# Multi-Spacecraft Tracking and Data Association Based on Uncertainty Propagation
## 3 Methodology of the Multi-Spacecraft Tracking
### Uncertainty Propagation
Moreover, the measurement equation in Equation (4) can be approximated by a second-order expression as:  
\[\delta\mathbf{z}_{Ti}^{p}(t_{k})=\mathbf{\Omega}_{Ti}^{p,l}(t_{k})\delta\mathbf{x}_{Ti}^ {l}(t_{k})+\frac{1}{2}\mathbf{\Xi}_{Ti}^{pJ_{1}l_{2}}(t_{k})\delta\mathbf{x}_{Ti}^{l_ {1}}(t_{k})\delta\mathbf{x}_{Ti}^{l_{2}}(t_{k}) \tag{14}\]  
where \(\mathbf{\Omega}_{Ti}(t_{k})\) and \(\mathbf{\Xi}_{Ti}(t_{k})\) are the first-order and second-order coefficients of the LOS measurement equation:  
\[\mathbf{\Omega}_{Ti}^{p,l}(t_{k})=\frac{\partial\mathbf{z}_{Ti}^{p}(t_{k})}{\partial \mathbf{x}_{Ti}^{l}(t_{k})} \tag{15}\]  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]0  
Subscribe Equations (14)-(16) into Equation (6) and neglect all the terms higher than second-order. Then, a second-order semi-analytical solution for measurement prediction is proposed as:  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]1 \[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]2  
The proposed second-order semi-analytical propagation can be further simplified as:  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]3where \(\mathbf{G}_{Ti}(t_{k},t_{k+1})\) and \(\mathbf{H}_{Ti}(t_{k},t_{k+1})\) are the corresponding STIs for measurement prediction:  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]4  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]5  
The second-order semi-analytical propagation in Equation (18) can predict the future measurement deviation \(\mathbf{z}_{Ti}(t_{k+1})\) based on the current orbit estimation \(\mathbf{x}_{Ti}(t_{k})\). To distinguish the STTs \(\mathbf{G}_{Ti}(t_{k},t_{k+1})\) and \(\mathbf{H}_{Ti}(t_{k},t_{k+1})\) with the STTs \(\mathbf{\Phi}_{Ti}(t_{k},t_{k+1})\) and \(\mathbf{\Psi}_{Ti}(t_{k},t_{k+1})\), in this paper, the STTs in Equations (7) and (8) are called the state transfer STTs and the STTs in Equations (19) and (20) are called the measurement prediction STTs. Let \(\mathbf{x}_{Ti}(t_{k})\) and \(\mathbf{P}_{Ti}(t_{k})\) be the estimated state and associated covariance matrix of the \(i\)-th target at the current epoch \(t_{k}\). According to [38], the predicted LOS measurement \(\mathbf{z}_{Ti}(t_{k+1})\) and associated covariance matrix \(\mathbf{P}(\mathbf{\hat{z}}_{Ti}(t_{k+1}))\) at the future epoch \(t_{k+1}\) can be obtained using the STTs in Equations (18)-(20) as:  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]6  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]7 \[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]8 \[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]9  
The predictions in Equations (21) and (22) are used to compare with the true measurements, and then decisions are made on whether they are matched. It can be seen from Equations (21) and (22) that the predicted LOS measurement \(\mathbf{\hat{z}}_{Ti}(t_{k+1})\) and associated covariance matrix \(\mathbf{P}(\mathbf{\hat{z}}_{Ti}(t_{k+1}))\) are only related to the current state of the \(i\)-th target. Thus, these predictions can be obtained in a decentralized computing framework to further reduce the computational overhead. The process of data association is detailed in the following subsection.