Leonardo Cament, Universidad de Chile; Martin Adams, Universidad de Chile; Pablo Barrios, Universidad de Chile
Keywords: Multi-target tracking, Space situational awareness, Random Finite Sets, Poisson Labeled Multi-Bernoulli (PLMB) filter
Abstract:
The efficient detection, tracking, and cataloging of orbiting Space Objects (SOs) are of paramount importance for improved Space Situational Awareness (SSA) and the demand for modern SO tracking applications to produce more accurate and more computationally efficient tracking capabilities is higher than ever.
Due to the random nature of various forces which act on SOs, their orbits can change significantly over time. Therefore, stochastic estimation methods have been popular for updating the track catalogs of SOs. Recent approaches model the multi-target state of SOs with a probability density function. Such approaches usually initialize the tracks of observed SOs using Initial Orbit Determination (IOD), where an initial estimate of the orbit is refined with observations using a non-linear optimization approach. The estimated states of the SOs are propagated via a recursive Bayesian filter based on subsequent observations. Bayesian approaches have been based on Multi-Hypothesis Tracking and more recently Random Finite Set (RFS) concepts, which jointly estimate target number as well as their state values.
To improve the important IOD component in SO tracking, recent work has advocated the use of a Probabilistic Admissible Region (PAR), which is an orbital Admissible Region (AR) for SOs that, combined with an Partially Uniform Birth (PUB) concept, adheres to the assumption of independence between newborn and surviving targets. Recent notable articles which have combined the use of Bayesian RFS tracking methods with the PAR, IOD approach in SSA, include those by B. Jones et al. in which an RFS based Cardinalized Probability Hypothesis Density (CPHD) filter is adopted. However, the CPHD filter does not identify individual tracks and, in SSA and in general, notable problems include “spooky action at a distance” and a lower multi-target tracking performance than their newer labeled counterparts. Therefore recent RFS Multi-Target Tracking (MTT) approaches have focused on labeled filters such as the Labeled Multi-Bernoulli (LMB) filter, which jointly estimates the state and unique identities for each target track. Although labeled RFS filters demonstrate a superior performance in various MTT applications over their unlabeled counterparts, they are not compatible with the PAR, PUB, IOD approach, since they require target births which are labeled and Bernoulli distributed.
Therefore, in this article, multi-target SO tracking is realized with the RFS based Poisson Labeled Multi-Bernoulli (PLMB) filter. It consists of a labeled multi-Bernoulli distribution propagated in time and a Poisson distribution that models SO births. The PLMB filter is capable of tracking and identifying targets with unique labels, and has been shown to out-perform the CPHD filter in a manner similar to the LMB filter. In contrast to other labeled RFS approaches however, it accommodates the PAR-PUB SO birth concept, allowing this powerful IOD method to be incorporated into the filtering framework.
In this article, the PUB density generates uniformly distributed potential SO births in the sensor FoV along with a Gaussian Mixture (GM) distribution of the semi-major axis length and eccentricity, constrained by physical energy considerations. The PAR approach allows the calculation of a particle distribution during the update which obeys the orbital energy constraints. In our application, this corresponds to a maximum value of the semi-major axis length and eccentricity values representing Low Earth Orbit (LEO) SOs. The new target particle distribution is then converted to a GM distribution using the Expectation Maximization (EM) algorithm for the filtering process.
From a practical stand point, many of the telescopes used in SSA have narrow FoVs, and SOs are therefore visible only for short time periods. The use of multiple telescopes is therefore necessary. In this work, a multi-sensor strategy is used in which measurements from multiple telescopes are used to perform the multi-target update in a centralized manner.
The spatial SO state is usually modelled by a 3D position and velocity GM in the Earth-Centered Inertial frame. It is however known that the orbital kinematic process evolves according to a “banana shaped” distribution, and that the GM fails to accurately predict SO states. Published solutions to this problem include splitting each component of the GM into several Gaussian components; however this substantially increases the computational complexity. Other solutions use a particle distribution, requiring many particles, the states of which are directly predicted by the kinematic model. However, since the kinematic model is complex the use of particles can be computationally expensive since the non-linear SO kinematic model not only includes the Earth’s central body gravity, but also other forces, including gravity due to the sun and moon, the non-spherical gravitational potential of the Earth, atmospheric drag and solar radiation pressure. Therefore in this article, the Lie Special Euclidean group SE(3) will be used to model the SO kinematics, since such models preserve the geometric characteristics of SO motion, permitting higher accuracy for extended periods of time, without increase of computational complexity. The former is particularly important due to the often low observational times of SOs. In particular an Unscented Kalman Filter (UKF) implementation of the SO Lie based prediction is developed for each single target component within the multi-target PLMB filter.
Experiments are demonstrated using simulated SO trajectories created from real Two-Line Element data, with simulated measurements from twelve telescopic observatories, which form part of the Falcon Telescope Network. A comparison between the proposed PLMB filter based on SO Lie group state models with a standard Monte Carlo particle distribution used for state prediction together with a linear GM model during the measurement update is given. Results are compared using the Optimal Sub-Pattern Assignment (OSPA) and OSPA(2) metrics and the Multi Object Tracking Precision (MOTP) and Multi Object Tracking Accuracy (MOTA) CLEAR MOT metrics.
The article will demonstrate that the combination of the PLMB filter and PAR, IOD process provides promising target tracking capabilities for multiple SOs, particularly under short time observation periods. An improvement in computational efficiency will also be demonstrated based on the Lie SE(3) group modelling and UKF single target estimation component, due to a vast reduction in the necessary number of particles (Sigma Points within the UKF) when compared with the MC prediction strategy.
Date of Conference: September 14-17, 2021
Track: Dynamic Tasking