## 2 The MISP Structure  
### Introduction of MISP System  
A structural diagram of the MISP is shown in Figure 1. The frame of the MISP is composed of a base assembly, a lateral outside gimbal, and an elevated inner gimbal. The base assembly is installed on the vehicle through four shock absorbers to help eliminate the high-frequency vibration from the vehicle. The mechanical bearing connects the lateral gimbal with the base so that the lateral gimbal rotates around the longitudinal axis of the vehicle. A magnetic bearing is placed in the elevation gimbal as a fine stage to isolate the residual angular movement of the inner gimbal, and the payload instrument is suspended by gimbals, which forms the magnetic levitation system. Motor-gear systems are used to drive the gimbles. Resolvers are mounted on the end of the axes to measure the gimbal angles, which are restricted in a sway of \(\pm 8^{\circ}\). Angular rate gyros are installed on the elevation gimbal in the corresponding directions to furnish angular velocity feedback for the stabilizing loops in both directions. Platform controllers are installed on the corners of the lateral gimbal.  
A conceptual drawing of the MISP is depicted in Figure 2. The payload is levitated in five degrees of freedom (DOF), including the axial translation along Z, radial translation along X' and Y', and the slight rotation around the axial X and axial Y. \(F_{A}\), \(F_{B}F_{c}\), and \(F_{d}\) are the magnetic force at four locations and adjusted by controllers. The degree of freedom to tilt around the radial direction of the magnetic bearing and the other DOFS are decoupled from each other, which can be referred to in [3,7].  
Thus, the elevation gimbal control and the lateral gimbal control can be generalized as a single-DOF dual-stage control system. The MISP control block diagram for the pile-up control strategy [4] is shown in Figure 3. Because the gimbal is driven by motor-gear systems, there is a resonance structure in the coarse control channels, which is represented in orange. The dual-stage stabilization model is written as follows  
\[\left\{\begin{array}{c}m\ddot{x}_{f}=u_{f}+k\Delta x+T_{f}\\ M\ddot{x}_{c}=u_{c}-u_{f}-k\Delta x+T_{c}\end{array}\right. \tag{1}\]  
where \(x_{f}\) and \(x_{c}\) denote the tilting angles of the fine and coarse stages, respectively. \(\Delta x\) is the relative angle and \(k\) is the tilting negative stiffness in the magnetic force. \(u_{f}\) and \(u_{c}\) are the corresponding control inputs and \(m\) and \(M\) are the moments of inertia. \(T_{f}\) and \(T_{c}\) are the disturbances acting on the fine and coarse stages, respectively, which are relatively low-frequency vibrations from the base and high-frequency nonlinear friction. \(C_{f}\) and \(C_{c}\) are the PID controllers in the fine and coarse stages, and take the forms  
\[C_{f}=K_{p}(1+Ds+1/s) \tag{2}\]  
Figure 1: The MISP structure.  
Figure 2: Conceptual drawing of MISP system.  
\[C_{f}=k_{p}(1+ds+i/s) \tag{3}\]  
where \(K_{p}\), \(D\), \(I\) are the proportional coefficient, differential coefficient and integral coefficient of \(C_{f}\), respectively. \(k_{p}\), \(d\), and \(i\) are the proportional coefficient, differential coefficient, and integral coefficient of \(C_{c}\). \(s\) is the variable in frequency system function.