Simplify expressions for smooth projection on convex polytopes
• Ensure distance vanishes when objects overlap

Innovation

• Simpler expressions for smooth projection
• Distance vanishes during object overlap
• Public Python package UAIBot implementation