# A Predefined-Time Control for the Laser Acquisition in Space Gravitational Wave Detection Mission
## 5 Numerical Simulations
### Initial Conditions  
The simulator consisted of three individual satellites and a ground station and every satellite contained a multibody system nonlinear dynamic model, a noise model and a controller, connected to the ground station via data communication. The nonlinear dynamic model for the platform with 10 DOFs included a satellite and telescopes, while the dynamics of the test masses were neglected which are regarded as initial sensors and provide high-precision attitude information for spacecraft. The interaction between the test masses and satellite was regarded as a disturbance to add to the dynamic model of the platform. The noise of actuators and sensors which was discussed in Section 2.3 was considered in the dynamic model. We assumed the uncertainty half-cone was 13.6 \(\upmu\)rad while the laser beam half-cone was about 1.5 \(\upmu\)rad [24]. Then we could obtain the guidance law for the T1 and T2 of satellite A. The detail of the guidance law is show in Figure 10.  
We assumed that the waiting time at each point was 6s which was enough for the response of the opposite satellite and the predefined stable time for the controller was 5s. The initial direction of SC for T1 of SA was [5.9 \(-\)2.8] \(\times\) 10\({}^{-6}\) rad and SB for T2 of SA was [\(-\)6.2 7.6] \(\times\) 10\({}^{-6}\) rad. The initial direction of SC for T2 of SC was [6 \(-\)10.4] \(\times\) 10\({}^{-6}\)\(\upmu\)rad. The total simulation time was 1000 s, and the step size was 0.1 s.  
Figure 10: Guidance law for T1 and T2 on satellite A. The ends of L2/L2\({}^{\prime}\) are the T2 of SA and T1 of SB, the same as L1/L1\({}^{\prime}\) link. The T2 of SA was the master end to scan the uncertainty cone of SB with the hexagonal scanning guidance law is shown in Figure 15. Figure 16 shows the angle error of T2 on SA. Figure 17 shows the T1 of SB as the slave end to stare the uncertainty cone of SA and to wait for the laser signal from the T2 of SA. When it received the laser signal, it controlled its T1 to  
Figure 16: Ponting Angle Error of T2 on SA. Figure 17: Ponting Angle of T1 on SB. Figure 18: Ponting Angle Error of T1 on SB. point the direction of the laser beam, this occurred at about 256 s from Figure 18. When the T2 of SA finished the scanning, it found the laser beam from T2 of SC and controlled its T2 to point SA, then the laser link between SA and SC namely L2/L2\({}^{\prime}\) was stablished at about 735 s which is same to the L1/L1\({}^{\prime}\). It can be observed from the partial enlargement in Figures 15 and 17 that the precision of the results was better than 0.1\(\upmu\)rad and the partial enlargements in Figures 16 and 18 indicate that the controller could meet the requirements where the settling time was less than 5 s.
3. The Establishment of L3/L3\({}^{\prime}\) The establishment of L3/L3\({}^{\prime}\) is showed in Figures 19-22. During the establishment of laser link, these three satellites were in constant communication with the ground station, so when SA received the laser signal, it sent the direction information of SB and SC to the ground station. The ground station processed the information and calculated the guidance law to cover the uncertainty cone of SC, then sent it to SB. The purpose of this operation was to reduce the acquisition time. Then SB controlled its T2 and scanned the processed uncertainty cone of SC with the new scanning guidance law while SC stared the uncertainty cone of SA waiting for the laser beam. Figure 19 shows the time series of the pointing angle of T2 on SB. We found that the guidance law was simpler and the acquisition time was shorter. Figure 21 shows the time for the T1 of SC to receive the incoming laser beam was about 765 s. The partial enlargements in Figures 19-22 indicate  
Figure 19: Ponting Angle of T2 on SB. Figure 20: Ponting Angle Error of T2 on SB. that the precision of the results and controller performance could satisfy the requirements of acquisition.