## 3 The Design of OCPSI Mode
### Single-Camera Constant Speed OCPSI Mode  
In the single-camera cone-scanning imaging mode, the blank part of the Semiring I formed by the current cone scan can be covered by the Semiring I of the next ring. As shown in Figure 10, after a reasonable selection of the satellite's cone scanning speed \(\omega_{s}\), the second ring can completely cover the blank area of the first ring. First, we calculate the sub-satellite velocity of the satellite. In Figure 10, \(V_{s}\) is the relative instantaneous velocity of the satellite sub-satellite point with respect to the ground and is calculated from the orbital height \(H\) as  
\[V_{s}=\sqrt{\frac{GM}{(R+H)}}\times\frac{R}{R+H} \tag{9}\]  
where \(GM\) is the gravitational constant. Second, we derive the satellite cone scanning velocity from the overlap between two cone scanning cycles and the satellite sub-satellite velocity. It can be seen from Figure 10 that when the satellite cone scanning moves once, the distance traveled by the satellite sub-satellite point is \(d_{sate}\). When the distance traveled by the sub-satellite point is equal to the width of the hoop--that is, \(d_{sate}=d_{hoop}\), the image can cover the ground target perfectly. Since the hoop belt is composed of image blocks and there are jagged protrusions  
Figure 10: Blank part filling diagram. Figure 9: Region division of a ring. on the edge of the hoop belt, the distance traveled by the satellite sub-satellite point is slightly less than the width of the hoop belt, that is  
\[d_{sate}=(1-\eta_{s})\times d_{hoop} \tag{10}\]  
where \(\eta_{s}\) is the overlap ratio of the width of the cone-scanning hoops. The distance \(d_{sate}\) traveled by the sub-satellite point that the satellite cone scanning has moved once is  
\[d_{sate}=V_{s}\times T_{s} \tag{11}\]  
where \(T_{s}\) is the period of the satellite cone scanning. By Equations (8)-(11), the coverage problem of the satellite cone scanning is formulated as  
\[V_{s}\times T_{s}=(1-\eta)\times(r_{N}-r_{1}) \tag{12}\]  
Therefore, the period of the satellite cone scanning and the angular velocity of the satellite cone scanning can be designed by solving the Equation (12) of the satellite cone scanning coverage problem. \[T_{s}=(1-\eta)\times\frac{(r_{N}-r_{1})}{V_{s}} \tag{13}\]  
\[\omega_{s}=\frac{2\pi}{T_{s}}=\frac{2\pi V_{s}}{(1-\eta)\times(r_{N}-r_{1})} \tag{14}\]  
Third, we deduce the camera pendulum scanning velocity according to the satellite sub-satellite velocity, the satellite cone scanning velocity, and the overlap between two pendulum scanning cycles. We first consider pendulum motion imaging separately. In Figure 10, the image coverage area formed by imaging in one pendulum scanning period is a square, as shown in detail in Figure 11.  
The number of image frames in a pendulum motion period is  
\[f_{pend}=2\times(N-1) \tag{15}\]  
Figure 10:0 Image coverage in 2 pendulum cycles. where \(N\) is the frame number of the hoop belt. Since different pendulum scanning angles are generated during scanning, the ground coverage in the vertical orbit direction of each frame of image is different, so we use angle overlap to plan ground coverage. Since the pendulum scanning direction is vertical to the circumference and the direction is always changing, when the pendulum scanning speed is in the positive and negative directions along the orbit, the image movement speed is at the maximum and minimum. We perform the analysis in the vertical orbit direction. In order to ensure complete coverage and sufficient overlap of the image, we define the pendulum scanning angle overlap ratio \(\eta_{frame}\):  
Figure 10:1  
where \(\theta_{overlap}\) is the overlap angle of two adjacent frames. From this, we can get the angle change of every two frames of images:  
Figure 10:2  
Then, the pendulum scanning angle of the nth frame image is expressed as  
\[\beta=