# Sliding Mode Control of Servo Feed System Based on Fuzzy Reaching Law
## 2 Flexible Dynamics Modeling of the Feed System  
The rigid body model can better describe the dynamic characteristics of the system in the low frequency band of the system, which is easy to implement the control method. However, under high-speed operating conditions, the ball screw drive sub of the servo feed system will inevitably undergo elastic deformation, making the system susceptible to shaft-torsion coupling vibration, which will have a great impact on the tracking performance of the system. Therefore, the flexible body characteristics exhibited by the system in the high frequency band must be fully considered to establish the flexible body dynamics model of the servo-feed system. Figure 1 shows the servo feed system model: the AC servo motor drives the screw to rotate through the coupling, the nut on the screw is fixed to the table by the nut sleeve, both ends of the platform are supported by rolling linear guides and the servo motor generates motor torque through a closed-loop servo control system, which controls the position of the table. Several two-dimensional models have been proposed in the literature [30] to simulate the rigid body model and the first vibration pattern of the ball screw drive. In these methods, the concentrated mass model can effectively capture the stiffness characteristics and the main vibration pattern on the ball screw feed drive system and be easily used for controller design [31]. According to the concentrated mass method, the servo-feed system model is equated to a two-degree-of-freedom mass-spring-damping system, as shown in Figure 2, and the physical significance of each parameter is shown in Table 1.  
Figure 1: Servo feed system model. Figure 2: Concentrated mass model of the servo feed system. In Figure 2, the equation of motion for the concentrated mass model can be written as:  
\[ \begin{array}{l}m_{1}\overset{\bullet\bullet}{x_{1}}=-b_{1}\overset{\bullet}{x _{1}}+k(x_{2}-x_{1})+c(\overset{\bullet}{x_{2}}-\overset{\bullet}{x_{1}})+u+d_ {1}\\ m_{2}\overset{\bullet\bullet}{x_{2}}=-b_{2}\overset{\bullet}{x_{2}}+k(x_{1}-x_{ 2})+c(\overset{\bullet}{x_{1}}-\overset{\bullet}{x_{2}})+d_{2}\end{array}  \tag{1}\]  
The transfer dynamics between the motor and the table can be derived from the equation of motion (1) and transformed by [PERSON] as:  
\[ \begin{array}{l}G_{M}(S)=\frac{X_{1}(S)}{U(S)}=\frac{m_{2}s^{2}+(c+b_{2})s+k}{ m_{1}m_{2}s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}}\\ G_{T}(S)=\frac{X_{2}(S)}{U(S)}=\frac{c+k}{m_{1}m_{2}s^{4}+a_{3}s^{3}+a_{2}s^{2}+ a_{1}s+a_{0}}\end{array}  \tag{2}\]  
Among them:  
\[ \begin{array}{l}a_{0}=0\\ a_{1}=(b_{1}+b_{2})k\\ a_{2}=b_{1}b_{2}+c(b_{1}+b_{2})+k(m_{1}+m_{2})\\ a_{3}=m_{1}b_{2}+m_{2}b_{1}+c(m_{1}+m_{2})\end{array} \]  
\(G_{M}(S)\) and \(G_{T}(S)\) are the transfer functions between motor torque to motor position and motor torque to table position, respectively. Equation (2) describes the overall transfer dynamics of the servo-feed system. Since the characteristic equations are of fourth order, they are converted into state-space form by selecting four state quantities. In servo feed system installations, the position and speed of the motor and table are measured by rotary encoder and scale systems, respectively [32]. Select the state vector as \(x= \begin{bmatrix}x_{1}&\overset{\bullet}{x_{1}}&x_{2}&\overset{\bullet}{x_{2} }\end{bmatrix} ^{T}\), then Equation (2) can be rewritten as the equation of state for the two-degree-of-freedom flexible body model:  
\[\overset{\bullet}{x}=A_{M}v+B_{M}u \tag{3}\]  
Among them:  
\[ \begin{array}{l}A_{M}=\begin{bmatrix}0&1&0&0\\ -\frac{k}{m_{1}}&-\frac{(c+b_{1})}{m_{1}}&\frac{k}{m_{1}}&\frac{c}{m_{1}}\\ 0&0&0&1\\ \frac{k}{m_{2}}&\frac{c}{m_{2}}&-\frac{k}{m_{2}}&-\frac{(c+b_{2})}{m_{2}} \end{bmatrix}\\ B_{M}=\begin{bmatrix}0&\frac{1}{m_{1}}&0&0\end{bmatrix}^{T}\\ x=\begin{bmatrix}x_{1}&\overset{\bullet}{x_{1}}&x_{2}&\overset{\bullet}{x_{2} }\end{bmatrix}^{T}\end{array} \]