## 2 Tracking Algorithms Overview
### Uncertainty Quantification  
In multi-sensor platforms, the uncertainty position of the RSO is influenced by the accuracy of the sensor positions and the sensor to target LOS. It is crucial to know each sensor location (x\({}_{\mathrm{1}}\), y\({}_{\mathrm{1}}\), z\({}_{\mathrm{1}}\)), (x\({}_{\mathrm{2}}\), y\({}_{\mathrm{2}}\), z\({}_{\mathrm{2}}\)) and the corresponding LOS vector's azimuth and elevation (\(\theta\) and \(\varphi\)), respectively. The tracked RSO position can be analysed by combining the tracking and the navigation uncertainty using the proposed mathematical framework. To obtain the covariance matrix corresponding to the navigation error, the reader is directed to [37] in which the steps are described in detail. Therefore, navigation measurements are assumed to be provided by an on-board GNSS system, and the corresponding uncertainty values are derived from a literature LEO GPS accuracy experiment [50]. The uncertainty in the tracking measurements can be expressed in terms of the covariances as follows:\[\text{Q}_{\text{TRK}}=\begin{bmatrix}\sigma_{\text{x}}^{2}&0&0\\ 0&\sigma_{\text{y}}^{2}&0\\ 0&0&\sigma_{\text{z}}^{2}\end{bmatrix} \tag{9}\]  
where \(\sigma_{\text{x}}^{2}\), \(\sigma_{\text{y}}^{2}\) and \(\sigma_{\text{z}}^{2}\) are obtained using Equations (5), (6) and (8a).  
To determine the total covariance matrix Gauss-Helmert formulation, [51; 52] are used in order to relate the sensor measurement errors \((\sigma_{\text{p1}},\sigma_{\text{p2}},\sigma_{\text{\varrho1}},\sigma_{ \text{\varrho2}},\sigma_{\text{\varphi1}},\sigma_{\text{\varphi2}})\) to the final RSO position \((\text{x}_{\text{t}},\ \text{y}_{\text{t}},\text{z}_{\text{t}})\). We define the \(\text{L},\text{X}\) vectors of estimated observations and estimated parameters, respectively:  
\[\text{L}=[\text{x}_{1},\text{y}_{1},\text{z}_{1},\text{\theta}_{1},\text{ \varphi}_{1},\text{x}_{2},\text{y}_{2},\text{z}_{2},\text{\theta}_{2},\text {\varphi}_{2}]^{\text{T}} \tag{10}\]  
\[\text{X}=[\text{x}_{\text{t}},\ \text{y}_{\text{t}},\text{z}_{\text{t}}]^{ \text{T}} \tag{11}\]  
The total covariance matrix can then be expressed as:  
\[\text{Q}_{\text{TOT}(3*3)}=\ \text{BC}_{\text{r}}\text{B}^{\text{T}} \tag{12}\]  
where \(\text{C}_{\text{r}}\) is the covariance matrix of the observations which can be written as a \(10*10\) diagonal matrix with elements \([\text{\sigma}_{\text{x1}},\text{\sigma}_{\text{y1}},\text{\sigma}_{\text{z1 }},\text{\sigma}_{\text{\varrho1}},\text{\sigma}_{\text{\varphi1}},\text{ \sigma}_{\text{\chi 2}},\text{\sigma}_{\text{y2}},\text{\sigma}_{\text{z2}},\text{ \sigma}_{\text{\theta2}},\text{\sigma}_{\text{\varphi2}}]\). For matrix B, we first define the function \(\text{F}(\text{X},\text{L})=0\) as:  
\[\text{x}_{\text{t}}-\frac{\text{x}_{2}\text{tan}\ (\text{\theta}_{1})-\text{x}_{ 1}\text{tan}\ (\text{\theta}_{2})+(\text{y}_{1}-\text{y}_{2})\ \text{tan}\ (\text{\theta}_{1})\ \text{tan}\ (\text{\theta}_{2})}{\text{tan}\ (\text{ \theta}_{1})-\text{tan}\ (\text{\theta}_{2})}=0\]  
\[\text{y}_{\text{t}}-\frac{\text{y}_{1}\text{tan}\ (\text{\theta}_{1})-\text{y}_{2} \text{tan}\ (\text{\theta}_{2})+(\text{x}_{2}-\text{x}_{1})}{\text{tan}\ (\text{\theta}_{1})-\text{tan}\ (\text{ \theta}_{2})}=0 \tag{13}\]  
\[\text{z}_{\text{t}}-\frac{\text{r}_{1}\text{tan}(\text{\varphi}_{1})+\text{z} _{1}+\text{r}_{2}\ \text{tan}(\text{\varphi}_{2})+\text{z}_{2}}{2}=0\]  
Then, matrix B is defined as \(\frac{\partial\text{F}}{\partial\text{L}}\) and, therefore, has a \(3\times 10\) dimension.