## 2 Optical Cone and Pendulum Scanning Imaging Method  
The principle of OCPSI (optical cone and pendulum scanning imaging) is illustrated in Figure 3, taking a single camera as an example. The OCPSI satellite operates on a circular low orbit, with \(V_{s}\). being the velocity of the satellite. The optical camera composed of single CMOS has an initial inclination angle before booting, and the optical axis and the nadir axis form an inclination angle \(\alpha\). During the imaging of the camera, the satellite rotates continuously around the nadir axis with an angular velocity \(\omega_{s}\). As shown in Figure 3, multiple frames of images are formed into a coverage area within the \(T_{1}\) time interval. The camera rotates continuously around the main point of the camera with an angular velocity \(\omega_{c}\) to form a coverage area. The imaging sequence of a coverage area is shown in Figure 4. The time interval \(T\) during which the satellite rotates once is the cone scanning period. The time interval \(T_{out}\) during which the camera oscillates once is the pendulum scanning period. Under the coupled motion condition of satellite rotation and camera pendulum, the camera ground trajectory of the OCPSI remote sensing satellite can completely cover the ground. There are many control strategies, which prove the possibility of attitude control of OCPSI satellites. For example, the quaternion-based feedback control law proposed by [PERSON] et al. and two reaction wheels mounted in a V shape proposed by [PERSON] can be used for attitude control of a satellite rotation [26; 27]. From the camera image plane to the ground target, it needs to go through four coordinate frame for mapping. In Figure 4, the image plane coordinate frame \(o_{p}x_{p}y_{p}z_{p}\) is defined as follows:  
1. The origin \(o_{p}\) is the center of the camera image plane;
2. The \(z_{p}\) axis coincides with the optical axis of the camera and points to the ground;
3. The \(y_{p}\) axis is perpendicular to the \(z_{p}\) axis in the camera image plane, pointing to the opposite of the satellite movement direction;
4. The \(x_{p}\) axis is perpendicular to the plane formed by the camera optical axis and the \(y_{p}\) axis and forms a right-handed coordinate frame with the \(y_{p}\) axis and the \(z_{p}\) axis. The camera coordinate frame \(o_{c}x_{c}y_{c}z_{c}\) is defined as:  
1. The origin \(o_{c}\) is the intersection of the camera optical axis and the nadir axis;
2. The \(z_{c}\) axis coincides with the camera optical axis and points to the ground;
3. The \(y_{c}\) axis is perpendicular to the \(z_{c}\) axis and points to the opposite direction of the satellite movement direction;
4. The \(x_{c}\) axis is perpendicular to the plane composed of the camera optical axis and the \(y_{c}\) axis, forming a right-hand coordinate frame with the \(y_{c}\) axis and the \(z_{c}\) axis. The satellite coordinate frame \(o_{s}x_{s}y_{s}z_{s}\) is defined as:  
1. The origin \(o_{s}\) is the intersection point of the camera optical axis and the nadir axis;
2. The \(z_{s}\) axis coincides with the nadir axis and points to the ground;  
Figure 4: OCPSI axis mapping. Figure 3: Illustration of an OCPSI satellite. The \(y_{s}\) axis is perpendicular to the \(z_{s}\) axis and points to the opposite direction of the satellite movement direction;
4. The \(x_{s}\) axis is perpendicular to the plane formed by the camera optical axis and the \(y_{s}\) axis, forming a right-handed coordinate frame with the \(y_{s}\) axis and the \(z_{s}\) axis. The geographic coordinate frame \(o_{G}x_{G}y_{G}z_{G}\) is defined as: 1. The origin \(o_{G}\) is the intersection of the nadir axis and the ground; 2. The \(z_{G}\) axis coincides with the zenith axis and points to the zenith; 3. The \(y_{G}\) axis is perpendicular to the \(z_{G}\) axis and points to the direction of satellite motion; 4. The \(x_{G}\) axis is perpendicular to the plane formed by the optical axis of the camera and the \(y_{G}\) axis and forms a right-hand coordinate frame with the \(y_{G}\) axis and the \(z_{G}\) axis.