Collision detection is a fundamental problem in various domains, such as robotics, computational physics, and computer graphics. In general, collision detection is tackled as a computational geometry problem, with the so-called Gilbert, Johnson, and Keerthi (GJK) algorithm being the most adopted solution nowadays. While introduced in 1988, GJK remains the most effective solution to compute the distance or the collision between two 3-D convex geometries. Over the years, it was shown to be efficient, scalable, and generic, operating on a broad class of convex shapes, ranging from simple primitives (sphere, ellipsoid, box, cone, capsule, etc.) to complex meshes involving thousands of vertices. In this article, we introduce several contributions to accelerate collision detection and distance computation between convex geometries by leveraging the fact that these two problems are fundamentally optimization problems. Notably, we establish that the GJK algorithm is a specific subcase of the well-established Frank–Wolfe (FW) algorithm in convex optimization. By adapting recent works linking Polyak and Nesterov accelerations to FW methods, we also propose two accelerated extensions of the classic GJK algorithm. Through an extensive benchmark over millions of collision pairs involving objects of daily life, we show that these two accelerated GJK extensions significantly reduce the overall computational burden of collision detection, leading to computation times that are up to two times faster. Finally, we hope this work will significantly reduce the computational cost of modern robotic simulators, allowing the speedup of modern robotic applications that heavily rely on simulation, such as reinforcement learning or trajectory optimization.

中文翻译：

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GJK++：利用加速方法实现更快的碰撞检测

碰撞检测是机器人学、计算物理和计算机图形学等各个领域的一个基本问题。一般来说，碰撞检测作为计算几何问题来解决，所谓的吉尔伯特、约翰逊和克尔蒂 (GJK) 算法是当今最常用的解决方案。尽管 GJK 于 1988 年推出，但它仍然是计算两个 3-D 凸几何图形之间的距离或碰撞的最有效解决方案。多年来，它被证明是高效、可扩展和通用的，可在广泛的凸形状上运行，范围从简单的基元（球体、椭球体、盒子、圆锥体、胶囊等）到涉及数千个顶点的复杂网格。在本文中，我们介绍了利用这两个问题本质上是优化问题这一事实来加速凸几何形状之间的碰撞检测和距离计算的一些贡献。值得注意的是，我们确定 GJK 算法是凸优化中成熟的 Frank-Wolfe (FW) 算法的特定子情况。通过改编最近将 Polyak 和 Nesterov 加速与 FW 方法联系起来的工作，我们还提出了经典 GJK 算法的两种加速扩展。通过对涉及日常生活对象的数百万个碰撞对进行广泛的基准测试，我们表明这两个加速的 GJK 扩展显着减少了碰撞检测的整体计算负担，从而使计算时间加快了两倍。最后，我们希望这项工作将显着降低现代机器人模拟器的计算成本，从而加速严重依赖于模拟的现代机器人应用程序，例如强化学习或轨迹优化。