## 2 Modeling of a Mobile Satcom Antenna
### Dynamic of the Azimuth Gimbal  
The gimbals in this system are all rigid, therefore the basic Newton-Euler rotation equation can be written as follows:  
\[\begin{array}{l}\sum M=\dot{H}+\omega\times H\\ H=J\omega\end{array} \tag{3}\]  
in which \(\sum M\) is the resultant moment of the force added to the rigid body; \(H\) is the inertial angular momentum; \(J\) is the inertial moment; \(\omega\) is the absolute angular rate.  
Since the structures of the two gimbals are similar (the difference is that the azimuth gimbal is effected by coupling of elevation gimbal), only the azimuth gimbal is considered in this paper.  
According to the Equation (3), the dynamic model of azimuth gimbal can be obtained as follows:  
\[(J_{az}+J_{ez})(\dot{r}+\ddot{\theta}_{a})=M_{am}-M_{ad}-M_{aez}-(J_{ay}+J_{ey} -J_{ax}-J_{ex})\omega_{iax}^{a}\omega_{lay}^{a} \tag{4}\]  
where, \(M_{ad}\) is the disturbance torque added to elevation gimbal; \(M_{am}\) is the driving torque of elevation motor; \(M_{aez}\) is the coupling torque that effected by elevation gimbal. They are given by Equations (5) and (6):  
\[M_{aez}=J_{ez}\dot{\omega}_{iez}^{e}\sin\theta_{e}+\left(J_{ex}-J_{ey}\right) \omega_{iex}^{e}\omega_{iey}^{e}\sin\theta_{e}+J_{ez}\dot{\omega}_{iez}^{e} \cos\theta_{e}+\left(J_{ex}-J_{ey}\right)\omega_{iex}^{e}\omega_{iey}^{e} \cos\theta_{e} \tag{5}\]  
\[M_{am}=\frac{k_{at}}{R_{aa}}\Big{(}N_{a}u_{aa}-N_{a}^{2}k_{ab}\dot{\theta}_{a} \Big{)}-\Big{(}N_{a}^{2}J_{ad}+J_{af}\Big{)}\ddot{\theta}_{a}-\Big{(}N_{a}^{2} f_{ad}+f_{af}\Big{)}\dot{\theta}_{a} \tag{6}\]  
\(J_{ax},J_{ay},J_{az}\) are the moments of inertia of the azimuth gimbal related to the \(X,Y,Z\) axes in azimuth coordinates, respectively; \(J_{ex},J_{ey},J_{ez}\) are the moments of inertia of the elevation gimbal related to the \(X,Y,Z\) axes in the elevation gimbal, respectively; \(k_{at}\) is the azimuth motor torque constant; \(k_{ab}\) is the back-EMF coefficient of azimuth motor; \(N_{a}\) is the azimuth gear ratio; \(R_{aa}\) is the motor resistance; \(J_{ad}\) is the moment of inertial of driver gear; \(f_{ad}\) is the viscous friction coefficient of driver gear; \(J_{af}\) is the moment of inertial of passive gear; \(f_{af}\) is the viscous friction coefficient of passive gear; \(u_{aa}\) is the input driven voltage of the motor.  
Based on the preceding analysis, system (4) can be simplified into a standard second order control system:  
\[\left\{\begin{array}{l}\dot{x}_{a1}(t)=x_{a2}(t)\\ \dot{x}_{a2}(t)=A_{a}x_{a2}(t)+B_{a}u_{a}(t)+C_{a}d_{a}(t)+D_{a}w_{a}(t)\end{array}\right. \tag{7}\]  
Figure 1: Coordinate definition and transformation.  
in which, \(x_{a1}=\theta_{a}\), \(x_{a2}=\dot{\theta}_{a}\), \(d_{a}(t)\) is external disturbance and \(w_{a}(t)\) is the coupling effect torque. To further simplify the analysis, we define that \(|C_{a}d_{a}(t)+D_{a}w_{a}(t)|\leq l_{d}\). Then the proposed control block diagram can be obtained as follows:  
\[A_{a}=\left[\begin{array}{cc}0&1\\ 0&-\frac{k_{at}k_{ab}N_{a}^{2}+R_{aa}N_{a}^{2}f_{ad}+R_{au}f_{af}}{(I_{ax}+I_{ ex}+N_{a}^{2}I_{ad}+I_{af})R_{aa}}\end{array}\right],\,B_{a}=\left[\begin{array}{cc}0 \\ \frac{k_{at}N_{a}}{(I_{ax}+I_{ex}+N_{a}^{2}I_{ad}+I_{af})R_{aa}}\end{array}\right]\]  
\[C_{a}=\left[\begin{array}{cc}0\\ -\frac{1}{I_{ax}+I_{ex}+N_{a}^{2}I_{ad}+I_{af}}\end{array}\right],\,D_{a}=\left[ \begin{array}{cc}0\\ -\frac{1}{I_{ax}+I_{ex}+N_{a}^{2}I_{ad}+I_{af}}\end{array}\right]\]  
and \(d_{a}(t)=M_{ad}+(I_{az}+I_{ez})\dot{r}+(I_{ay}+I_{cy}-I_{ax}-I_{ex})\omega_{iax }^{a}\omega_{iay}^{a}\).  
Then the control block diagram shown in Figure 2 can be obtained.  
The NFTSM control applied in this paper can guarantee that the system state arrives at the equilibrium point in a finite time with fast response and higher precision. In the following section, a new NFTSM controller is proposed and proved.