# Active Control Loop of the BOROWIEC SLR Space Debris Tracking System
## 4 Experiment
### Photon Event Active Control Loop  
Due to the unavailability of a scientific laser source and a photon detector, the photon event active control loop described in Section 3.2 was partially simulated. For this purpose, a demonstration CW laser was used to simulate the emission of a pulsed scientific laser. The use of a laser beam divergence corrector and a photon event dataset were also simulated numerically. An illuminated object from the LEO region in the form of the CryoSat-2 satellite was intentionally selected for the experiment. Figure 10 shows the object tracking process with the simultaneous use of two SLR systems located at the SRC PAS Borowiec Astrogeodynamic Observatory.  
The control algorithm uses the properties of the Archimedean spiral defined by the formula in polar coordinates \((r,\varphi)\):  
\[r=a+b*\varphi, \tag{6}\]  
where \(a\) moves the center point of the spiral outward from the origin (positive \(a\) toward \(\varphi=0\), negative \(a\) toward \(\varphi=\pi\)), while \(b\) controls the distance between spiral loops. It follows from the above that the position of the next spiral point is proportional to the angle \(\varphi\) over time. Using this dependence, it was possible to make a modification consisting of taking into account the value of the beam divergence (laser spot size for a given altitude) and the radius (\(r_{S}\)) of the search area of the orbital object \(\kappa\). In this way, it was possible to define the number of loops depending on the search area and the degree of coverage by the laser spot size. Such a dependency in the local polar coordinate system \(\mathbb{R}^{2}\) can be described as:  
\[X=X_{0}(Az_{\kappa}(t_{n}))+cos(\varphi+\mathcal{R})*\left[\varphi*\frac{r_{S} }{\left(\frac{r_{S}}{2\pi r_{S}\varphi}\right)}\right] \tag{7}\]  
Figure 10: Laser tracking of the CryoSat-2 satellite using two SLR systems at CBK PAN.  
\[Y=Y_{0}(El_{\kappa}(t_{n}))+sin(\varphi+\mathcal{R})*\left[\varphi*\frac{r_{S}}{ \left(\frac{r_{S}}{2\pi r_{S}}\right)}\right] \tag{8}\]  
where \(X_{0},Y_{0}\) are the origin of the coordinate system, \(Az_{\kappa}(t_{n}),El_{\kappa}(t_{n})\) are the theoretical angle values of the object \(\kappa\) in the \(t_{n}\) moment of time, \(\varphi\) is the the processed angle of the spiral, \(\mathcal{R}\) is the possible rotation of the spiral, \(r_{S}\) is the search area radius, and \(S_{\theta}\) is the spot size radius at the orbital altitude. Each successive iteration of the algorithm is dependent on an offset proportional to the spot size at the orbital altitude. The steps of the algorithm also depend on the time interval \(\Delta t_{R_{\kappa}}\), which is necessary to obtain the photon trace (\(E_{x}\)) scattered from the orbital object. The operation of the algorithm was interrupted when a photon event was detected, according to the procedure described in Section 3.2. The operation of such a modified algorithm is shown in Figure 11.  
Using the above modification, it was possible to define the search area radius (\(r_{S}\)) of the object \(\kappa\) depending on the \(\zeta_{x}\) ephemeris uncertainty expressed in \(arcmin\). The scaling of the number of loops and the angle offset were performed automatically according to Formulas (7) and (8), so it is possible to use this solution regardless of the orbital region knowing only the distance to the orbital object and the laser beam divergence. The advantage of this solution is the duality of using the algorithm to correct the telescope's movement during the object tracking process or to control of movable prismatic head in the XY plane system, which can be used behind the laser source.