A Differentiable Distance Metric for Robotics through Generalized Alternating Projection
Vinicius Mariano Gonçalves, Shiqing Wei, Eduardo Malacarne Soeiro de Souza, Prashanth Krishnamurthy, Anthony Tzes, Farshad Khorrami
AI summary
Problem
Traditional Euclidean distance lacks guaranteed differentiability, which breaks optimization-based controllers like Control Barrier Functions, while existing smooth distance metrics are computationally heavy and fail to vanish when objects overlap.
Approach
The authors derive simpler, integral-free expressions for k-times differentiable projections on convex polytopes and integrate them into a generalized alternating projection algorithm that guarantees the distance metric vanishes upon overlap.
Key results
- Integral-free, practical projection formulas for general convex polytopes
- Guaranteed vanishing distance metric when objects intersect
- Mean computation time of 40 µs, significantly faster than prior smooth methods
- Successful CBF-based collision avoidance demonstrated on a Franka Emika robot
Why it matters
Provides a computationally efficient and mathematically robust distance function essential for real-time, optimization-based robot motion planning and control.
Abstract
In many robotics applications, it is necessary to com- pute 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, meaning it is not guaranteed to be differentiable everywhere, 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.