## 4 Image Motion Modeling and Analysis
### Residual of Image Motion Compensation  
#### 4.2.1 Residual Model Establishment  
The linear velocity of the image motion will vary with the position on the sensor plane, while the target roll angle or pitch angle is a nonzero value. The residual of the IMC had to be analyzed quantitatively so that the exposure parameters could be set accordingly to limit the residual of the IMC. According to the image motion model established in the previous section, the component of the image motion velocity along the \(x_{S}\) axis plays the  
Figure 8: Maximum deviation of the IMLV with the target roll and pitch angles.  
Figure 9: Magnitude error of the maximum deviation of the IMLV with the target roll and pitch angles.  
dominant role. Therefore, the servo system of the three-axis ISP will rotate the pitch axis to compensate the image motion on the \(x_{S}\) axis.  
The image motion angular rate of \(O_{S}\) about the pitch axis can be expressed as in Equation (14). \(\omega_{F}\) is set as the given value for the pitch axis servo system during the imaging process to compensate for the image motion:  
\[\omega_{F}=\frac{r_{X}\cdot\omega_{D}}{\left(\left|r_{O_{d}}^{S}\right|\cdot \sin(\theta_{O_{S}})\right)} \tag{14}\]  
where, \(\omega_{F}\) is the image motion angular rate of \(O_{S}\) about the pitch axis, \(r_{O_{d}}^{S}\) is the direction vector of the image motion's linear velocity of \(O_{S}\), \(r_{O_{d}}^{S}=v^{S}\times r_{O_{S}}^{S}\times\left[\begin{array}{ccc}0&0&1 \end{array}\right]^{T}=\left[\begin{array}{ccc}r_{X}&r_{y}&r_{z}\end{array} \right]^{T}\), \(r_{O_{S}}^{S}\) is the direction vector of \(O_{L}\overset{\rightarrow}{O_{S}}\), \(\theta_{O_{S}}\) is the intersection angle for the image motion's linear velocity of \(O_{S}\) and \(\overset{\rightarrow}{O_{L}}O_{S}\) which can be computed in analogy with \(\theta_{A_{S}}\), and \(\omega_{D}\) is the image motion angular rate of object D which can be computed by analogy with \(\omega_{A}\).  
The angular rate of \(A_{S}\) while rotating the pitch axis at \(\omega_{F}\), represented by \(\omega_{A_{S}}^{IMC}\), can be described as in the formula below:  
\[\omega_{A_{S}}^{IMC}=\left[\begin{array}{ccc}\omega_{X}&\omega_{y}&\omega_{ z}\end{array}\right]^{T}=R_{y}^{pitch}(\beta)\cdot R_{x}^{roll}(\alpha)\cdot \left[\begin{array}{ccc}0&\omega_{F}&0\end{array}\right]^{T} \tag{15}\]  
The linear velocity of \(A_{S}\) corresponding to \(\omega_{A_{S}}^{IMC}\), represented by \(v_{A_{L}}^{S}\), is expressed as  
\[v_{A_{L}}^{S}=R_{x}^{roll}(\alpha)^{T}\cdot R_{y}^{pitch}(\beta)^{T}\cdot \left[\begin{array}{ccc}\omega_{y}\cdot\left|O_{L}^{\overset{\rightarrow}{ \rightarrow}{A_{S}}}\right|&\omega_{x}\cdot\left|O_{L}^{\overset{\rightarrow}{ \rightarrow}{A_{S}}}\right|&0\end{array}\right]^{T} \tag{16}\]  
According to the vector cross product law, the direction of the linear velocity of the IMC for \(A_{S}\), represented by \(r_{A_{F}}^{S}\), can be given as follows:  
\[r_{A_{S-F}}^{S}=v_{A_{L}}^{S}\times r_{A_{S}}^{S}\times\left[\begin{array}{ ccc}0&0&1\end{array}\right]^{T} \tag{17}\]  
The intersection angle between \(r_{A_{F}}^{S}\) and \(r_{A_{S}}^{S}\), represented by \(\theta_{A_{F}}\), is calculated using the vector dot product law:  
\[\theta_{A_{F}}=\cos^{-1}\left(\left(r_{A_{S}}^{S}\cdot r_{A_{F}}^{S}\right)/ \left(\left|r_{A_{S}}^{S}\right|\cdot\left|r_{A_{F}}^{S}\right|\right)\right) \tag{18}\]  
The linear velocity of the IMC for \(A_{S}\), represented by \(v_{A_{F}}^{S}\), is calculated as in Equation (19):  
\[v_{A_{F}}^{S}=r_{A_{F}}^{S}\cdot\left|v_{A_{L}}^{S}\right|/\left(\left|r_{A_{ F}}^{S}\right|\cdot\sin(\theta_{A_{F}})\right) \tag{19}\]  
Then, the linear velocity residual of the IMC for \(A_{S}\), represented by \(v_{A_{E}}^{S}\), can be calculated according to Equations (13) and (19):  
\[v_{A_{E}}^{S}=\frac{r_{A_{d}}^{S}\cdot f\cdot\omega_{A}}{\left|r_{A_{d}}^{S} \right|\cdot\cos(\alpha)\cdot\cos(\beta)\cdot\sin(\theta_{A_{S}})}-\frac{r_{A _{F}}^{S}\cdot\left|v_{A_{L}}^{S}\right|}{\left|r_{A_{F}}^{S}\right|\cdot\sin (\theta_{A_{F}})} \tag{20}\]