# Distance and Collision Probability Estimation from Gaussian Surface Models

arXiv: https://arxiv.org/abs/2402.00186 · HTML: https://mlsanity.com/paper/2402.00186/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 62 references that are in the map.
- First posted on arXiv 31 Jan 2024.

## Builds on (most cited first, 24 of 24)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Voxblox: Incremental 3D Euclidean Signed Distance Fields for On-Board MAV Planning](https://mlsanity.com/paper/1611.03631/) (2016)
- [iSDF: Real-Time Neural Signed Distance Fields for Robot Perception](https://mlsanity.com/paper/2204.02296/) (2022)
- [Approximate Differentiable Rendering with Algebraic Surfaces](https://mlsanity.com/paper/2207.10606/) (2022)
- [FIESTA: Fast Incremental Euclidean Distance Fields for Online Motion Planning of Aerial Robots](https://mlsanity.com/paper/1903.02144/) (2019)
- [Search-based Motion Planning for Aggressive Flight in SE(3)](https://mlsanity.com/paper/1710.02748/) (2017)
- [FASTER: Fast and Safe Trajectory Planner for Navigation in Unknown Environments](https://mlsanity.com/paper/2001.04420/) (2020)
- [Motion Primitives-based Navigation Planning using Deep Collision Prediction](https://mlsanity.com/paper/2201.03254/) (2022)
- [Confidence-rich grid mapping](https://mlsanity.com/paper/2006.15754/) (2020)
- [Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces](https://mlsanity.com/paper/2010.11487/) (2020)
- [Efficient Path Planning in Narrow Passages for Robots with Ellipsoidal Components](https://mlsanity.com/paper/2104.04658/) (2021)
- [Probabilistic Point Cloud Modeling via Self-Organizing Gaussian Mixture Models](https://mlsanity.com/paper/2302.00047/) (2023)
- [GMMap: Memory-Efficient Continuous Occupancy Map Using Gaussian Mixture Model](https://mlsanity.com/paper/2306.03740/) (2023)
- [Differentiable Collision Detection for a Set of Convex Primitives](https://mlsanity.com/paper/2207.00669/) (2022)
- [Incremental Multimodal Surface Mapping via Self-Organizing Gaussian Mixture Models](https://mlsanity.com/paper/2309.10900/) (2023)
- [On the Parameterized Computation of Minimum Volume Outer Ellipsoid of Minkowski Sum of Ellipsoids](https://mlsanity.com/paper/1803.08157/) (2018)
- [Log-GPIS-MOP: A Unified Representation for Mapping, Odometry and Planning](https://mlsanity.com/paper/2206.09506/) (2022)
- [Exact and Bounded Collision Probability for Motion Planning under Gaussian Uncertainty](https://mlsanity.com/paper/2110.06348/) (2021)
- [Accurate Gaussian-Process-based Distance Fields with applications to Echolocation and Mapping](https://mlsanity.com/paper/2302.13005/) (2023)
- [NanoMap: Fast, Uncertainty-Aware Proximity Queries with Lazy Search over Local 3D Data](https://mlsanity.com/paper/1802.09076/) (2018)
- [Functional Path Optimisation for Exploration in Continuous Occupancy Maps](https://mlsanity.com/paper/1805.01079/) (2018)
- [Lambda-Field: A Continuous Counterpart of the Bayesian Occupancy Grid for Risk Assessment](https://mlsanity.com/paper/1903.02285/) (2019)
- [Tight Collision Probability for UAV Motion Planning in Uncertain Environment](https://mlsanity.com/paper/2303.02607/) (2023)

## Built on by (most cited first, 3 of 3)

- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Photon Splatting: A Physics-Guided Neural Surrogate for Real-Time Wireless Channel Prediction](https://mlsanity.com/paper/2507.04595/) (2025)
- [SplatCtrl: Perception-Action Coupling via Gaussian Scene Representations and Reactive Robot Control](https://mlsanity.com/paper/2607.08948/) (2026)

## Abstract (arXiv)

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