## 3 Dynamics Modeling  
### Coarse-Fine Composite Analysis  
As shown in Figure 1, the structure of the two-axis four-gimbal electro-optical pod is more complicated. It is therefore an effective solution to study the coarse-fine composite structure first. The working principle of the coarse-fine composite structure involves the definition of multiple coordinate systems, which are respectively explained as follows. 1. Inertial coordinate system (\(\{i\}\), \(O_{i}X_{i}Y_{i}Z_{i}\))
2. UAV coordinate system (\(\{d\}\), \(O_{d}X_{d}Y_{d}Z_{d}\))
3. Coarse motor stator coordinate system (\(\{u\}\), \(O_{u}X_{u}Y_{u}Z_{u}\))
4. Fine motor rotor coordinate system (\(\{g\}\), \(O_{g}X_{g}Y_{g}Z_{g}\))  
The coarse motor is fixedly connected with the guide rail through the threaded connection, without considering the damping effect between the structures. There is geometric eccentricity \(e_{r}\) in the shafting structure of the coarse-fine stage mechatronic system, which will cause coaxiality error and affect the high precision control performance of the electro-optical pod. As shown in Figure 8a, the geometric eccentricity of the shafting structure is caused by force deformation, uneven cutting force, and chip formation of the cutting edge. In Figure 8b, according to the transformation matrix of fixed-point rotation in the Cartesian coordinate system, the Euler transformation [17] is analyzed as shown in Figure 8b, and the Euler angle is \(\theta,\phi,\varphi\). The first step is to rotate the \(\theta\) angle about the \(k\) axis, so that the \(i\) axis rotates to the \(m\) position and the \(j\) axis rotates to the \(n\) position; Cartesian coordinate system \(Oijk\to Omnk\). The second step is to rotate the \(\phi\) angle about the \(m\) axis, so that the \(n\) axis rotates to the \(q\) position and the \(k\) axis rotates to the \(p\) position; Cartesian coordinate system \(Omnk\to Omqp\). The third step is to rotate the \(\varphi\) angle about the \(p\) axis, so that the \(m\) axis rotates to the \(r\) position and the \(q\) axis rotates to the \(s\) position; Cartesian coordinate system \(Omqp\to Orsp\). Finally, the Euler transformation of fixed-point rotation is completed. In the Cartesian coordinate system, after the system \(\{u\}\) rotates the \(\theta_{1}\) angle around the \(X_{i}\) axis, the system \(\{i\}\) is used as the reference system to observe the position of the system \(\{u\}\). Then, the system \(\{u\}\)  
Figure 8: (**a**) Geometric eccentricity error of coarse–fine composite structure; (**b**) Euler transformation diagram for fixed-point rotation. rotates the \(\theta_{2}\) angle around \(Z_{i}\) axis. Euler transformation of coarse-fine composite structure can be calculated as the rotation transformation matrix, denoted as  
\[E^{X\theta}=A^{\theta}=\left( \begin{array}{ccc}1&0&0\\ 0&\cos\theta_{1}&-\sin\theta_{1}\\ 0&\sin\theta_{1}&\cos\theta_{1}\end{array} \right), \tag{7}\]  
\[E^{Z\theta}=A^{\theta}=\left( \begin{array}{ccc}\cos\theta_{2}&-\sin\theta_{2 }&0\\ \sin\theta_{2}&\cos\theta_{2}&0\\ 0&0&1\end{array} \right). \tag{8}\]  
According to Euler transformation law of rigid body fixed point rotation, it can be obtained from Equations (7) and (8) that  
\[E=E^{k\theta}\cdot E^{m\psi}\cdot E^{pq}=\left( \begin{array}{ccc}\cos\theta &-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{array} \right)\left( \begin{array}{ccc}1&0&0\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi&\cos\phi\end{array} \right)\cdot\left( \begin{array}{ccc}\cos \varphi&-\sin\varphi&0\\ \sin\varphi&\cos\varphi&0\\ 0&0&1\end{array} \right) \tag{9}\]  
Since the stator of the coarse motor is connected with the guide rail by thread, there is no fixed-point rotation for the system \(\{u\}\) against the system \(\{i\}\). Only the installation error of rotation along the \(Y\)-axis exists. The kinematic coupling equation shows that  
\[\omega_{u}=\left( \begin{array}{c}\omega_{ux}\\ \omega_{uy}\\ \omega_{uz}\end{array} \right)=\left( \begin{array}{ccc}\cos\theta_{u}&0& \sin\theta_{u}\\ 0&1&0\\ -\sin\theta_{u}&0&\cos\theta_{u}\end{array}