## 2 Methods  
The photoelectric pod with two axes and four frame structure can expand the stable tracking range and improve the tracking accuracy [24]. Due to the connection of the mechanical structure, there will be a coupling effect between the internal and external frames. However, with the improvement of the platform production process and the increasingly rich structural design, the coupling effect between the internal and external frames is negligible. With the stable loop as the internal loop of the system, its quality directly determines the tracking performance of the photoelectric pod. Therefore, this paper does not focus on kinematics and dynamics derivation, but mainly takes the heading axis stability loop in the frame of the photoelectric platform as an example to establish the mathematical model. The mathematical model of photoelectric pod is based on the conventional control system using the double closed-loop system of speed and position. The control loop usually takes the second-order inertial element and the oscillation element as the control objects. The second-order inertial element includes the mechanical and electrical characteristics of the motor. The oscillation element represents the mechanical resonance characteristics inside the system platform. The low frequency gain of the speed loop is limited by the phase margin of the system, which then affects the tracking accuracy of the position loop. If a current feedback sensor and a current feedback control network are added to the speed loop to form a current closed-loop system, the characteristics of the control object can be changed and the accuracy of the control system can be improved. Many existing research results show that by adding current loop on the basis of position loop and speed loop, the control system has the advantages of good dynamic response, strong anti-interference ability, and high precision [25]. Under the rated excitation conditions, the armature current of the motor is continuous, so the dynamic voltage balance equation of the armature circuit is shown in Equation (1) [26]. \[U_{d0}=E+I_{a}R_{0}+L\bigg{(}\frac{dL_{a}}{d_{t}}\bigg{)} \tag{1}\]  
where \(U_{d0}\) = motor armature voltage; \(I_{a}\) = motor armature current; \(R_{0}\) = total circuit resistance; and \(L\) = armature inductance. \[E=C_{e}\cdot n \tag{2}\]  
where \(E\) = anti-electromotive force of the motor; \(C_{e}\) = motor reverse electric type coefficient; and \(n\) = motor speed. Equation (3) performs the Laplace transformation by Equation (1):  
\[U_{d0}(s)-E(s)=R_{0}\bigg{[}I_{a}(s)+\frac{L\cdot I_{a}(s)}{R_{0}}s\bigg{]} \tag{3}\]  
Make the current loop time constant \(T_{a}\) = \(L/R_{0}\), and Equation (3) is reduced to Equation (4):  
\[\frac{I_{a}(s)}{U_{d0}(s)-E(s)}=\frac{1}{R_{0}(T_{a}s+1)} \tag{4}\]  
At the rated voltage, the torque and the armature current of the DC motor are in a linear relationship. When the armature current is unchanged, the torque also does not change. According to the equation of motion, the rated torque of motor and the load torque can be expressed by Equations (5)-(7): [26]  
\[T_{em}-T_{L}=J\frac{dn}{dt} \tag{5}\]\[T_{em}=C_{m}\cdot I_{a} \tag{6}\]  
\[T_{L}=C_{m}\cdot I_{aL} \tag{7}\]  
where \(T_{em}\) = rated torque of the motor; \(T_{L}\) = load torque; \(C_{m}\) = motor torque coefficient; \(J\) = total moment of inertia; and \(I_{aL}\) = load-back current. \[C_{m}[I_{a}(s)-I_{aL}(s)]=J\cdot n(s)\cdot s \tag{8}\]  
The transfer function between the armature antipotential and the armature current can be obtained from Equations (2) and (8):  
\[\frac{E(S)}{I_{a}(s)-I_{aL}(s)}=\frac{R_{0}}{T_{m}s} \tag{9}\]  
The electromechanical time constant \(T_{m}=JR_{0}/C_{m}C_{e}\). Combined with Equations (1)-(9), the motor dynamic structure block diagram can be obtained as shown in Figure 1.  
Using the current loop control the system, the current loop input signal is at point A and the feedback signal input is at point B. The speed loop affects the current loop through the anti-electromotive force of the motor \(E(s)\).