Hyperplane Intersection Methods

The hyperplane intersection methods determine the association of two tracklets by trans- forming all hypothetical orbits to a comparison space to assess the intersection of their hyper- planes. Maruskin et al. [28] proposed a recursive intersection method to determine the unique orbit of two tracklets in the Delaunay space. However, it needs to be performed manually, which is difficult for processing a large amount of data. Maruskin et al. [29] also investigated the strategy of employing a distance metric to determine the correlation between unperturbed Kepler hypothetical orbits. However, it is mainly suitable for the case of zenith observations. Fujimoto et al. [19, 95] transformed the uniform PDF of the admissible region to the Poincar´e space, and the association is determined by assessing the overlap of the hypercubes (or bins) in the Poincar´e space. This approach is referred to the BIN method in this dissertation. Compared to the intersection approach proposed by Maruskin, the BIN method is more efficient, and it is able to automatically process a large numbers tracklets to determine association.

The core of the BIN method is to map the PDF of an admissible region to the 6-dimensional Poincar´e space, and the association is confirmed if two Poincar´e PDFs are overlapped. Given two Poincar´e PDFs f and g from two admissible regions, which are propagated to the same epoch, their intersection region is represented by h, where h > 0 for a bin in the case that both f > 0 and g > 0. The association of two tracklets can be determined by checking if h > 0 over all bins. In addition, multiple tracklets can be grouped together according to the Bayes’ theorem.

Another contribution made by Fujimoto is the development of a linear map from the ad- missible region to the Poincar´e space. In order to ensure a high accuracy of association, a large number of VPs needs to be mapped to the Poincar´e space and propagated to the same epoch. The transformation of a discrete VP starts from topocentric spherical coordinates (i.e.,

(a, e, i, Ω, ω, M ), and finally to Poincar´e variables (L, l, G, g, H, h).

For the unperturbed Keplerian orbit, the orbit propagation of the Poincar´e elements from

time t0 to t is expressed by Ψpoi(t, t0) : (L(t0), l(t0), G(t0), g(t0), H(t0), h(t0)) → (L(t0), l(t0) + µ2 L(t0)3 (t − t0), G(t0), g(t0), H(t0), h(t0)). (2.2.10)

The transformation and orbit propagation process can be linearised, and the detailed derivation can be found in Ref. [19].

Mapping numerous points exactly results in large computational demands. Alternatively, a group of uniformly distributed VPs can first be selected, and the linearisation algorithm applied to map the vicinity of each VP to the comparison space. In other words, the algorithm linearly maps a sub-plane in the admissible region to the comparison space with fast speed, and it is able to obtain all bins as the exact map does. The flowchart of the BIN method is shown in Fig. 2.2.5.

The BIN method is evaluated using a test scenario of associating two tracklets from the same object (NORAD ID: 815), and associating two tracklets from two objects (NORAD ID: 815 and 1430) respectively. The former is shown in the top three subfigures in Fig. 2.2.6, and the latter is given in the bottom subfigures. The bins are expressed in three 2-dimensional subspaces of the Poincar´e space. In each subfigure, the blue pluses and red stars represent the bins of corresponding tracklets that are propagated to the same epoch, and the black circles represent the overlapped bins between the two tracklets. Results of the top subfigures show 201 bins are overlapped, indicating that these two tracklets are from the same object. In addition, the bottom three subfigures show no overlap bins, and actually, the two tracklets are from different objects. This test case illustrates that the BIN method is able to associate tracklets from the same target, and it is able to distinguish tracklets from different targets.

Figure 2.2.5: The flowchart of the BIN method

Figure 2.2.6: Bins of two pairs of tracklets in the Poincar´e space, the top three subfigures are the results of tracklets from the same object (NORAD ID: 815), the bottom ones are the results

cretisation size of the bins in the comparison space. Decreasing the bin size will result in a more accurate association, but significantly increase the computation time and vice versa. A possible solution is to refine the discretisation size only for the regions that two PDFs overlap. However, how to adaptively tune the discretisation size to balance the accuracy and efficiency of association still needs further investigation.

Generally, compared to the BIN method, the IVP and BVP optimisation methods yield less computational demand. A detailed comparison of the computational efficiencies can be found in Ref. [17]. Therefore, further improvement based on the optimisation method is one major objective of this thesis.

Note that both the hyperplane method and the optimisation method are able to determine a reliable IOD solution for two tracklets after association. Fujimoto et al. [25] validated the feasibility of combining the tracklet association method with a Bayesian LS estimator to refine the orbital state estimation. However, tracklet association methods have not been considered for incorporation with multi-target Bayesian recursive estimation methods for space object track- ing. Thus, this thesis seeks to explore the use of tracklet association methods for initialisation of multi-target trackers to achieve an improved computational efficiency. In the following section, a summary of the RFS theory is given, along with an overview of the multi-target Bayesian estimation, four latest labelled RFS filters, and a comparison of their performance for space objects tracking.