## 2 The MISP Structure
### Transmission Structure  
The gimbal frame of the MISP is driven by a motor, shaft coupler, gear, and racks, which are regarded as servo systems, and its structure is shown in Figure 4. It is equivalent to a double-inertia transmission system with clearance, as depicted in Figure 5.  
A dynamic model of the MISP transmission structure can described in the following forms  
\[\left\{ \begin{array}{c}m_{0}\ddot{x}_{0}=T_{e}-C_{m}\dot{x}_{0}-T_{w}\\ M\ddot{x}_{c}=T_{w}-C_{l}\dot{x}_{c}-T_{l}\end{array} \right. \tag{4}\]  
where \(m_{0}\) is the moment of inertia of the motor and \(M\) is the moment of inertia of the gimbal frame. \(C_{m}\), \(C_{l}\), and \(C_{w}\) are the damping coefficients of the motor, gimbal, and drive shaft, respectively. Accordingly, \(x_{0}\) and \(x_{c}\) represent the angles of the motor and  
Figure 4: The transmission structure of gimbal frame. Figure 5: The transmission diagram. Figure 3: The MISP control diagram with pile-up control strategy. gimbal, respectively, and \(T_{e}\), \(T_{l}\), and \(T_{w}\) are the torques of the motor, gimbal, and drive-shaft, respectively, where \(K\) denotes the stiffness of the transmission shaft. The backlash model is used to describe the effect of gear clearance after approximate linearization, and the shaft moment \(T_{w}\) is given by  
\[T_{\rm w}=\left\{ \begin{array}{c}K(x_{0}-x_{c}-b)+C_{w}(\dot{x}_{0}-\dot{x}_ {c}),x_{0}-x_{c}>b\\ 0,-b\leq x_{0}-x_{c}\leq b\\ K(x_{0}-x_{c}+b)+C_{w}(\dot{x}_{0}-\dot{x}_{c}),x_{0}-x_{c}<-b\end{array} \right. \tag{5}\]  
where b is related to gear model. The relationship between the amplitude of the angle error and the clearance is used as the description function \(N(A)\) of the backlash model [11], taking the form  
\[N(A)=1-\frac{2}{\pi}\left(\arcsin(\frac{b}{A})+\frac{b}{A}\sqrt{1-\left(\frac {b}{A}\right)^{2}}\right),b<A \tag{6}\]  
where \(A\) is the amplitude of the angular error. The range of \(N(A)\) is 0-1. With approximate linearization of \(N(A)\) in a certain range, the elastic coefficient of the model with clearance can be equivalent to the product of \(K\) and \(N(A)\) without clearance, expressed by  
\[K^{*}=K\times N(A) \tag{7}\]  
\(K^{*}\) can be regarded as the equivalent elastic coefficient of a model with clearance. When \(C_{m}\) and \(C_{l}\) are too small to be neglected, the transfer function from the motor output torque \(T_{e}\) to its angular velocity \(\dot{x}_{0}\) is deduced as  
\[G_{m}(\mathrm{s})=\frac{1}{(m_{0}+M)s}\frac{Ms^{2}+C_{w}s+K^{*}}{m_{p}s^{2}+C_ {w}s+K^{*}}=\frac{1}{(m_{0}+M)s}G^{\prime}_{m}(\mathrm{s}) \tag{8}\]  
where \(m_{p}\) is equivalent inertia, stated as  
\[m_{p}=\frac{m_{0}M}{m_{0}+M} \tag{9}\]  
\(G^{\prime}_{m}\) is the link leading to mechanical resonance, denoted as  
\[G^{\prime}_{m}(\mathrm{s})= \frac{Ms^{2}+C_{w}s+K^{*}}{m_{p}s^{2}+C_{w}s+K^{*}}=\frac{(\frac {s}{\omega_{z}})+2\xi_{z}\omega_{z}s+1}{(\frac{s}{\omega_{p}})+2\xi_{p}\omega_ {p}s+1} \tag{10}\]  
where \(\omega_{p}\) and \(\omega_{z}\) are the resonant and antiresonant frequencies, respectively. \(\xi_{z}\) and \(\xi_{p}\) represent resonant damping and antiresonant damping, respectively. They are expressed by  
\[\left\{ \begin{array}{c}\omega_{p}=\sqrt{\frac{K^{*}}{m_{p}}},\;\omega_{z}= \sqrt{\frac{K^{*}}{M}}\\ \\ \xi_{p}=\frac{C_{in}}{2\sqrt{m_{p}K^{*}}},\;\xi_{z}=\frac{C_{in}}{2\sqrt{MK^{ *}}}\end{array} \right.