Risk-Sensitive Orbital Debris Collision Avoidance using Distributionally Robust Chance Constraints

TL;DR

This work tackles risk-sensitive collision avoidance for satellites amid rising orbital debris by formulating a distributionally robust chance-constrained MPC. It replaces the intractable distributional chance constraint with a CVaR surrogate under a moment-based ambiguity set defined by mean and covariance, yielding a closed-form DR-CVaR bound $-1 + \frac{1}{\varepsilon}\text{Tr}\{\Sigma_d^k E^k\}$. The reformulated problem is solved with a constrained Cross-Entropy Method, guided by a Trajectory_Risk metric, and validated on a real-world-inspired near-Earth collision scenario using three uncertainty propagation methods (Linear Gaussian, Unscented Transform, and Monte Carlo). Results show the approach conservatively enforces safety while remaining agnostic to the uncertainty propagation method, with the risk parameter $\varepsilon$ modulating trade-offs between safety and fuel use. The methodology offers a scalable, distributionally robust framework for autonomous collision avoidance under non-Gaussian uncertainty in nonlinear orbital dynamics.

Abstract

The exponential increase in orbital debris and active satellites will lead to congested orbits, necessitating more frequent collision avoidance maneuvers by satellites. To minimize fuel consumption while ensuring the safety of satellites, enforcing a chance constraint, which poses an upper bound in collision probability with debris, can serve as an intuitive safety measure. However, accurately evaluating collision probability, which is critical for the effective implementation of chance constraints, remains a non-trivial task. This difficulty arises because uncertainty propagation in nonlinear orbit dynamics typically provides only limited information, such as finite samples or moment estimates about the underlying arbitrary non-Gaussian distributions. Furthermore, even if the full distribution were known, it remains unclear how to effectively compute chance constraints with such non-Gaussian distributions. To address these challenges, we propose a distributionally robust chance-constrained collision avoidance algorithm that provides a sufficient condition for collision probabilities under limited information about the underlying non-Gaussian distribution. Our distributionally robust approach satisfies the chance constraint for all debris position distributions sharing a given mean and covariance, thereby enabling the enforcement of chance constraints with limited distributional information. To achieve computational tractability, the chance constraint is approximated using a Conditional Value-at-Risk (CVaR) constraint, which gives a conservative and tractable approximation of the distributionally robust chance constraint. We validate our algorithm on a real-world inspired satellite-debris conjunction scenario with different uncertainty propagation methods and show that our controller can effectively avoid collisions.

Figures (5)

• Figure 1: Based on the satellite position $\mathbf{r}_s$, the region outside of the collision threshold $d_{thres}$ is defined as the collision-free set $\mathcal{R}_{free}$ for debris.
• Figure 2: Under approximation of the collision-free set $\mathcal{R}_{free}^k$ as an ellipsoid. We assume the ellipsoidal set $\mathcal{R}_{free}^k$ is centered on mean $\mu_d^k$ of debris position $\mathbf{r}_d^k$.
• Figure 3: Debris position covariance and satellite position at Time of Closest Approach. We observe that while unscented transform under-estimate debris position uncertainty compared to Monte Carlo estimation, it can capture covariance shape similarly to Monte Carlo estimation. However, linear Gaussian propagation leads to a different shape of covariance which yields a different avoidance maneuver.
• Figure 4: Minimum distance between the debris and satellite during operation and total $\Delta v$ used for collision avoidance maneuvers with different uncertainty propagation methods and different chance constraint probability $\varepsilon$. We observe that enforcing lower collision probability leads to a conservative behavior, which maintains a larger minimum distance and incurs a larger $\Delta v$ cost.
• Figure 5: Minimum distance between the debris and satellite during operation and total $\Delta v$ used for collision avoidance maneuvers with different debris uncertainty $Q$ for $\mathbf{w}_d \sim \mathcal{N}(0, Q)$. Our observations indicate that the controller tends to maintain a smaller minimum distance and consume less fuel as the uncertainty in debris dynamics decreases, corresponding to smaller values of $Q$.

Theorems & Definitions (6)

• Definition 1: Chance constraints for collision avoidance
• Definition 2: Safety cost
• Definition 3: Value-at-Risk (VaR)
• Definition 4: Conditional Value-at-Risk (CVaR)
• Theorem 1
• proof