# Constant Force Control of Centrifugal Pump Housing Robot Grinding Based on Pneumatic Servo System
## 4 Modeling of the Pneumatic Servo System
### Force Balance Equation of the Cylinder  
The force of the cylinder of the pneumatic system in the robot grinding and polishing device was analyzed according to [PERSON]'s second law and the force balance equation of the cylinder was obtained as  
\[P_{1}A_{1}-P_{2}A_{2}-F_{f}-F_{L}=M\frac{d^{2}y}{dt^{2}}+f\frac{dy}{dt}+K_{p}y \tag{19}\]  
where \(A_{1},A_{2}\) is the area of the piston in the two chambers of the cylinder, m\({}^{2}\); \(P_{1},P_{2}\) is the pressure of the two chambers of the cylinder, Pa; \(F_{L}\) is the actual grinding and polishing force, N; \(F_{f}\) is the friction force on the cylinder, N; \(M\) is the total mass of all parts of the cylinder, kg;\(f\) is the viscous damping coefficient; \(K_{P}\) is the cylinder spring stiffness; and \(y\) is the output displacement of the cylinder.  
Establishment of a Mathematical Model of the Pneumatic System of the Robot Grinding and Polishing Device  
By applying the Laplace transformation to Equations (11) and (18), we get  
\[\left\{\begin{array}{l}Q_{m_{1}}(s)=\frac{1}{K_{10}}\frac{V_{1}}{k}sP_{1}(s) \\ Q_{m_{2}}(s)=\frac{1}{RT_{0}}\frac{V_{2}}{k}sP_{2}(s)\end{array}\right. \tag{20}\]  
\[\left\{\begin{array}{l}Q_{m_{1}}(s)=K_{1}U(s)\\ Q_{m_{2}}(s)=K_{3}U(s)\end{array}\right. \tag{21}\]  
Simultaneously, for Equations (20) and (21), we get  
\[\left\{\begin{array}{l}\frac{P_{1}(s)}{U(s)}=\frac{RT_{0}K_{1}}{V_{1}s}\\ \frac{P_{2}(s)}{U(s)}=\frac{RT_{0}K_{2}}{V_{2}s}\end{array}\right. \tag{22}\]From the Laplace transformation of the cylinder force balance equation, Equation (19), we get  
\[M\!s^{2}Y(s)+fsY(s)+K_{p}Y(s)=P_{1}(s)A_{1}-P_{2}(s)A_{2}-F_{f}(s)-F_{L}(s) \tag{23}\]  
If we ignore the friction and let \(F_{L}(s)=0\), the transfer function of the cylinder piston displacement and the proportional valve input voltage obtained from Equations (20)-(23) is  
\[\frac{Y(s)}{U(s)}=\frac{b_{0}}{a_{3}s^{3}+a_{2}s^{2}+a_{1}s} \tag{24}\]  
where  
\[\left\{\begin{array}{l}b_{0}=RT_{0}k(K_{1}A_{1}V_{2}-K_{3}A_{2}V_{1})\\ a_{3}=MV_{1}V_{2}\\ a_{2}=fV_{1}V_{2}\\ a_{1}=K_{p}V_{1}V_{2}\\ K_{p}=\frac{4 kP_{1}A_{1}}{l}\end{array}\right.\]  
Let \(U(s)=0\); the transfer function between the cylinder piston displacement and the input load force is  
\[\frac{Y(s)}{F_{n}(s)}=\frac{-1}{M\!s^{2}+fs+K_{p}} \tag{25}\]  
The transfer function between the cylinder piston displacement and the grinding force is  
\[\frac{Y(s)}{F_{n}(s)}=\frac{1}{M\!s^{2}+fs+K_{p}} \tag{26}\]  
If the input load force value of the pneumatic system is equal to the output force value of the pneumatic system, and the direction of the force is opposite, the transfer function between the load force of the grinding and polishing device and the input voltage of the proportional valve is  
\[\frac{F(s)}{U(s)}=\frac{\left(M\!s^{2}+fs+K_{p}\right)b_{0}}{a_{3}s^{3}+a_{2}s ^{2}+a_{1}s}=\frac{\left(M\!s^{2}+fs+K_{p}\right)b_{0}}{s(a_{3}s^{2}+a_{2}s+a_ {1})} \tag{27}\]  
According to Equation (27), the aerodynamic control system of the robot grinding and polishing device is composed of the proportional link, second-order differential link, integral link, and second-order oscillation link.  
Based on Equations (22) and (23), a block diagram representing the transfer function of the pneumatic system of the grinding device is shown in Figure 5.  
Figure 5: Block diagram of the open-loop transfer function of the pneumatic servo system.  
In practical engineering applications, the force sensor is used to detect the load force, and the displacement sensor is used to detect the movement displacement of the cylinder piston, so as to realize closed-loop control. The block diagram is shown in Figure 6.