Sehyun Yun, The University of Texas at Austin; Nicholas Ravago, The University of Texas at Austin; Benjamin L. Reifler, The University of Texas at Austin; Renato Zanetti, University of Texas at Austin; Brandon A. Jones, The University of Texas at Austin
Keywords: Multi-target tracking, Generalized labeled multi-Bernoulli filter, Kernel density estimation, Particle filter, Gaussian mixture model, Bi-fidelity propagation, Adaptive algorithm
Abstract:
Space situational awareness (SSA) refers to the knowledge of the near-space environment, including the ability to track and predict the states of space objects (SOs) orbiting Earth and Cislunar space. The majority of SOs in Earth orbits are debris objects, such as used satellites and rocket fragments. As the space domain becomes more congested due to the launching of new objects, it poses a serious threat to newly launched satellites and the risk of collision between SOs considerably increases. Out of the more than 40,000 objects larger than 10cm in diameter tracked by the Joint Space Operations Center, only about 4,000 are active payloads and a limited number of sensors are available and used to estimate the states of these SOs. A multi-target tracking algorithm is proposed for sparse-data tracking by using an efficient data association algorithm and an accurate and computationally efficient nonlinear estimation algorithm for orbit determination. In this study, the modified kernel-based ensemble Gaussian mixture filtering (EnGMF) is used in combination with the generalized labeled multi-Bernoulli (GLMB) multi-target tracking filter to track and identify multiple SOs.
The GLMB filter is designed to provide estimates of object trajectories based on the labeled random finite set (RFS) theory. An RFS is a marked finite set containing a random number of random variables, and a labeled RFS includes a unique label on each element to indicate identity. The GLMB density is a special case of labeled RFS density, which provides an analytical solution to the Bayes multi-target filter recursion, and in the GLMB filter, multi-target state and measurement set are modeled as an RFS. One of the challenges existing in the GLMB filter (and any hypothesis-based multi-target tracker) is high computational cost due to the exponential increase of the number of components in the filtering process; therefore, it requires a truncation strategy to remain computationally tractable. To reduce the computational cost, in this work the EnGMF is used as the single-target filter to reduce the overall computational burden when compared to other state-of-the-art nonlinear estimators.
The EnGMF is developed to efficiently track SOs with short and sparse observation data (i.e., tracking passes are short and sparse). As a recursive algorithm, the knowledge of the distribution at the prior time is assumed and approximated by N independent and identically distributed (i.i.d.) samples. As in the bootstrap particle filter (BPF), in the time update step, a set of samples at the next time step is obtained using the Markov transition kernel. Then, the propagated samples are converted into Gaussian mixtures using kernel density estimation (KDE). In other words, each and every sample is considered as a Gaussian component with the same covariance (i.e., bandwidth) matrix. The bandwidth matrix can be selected by numerically solving an optimization problem; however, in this study, Silverman’s rule of thumb is employed to estimate the bandwidth matrix to reduce the KDE computational cost. When the sampling distribution is Gaussian, the optimal bandwidth matrix is obtained by Silverman’s rule of thumb based on the mean integrated squared error (MISE) as a performance criterion. Moreover, in our previous work, we demonstrated that Silverman’s rule will result in conservative (large) estimates even though the sampling distribution is not close to Gaussian, which means inaccuracies make the estimator conservative rather than divergent.
In addition, a bi-fidelity approach to propagation and an adaptive algorithm are applied to the EnGMF to reduce its computational cost with an acceptable loss in accuracy. The bi-fidelity approach employs both low- and high-fidelity models to minimize computational cost for the propagation of the EnGMF, while maximizing the accuracy of orbit uncertainty propagation. Choosing the appropriate number of particles is also one key parameter of the EnGMF to improve its computational complexity. Thus, an adaptive algorithm is employed to select an appropriate number of particles of the EnGMF based on the convergence assessment using a predictive observation probability density function (PDF).
The performance of the GLMB filter with the EnGMF is evaluated through simulations of multi-target tracking problem with sparse observation data. Multiple SOs in geosynchronous Earth orbit (GEO) are considered in the simulations and some of them are close in the measurement space, which causes many data association hypotheses. We demonstrate and assess the estimator’s ability to handle multi-target tracking problem with short and sparse observation data. We also show that the GLMB filter with a linear filter such as the extended Kalman filter or unscented Kalman filter fails under this challenging scenario. Moreover, we compare the performances of the GLMB when using: the EnGMF, the EnGMF with bi-fidelity propagation, and the adaptive EnGMF with bi-fidelity propagation. We demonstrate that the GLMB with the EnGMF with bi-fidelity propagation and the adaptive EnGMF with bi-fidelity propagation can significantly reduce computational time compared to the GLMB with EnGMF with an acceptable loss in accuracy.
Date of Conference: September 27-20, 2022
Track: Astrodynamics