## 3 Dynamics Modeling
### Two-Axis Four-Gimbal Structure
\right)^{T}=\) moment of momentum; \(\frac{dt}{dt}=\) absolute derivatives (rate of change) of vector H; \(\frac{\partial H}{dt}=\frac{dH_{X}}{dt}i+\frac{dH_{Y}}{dt}j+\frac{dH_{Z}}{dt}k\) = relative derivatives of vector H; and \(i,j\), \(k\) = unit vectors of coordinate axis of the body reference system, respectively. According to the moment of momentum theorem,  
\[\frac{\mathrm{d}H}{dt}=M, \tag{16}\]  
where \(M=\left( \begin{array}{cc}M_{X}&M_{Y}&M_{Z}\end{array} \right)^{T}=\) the external addition torque vector of the rigid body. Under the assumption that the all three axes are principal axes of inertia, the following equation can be established:  
\[\left\{ \begin{array}{l}I_{X}\dot{\omega_{X}}+(I_{Z}-I_{Y})\omega_{Y}\omega_{ Z}=M_{X}\\ I_{Y}\dot{\omega_{Y}}+(I_{X}-I_{Z})\omega_{X}\omega_{Z}=M_{Y}\\ I_{Z}\dot{\omega_{Z}}+(I_{Y}-I_{X})\omega_{X}\omega_{Y}=M_{Z}\end{array} ,\right. \tag{17}\]  
where \(I_{X},I_{Y},I_{Z}\) = the moment of inertia of the rigid body around the coordinate axis of the follower reference system. The moment of momentum theorem is applicable to the calculation of larger angular velocity. However, the electro-optical pod of two-axis four-gimbal structure is compact and ultralight, with a mass less than 1 kg and a stabilization precision is 20 \(\upmu\)rad. When the design is carried out in combination with the actual situation, the values of some parameters are ignored. Based on the space dynamics [18], the coarse-fine composite self-correction drive equation are derived. **A. Inner Pitch Gimbal**  
\[M_{Ye}\approx\Big{(}x_{e}\times\dot{v_{b}}\Big{)}m_{e}+\Big{[}x_{e}\times\big{(} \dot{\omega_{a}}\times y_{e}\big{)}\Big{]}m_{e}+\Big{[}x_{e}\times\big{(}\dot{ \omega_{E}}\times y_{a}\big{)}\Big{]}m_{e}+\big{[}x_{e}\times\big{(}\dot{\omega _{A}}\times y_{E}\big{)}\Big{]}m_{e}+\big{[}y_{e}\times\dot{v_{b}}\big{)}m_{e} \leavevmode\
obreak\ \leavevmode\
obreak\, \tag{18}\]  
**B. Inner Roll Gimbal**  
\[ \begin{array}{l}M_{Xa}\approx\Big{(}x_{a}\times\dot{v_{b}}\Big{)}m_{a}+ \Big{[}x_{a}\times\big{(}\dot{\omega_{E}}\times y_{a}\big{)}\Big{]}m_{a}+\Big{[} x_{a}\times\big{(}\dot{\omega_{A}}\times y_{E}\big{)}\Big{]}m_{a}+\Big{(}y_{e} \times\dot{v_{b}}\Big{)}m_{e}\\ +\Big{[}y_{e}\times\big{(}\dot{\omega_{a}}\times y_{e}\big{)}\Big{]}m_{e}+ \Big{[}y_{e}\times\big{(}\dot{\omega_{e}}\times y_{e}\big{)}\Big{]}m_{e}+ \big{[}x_{a}\times\big{(}1_{Xa}\omega_{a}\big{)}\big{]}+M_{Ye}\leavevmode \
obreak\ \leavevmode\
obreak\,\end{array}  \tag{19}\]  
**C. Outer Pitch Gimbal**  
\[ \begin{array}{l}M_{Y\Sigma}\approx\Big{(}x_{E}\times\dot{v_{b}}\Big{)}m_{E} +\Big{[}x_{E}\times\big{(}\dot{\omega_{A}}\times y_{E}\big{)}\Big{]}m_{E}+ \Big{(}y_{a}\times\dot{v_{b}}\Big{)}m_{a}+\Big{[}y_{a}\times\big{(}\dot{\omega _{E}}\times y_{a}\big{)}\Big{]}m_{a}\\ +\Big{[}y_{a}\times\big{(}\dot{\omega_{a}}\times y_{a}\big{)}\Big{]}m_{a}+ \Big{(}y_{a}\times\dot{v_{b}}\Big{)}m_{e}+\Big{[}y_{a}\times\big{(}\dot{\omega _{E}}\times y_{e}\big{)}\Big{]}m_{e}+\big{[}y_{a}\times\big{(}\dot{\omega_{a }}\times y_{e}\big{)}\Big{]}m_{e}\leavevmode\
obreak\ \leavevmode\
obreak\,\end{array}  \tag{20}\]  
**D. Outer Roll Gimbal**  
\[ \begin{array}{l}M_{XA}\approx\Big{(}x_{A}\times\dot{v_{b}}\Big{)}m_{A}+ \Big{(}y_{E}\times\dot{v_{b}}\Big{)}m_{E}+\Big{[}y_{E}\times\big{(}\dot{ \omega_{A}}\times y_{E}\big{)}\Big{]}m_{E}+\Big{[}y_{E}\times\big{(}\dot{ \omega_{E}}\times y_{E}\big{)}\Big{]}m_{E}\\ +\Big{(}y_{E}\times\dot{v_{b}}\Big{)}m_{a}+\Big{[}y_{E}\times\big{(}\dot{ \omega_{A}}\times y_{a}\big{)}\Big{]}m_{a}+\Big{[}y_{E}\times\big{(}\dot{\omega _{E}}\times y_{a}\big{)}\Big{]}m_{a}+\Big{[}y_{E}\times\big{(}\dot{\omega_{a }}\times y_{a}\big{)}\Big{]}m_{a}\\ +\Big{(}y_{E}\times\dot{v_{b}}\Big{)}m_{e}+\Big{[}y_{E}\times\big{(}\dot{ \omega_{A}}\times y_{e}\big{)}\Big{]}m_{e}+\Big{[}y_{E}\times\big{(}\dot{ \omega_{E}}\times y_{e}\big{)}\Big{]}m_{e}+\Big{[}y_{E}\times\big{(}\dot{ \omega_{a}}\times y_{e}\big{)}\Big{]}m_{e}\leavevmode\
obreak\ \leavevmode\
obreak\,\end{array}