# Exact and Bounded Collision Probability for Motion Planning under Gaussian Uncertainty

arXiv: https://arxiv.org/abs/2110.06348 · HTML: https://mlsanity.com/paper/2110.06348/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 42 references that are in the map.
- First posted on arXiv 12 Oct 2021.

## Builds on (most cited first, 9 of 9)

- [PAMPC: Perception-Aware Model Predictive Control for Quadrotors](https://mlsanity.com/paper/1804.04811/) (2018)
- [Monte Carlo Motion Planning for Robot Trajectory Optimization Under Uncertainty](https://mlsanity.com/paper/1504.08053/) (2015)
- [Provably Safe Robot Navigation with Obstacle Uncertainty](https://mlsanity.com/paper/1705.10907/) (2017)
- [Evaluating Trajectory Collision Probability through Adaptive Importance Sampling for Safe Motion Planning](https://mlsanity.com/paper/1609.05399/) (2016)
- [A Scalable Framework For Real-Time Multi-Robot, Multi-Human Collision Avoidance](https://mlsanity.com/paper/1811.05929/) (2018)
- [Towards Multi-Robot Task-Motion Planning for Navigation in Belief Space](https://mlsanity.com/paper/2010.00780/) (2020)
- [Efficient motion planning for problems lacking optimal substructure](https://mlsanity.com/paper/1703.02582/) (2017)
- [Collision Probabilities for Continuous-Time Systems Without Sampling [with Appendices]](https://mlsanity.com/paper/2006.01109/) (2020)
- [An Integrated Localisation, Motion Planning and Obstacle Avoidance Algorithm in Belief Space](https://mlsanity.com/paper/2101.11566/) (2021)

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

- [Distance and Collision Probability Estimation from Gaussian Surface Models](https://mlsanity.com/paper/2402.00186/) (2024)
- [Safe and Efficient Path Planning under Uncertainty via Deep Collision Probability Fields](https://mlsanity.com/paper/2409.04306/) (2024)
- [Robust Planning for Autonomous Driving via Mixed Adversarial Diffusion Predictions](https://mlsanity.com/paper/2505.12327/) (2025)

## Abstract (arXiv)

Computing collision-free trajectories is of prime importance for safe navigation. We present an approach for computing the collision probability under Gaussian distributed motion and sensing uncertainty with the robot and static obstacle shapes approximated as ellipsoids. The collision condition is formulated as the distance between ellipsoids and unlike previous approaches we provide a method for computing the exact collision probability. Furthermore, we provide a tight upper bound that can be computed much faster during online planning. Comparison to other state-of-the-art methods is also provided. The proposed method is evaluated in simulation under varying configuration and number of obstacles.