AI 中文总结
该研究提出一种适用于凸多面体的可微符号距离，通过引入 Hölder 最小/最大算子替换经典 SDF 的 min-max 算子，实现闭式计算且可 GPU 并行，经实验验证其在机器人控制领域的适用性。
AI 中文摘要
在机器人运动规划与控制中，集合间距离的计算至关重要，可微梯度能支持实时优化。然而，欧几里得符号距离函数（SDF）并非处处可微，现有替代方案往往会牺牲可微性、符号信息或计算效率。本文提出了一种适用于凸多面体的新型可微符号距离。为此，我们首先提出了可微的最小值和最大值算子，分别称为 Hölder 最小值和 Hölder 最大值。随后，我们用这些算子替换经典 SDF 公式中的原始 min-max 算子，得到 Hölder 符号距离。与依赖迭代算法的现有可微距离公式不同，我们的方法以闭式形式计算，消除了收敛问题，同时天然适用于 GPU 并行化。我们通过与现有方法的运行时比较，验证了该距离的实际优势和计算性能；还开展了机械臂实验，证明其适用于控制相关应用。
英文摘要
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.
