# Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments

arXiv: https://arxiv.org/abs/1810.01577 · HTML: https://mlsanity.com/paper/1810.01577/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Oct 2018.

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

- [Calibrating Perception Uncertainty for Autonomous Driving](https://mlsanity.com/paper/2211.04340/) (2022)
- [Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures](https://mlsanity.com/paper/2109.09975/) (2021)
- [Convex Risk Bounded Continuous-Time Trajectory Planning and Tube Design in Uncertain Nonconvex Environments](https://mlsanity.com/paper/2305.17291/) (2023)

## Abstract (arXiv)

In this paper, we address the risk estimation problem where one aims at estimating the probability of violation of safety constraints for a robot in the presence of bounded uncertainties with arbitrary probability distributions. In this problem, an unsafe set is described by level sets of polynomials that is, in general, a non-convex set. Uncertainty arises due to the probabilistic parameters of the unsafe set and probabilistic states of the robot. To solve this problem, we use a moment-based representation of probability distributions. We describe upper and lower bounds of the risk in terms of a linear weighted sum of the moments. Weights are coefficients of a univariate Chebyshev polynomial obtained by solving a sum-of-squares optimization problem in the offline step. Hence, given a finite number of moments of probability distributions, risk can be estimated in real-time. We demonstrate the performance of the provided approach by solving probabilistic collision checking problems where we aim to find the probability of collision of a robot with a non-convex obstacle in the presence of probabilistic uncertainties in the location of the robot and size, location, and geometry of the obstacle.