## 2 Tracking Algorithms Overview
### Tracking Algorithm  
In order to compute the RSO position estimate, it is necessary to know the location of each sensor \((\mathrm{x_{i}},\mathrm{y_{i}},\mathrm{z_{i}})\), as well as the line of sight (LOS) azimuth and elevation pointing angles \((\theta,\Phi)\) from each sensor to the target. An inaccuracy in determining these 10 parameters will result in an error in the target position estimate. Furthermore, the relationship between the measurement error and the errors in the estimated target location is a function of the sensor-target-sensor geometry, where a sensor-target-sensor separation of 90\({}^{\circ}\) results in the lowest error sensitivity, while a sensor separation of 0\({}^{\circ}\) or 180\({}^{\circ}\) results in impossible solutions (as seen from the target). To define the triangulation problem, we need to define the positions of the sensors and the RSO in a right-handed coordinate system as illustrated in Figure 4.  
The x and y axes form the horizontal plane, and the z axis points out of plane vertically. \(\theta_{1}\) and \(\theta_{2}\) are the azimuth angles of the corresponding sensors measured clockwise from positive y axis towards positive x axis. The corresponding elevation angles are denoted by \(\phi_{1}\) and \(\phi_{2}\) which increase from 0\({}^{{}^{\circ}}\) in the xy plane to 90\({}^{{}^{\circ}}\) pointing vertically. The separation angle \(\theta_{\mathrm{sep}}\) is measured from sensor 1 through the RSO to sensor 2. The sensor positions and the LOS from sensor to target allows the 3D target position computation. The equations to calculate the position estimates are defined below [47]:  
\[\mathrm{x_{t}}=\frac{\mathrm{x_{2}tan\ (\theta_{1})-x_{1}tan\ (\theta_{2})+(y_{1}-y_{2})tan\ (\theta_{1})tan\ (\theta_{2})}}{\tan\ (\theta_{1})-\tan\ (\theta_{2})} \tag{1}\]  
\[\mathrm{y_{t}}=\frac{\mathrm{y_{1}tan\ (\theta_{1})-y_{2}\ tan\ (\theta_{2})+(x_{2}-x_{1})}}{ \tan\ (\theta_{1})-\ \tan\ (\theta_{2})} \tag{2}\]  
\[\mathrm{r_{i}}=\sqrt{\left(\mathrm{x_{i}-x_{t}}\right)^{2}+\left(\mathrm{y_{i }-y_{t}}\right)^{2}} \tag{3}\]  
Figure 4: (**a**) Multi sensor RSO tracking for space-based scenario, (**b**) Multi sensor RSO tracking for ground-based scenario. Figure 3: Simplified conceptual illustration for ground-based scenario. \[z_{t}=\frac{r_{1}\tan(\varphi_{1})+z_{1}+r_{2}\tan(\varphi_{2})+z_{2}}{2} \tag{4}\]  
where:  
(x\({}_{\mathrm{L}}\)y\({}_{\mathrm{L}}\)z) = target RSO position coordinates,  
(x\({}_{\mathrm{L}}\)y\({}_{\mathrm{L}}\)z) = corresponding sensor position coordinates  
r\({}_{\mathrm{i}}\) = horizontal ranges from the x - y components of the corresponding sensor to the x and y components of the RSO,  
\(\varphi_{1}\) = corresponding sensor to RSO elevation angle,  
\(\theta_{1}\) = corresponding sensor to RSO azimuth angle. i = 1, 2 (number of sensors used to perform triangulation)  
These equations relate the target location to measurements of the sensor position and LOS from the sensors to the target, aiding the target position estimation. The error propagation equations corresponding to the respective position coordinates are derived in [48], with a key assumption that the error generated by each sensor follows a Gaussian distribution.