# Line of Sight and Image Motion Compensation for Step and Stare Imaging System
## 3 LOS and Motion Compensation Kinematics and Algorithm
### Image Motion Compensation  
Three factors cause the image motion in the system: the linear speed of the aircraft, the angular rate of the aircraft, and the angular rate of the gimbals. As mentioned above, the image motion compensation is implemented by the mirror rotating around the corresponding axis. Therefore, it is necessary to convert various types of image motions into angular rates of the corresponding axes in the mirror coordinate. According to the principle that the linear speed is vertical to the LOS of a point that lies on the LOS vector, the equations are developed for the linear speed of the aircraft. Dividing the orthogonal components by the range to the point, then yields the two angular LOS rates orthogonal to the LOS that are required for control [18].M3 and M4 are fixed mirrors, which are used to reflect the LOS into the detector and are not used for compensation. So the coordinate transformation here can only be converted to the M2 mirror, that is T3 and T4 were ignored in Equation (5). Converting the linear speed of the aircraft, (V\({}_{\text{Nx}}\), V\({}_{\text{Ny}}\), V\({}_{\text{Nz}}\))\({}^{\text{T}}\) in the geographic coordinate frame to that in the mirror coordinate system is:  
\[\left( \begin{array}{c}\text{V}_{\text{Mx}}\\ \text{V}_{\text{My}}\\ \text{V}_{\text{Mz}}\end{array} \right)=C_{\text{G}}^{\text{M2}}\times C_{\text {b}}^{\text{G}}\times C_{\text{N}}^{\text{b}}\left( \begin{array}{c}\text{V}_ {\text{Nx}}\\ \text{V}_{\text{Ny}}\\ \text{V}_{\text{Nz}}\end{array} \right) \tag{15}\]  
The resulting angular rates in the mirror system are:  
\[\omega_{\text{VMy}}=\frac{\text{V}_{\text{Mx}}}{2\text{L}} \tag{16}\]  
\[\omega_{\text{VMx}}=\frac{-\text{V}_{\text{My}}}{2\text{L}} \tag{17}\]  
The transformation of the angular rates of the aircraft and the gimbal to the mirror coordinate can be expressed as follows:  
\[\left( \begin{array}{ccc}1&0&0\\ 0&-\sin 2\hat{\delta}&\cos 2\hat{\delta}\\ 0&\cos 2\hat{\delta}&\sin 2\hat{\delta}\end{array} \right)\left(\left( \begin{array}{ ccc}1&0&0\\ 0&\cos\hat{\beta}&-\sin\hat{\beta}\\ 0&\sin\hat{\beta}&\cos\hat{\beta}\end{array} \right)\left(\left( \begin{array}[] {ccc}\cos\alpha&0&-\sin\alpha\\ 0&1&0\\ \sin\alpha&0&\cos\alpha\end{array} \right)\left( \begin{array}{c}\omega_{\text {bx}}\\ \omega_{\text{bx}}\\ \omega_{\text{bx}}\end{array} \right)+\left( \begin{array}{c}0\\ \hat{\alpha}\\ 0\end{array} \right)\right)+\left( \begin{array}{c}\hat{\beta}\\ 0\\ 0\end{array} \right)\right) \tag{18}\]  
where \(\hat{\alpha}\) and \(\hat{\beta}\) are the angle rates of the gimbal, and \(\hat{\delta}\) is the angle rate of the M1 mirror.\(\omega_{\text{bx}}\), \(\omega_{\text{by}}\), and \(\omega_{\text{bx}}\) are the angle rates of roll, pitch and yaw respectively in the aircraft body coordinate frame. After calculation, the compensation velocities of M1 and M2 mirrors are as follows:  
\[\omega_{\text{Mlx}}=\dot{\beta}+\omega_{\text{bx}}\cos\alpha+\omega_{\text {bz}}\sin\alpha+2\dot{\delta} \tag{19}\]  
Figure 5: Block diagram of gimbaled control. \[\omega_{\rm M2r}=2\dot{\lambda}+\cos 2\delta\big{[}\cos\beta\big{(}\dot{\alpha}+ \omega_{\rm yy}\big{)}+\sin\beta(\omega_{\rm bra}\cos\alpha+\omega_{\rm x}\sin \alpha)\big{]}-\sin 2\delta\big{[}\sin\beta\big{(}\dot{\alpha}+\omega_{\rm yy}\big{)}- \cos\beta(\omega_{\rm bra}\cos\alpha+\omega_{\rm x}\sin\alpha)\big{]} \tag{20}\]  
where \(\dot{\lambda}\) is the angle rate of the mirror of M2. To ensure the image speed on the focal plane is zero, it is only necessary to ensure that Equations (19) and (20) are equal to zero.