Thomas Dearing, Arka Group; Jacob Griesbach, Arka Group; Piyush Mehta, West Virginia University
Keywords: CARMHF, Multi-Hypothesis Filter, Constraint Admissible Region
Abstract:
A fundamental first step for any Kalman Initial Orbit Determination (IOD) strategy is the generation of an initial orbit estimate used for Kalman Filter initialization: a hypothesis necessarily based on sparse and imprecise information. A Multi-Hypothesis Filter (MHF) approaches this initialization problem robustly by considering a family of initial orbit hypotheses that sample the measurement uncertainty space. Functionally, these hypotheses must sample the state space comprehensively and finely enough to guarantee that the true state lies within at least one hypotheses’ convergence radius (as governed by the Kalman filter). However, the computational impact of these hypotheses is also considerable: each must be propagated and compared against subsequent observations and new hypotheses must be generated whenever the MHF encounters a new unassociated observation. Thus, the hypothesis set must be carefully chosen to be large enough to contain the true solution, but small enough to remain computationally tractable and scalable while processing thousands of RSO’s.
Initializing the MHF hypothesis set becomes particularly challenging when considering Electro-Optical (EO) measurements. Such observations are extracted from short arcs (streaks) in the image plane and nominally provide Right-Ascension/Declination angle pairings, with angle-rates either measured natively or inferred by differencing successive measurements. However, angle/angle-rates only encompass 4 of the 6 degrees of freedom (DOFs) necessary to compute a full position/velocity orbital state. That is, the range/range-rate space is entirely unobservable over these arcs. To capture the remaining 2 DOF using a finite MHF hypothesis set, the feasible range/range-rate space must be refined to a finite subset or Admissible Region (AR). Two major approaches for defining an AR currently exist in the literature. The first approach is called the Constrained AR (CAR) and uses a-priori orbital constraints to include only the orbital families of interest (for example, orbits within an assumed eccentricity range). This approach rapidly converts a set of observation parameters and orbital constraints into a compact region that can be easily subsampled to return hypotheses for the MHF. The second approach is called the Probabilistic AR (PAR) and is constructed using gaussian particle clouds to directly estimate the measurement uncertainty distribution in the range/range-rate space. Critically, this inclusion of measurement uncertainty in the construction of the PAR makes it notably larger than the equivalent CAR, but the associated distribution also enables users to sample the PAR more strategically. As a result, practical selection between these approaches simplifies to the application’s prioritization of computational efficiency and robustness.
In furtherance of IARPA’s SINTRA debris detection and tracking program, this work examines alternative strategies for incorporating parameter uncertainty in the CAR generation algorithm to address practical issues encountered in large scale MHF applications. Specifically, this work first reproduces and expands upon the differential inflation approach developed by Holzinger et al. to include both prevalent CAR constraint bounds on orbital eccentricity and semi-major axis. Strategies are developed to reduce intrinsic numerical ill-conditioning of this approach under these constraints, addressing key shortcomings reported in the original literature. To examine the higher order moments of the parameter-uncertainty inflated CAR, this work also adapts the Unscented Transform (UT) to measure nonlinear distortions in the CAR boundary using synchronized parametrizations of the solution contours. The accuracy and numerical efficiency of the above approaches are then compared against Monte-Carlo simulations in both the LEO and GEO orbital regimes. This comparison yields the following key conclusions: (1) the size and shape of the CAR is non-negligibly affected by the inclusion of parameter error in both LEO and GEO regimes, (2) the UT approach reproduces the nonlinear inflation of the CAR more accurately than the differential approach, but is more computationally intensive, and (3) results from these two approaches mainly diverge near saddle points in the constraint functions. These results highlight the importance of parameter error in admissible region generation and present multiple approaches to combine the respective robustness and computational efficiency benefits of the PAR and CAR approaches in practical applications.
Date of Conference: September 17-20, 2024
Track: Astrodynamics