# A Predefined-Time Control for the Laser Acquisition in Space Gravitational Wave Detection Mission  
[PERSON]  
1 School of Aeronautics and Astronautics, Sun Yat-sen University, Shenzhen 518038, China 11 School of Physics and Astronomy, Zhuhai Campus, Sun Yat-sen University, Zhuhai 519082, China 11 School of Physics and Astronomy, Zhuhai Campus, Sun Yat-sen University,  
We collected some main GW detection programs all over the world which are listed in Table 1.  
From Table 1 we can see that these space GW detection programs consist of at least three satellites forming an approximate equilateral triangle constellation with very large arm lengths. That is to say, there are three two-way laser links which need to be established. With the development of space technology, the establishment of inter-satellite laser links has become a research hotspot in the field of space laser communication, gravity measurement and space GW detection. There are many studies on the acquisition phase of inter-satellite laser links. [PERSON] [17] proposed four acquisition modes for satellite laser communication, namely: Stare/Stare mode, Stare/Scan mode, Scan/Scan mode and Scan/Stare mode. They are distinguished by the relationship among the laser divergence angle (LDA), uncertainty cone angle (UCA) and field of view (FOV), see Table 2. Stare/Stare mode is based on the condition that the LDA and FOV all are bigger than the UCA so that the acquisition sensor will detect the signal immediately when the laser is on. In, this mode, it is easy to establish the laser link, but it requires large laser power and ca not be used for long distances, so it is rarely used in practice. The Stare/Scan mode is based on the condition that the LDA is smaller than UNA and the FOV is bigger than the UCA, so the laser beam of the master one needs to scan the uncertainty cone with the guidance law and the slave one needs to keep still and stare at the uncertainty cone to wait for the signal. This is the most commonly used mode in practice. The Scan/Stare mode is when the LDA is bigger than UCA and UCA is bigger than FOV, so the laser beam of the master one needs to keep still and the slave one scans the uncertainty cone to search the signal. This mode also needs large laser power and is rarely used in practice. The Scan/Scan mode is when the laser beam of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Orbit Type** & **Mission Concept** & **Number of Satellites** & **Orbit Configuration** & **Arm Length** \\ \hline \multirow{5}{*}{Solar Orbit} & LISA [5] & 3 & Earth-like solar orbits with 20\({}^{\circ}\) lag & 2.5 \(\times\) 10\({}^{6}\) km \\  & ASTROD-GW [8] & 3 & Near Sun-Earth L3, L4, L5 points & 2.6 \(\times\) 10\({}^{8}\) km \\  & DECIGO [9] & 12 & Earth-like solar orbits & 1 \(\times\) 10\({}^{3}\) km \\  & Big Bang Observer [10] & 12 & Earth-like solar orbits & 5 \(\times\) 10\({}^{4}\) km \\  & ALIA [11] & 3 & Earth-like solar orbits & 5 \(\times\) 10\({}^{5}\) km \\  & TAJII [12] & 3 & Earth-like solar orbits & 3 \(\times\) 10\({}^{6}\) km \\ \hline \multirow{5}{*}{Earth Orbit} & OMEGA [13] & 6 & 6 \(\times\) 10\({}^{5}\) km height earth orbit & 1 \(\times\) 10\({}^{6}\) km \\  & gLISA/GEOGRAWI [14] & 3 & Geostationary orbit & 7.3 \(\times\) 10\({}^{4}\) km \\ \cline{1-1}  & GADFLI [15] & 3 & Geostationary orbit & 7.3 \(\times\) 10\({}^{4}\) km \\ \cline{1-1}  & LAGRANGE [16] & 3 & Earth-Moon L3, L4, L5 points & 6.6 \(\times\) 10\({}^{4}\) km \\ \cline{1-1}  & TIANQIN [6] & 3 & 1 \(\times\) 10\({}^{5}\) km height earth orbit & 1.7 \(\times\) 10\({}^{5}\) km \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main space GW detection programs all over the world. Figure 1: Orbital configuration of the LISA mission and the TianQin project. (**a**) LISA mission [5]. (**b**) TianQin project. the master one and the slave one all need to perform scans to search for each other. This mode will reduce the probability of acquisition and is rarely used in practice.