Advances in remote sensing technologies and privatization of space access has paved the way to low-cost Earth orbital missions, predominantly concentrated around the Low Earth Orbital region (LEO). The majority of these missions have academic, research or technology demonstration interests, but some companies have identified a market niche in communications infrastructure based on the ever decreasing costs of launching satellites in the sub-500 kg range. This idea is not new as the Iridium satellite constellation [1] has been providing voice and data information coverage since 1998. However, recent constellation designs aim at providing a broader range of services [2] such as low-latency internet coverage, seeking to optimize the operating expenses by deploying a significant amount of cost-effective satellites, on the order of thousands.

The majority of these constellations are built upon the requirement that multiple spacecraft can be injected simultaneously, and thus are able to autonomously acquire their respective operational regime. The latter translates into high $\Delta V$ transfers, which can have a huge impact on the mass budget in case traditional chemical thrusters are employed. Electric propulsion engines, based on ion acceleration through electromagnetic fields, are well suited for this type of missions since they can deliver high total impulse magnitudes with significantly lower mass consumption. This comes at the cost of high power requirements and low thrusting forces $F/m<0.5\phantom{\rule{1em}{0ex}}\mathrm{mN}/\mathrm{kg}$ (e.g. $0.2\phantom{\rule{1em}{0ex}}\mathrm{mN}/\mathrm{kg}$ for SMART-1 [3]), resulting in orbital transfers with a duration on the order of months.

Large constellations based on autonomous orbital acquisition are seeing a rapid deployment, and their threat to space sustainability is a growing concern. To prevent catastrophic events as the Kosmos-Iridium 2009 in-orbit collision, it is necessary to build-up and maintain a space object catalog, preferably with an accuracy that enables realistic collision risk assessment. Such catalogs are generally built based on surveillance data acquired by sensors located on ground, predominantly telescopes and radars. Measurements gathered from these sensors are compared against orbits predicted from the catalog in order to update the state of known objects and detect new targets. These predictions are usually based on the already complex dynamical behavior of Earth orbiting targets, disregarding controlled or forced dynamics. Hence, targets with maneuvering capabilities present a major threat to space object catalog maintenance in the absence of a priori knowledge on the maneuver plan, even though scientific and commercial operators should in principle share their ephemeris predictions considering planned maneuvers.

To address this potential source of ambiguity, several maneuver detection and maneuver target tracking methods have been proposed in the literature, principally aimed at maintaining custody of active spacecraft despite unknown control inputs. Some of these works rely on continuous tracking radar data: Ko and Scheeres [4], [5] propose to parameterize any type of control input as a truncated Fourier series (retaining the 14 coefficients with the highest impact on secular orbital variations) and use them to perform orbit determination. In their analysis they do not perform orbit determination during thrusting intervals and present test cases with a $\sim $2 h observation gap and a measurement latency of 60 s. An alternative approach by Guang et al. [6] apply linear estimation theory to approach unknown dynamics, in this case with an even higher measurement frequency $\sim $5 Hz. Due to the timeliness of the observations considered therein, linear approximations are shown to provide a valid framework for maneuver estimation, and even in the former case with a lower latency, custody maintenance does not seem to be an issue provided the observation gap and duration of the maneuvers themselves. Note however that if the target were continuously maneuvering then the proposed parameterization may fail to approximate the true control sequence. The current work is nonetheless concerned about data sparse scenarios, characteristic of space surveillance operations, in which revisit times can be on the order of days with observation arcs spanning a few minutes, thus rendering data association and state estimation a challenging task. Lubey [7] developed a sequential estimator based on optimal control which can be used to efficiently estimate post-maneuver states conditioned on measurement association hypotheses. Under the assumption of optimal transfers from a prior Gaussian to certain observables, the proposed estimator builds on a dynamical linearization around the reference optimal control solution, which in general presents biasedness towards energy efficient (or otherwise optimal) transfers, especially for reduced measurement information content. The latter can be traduced to favoring in plane phasing maneuvers for optical observations near the equator, as they are more efficient than plane changes, or eccentric estimates during orbit raising with surveillance radar data, where velocity information is rather limited compared to position (generally $\ge $50 m/s and $\le $10 km for initial orbit determination methods). A somewhat similar approach has been recently proposed by Pastor et al. [8] and Porcelli et al. [9] considering the use of a batch least squares estimator for characterizing impulsive (high thrust) maneuvers within an observation (tracklet) sequence. Therein, they propose to estimate the optimal impulsive burn that minimizes the observation residuals, and in analogy to the approach in [7], perform data association based on control (in this case delta-V) thresholding. Note the estimates provided by this method can fall short for spacecraft performing continuous maneuvers since (1) single burn transfers require some intersection between the initial and final orbit and (2) impulsive burns present different dynamical characteristics when compared to low thrust, i.e. they are more efficient for out-out-plane and eccentricity change maneuvers. Other works have focused on the use of machine learning methods [10], emphasizing on the maneuver detection and prediction yet lacking consistent post-maneuver state estimation. In this regard, Siminski et al. [11] propose to combine optimal control based data association with post-maneuver state estimates using historical data, a line of research that has been continued in [12] using an impulsive metric and a feedback-loop maneuver estimator. Besides, different control distance metrics have been applied to the particular problem of data association [13], [14], [15], focusing on analyzing the distribution of the control metric with the aim of identifying correct measurement to object correlations in the event of maneuvers. Note these approaches do not seem suitable for objects performing long duration maneuvers, either due to convergence issues stemming from biased sequential estimation, or the provision of models that are unable to capture the dynamical characteristics of uncooperative low-thrust maneuvering satellites. As a corollary, long term orbital transfers pose a great challenge to Space Surveillance and Tracking (SST) systems since the *dynamical link* between the object and the observation is lost for a sequence of observations, so that it is in principle necessary to jointly estimate the state and the control profile.

The maneuvering target tracking literature is rather vast [16], [17], [18], [19], [20], and there are contributions not only from domain-specific areas but also from a broader problem type perspective, for instance dealing with the identifiability of maneuver models. Tracking methods applied to maneuvering targets can be generally framed under two categories: (a) adjustable level process noise, Input Estimation (IE) and Variable State Dimension (VSD) filters; and (b) multiple model formulations. The former require some means to detect maneuver onset (and termination) in order to switch to the *maneuvering* filter mode, which characterizes maneuvers in terms of an unknown acceleration sequence applied to the ballistic or unperturbed dynamics, and approximates them as either an increase in process noise [7], an input estimation problem [6] or an augmentation of the state dimension [5], [8], respectively. Multiple model methods (c.f. [11], [12]) regard the target as a state machine whose physical state evolves according to one of many different (yet finite) set of dynamical models. Therefore, this type of filters assumes the true dynamics to be sufficiently represented by one of the proposed models, thus dropping the need for maneuver detection in favor of model selection or estimation. This presents additional advantages in multi target environments as maneuvering capabilities for each target are not inferred conditioned on a posteriori information (this is, data association hypotheses) but rather included directly in the definition of the target prior distribution. Thus, in the interest of further research aligned with multi-object tracking and data association, it seems more natural to adopt a multiple model framework with maneuver models that allow for robust and unbiased Bayesian filtering.

Within this paper, the authors propose a methodology to address the problem of tracking an uncooperative spacecraft performing long duration low-thrust maneuvers provided the above considerations. The developed method is based on the former approach in [12], which focused on high-thrust station keeping maneuvers within the GEO regime. Still, the target is modeled as a hybrid stochastic system, and the maneuvering dynamics are derived from certain expected controllability bounds. A novel control metric, $P$, specifically tailored to limited thrust transfers, is derived from the Thrust Fourier Coefficients (TFC) formulation (see [21]) and represents the maximum instantaneous control acceleration required to perform certain orbital transfer in a fuel optimal sense. As suggested in [12], one can define an admissible control region conditioned on a post-maneuver observation provided certain bounds on the maximum expected control acceleration. Note the definition of such admissible region is highly dependent on the prior state, hence a robust algorithm requires the implementation of capable hypothesis management techniques. In essence, we propose to define a vacuous prior in the control metric space for the maneuver mode, which effectively reduces to the admissible control region for single target tracking purposes, an approach that conceptually resembles the one devised by Hall et al. [22] for space-based uncooperative target tracking (whose application considers more timely data and impulsive maneuvers).

The paper is organized as follows: Section 2 presents the problem formulation and discusses on the system characteristics and observed quantities; a means to characterize thrust-limited orbital transfers conditioned on a surveillance radar tracklet is discussed on Section 3; Section 4 presents the developed filter for tracking non-cooperative targets with surveillance radar data, together with some background on sampling-based estimation methods; Section 5 describes the simulated test scenario and presents an analysis of the algorithm performance; finally, concluding remarks are drawn on Section 6.

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