Distance and Collision Probability Estimation from Gaussian Surface Models

2024-01-31

arXiv.org

AI Summary

Existing robot collision-avoidance methods predominantly rely on spherical robot models and raw point clouds, struggling to simultaneously achieve accuracy, computational efficiency, and principled uncertainty modeling. This paper addresses the interaction between ellipsoidal robots and environments represented by Gaussian surface models, proposing the first continuous-space algorithm that jointly estimates collision probability, Euclidean distance, and analytically differentiable gradients. Key contributions include: (1) the first extension of ellipsoid–ellipsoid distance and collision-probability estimation to Gaussian mixture and splat-based surface representations; (2) a geometric fusion strategy that improves probabilistic estimation accuracy; and (3) support for hybrid Gaussian modeling accommodating both free space and obstacles. By bypassing dense point-cloud computations, the algorithm achieves microsecond-scale per-ellipsoid-pair inference—enabling deployment on embedded, low-power platforms. Extensive 2D/3D experiments on real point-cloud data demonstrate significant performance gains over direct point-cloud processing methods.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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. Methods for efficient calculation of these quantities are demonstrated to execute within a few microseconds per ellipsoid pair using a single-thread on low-power CPUs of modern embedded computers

Problem

• Estimates collision probability for ellipsoidal robots in Gaussian-modeled environments
• Computes Euclidean distance and gradient from Gaussian surface models efficiently
• Enables navigation in cluttered spaces using continuous-space occupancy estimation

Innovation

• Ellipsoid-to-Gaussian distance and gradient estimation
• Geometric blending for collision probability
• Efficient computation on embedded CPUs