## 2 Model of Electromechanical Servo System
### Dynamic Model of Planetary Roller Screw  
In this paper, the three-stage planetary roller screw is used as the actuator of EMA to measure the performance indexes of the three-stage planetary roller screw, mainly including displacement acceleration, velocity, and displacement. \[l=\int_{0}^{t}n(P_{h1}+P_{h2}+P_{h3})dt=\int_{0}^{t}\frac{\omega(t)}{2\tau \cdot i}(P_{h1}+P_{h2}+P_{h3})dt \tag{3}\]where \(n\) is the speed of the main screw, \(t\) is the working time, \(i\) is the reduction ratio, and \(\omega(t)\) is the rotational angular speed of the motor. By deriving the displacement, the velocity \(v\) of the three-stage planetary roller screw is obtained as:  
\[v=\frac{dl}{dt}=\frac{\omega(t)}{2\pi\cdot i}(P_{h1}+P_{h2}+P_{h3}) \tag{4}\]  
By deriving the velocity, the expression of acceleration, \(a\), is:  
\[a=\frac{dv}{dt}=\frac{\alpha(t)}{2\pi\cdot i}(P_{h1}+P_{h2}+P_{h3}) \tag{5}\]  
where \(\alpha(t)\) is the rotational angular acceleration of the motor. For the bearing problem of the three-stage planetary roller screw, the screws at all levels are distributed the load according to the structure. The load distribution law inside a certain level of screw is the same as that of the single-stage planetary roller screw. The axial load of the screw at all levels is equal, which is equal to the load F of the screw. Then, we obtain:  
\[F_{1}=F_{2}=F_{3}=F \tag{6}\]  
The three-stage hollow screw only needs to drive the nut of this stage to transmit motion, and the transmission efficiency is \(\eta_{3}\). The torque of the three-stage hollow screw is:  
\[T_{3}=\frac{FP_{h3}}{2\pi\eta_{3}} \tag{7}\]  
The two-stage hollow screw not only provides torque for the axial movement of the two-stage nut, but also drives the rotation of the three-stage hollow screw. It can be regarded as the screw transmission with \(P_{h2}+P_{h3}\), and its transmission efficiency is \(\eta_{2}\). Then, the torque of the secondary hollow screw is:  
\[T_{2}=\frac{F(P_{h2}+P_{h3})}{2\pi\eta_{2}} \tag{8}\]  
Similarly, it can be seen that the primary main screw can be regarded as the screw transmission with lead of \(P_{h1}+P_{h2}+P_{h3}\), and its transmission efficiency is \(\eta_{1}\). The torque of the primary main screw is:  
\[T_{1}=\frac{F(P_{h1}+P_{h2}+P_{h3})}{2\pi\eta_{1}} \tag{9}\]  
The mathematical model of mechanical transmission part is mainly divided into two parts: the linear speed of screw and the thrust of electromechanical actuator. Let the reduction ratio of the reducer be \(i\), the lead of the planetary roller screw be \(p\), and the angular speed of the motor rotor be \(\omega_{m}\). The speed of the screw can be expressed as:  
\[v=\frac{ip\omega_{m}}{2\pi} \tag{10}\]  
Set the thrust of electromechanical actuator as \(F_{m}\), \(\eta\) is the efficiency of the actuator, and the torque of the motor can be expressed as:  
\[T_{m}=\frac{F_{m}p}{2\pi i\eta} \tag{11}\]  
In this paper, large vertical equipment is analyzed as the application object of EMA. Figure 1 shows the schematic diagram of vertical equipment. Taking this working condition as an example, the mathematical models of load torque and wind load torque of vertical equipment are analyzed and established. When the system is in state 1, set the angle between the line between the load center of mass \(O_{4}\) and the vertical rotation point \(O_{1}\) and the horizontal plane as \(\theta_{0}\), then when the vertical mechanism is in state 2 at time \(t\), the angle between the line between the load center of mass \(O_{4}\)' and the vertical rotation point \(O_{1}\) and the horizontal plane:  
\[\theta^{\prime}(t)=\theta_{0}+\theta(t) \tag{12}\]  
At this time, the load gravity torque \(M_{G}(t)\) that the load acting on the mechanism rotation point \(O_{1}\) due to its own gravity is:  
\[v=\frac{dl}{dt}=\frac{\omega(t)}{2\pi\cdot i}(P_{h1}+P_{h2}+P_{h3}) \tag{4}\]0  
where \(G\) is the load gravity, and \(L_{G}\) is the length of \(O_{1}\)\(O_{4}\) from the load centroid to the vertical rotation point.