Thomas Dearing, ARKA Group; Isaiah Owsley, ARKA Group; Piyush Mehta, West Virginia University
Keywords: Astrodynamics, Unscented Transform, Constraints, Unscented Kalman Filtering
Abstract:
As the density and congestion of popular orbital regimes has continued to grow over the last decade, the detection, tracking, and prediction of small space debris has evolved into a critical issue for operational safety as well as a broad and compelling research topic. In particular, the combination of the low mass and surface area of small debris promotes a high sensitivity to perturbing forces and a low signal-to-noise ratio for detections, resulting in rapidly growing uncertainties between sparse detections. To accommodate these challenges, a suitable associator and tracker require an efficient and reliable propagator which can accurately and robustly estimate long-term covariance growth of many thousands of objects. While numerous strategies exist, a prevalent approach is to combine a high-fidelity propagator (e.g. Orekit, STK, etc.) with a representative set of sample points for modelling the estimated state uncertainty, such as particle filtering, Gaussian Mixture Model (GMM) point clouds, or the sigma points for the Unscented Kalman Filter (UKF). Combining these approaches with a Multi-Hypothesis Filter (MHF) or other soft (probabilistic multi-hypothesis) association scheme creates a robust tracking solution.
The use of covariance sampling or unscented sigma points in propagating the orbital state uncertainty presents many advantages over traditional linear approximations made by Kalman and Extended Kalman Filtering (EKF). While propagating a complete sample family is more computationally intensive than only propagating the orbit mean and state transition matrix, the improved accuracy from measuring high-order nonlinear effects in the dynamical model and higher-order moments in the state distribution are compelling to address small space debris tracking challenges. Unfortunately, the underlying requirement to successfully propagate all sample/sigma points presents a new challenge when considering modeling constraints enforced by the propagator. For example, the popular Holmes-Featherstone model for gravitational acceleration is defined on a basis of spherical harmonics, making the model inaccurate or even divergent below the altitude threshold defined by the Earth’s Brillouin sphere. As such, many high-order propagators will reject any track (or sigma point) entering this volume: an issue that becomes particularly relevant for highly uncertain or poorly observed tracks in LEO. Existing solutions to this problem assume that the entire distribution must remain feasible (as with traditional state constraints) and involve projecting, replacing, or ignoring infeasible sigma points. In the context of space debris detection however, it is desirable to model an uncertainty distribution which allows for partial intersection with relevant model constraints without objection from the propagator.
This work examines a novel approach for accurately propagating orbit estimates whose uncertainty distributions intersect implicit model or domain constraints. This approach addresses practical challenges for tracking small debris in LEO where position uncertainty expands below the altitude floor of popular gravity and atmospheric drag models during periods of poor observability. To accurately and feasibly estimate these distributions, our approach actively adjusts the sampling pattern (sigma points) of the Unscented Transform (UT) to constrain propagated trajectories to the feasible ranges of each force model. This Model-Constrained Unscented Transform (MCUT) offers several advantages over traditional linearization and projection approaches applied in constrained Kalman Filtering. Firstly, by ensuring that the representative trajectories for each distribution remain strictly feasible under each force model, the MCUT avoids approximation errors from extrapolating those models (potentially nonphysically) beyond the constraint boundary while maintaining the nonlinear estimation capabilities of the classic UT. Additionally, the MCUT enables the definition of probabilistic dismissal criteria, allowing for the potential recovery of classically infeasible orbit estimates when new observations are acquired. To evaluate the accuracy, robustness, and numerical cost of this approach, the MCUT is compared against prevalent estimation approaches for an example LEO orbit intersecting a variable altitude floor constraint. This comparison yields the following key conclusions: (1) the MCUT successfully propagates all test distributions whose mean remains feasible, (2) the estimation error of the MCUT is bounded between that of the linear (state-transition-matrix) projection and the (unconstrained) UT, and (3) the estimation error of the MCUT increases as the distribution mean approaches the constraint boundary. These results form a compelling argument for the application of the MCUT in LEO debris tracking applications.
Date of Conference: September 16-19, 2025
Track: Astrodynamics