Probability of collision in nonlinear dynamics by moment propagation

TL;DR

The paper tackles the problem of estimating collision probabilities between spacecraft under nonlinear dynamics and non-Gaussian initial uncertainties. It introduces a semi-analytical framework that propagates initial-state moments to the closest-approach event manifold using a high-order Taylor map and then reconstructs a univariate PDF from these moments via orthogonal polynomials; the collision probability is obtained by integrating the PDF of the squared distance up to $R^2$, with closed-form results for common reference distributions. The approach is general, avoiding Gaussian or linearization assumptions, and is evaluated against Akella and Taylor Monte Carlo methods on real-world January 2022 conjunctions, showing robust accuracy and efficiency. The method has broad applicability to uncertainty quantification in deterministic dynamical systems and offers potential extensions to multivariate PDFs and higher-dimensional event manifolds.

Abstract

Estimating the probability of collision between spacecraft is crucial for risk management and collision-avoidance strategies. Current methods often rely on Gaussian assumptions and simplifications, which can be inaccurate in highly nonlinear scenarios. This paper presents a general and efficient approach for computing collision probabilities without relying on such assumptions. Using high-order multivariate Taylor polynomials, we propagate statistical moments of initial uncertainties to the point of closest approach between the spacecraft. To compute the probability of collision, we derive a semi-analytical expression for the probability density function (PDF) of the closest approach distance, inferred from the propagated moments using orthogonal polynomials. Tested on various short-term and long-term encounters in low-Earth orbit, our method accurately handles nonlinear dynamics, non-Gaussian uncertainties, and irregular distributions. This versatile framework advances space situational awareness by providing precise collision probability estimates in complex dynamical environments. Moreover, our methodology applies to any dynamical system with uncertainty in its initial state and is therefore not restricted to collision probability estimation.

Figures (6)

• Figure 1: Propagation of the state to an event manifold (dashed line). The state $\vec{x}_f$ is reached at time $t_f$, but this final state may not lie on the event manifold anymore. The adapted Taylor map accounts for this by extending the propagation to a slightly later time, ensuring the final state lies on the manifold.
• Figure 2: Example of PDF estimation, using the first 12 moments of a mixture of Gaussian distributions (solid line). The reference distribution (dotted line) is a generalized beta distribution on $[-1,1]$ with $\alpha=12.7$ and $\beta=6.4$. The estimated PDF is shown with the dashed line.
• Figure 3: Flow diagram of our proposed approach for estimating the probability of collision. Inputs are indicated with bold borders, while the output is shown with a dashed border.
• Figure 4: Comparison of true and predicted probability of collision for all three methods in the benchmark. We consider only encounters with a nonzero true probability of collision, as determined with Monte Carlo sampling. Shaded areas indicate predictions within the correct order of magnitude.
• Figure 5: Mean squared error of each method as a function of integration time and the type of uncertainty distribution in the initial conditions. The error of the Taylor Monte Carlo approach increases with longer integration times, primarily due to cases where the true probability of collision is zero, but TMC predicts a nonzero probability.

Theorems & Definitions (3)

• Theorem 1
• proof
• proof