# Multi-Spacecraft Tracking and Data Association Based on Uncertainty Propagation
## 1 Introduction  
Tracking of the space objects is a vital task in space missions as accurate orbit determination of the spacecraft is the basis of the subsequent operations like trajectory planning, guidance, and control [1, 2, 3]. A lot of techniques have been developed for space objects tracking, such as the least square method (LSM) and its variants, the Kalman filter (KF), and its variants [4, 5, 6]. The LSM and KF can provide optimal orbit estimations for the space objects by combining the dynamic equations and measurement information. It is usually easy to track a single space object as the association between the collected measurements, and the target is determined [7, 8, 9, 10]. However, tracking a variety of spacecraft at the same time is a complicated problem. Multi-spacecraft tracking suffers from lots of challenges, which can be broadly catalogued into two aspects. The first challenge is that the number of the targets is usually unknown and time-varying [4, 5, 11]. The second one is that the association between the targets and the measurements are ambiguous [12, 13]. The data association is the major difficulty of the multi-spacecraft tracking [13, 14, 15]. The aim of data association is to associate each target with its appropriate measurements. Several classic data association methods have been developed such as the Nearest Neighbor (NN), Probability Data Association (PDA), and Joint Probability Data Association (JPDA). The NN method performs data association by selecting the closest collected measurements in the target space. The NN method is simple and easy-to-operate, but its performance relies on accurate prediction of the measurement [16, 17, 18]. The PDA method set up the extension gate for each target based on the measurement prediction, and the measurement in the extension gate is selected for association. The PDA method assumes that the extensiongates of different targets are uncrossed. Its performance seriously degrades when the targets are close to each other [19]. The defect of the PDA method can be well addressed using the JPDA method [20; 21]. JPDA is an extension of PDA. The JPDA calculates the joint probability of all the feasible associations of all the targets. The association at the highest probabilities is selected, and no two measurements are associated with the same target [22]. The JPDA method, as well as its variants, have been successfully applied into solving multi-target tracking problems such as the traffic management [23]. Although the JPDA method is considered as reliable and accurate, its complexity and computational burden are usually high [12]. The above-mentioned methods are mainly based on the probability and statistics theory, and few or even no dynamic information is used. For multi-target tracking cases like the traffic management, the above methods are feasible as the dynamics are unavailable [24; 25]. However, for the multi-spacecraft tracking problems where the motions of the targets are driven by the highly nonlinear Keplerian equation, the probability and statistics theory is inadequate for accurately predicting the measurement [26; 27; 28; 29]. Thus, the extension gate can be incorrectly set up, and the performance of the mentioned methods are impacted. Recently, the issue of the orbit uncertainty propagation has been widely investigated. The orbit uncertainty propagation is to characterize the orbit probability based on the dynamic model [30]. The state-of-the-art approaches for orbit uncertainty propagation can be divided into linear methods and nonlinear methods. The linear methods include the state transform matrix (STM) and the Covariance Analysis Description Equation Technique (CADET) [31]. These linear methods are computationally efficient but suffer from drawbacks related to poor accuracy. To improve the propagation accuracy, many methods have been developed for the nonlinear methods, such as the Unscented Transformation (UT) [32], the Polynomial Chaos (PC) [33], the State Transition Tensors (STT) [34; 35; 36], and the Gaussian Mixture Model (GMM) [37].