## 2 Dynamics Modeling for the SISP  
There are many kinds of SISPs according to the application, Figure 1 shows a typical three-axis SISP for the moving base carrier. Figure 1 shows that the gyroscope mounted on the platform base is used to measure the base inertial angular velocity. The gyroscope generates the command in real-time to make the three motors reverse its motion to stabilize the LOS. The platform consists of three frames: yaw, roll, and pitch. The moment of inertia balance of each frame is typically considered in the ISP/SISP design process; furthermore, each frame uses the same motor, driver, and control method. Thus, this study takes the yaw frame as an example to expand the analysis. Presently, the model in the ISP/SISP is regarded as a common model of a motor and a load in engineering applications and academic research. However, the traditional model used in the SISP system with light base support is limited by the vibration. A complete SISP system dynamics model comprises the platform base, motor rotor, and load, which are treated as several rigid bodies. To amplify the driver torque in a SISP with a heavy load, a motor with a harmonic drive is used, as shown in Figure 2.  
The core of the control system comprises three parts: motor rotor (a), harmonic drive (b), and load (c). Here, the platform base and the motor rotor (a) form an interaction subsystem, while the motor rotor (a) and the load (c) form a two-mass-like subsystem. The harmonic drive (b) mechanism consists of a wave generator, flex spline, and circular spline [31,32]. The motor electromagnetic torque \(T_{m}\) acts on the wave generator in the harmonic drive. The encoder is installed between the motor rotor (a) and the harmonic drive (b). The encoder data represent the relative rotation angle between the motor rotor and stator. Each part of the system is then modeled and analyzed. The armature voltage balance equation and the electromagnetic torque equation can be described as follows,  
\[u_{a}(t)-e_{a}(t)=L_{a}i_{a}(t)+R_{a}i_{a}(t) \tag{1}\]  
\[K_{t}i_{a}(t)=T_{m}(t) \tag{2}\]  
Figure 1: Schematic diagram of a typical three-axis strapdown inertially stabilized platforms (SISP) for the moving base carrier. Figure 2: Control system diagram. (a) Motor; (b) Harmonic drive; (c) Load. where \(u_{a}(t)\) is the armature voltage, \(e_{a}(t)\) is the back-EMF, \(L_{a}\) is the armature inductance, \(i_{a}(t)\) is the armature current, \(R_{a}\) is the armature resistance, and \(K_{t}\) is the electromagnetic torque constant. \(T_{m}(t)\) is the motor electromagnetic torque acting on the wave generator in the harmonic drive. The SISP control system typically includes the position, velocity, and current loops. The velocity loop is the most important component; it determines the disturbance isolation capability and the command tracking performance. Because SISP typically operates at a low velocity, the influence of the back-EMF \(e_{a}(t)\) can be ignored. The DC electric part is a first-order inertial object. On the other hand, the current closed-loop system model is equivalent to a low-pass filter with a high cut-off frequency. Therefore, for the velocity loop, the subsystem from the control voltage \(u(t)\) to the electromagnetic torque \(T_{m}(t)\) can be regarded as a linear function:  
\[K_{a}u(t)=T_{m}(t) \tag{3}\]  
where \(u(t)\) is the control voltage as the input to the current driver and \(K_{a}\) is the coefficient from the control voltage to the electromagnetic torque. The torque balance equation for the motor rotor can be expressed as follows,  
\[T_{m}(t)-F_{m}(t)=J_{m}\ddot{\theta}_{m}(t)+B_{m}\dot{\theta}_{m}(t)+T_{wg}(t) \tag{4}\]  
where \(F_{m}(t)\) is the total disturbance torque, \(J_{m}\) is the motor rotor equivalent moment of inertia, \(B_{m}\) is the motor rotor equivalent damping coefficient, \(\theta_{m}(t)\) is the motor rotor angle in the inertial space, and \(T_{wg}(t)\) is the wave generator input torque.