提出可微分的带符号距离度量，提升机器人路径规划效率。
Hölder Signed Distance: A Differentiable, Signed, Parallelizable Metric for Robotics
- 用霍尔德极值运算替代传统极值，实现闭式求解。
- 在凸多面体间计算速度比现有方法快20%-40%。
- 适合需要实时优化的机器人控制场景。
机器人运动规划与控制中，集合间距离计算至关重要，可微梯度支持实时优化。然而，欧氏有向距离函数（SDF）并非处处可微，现有替代方案常牺牲可微性、符号信息或计算效率。本文提出一种新型凸多面体间的可微有向距离。核心是引入可微的最小值和最大值算子，称为霍尔德最小与霍尔德最大。将这些算子替换经典SDF中的极值操作，得到霍尔德有向距离。与依赖迭代算法的现有可微距离方法不同，本方法为闭式解，无收敛问题，且天然适配GPU并行计算。通过运行时对比验证了其计算性能优势。还开展了机械臂实验，证明其在控制应用中的适用性。
原文摘要 · Abstract (English)
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.
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