A Ray Intersection Algorithm for Fast Growth Distance Computation Between Convex Sets

Computing distances between sets is essential in robotic motion planning and control, where differentiable gradients enable real-time optimization. The Euclidean Signed Distance Function (SDF), however, is not differentiable everywhere, and existing alternatives often sacrifice differentiability, sign information, or computational efficiency. In this letter, we introduce a novel differentiable signed distance between convex polyhedra. To this end, we first propose differentiable versions of the minimum and maximum operators, termed the Hölder minimum and Hölder maximum. We then replace the original min-max operators in the classical SDF formulation, yielding the Hölder signed distance. Unlike prior differentiable distance formulations that rely on iterative algorithms, our approach is computed in closed form, eliminating convergence issues while remaining naturally amenable to GPU parallelization. We validate the practical advantages and computational performance of the proposed distance through runtime comparisons with existing approaches. We also present a robotic manipulator experiment, demonstrating its suitability for applications in control.

We study unlabeled MRMP for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Banyassady et al.(SoCG'22) guarantee feasibility in simple polygons under start--start and target--target distances of at least $4$, and start--target distances of at least $3$, but without optimality guarantees. Solovey et al.(RSS'15) provide a near-optimal solution in general polygonal domains, under stricter conditions: start/target positions must have pairwise distance at least $4$, and at least $\sqrt{5}\approx2.236$ from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different tradeoffs on the robots-separation $ρ$ and obstacles-separation $ω$, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) $ρ=2\frac{2}{3}$ and $ω=1\frac{2}{3}$, or (ii) $ρ\approx3.291$ and $ω\approx1.354$. These solutions are weakly-monotone; we also provide a monotone solution requiring $ω=\approx1.614$ and $ρ=4$. We prove that monotone plans may not exist when $ω<1.614$, and weakly-monotone plans may not exist when $ω<1.354$. We then present tradeoffs between the separation bounds and the approximation factor, specifically achieving an (almost) optimal bound of $ρ=2$ at the cost of a linear approximation factor and requiring $ω=2$. This applies also for the labeled variant of MRMP, in which case we show a tight bound on $ω$. Finally, we show that without any robots-separation assumption, obstacles-separation of at least $1.5$ may be necessary for a solution to exist.

Planners on graphs of convex sets return trajectories that are collision-free by construction, provided the convex regions are collision-free. The region generator only promises that property probabilistically, and no planner in the family verifies it. We report the first measurement of what the gap costs. On a scaled 14-DOF bimanual library, $3.2\%$ of interface samples are in collision, and a search-based GCS planner (\gcsstar) turns that volume error into a $62\%$ answer error: $18$ of $29$ pick-and-place queries return trajectories that drive the arms through the shelves, up to $91$,mm deep, reported as successes. Repairing the library does not work; a ten times stricter acceptance contract, sums-of-squares certified regions, and uniform margins each destroy the connectivity planning needs before they deliver soundness. We instead build a planner that certifies its answers. It samples the overlaps and shared faces of the decomposition, prunes with an admissible informed bound, and verifies the one candidate each search round proposes, continuously, by a chain of clearance certificate balls with no resolution parameter; failures are repaired with local in-region detours, and the convex polish is re-verified. Head-to-head on all $29$ task queries it delivers zero invalid answers against $21$ for the reference, reaches its first certified answer in $0.11$,s against $1.59$,s for the reference's unverified one, and reproduces the reference optimum exactly on every query whose reference answer is physically valid.