## 3 The Design of OCPSI Mode
### Derivation of Exposure Time  
In order to reduce the influence of image shift, we need to plan the exposure time reasonably. Cone and pendulum scanning imaging uses motion to expand ground coverage, but this has a greater impact on image motion. The cone and pendulum scanning imaging mode adopts CMOS area array imaging. During the exposure time, the image motion amount is small enough, and the image motion influence can be ignored. The image motion amount needs to satisfy  
\[t_{e}V_{f}<\mu \tag{31}\]  
where \(t_{e}\) is the exposure time, and \(V_{f}\) is the image motion speed. The orbital motion, pendulum scanning motion, cone scanning motion, and the Earth's rotation in the cone and pendulum scanning imaging mode all affect the image motion speed. If the image motion of the outermost image satisfies the requirement, then the image motion of entire images satisfies the requirement. Assuming that Figure 18 is the outermost imaging time, the outermost point of the coverage area is \(A\), and the pixel point on the corresponding detector image surface is \(A\). Then, the image motion velocity can be divided into the image motion component velocity \(V_{A^{\prime}V_{s}}\) caused by orbital motion, the image motion component velocity \(V_{A^{\prime}\omega_{s}}\) caused by cone scanning motion, the image  
Figure 16: The ground coverage image at time \(t_{n}\). Figure 17: Dual-camera variable Speed OCPSI Mode ground overlay schematic diagram: improvement. motion component velocity \(V_{A^{\prime}\omega_{c}}\) caused by pendulum scanning motion, and the image motion component velocity \(V_{A^{\prime}V_{c}}\) of the image motion caused by the Earth's rotation. The velocity of orbital motion at the sub-satellite point is \(V_{s}\). The speed and direction of the orbital motion of point \(A\) in the coverage area are the same as the satellite sub-satellite point speed \(V_{s}\). The relationship between the component velocity of image motion caused by orbital motion and the velocity caused by orbital motion of point \(A\) in the coverage area is  
\[\frac{V_{A^{\prime}V_{s}}}{V_{s}}=\frac{f}{H/\cos(\gamma+\frac{FOV}{2})} \tag{32}\]  
Then, the component velocity of image motion caused by orbital motion is  
\[V_{A^{\prime}V_{s}}=\frac{f\cos(\gamma+\frac{FOV}{2})}{H}\times V_{s} \tag{33}\]  
The direction is the opposite direction along the orbit. The angular velocity of the cone scanning motion corresponds to the linear velocity of point \(A\) in the coverage area:  
\[V_{A\omega_{s}}=\omega_{s}\times r_{N} \tag{34}\]  
Similarly, it can be deduced from the relationship between Equations (27) and (28) that the component velocity of the image motion caused by the cone scanning motion is  
\[V_{A^{\prime}\omega_{s}}=\frac{f\cos(\gamma+\frac{FOV}{2})}{H}\times V_{A \omega_{s}}=\omega_{s}f\times\frac{r_{N}\cos(\gamma+\frac{FOV}{2})}{H}=\omega _{s}fsin(\gamma+\frac{FOV}{2}) \tag{35}\]  
The direction is the circular tangent direction of the cone scanning. Figure 18: Image velocity analysis. Figure 19 is a cross-sectional view of the image plane \(A^{\prime}\) point. The instantaneous linear velocity at the point \(A^{\prime}\) of the image plane during the pendulum scanning motion is  
\[V^{\prime}_{\omega_{c}}=\omega_{c}\times\frac{f}{\cos(\frac{FOV}{2})} \tag{36}\]  
The relationship between the image motion velocity caused by the pendulum scanning motion and the instantaneous linear velocity caused by the pendulum scanning motion is shown in Figure 14, which is expressed as  
\[V_{A^{\prime}\omega_{c}}=V^{\prime}_{\omega_{c}}\times\cos(\frac{FOV}{2})= \omega_{c}f \tag{37}\]  
The linear velocity of the Earth's rotation at point \(A\) in the coverage area is \(V_{e}=0.463\:\mathrm{km/s}\), and the component velocity of the image movement caused by the Earth's rotation is  
Figure 16:0  
The direction is the vertical orbit direction. According to Equations (27)-(33), the four image motion component velocities are combined into two components along the orbit and the vertical orbit.