Enrico Zucchelli, University of Texas at Austin; Zachary McLaughlin, University of Texas at Austin; Brandon Jones, University of Texas at Austin
Keywords: Multi-Fidelity, Interacting Multiple Model Filters, Maneuvering Targets
Abstract:
Maneuvers introduce several systematic uncertainties to the target tracking problem. Models for maneuver time, magnitude, and direction are not usually available and bias prediction. Additionally, small maneuvers, such as low-thrust maneuvers, may have magnitudes so small that their effect is comparable to that of process noise and their integrated effect may be noticeable after some period of time. This is additionally a problem as low-thrust maneuvers transition to being an operational norm. Hence, accurate space object tracking must be able to detect low-thrust maneuvers. Since the space of possible maneuvers is virtually endless, and high-fidelity dynamics propagation is computationally demanding, the problem requires large computational capabilities. For these reasons, we propose to employ Multi-Fidelity (MF) methods.
MF methods leverage the accuracy of high-fidelity dynamics with the computational advantage of low-fidelity models, providing a solution that is faster to obtain than the former, and more accurate than the latter. An MF method consists of the following four operations: (i) low-fidelity propagation of all required samples, (ii) selection of important samples, (iii) high-fidelity propagation of important samples, and (iv) correction of the propagated low-fidelity samples. Through such an approach, several trajectory models can be propagated for a relatively small computational cost. These models may encompass uncertainty in the initial state and/or the maneuver time, direction, and magnitude.
MF methods can either be used for the propagation of particles in a particle filter, or for the propagation of the sigma points in a Gaussian Mixture Model (GMM) filter in which the mixands are propagated via the Unscented Transform (UT). In previous work from one of the coauthors, it was shown that when the low-fidelity model only has a J2 perturbation, and the high-fidelity model includes drag, solar radiation pressure (SRP), and a 70×70 gravity field, the computational speed-up with respect to the use of just the high-fidelity model is of about 100 times. For propagating the sigma points or particles, the computational cost is for all practical purpose equivalent to that of propagating all points via a low-fidelity solver, as the other three operations have negligible computational cost in comparison. To reduce this cost, we use a parallelized implementation on a GPU to evaluate the low-fidelity propagations. For several initial cases, the multi-fidelity model improves the accuracy of the low-fidelity solution by a factor of 100 to 1,000 times. The remaining error can be estimated via cross-validation, which provides a full matrix, similar to that of process noise. When adding the cross-validation matrix to the process noise matrix, the full high-fidelity and the MF tracking solutions provide equivalent tracking capability, the latter being about two orders of magnitude faster to compute before parallelization is enabled.
Interacting Multiple-Model (IMM) filters employ a finite number of models to estimate maneuvers and corresponding trajectories. Given that the space of possible maneuvers is virtually infinite, we group them into families with each one of them corresponding to a different model. Maneuvers may be classified according to thrust direction, thrust magnitude, and time of occurrence. The filter is provided a state transition kernel, which describes the probability of the spacecraft of switching from one model to any other one (including the probability of not switching). In other words, the maneuvering target is described as a Jump Markov System (JMS). This is necessary to avoid the fitting of maneuvers to perfectly match the measurements, and consequently produce a bias in the solution. Based on the state transition kernel and on the likelihood of the measurement for each of the models, the filter then decides whether (and which kind of) a maneuver has occurred and, if it has, estimates it together with the updated state.
Using MF propagation, we aim to reduce the computational cost of IMM filters. As a consequence, the accuracy is expected to be improved as more maneuver models can be included. This paper is a first step towards developing a framework for efficient and tractable tracking of multiple low-thrust maneuvering spacecraft via multi-fidelity modeling. One of the coauthors previously developed a multi-trarget tracking filter for maneuvering satellites; the progress from this paper is meant to then be adapted to that solution, making it more tractable. In a multi-target tracking problem the trajectory propagation time is additionally important, as hundreds or thousands of objects are being tracked at once.
Our planned test case is the tracking of a GEO satellite performing multiple low-thrust maneuvers. Different scenarios are simulated to establish the performance of the filter. Preliminary results for this paper include the use of IMM filters with MF propagation to estimate instantaneous (high-thrust) maneuvers. Said results were obtained using the same low-fidelity samples over multiple models, and each maneuver model leveraged a different high fidelity correction to those samples. This capability results from the MF correction’s ability to account for even large errors in the low-fidelity model, as long as the high fidelity model introduces no new random inputs. This further motivates the use of a GPU to reduce low-fidelity propagation time since, as previously mentioned, that cost is often the runtime bottleneck of MF propagation.
Date of Conference: September 15-18, 2020
Track: Astrodynamics