# Multi-Spacecraft Tracking and Data Association Based on Uncertainty Propagation
## 3 Methodology of the Multi-Spacecraft Tracking
### Uncertainty Propagation  
In this paper, the purpose of the uncertainty propagation is to predict the uncertainty distribution of the measurement at a future epoch, on which to base the optimal association model, is established for data association. The concept of uncertainty propagation is originally put forward for predicting the orbital uncertainty. In this paper, the concept of uncertainty propagation is extended into the measurement uncertainty prediction, and a semi-analytical prediction method is proposed based on the STT technique. According to [34], the relationship between state deviations \(\delta\mathbf{x}_{Ti}(t_{k})\) and \(\delta\mathbf{x}_{Ti}(t_{k+1})\) can be represented by a second-order STT as follows:  
\[\delta\mathbf{x}_{Ti}^{p}(t_{k+1})=\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})\delta\mathbf{x }_{Ti}^{l}(t_{k})+\frac{1}{2}\mathbf{\Psi}_{Ti}^{p,l_{1}l_{2}}(t_{k},t_{k+1}) \delta\mathbf{x}_{Ti}^{l}(t_{k})\delta\mathbf{x}_{Ti}^{l}(t_{k}) \tag{6}\]  
where the superscripts '\(p\)', '\(l_{1}\)' and '\(l_{2}\)' denote the \(p\)-th, \(l_{1}\)-th and \(l_{2}\)-th elements, respectively; \(\mathbf{\Phi}_{Ti}(t_{k},t_{k+1})\) and \(\mathbf{\overline{\Psi}}_{Ti}(t_{k},t_{k+1})\) are the first-order and second-order STTs, given by:  
\[\mathbf{\Phi}_{Ti}^{p,l}(t_{k},t_{k+1})=\frac{\partial\mathbf{x}_{Ti}^{p}(t_{k+1})}{ \partial\mathbf{x}_{Ti}^{l}(t_{k})} \tag{7}\]\[\mathbf{\Psi}_{Ti}^{pJ_{1}l_{2}}(t_{k},t_{k+1})=\frac{\partial^{2}\mathbf{x}_{Ti}^{p}(t_{k+ 1})}{\partial\mathbf{x}_{Ti}^{J_{1}}(t_{k})\partial\mathbf{x}_{Ti}^{J_{2}}(t_{k})} \tag{8}\]  
Note that, for brevity, summation convention is used in Equation (6). As an example, for the second-order term in Equation (6), the summation convention is expressed as:  
\[\mathbf{\Psi}_{Ti}^{pJ_{1}l_{2}}(t_{k},t_{k+1})\delta\mathbf{x}_{Ti}^{l_{1}}(t_{k}) \delta\mathbf{x}_{Ti}^{l_{2}}(t_{k})=\sum_{l_{1}=1}^{6}\sum_{l_{2}=1}^{6}\mathbf{\Psi} _{Ti}^{pJ_{1}l_{2}}(t_{k},t_{k+1})\delta\mathbf{x}_{Ti}^{l_{1}}(t_{k})\delta\mathbf{x }_{Ti}^{l_{2}}(t_{k}) \tag{9}\]  
The first-order STT \(\mathbf{\Phi}_{Ti}(t_{k},t_{k+1})\) and the second-order STT \(\mathbf{\[PERSON]}_{Ti}(t_{k},t_{k+1})\) can be computed by integrating the following differential equations:  
\[\mathbf{\[PERSON]}_{Ti}^{pJ}(t_{k},t)=f_{*}^{p,\alpha}\mathbf{\[PERSON]}_{Ti}^{q,l}(t_{k},t) \tag{10}\]  
\[\mathbf{\[PERSON]}_{Ti}^{pJ_{1}l_{2}}(t_{k},t)=f_{*}^{p,\alpha}\mathbf{\[PERSON]}_{Ti}^{q,l_{1}l _{2}}(t_{k},t)+f_{*}^{p,\alpha\beta}\mathbf{\Phi}_{Ti}^{q,l_{1}}(t_{k},t)\mathbf{\Phi} _{Ti}^{\beta J_{2}}(t_{k},t) \tag{11}\]  
where:  
\[f_{*}^{p,\alpha}=\frac{\partial\mathbf{x}_{Ti}^{p}(t)}{\partial\mathbf{x}_{Ti}^{q}(t )} \tag{12}\]  
\[f_{*}^{p,\alpha\beta}=\frac{\partial^{2}\mathbf{x}_{Ti}^{p}(t)}{\partial\mathbf{x}_{ Ti}^{\alpha}(t)\partial\mathbf{x}_{Ti}^{\beta}(t)} \tag{13}\]  
The details on how these differential equations are obtained are well provided by [PERSON] and [PERSON] (2018). For brevity, they are not given in this paper, and any reader with particular interest on this issue can refer to (2018). In this paper, for the dynamics mentioned in Equations (1) and (2), the partials in Equations (12) and (13) are analytically computed using the MATLAB's symbolic toolbox (MATLAB version R2019a, The MathWorks, Inc., Natick, MA, USA).