# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 2 Kinematic Modeling of Three-Axis SOTM Antenna  
The three-axis SOTM antenna structure under study in this paper is illustrated in Figure 1. The entire system is installed on a fixed base, which is firmly attached to the carrier. Typically, bearings are employed to drive the base unit for rotational movement and mechanical linkage. Figure 2 displays a physical image of a three-axis SOTM antenna manufactured by SATPRO M&C Tech Co., Ltd. (located in Xi'an, China). The three-axisantenna system includes an azimuth sub-system, a roll sub-system and a pitch sub-system. Each of the three sub-systems has a rotating axis, which is used to adjust the antenna attitude, isolate the carrier disturbance and keep the antenna stable. Due to the limitation of the mechanical structure of the SOTM system, there is an angle less than 90\({}^{\circ}\) between the roll axis and the system base (generally 30\({}^{\circ}\)-35\({}^{\circ}\)) [23]. Brushless DC motors are usually used to drive the movement of each axis through gears or belt drives. The antenna used to transmit and receive satellite signals is installed on the pitch sub-system and moves with the pitch axis. The system uses a high-gain antenna to achieve wideband multimedia satellite communication. When the SOTM antenna tracking satellite error exceeds a certain range, it may interfere with adjacent satellites, causing the antenna to fail to transmit radio frequency signals. Therefore, the antenna pointing fluctuation must be kept within a sufficiently small range [24]. When measuring the antenna direction, a sensing system with sufficiently high precision and high sampling rate must be selected to quickly and accurately feed location and attitude information to the controller. The controller adjusts the antenna attitude by driving the motor, thereby ensuring that the antenna always points to the target satellite [25]. During continuous communication with the satellite, the servo system needs to continuously calculate the rotation angles of the three motors. Figure 1: Structure of the three-axis SOTM antenna. Figure 2: Three-axis SOTM antenna manufactured by SATPRO (located in Xi’an, China). The three-axis SOTM antenna structure has three rotating axes, namely the azimuth axis, the pitch axis and the roll axis. In this section, we introduce an NMDH method based on the modified DH method [26; 27; 28; 29] to establish the kinematic model of the three-axis SOTM antenna. Based on the mechanical structure of the three-axis SOTM antenna, the antenna coordinate system configuration is obtained as shown in Figure 3. Among them, the coordinate system 0 is the reference coordinate system, which is firmly attached to the base. When the rotation angle of the azimuth axis is 0, the coordinate system 1 coincides with the reference coordinate system, the z-axis of the reference coordinate system coincides with the axis of the azimuth axis and \(Z_{1}\) represents the azimuth axis. Beta is the fixed angle between the roll axis and the base. The existence of this angle can improve the rigidity of the mechanical structure to improve the mechanical performance of the antenna, but it will increase the difficulty of modeling the antenna kinematic model. Since beta (\(\beta\)) is a fixed angle, it is difficult to directly use the modified DH method for system modeling. For this reason, a virtual coordinate system 2 is added in the coordinate system shown in Figure 3. In addition, due to the spatial relationship characteristics between \(Z_{2}\) and \(Z_{4}\), \(Z_{4}\) and \(Z_{6}\) in Figure 3, the correct link length cannot be obtained along \(Z_{4}\) and \(Z_{6}\) respectively. Therefore, auxiliary coordinate system 3 and 5 are added in Figure 3 respectively. In Figure 3, the origins of the virtual coordinate system 2 and the auxiliary coordinate system 3 coincide, and the auxiliary coordinate system 5 and the origins of the coordinate system 6 coincide.