## 2 Theoretical Basis
### Image Motion Compensation Model of Secondary Mirror Tilt Motion  
The changes of camera position and field of view during exposure time are shown in Figure 1, where a, b, c and d are the object points on the ground. The camera moves from position 1 to position 2 during the exposure time. The solid black line is the light path from the object point to the camera before exposure, and the blue dotted line is the light path from the object point to the camera after exposure.For the same object point, there is a deviation of the field of view from the optical system before and after exposure. For example, point b is the on-axis field of view for the optical system before exposure, and the off-axis field of view for the optical system after exposure, so the position of the image point relative to the detector is shifted after exposure. In order to ensure that the positions of the same object point imaging relative to the detector before and after the camera movement are the same, it is necessary to ensure that the exit angles of the light rays passing through the optical system of the same object point before and after exposure are the same.  
An image motion compensation model of secondary mirror tilt motion is shown in Figure 2. The black light in Figure 2 is the light incident to the optical system from the object point before exposure, and the blue light is the light incident to the optical system from the same object point after exposure. PM is the primary mirror, SM is the secondary mirror, \(u\) is the angle between the exit ray and the optical axis ray (OAR), d is the surface distance, \(y\) is the height of the light on the surface, and i is the inclination angle of the secondary mirror. If the change of the field of view angle of the same object point before and after exposure is \(\alpha\), and the angular magnification of the front correction lens is \(\beta_{1}\), then the angle of the light on the primary mirror changes as \(\alpha\cdot\beta_{1}\). When the image motion compensation is not performed, the secondary mirror is at pose 1, the exit angles of the black light and the blue light through the optical system are different, that is, \(u_{1}\
eq u_{2}\), and the distance of the exit lights (blue ray and orange ray) on the image plane is the amount of image shift \(\delta\). The secondary mirror is tilted so that the lights emitted from the same object point through the optical system before and after exposure are the same, that is, the blue light in the Figure 2, and the image motion compensation is realized. Therefore, the amount of image shift caused by the movement of the camera from position 1 to position 2 during the exposure time can be compensated by the tilt movement of the secondary mirror from pose 1 to pose 2.  
Based on the above analysis, it can be concluded that the current principle of image movement compensation is that the light from different fields of view is imaged at the same position on the image plane by adjusting the pose of the secondary mirror. In order to ensure the image quality of the camera, it is necessary to reduce the difference of the image  
Figure 1: The changes of camera position and field of view during exposure time.  
Figure 2: Image motion compensation model of secondary mirror tilt motion.  
quality of different fields of view. However, in the process of image motion compensation, the pose adjustment of the secondary mirror will change the aberration distribution in the effective field of view. Therefore, analyzing the aberration field distribution during the pose adjustment process is essential for optimizing the catadioptric optics with low sensitivity of the secondary mirror.