This paper describes continuous-space methodologies to estimate the collision probability, Euclidean distance and gradient between an ellipsoidal robot model and an environment surface modeled as a set of Gaussian distributions. Continuous-space collision probability estimation is critical for uncertainty-aware motion planning. Most collision detection and avoidance approaches assume the robot is modeled as a sphere, but ellipsoidal representations provide tighter approximations and enable navigation in cluttered and narrow spaces. State-of-the-art methods derive the Euclidean distance and gradient by processing raw point clouds, which is computationally expensive for large workspaces. Recent advances in Gaussian surface modeling (e.g. mixture models, splatting) enable compressed and high-fidelity surface representations. Few methods exist to estimate continuous-space occupancy from such models. They require Gaussians to model free space and are unable to estimate the collision probability, Euclidean distance and gradient for an ellipsoidal robot. The proposed methods bridge this gap by extending prior work in ellipsoid-to-ellipsoid Euclidean distance and collision probability estimation to Gaussian surface models. A geometric blending approach is also proposed to improve collision probability estimation. The approaches are evaluated with numerical 2D and 3D experiments using real-world point cloud data.

翻译：本文描述了在连续空间中的方法，用于估计椭球机器人模型与环境表面（建模为高斯分布集合）之间的碰撞概率、欧氏距离及其梯度。连续空间碰撞概率估计对于考虑不确定性的运动规划至关重要。大多数碰撞检测与规避方法假设机器人模型为球体，但椭球表示能提供更紧密的近似，从而支持在杂乱且狭窄的空间中导航。现有先进方法通过处理原始点云来推导欧氏距离和梯度，这在大型工作空间中计算成本较高。高斯表面建模（如混合模型、高斯泼溅）的最新进展实现了压缩且高保真的表面表示。目前仅有少数方法能从这些模型中估计连续空间的占据状态，且它们要求高斯模型表征自由空间，无法为椭球机器人估计碰撞概率、欧氏距离及梯度。本文提出的方法通过将椭球间欧氏距离与碰撞概率估计的先前工作扩展到高斯表面模型，填补了这一空白。此外，还提出了一种几何融合方法以改进碰撞概率估计。这些方法通过使用真实世界点云数据的二维和三维数值实验进行了评估。