## 3 Imaging Attitude Control
### Imaging Attitude Analysis  
#### 3.2.1 Gimbal Angle Calculation  
During the sweep imaging process, the attitude of each image should be controlled to satisfy the overlap rate and gap-free coverage requirement. For a three-axis platform, the attitude of the area camera can be adjusted by rotating the gimbal axes. Let us define \(R_{K}^{M}\) as the matrix that transforms the vector from the K frame to the M frame. Then, \(R_{K}^{AC}\) represents the matrix that transforms the vector from the S frame to the AC frame. Figure 3: Fundamental coordinate frames. According to the mechanism design of the three-axis ISP, the \(x\) gimbal axis and y gimbal axis rotate with the \(z\) gimbal axis, and the \(y\) gimbal axis rotates with the \(x\) gimbal axis. The transformation order from the S frame to the AC frame is pitch, roll, and azimuth. Therefore, \(R_{S}^{AC}\) can be calculated as follows:  
\[ \begin{array}{c}R_{S}^{AC}=R_{z}^{yaw}(\kappa)^{T}\cdot R_{x}^{roll}(\varphi)^ {T}\cdot R_{y}^{pitch}(\omega)^{T}=\\ \left[\begin{array}{cc}\cos{(\kappa)}\cos{(\omega)}-\sin{(\varphi)}\sin{( \omega)}\sin{(\kappa)}&-\sin{(\kappa)}\cos{(\varphi)}&\cos{(\kappa)}\sin{( \omega)}-\sin{(\kappa)}\sin{(\varphi)}\sin{(\omega)}\\ \sin{(\kappa)}\cos{(\omega)}+\cos{(\kappa)}\sin{(\varphi)}\sin{(\omega)}&\cos{( \kappa)}\cos{(\varphi)}&\sin{(\kappa)}\sin{(\omega)}-\cos{(\kappa)}\sin{( \varphi)}\cos{(\omega)}\\ -\cos{(\varphi)}\sin{(\omega)}&\sin{(\varphi)}&\cos{(\varphi)}\cos{(\omega)} \end{array}\right]\\ \end{array}  \tag{2}\]  
where \(R_{x}^{roll}\) is the direction cosine matrix (DCM) for the \(x\) axis; \(R_{y}^{pitch}\) is the DCM for the \(y\) axis; \(R_{z}^{yaw}\) is the DCM for the \(z\) axis [23]; \(\Psi\), \(\Theta\), and \(\Phi\) are the azimuth angle, pitch angle and roll angle from the POS, respectively; and \(\kappa\), \(\omega\), and \(\varphi\) are the gimbal angles of the \(z\) axis encoder, \(y\) axis encoder, and \(x\) axis encoder, respectively. The matrix \(R_{N}^{AC}\), which transforms the vector from the N frame to the AC frame, can be defined using Equation (3):  
\[R_{N}^{AC}=R_{S}^{AC}\cdot\underbrace{R_{x}^{roll}(\Phi)\cdot R_{y}^{pitch}( \Theta)\cdot R_{z}^{yaw}(\Psi)}_{N->S} \tag{3}\]  
The projection of \(Y_{S}\) on the ground should be perpendicular to the aircraft velocity's direction to achieve gap-free coverage, which is essential for reconnaissance. Therefore, the transform matrix from the T frame to the N frame should be calculated using Equation (4), where \(\Psi_{V}\) stands for the aircraft velocity's direction, \(\Theta_{T}\) is the target pitch angle, and \(\Phi_{T}\) is the target roll angle. The LOS first rotates about the \(y\) axis by \(\Theta_{T}\) and then rotates about the \(x\) axis by \(\Phi_{T}\) in turn:  
\[R_{T}^{N}=R_{z}^{yaw}(\Psi_{V})^{T}\cdot R_{y}^{pitch}(\Theta_{T})^{T}\cdot R _{x}^{roll}(\Phi_{T})^{T} \tag{4}\]  
The unit vector of the \(x\) axis of the T frame can be transformed to vector \(r_{X}^{AC}\) in the AC frame, and the unit vector of the \(y\) axis of the T frame can be transformed to vector \(r_{Y}^{AC}\) in the AC frame. After rotating about the platform's azimuth gimbal axis by \(\kappa\), the roll gimbal axis by \(\varphi\), and the pitch gimbal axis by \(\omega\) simultaneously, the unit vector of the \(x\) axis of the S frame should be equal with \(r_{X}^{AC}\), and the unit vector of the \(y\) axis of the S frame should be equal with \(r_{Y}^{AC}\). Therefore, the following equation set can be established:  
\[\left\{ \begin{array}{c}R_{S}^{AC}\cdot\left[\begin{array}{ccc}1&0&0\\ 0&1&0\end{array}\right]^{T}=\left[\begin{array}{ccc}u_{1}&u_{2}&u_{3}\\ v_{1}&v_{2}&v_{3}\end{array}\right]^{T}_{T}\\ \end{array} \right. \tag{5}\]  
where  
\[\left[ \begin{array}{ccc}u_{1}&u_{2}&u_{3}\end{array} \right]^{T}=r_{X}^{AC} =R_{N}^{AC}\cdot R_{T}^{N}\cdot\left[ \begin{array}{ccc}1&0&0\\ 0&1\end{array}