# Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

arXiv: https://arxiv.org/abs/2101.12490 · HTML: https://mlsanity.com/paper/2101.12490/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2021.

## Built on by (most cited first, 6 of 6)

- [Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures](https://mlsanity.com/paper/2109.09975/) (2021)
- [Universal Learning of Stochastic Dynamics for Exact Belief Propagation using Bernstein Normalizing Flows](https://mlsanity.com/paper/2509.15533/) (2025)
- [Non-Gaussian Uncertainty Minimization Based Control of Stochastic Nonlinear Robotic Systems](https://mlsanity.com/paper/2303.01628/) (2023)
- [Real-Time Tube-Based Non-Gaussian Risk Bounded Motion Planning for Stochastic Nonlinear Systems in Uncertain Environments via Motion Primitives](https://mlsanity.com/paper/2303.01631/) (2023)
- [Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation](https://mlsanity.com/paper/2604.07525/) (2026)
- [The Score Kalman Filter](https://mlsanity.com/paper/2605.16644/) (2026)

## Abstract (arXiv)

In this paper, we address the problem of uncertainty propagation through nonlinear stochastic dynamical systems. More precisely, given a discrete-time continuous-state probabilistic nonlinear dynamical system, we aim at finding the sequence of the moments of the probability distributions of the system states up to any desired order over the given planning horizon. Moments of uncertain states can be used in estimation, planning, control, and safety analysis of stochastic dynamical systems. Existing approaches to address moment propagation problems provide approximate descriptions of the moments and are mainly limited to particular set of uncertainties, e.g., Gaussian disturbances. In this paper, to describe the moments of uncertain states, we introduce trigonometric and also mixed-trigonometric-polynomial moments. Such moments allow us to obtain closed deterministic dynamical systems that describe the exact time evolution of the moments of uncertain states of an important class of autonomous and robotic systems including underwater, ground, and aerial vehicles, robotic arms and walking robots. Such obtained deterministic dynamical systems can be used, in a receding horizon fashion, to propagate the uncertainties over the planning horizon in real-time. To illustrate the performance of the proposed method, we benchmark our method against existing approaches including linear, unscented transformation, and sampling based uncertainty propagation methods that are widely used in estimation, prediction, planning, and control problems.