## Moment-Sum-of-Squares Approach for Fast Risk Estimation in Uncertain Environments

##### Author(s)

Jasour, Ashkan M.; Hofmann, Andreas; Williams, Brian C
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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 nonconvex 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. Keywords: Probability distribution; Uncertainty; Chebyshev approximation; Optimization; Probabilistic logic; Estimation; Robots; collision avoidance; least squares approximations; mobile robots; optimisation; polynomials; set theory; statistical distributions

##### Date issued

2019-01-21##### Department

Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory##### Publisher

Institute of Electrical and Electronics Engineers

##### Citation

Jasour, Ashkan M. et al. "Moment-Sum-of-Squares Approach for Fast Risk Estimation in Uncertain Environments," 2018 IEEE Conference on Decision and Control (CDC), Miami Beach, FL, December 2018, Institute of Electrical and Electronics Engineers, 2019.

Version: Original manuscript

##### ISBN

9781538613955

9781538613948

9781538613962

##### ISSN

2576-2370

0743-1546