h-NESDF: An Infinitely Differentiable Signed Distance Between Two Sets with Robotic Applications

Computing distances between sets is important in many robotic applications, such as motion planning and control. In many applications, derivatives of the distance are needed as well. It is well known that the Euclidean distance is not always differentiable, which create issues to algorithms that need a derivative. In this paper, we introduce NESDF, the Normal Euclidean Signed Distance Function between two general (with mild regularity assumptions but possibly non-convex) sets. This metric is still not differentiable, but it allows us to introduce a generalization, $h$-NESDF, which has a smoothing parameter $h$ such that the metric is infinitely differentiable for all $h>0$ and that reduces to the NESDF when $h$ goes to $0$. The metric is amenable to GPU parallelization, and thus can be computed efficiently. We show extensive simulation results to evaluate its viability in motion planning.