# A Predefined-Time Control for the Laser Acquisition in Space Gravitational Wave Detection Mission
## 2 Acquisition Scheme
### Acquisition Procedure  
* Scanning for Uncertainty Cone  
The first step of acquisition is to scan the uncertainty cone of the opposite satellite. This configuration we proposed above has a huge advantage for the acquisition phase which is that the two telescopes can scan their corresponding uncertainty cone at the same time. That means a much shorter acquisition time. SA will turn on its laser and use the ground-provided navigation data to control T1 to scan the uncertainty cone of SC meanwhile controlling T2 to scan the uncertainty cone of SB with the same hexagonal scanning guidance law but a different scan direction. The hexagonal scanning guidance law has been proved to be the most efficient discrete stepping scan method [19]. The purpose of the same law but a different direction is hopefully to reduce the disturbance of the satellite which is caused by the moving of telescope as much as possible. The detailed description is in Figure 4.  
* The Establishment of Two-way Laser Links  
SA will perform scanning until the laser beam covers all the uncertainty cones of SB and SC. The SB and SC will keep their corresponding telescope staring at the uncertainty cone of SA at the same time and wait for the arrival of the laser beam from SA. Obviously, the FOV of the array detector must cover the whole uncertainty cone of SA to ensure that the laser signal from SA can be received. The laser beam will arrive the array detector at a certain time, then SB and SC will control their telescopes to point the direction of the laser beam according to the displacement information of the laser spot detected by the array detector and turn on their laser to illuminate SA. The laser spot on the array detector is shown in Figure 5. Because of the long distance between the three satellites, the laser beam will be very weak when arriving at the array detector of SA, so the signal will be drowned in noise. SA will turn off its laser when it finishes the scanning so that the laser signal from SB and SC can be detected. Then SA will control its two telescopes towards SB and SC,respectively, according to the direction of the incoming laser beam and turn on its laser. Finally, the two two-way laser links L1/L1\({}^{\prime}\) and L2/L2\({}^{\prime}\) will be established. Figure 4: Schematic diagram of scanning for the uncertainty cone. Where initial LOSA2C and LOSA2B denote the initial line of sight of the T1 and T2 of SA towards SC and SB, respectively; initial LOSC2A and initial LOSB2A denote the initial line of sight of the T2 of SC and T1 of SB towards SA; \(\theta_{ABS}\) and \(\theta_{ACS}\) denote the angle offset between initial LOS of SA and actual direction toward SC and SB, respectively; \(\theta_{CAS}\) denotes the angle offset between the initial LOS of SC and the actual direction toward SA; \(\theta_{BAS}\) denotes the angle offset between the initial LOS of SB and the actual direction toward SA. Figure 5: Signal detection on the array detector. The \(\theta_{ACS}\) and \(\theta_{ABS}\) can be orthogonally decomposed to (\(\theta_{ACp}\), \(\theta_{ACy}\)) and (\(\theta_{ABp}\), \(\theta_{ABy}\)), respectively. The relationship between them can be is represented as:  
\[ \begin{cases}\theta_{ABS}=\sqrt{\theta_{ABp}^{2}+\theta_{ABy}^{2}}\\ \theta_{ACS}=\sqrt{\theta_{ACp}^{2}+\theta_{ACy}^{2}}\end{cases}  \tag{1}\]  
Furthermore, according to Figure 5 we have:  
\[ \begin{cases}\tan\theta_{ABp}=\frac{x}{f}\\ \tan\theta_{ABy}=\frac{y}{f}\end{cases}  \tag{2}\]  
where \(f\) denotes the equivalent focal length of the telescope. Similarly, the \(\theta_{ABp}\) and \(\theta_{Aly}\) can also be calculated. * The Establishment of the Third Two-way Laser Links  
When the L1/L1\({}^{\prime}\) and L2/L2\({}^{\prime}\) are established, it is easier to establish the third one according to the high-accuracy attitude knowledge of the previous two links which is already available. As shown in Figure 6, according to Equations (1) and (2), we have:  
\[ \begin{cases}\varphi_{1}\approx\theta_{ACp}\\ \varphi_{1}\approx\theta_{ABp}\end{cases}