## 3 Dynamics Modeling
### Two-Axis Four-Gimbal Structure  
Based on the analysis of the transmission path and kinematics coupling compensation matrix of the coarse-fine composite structure, the dynamics modeling and theoretical study of the ultralight two-axis four-gimbal electro-optical pod are studied. The working principle of the two-axis four-gimbal electro-optical pod involves the definition of multiple coordinate systems, which are respectively explained as shown in Figure 10.  
The inner gimbal rotates in a small range, and the outer gimbal follows the macro-field control of the inner gimbal in a large range. At the same time, the feedback error of the outer gimbal is compensated by the inner gimbal so that the inner gimbal can offset the disturbance of rolling and pitching. Finally, the two-axis four-gimbal electro-optical pod maintains the stability of the visual axis to achieve high-precision coarse-fine composite control. **A. Direct Connection Stabilization**  
Figure 10: Simplified Cartesian coordinate system of the two-axis four-gimbal electro-optical pod. Figure 9: The coarse–fine composite structure of electro-optical pod kinematic coupling Piogram. The gyroscope is sensitive to the angular velocity of the inner pitch system {e} and the inner roll system {a} relative to the inertial system {i}. Therefore, make \(\omega_{Ye}=\omega_{Ze}=0\), and then the structure can keep the visual axis of the detector stabilization. \[ \begin{array}{l}\left(\begin{array}{c}\dot{\theta_{e}}\\ \dot{\theta_{a}}\end{array}\right)=\left(\begin{array}{c}\omega_{gyro\_Y}\\ \omega_{gyro\_X}\end{array}\right.=\left(\begin{array}{c}\sin\theta_{a}\cos \theta_{E}\\ \cot\theta_{e}\cos\theta_{a}\cos\theta_{E}+\sin\theta_{E}\end{array}\right) \omega_{ZA}\\ +\left(\begin{array}{cc}-\cos\theta_{a}&-\sin\theta_{E}\sin\theta_{a}\\ \cot\theta_{e}\sin\theta_{a}&-\cot\theta_{e}\cos\theta_{a}\sin\theta_{E}-\cos \theta_{E}\end{array}\right)\left(\begin{array}{c}\omega_{YE}\\ \omega_{XA}\end{array}\right)\end{array} , \tag{13}\]  
According to Equation (13), the structure can keep the visual axis of the stabilization. **B. Indirect Connection Stabilization**  
The angular velocity of gyroscope sensitive the outer pitch axis system {E} and the outer roll axis system {a} relative to the inertial system {i} is \(\omega_{gyro\_X}=\omega_{XA}\omega_{gyro\_Y}=\omega_{YE}\). \[ \begin{array}{l}\left(\begin{array}{c}\dot{\theta_{e}}\\ \dot{\theta_{a}}\end{array}\right)=\left(\begin{array}{cc}1&0\\ 0&-\sec\theta_{e}\end{array}\right)\left(\begin{array}{c}\omega_{YE}\\ \omega_{Ze}\end{array}\right)+\left(\begin{array}{c}\sin\theta_{a}\cos \theta_{E}\\ \cot\theta_{e}\cos\theta_{a}\cos\theta_{E}+\sin\theta_{E}\end{array}\right) \omega_{ZA}\\ +\left(\begin{array}{cc}-\cos\theta_{a}&-\sin\theta_{E}\sin\theta_{a}\\ \cot\theta_{e}\sin\theta_{a}&-\cot\theta_{e}\cos\theta_{a}\sin\theta_{E}-\cos \theta_{E}\end{array}\right)\left(\begin{array}{c}\omega_{gyro\_Y}\\ \omega_{gyro\_X}\end{array}\right)\end{array} , \tag{14}\]  
According to Equation (14), the structure can keep the visual axis of the stabilization. Assuming that, in the case of sensitive motion of pitch and roll gyroscopes, their sensitivity values are \(\omega_{gyro\_Y}\), \(\omega_{gyro\_Z}\), respectively, \(\dot{\theta_{a}}\) represents the angular velocity vector of the inner roll gimbal relative to the outer pitch gimbal, and \(\dot{\theta_{e}}\) represents the angular velocity vector of the inner roll gimbal relative to the inner roll gimbal; \(\omega_{A}\), \(\omega_{E}\), \(\omega_{a}\), \(\omega_{e}\), respectively, represent the angular velocity of the two-axis four-gimbal structure and the components of its three coordinate axes. According to the Euler dynamical theorem and Coriolis rotation law,  
\[\frac{\mathrm{d}H}{dt}=\frac{\partial H}{dt}+\omega\times H, \tag{15}\]  
where \(H=\left( \begin{array}{cc}H_{X}&H_{Y}&H_{Z}\end{array}