# Constant Force Control of Centrifugal Pump Housing Robot Grinding Based on Pneumatic Servo System
## 4 Modeling of the Pneumatic Servo System
### Cylinder Flow Continuity Equation  
According to the law of mass conservation, the mass flow of the gas in the cylinder cavity should be the mass change rate of the gas in the cylinder cavity, namely  
\[\left\{ \begin{array}{l}q_{m_{1}}=\frac{dm_{1}}{dt}\\ q_{m_{2}}=\frac{dm_{2}}{dt}\end{array} \right. \tag{2}\]  
where \(q_{m_{1}}\) is the gas mass flow in the rodless cavity of the cylinder, \(q_{m_{2}}\) is the gas mass flow in the rod cavity of the cylinder, \(m_{1}\) is the gas mass in the rodless cavity of the cylinder, and \(m_{2}\) is the gas mass in the rod cavity of the cylinder. The state equation of the ideal gas in equilibrium is  
\[pV=mRT \tag{3}\]  
Figure 4: Schematic diagram of the pneumatic control based on the VER servo valve. where \(p\) is the absolute pressure of gas, Pa; \(V\) is the gas volume, m\({}^{3}\); \(m\) is the gas mass, kg; \(R\) is the gas constant, dry air: \(R=287.1\) N \(\cdot\) m\(/\left(\text{kg}\cdot K\right)\); and \(T\) is the thermodynamic temperature of gas, \(K\). From Equation (3)  
\[m=\frac{pV}{RT} \tag{4}\]  
Equation (4) can be substituted into Equation (2) to obtain  
\[\left\{ \begin{array}{l}q_{m_{1}}=\frac{dm_{1}}{dt}=\frac{1}{RT}\left(p_{1} \frac{dV_{1}}{dt}+V_{1}\frac{dp_{1}}{dt}-\frac{p_{1}V_{1}}{T_{1}}\frac{dT_{1}}{ dt}\right)\\ q_{m_{2}}=\frac{dm_{2}}{dt}=\frac{1}{RT}\left(p_{2}\frac{dV_{2}}{dt}+V_{2} \frac{dp_{2}}{dt}-\frac{p_{2}V_{2}}{T_{2}}\frac{dT_{2}}{dt}\right)\end{array} \right. \tag{5}\]  
where \(q_{m_{1}},q_{m_{2}}\) is the mass flow of the two chambers of the cylinder, kg/s; \(\frac{dT_{1}}{dt},\frac{dT_{2}}{dt}\) is the temperature change rate of the two chambers of the cylinder, K/s; \(\frac{dp_{1}}{dt},\frac{dp_{2}}{dt}\) is the pressure change rate of the two chambers of the cylinder, Pa/s; \(\frac{dV_{1}}{dt},\frac{dV_{2}}{dt}\) is the volume change rate of the two chambers of the cylinder, m\({}^{3}\)/s; \(V_{1}\), \(V_{2}\) is the volume of the two chambers of the cylinder, m\({}^{3}\); \(p_{1}\), \(p_{2}\) is the pressure of the two chambers of the cylinder, Pa; and \(T\) is the thermodynamic temperature of the cylinder, K.  
According to the above assumptions, during the movement of the cylinder, the change process of the gas in the cylinder is an isentropic process, which is an adiabatic process, and the adiabatic equation is  
\[p^{k-1}T^{-k}=C \tag{6}\]  
where \(k\) is the gas isentropic index, for air: \(k=1.4\). By deriving Equation (6) from time, we get  
\[\frac{dT}{dt}=\frac{(k-1)T}{kp}\frac{dp}{dt} \tag{7}\]  
Substituting Equation (7) into Equation (5), we can find the mass flow of the two cylinder cavities, so that  
\[\left\{ \begin{array}{l}q_{m_{1}}=\frac{1}{RT_{1}}\left(p_{1}\frac{dV_{1}}{ dt}+\frac{V_{1}}{k}\frac{dp_{1}}{dt}\right)\\ q_{m_{2}}=\frac{1}{RT_{2}}\left(p_{2}\frac{dV_{2}}{dt}+\frac{V_{2}}{k}\frac{dp _{2}}{dt}\right)\end{array} \right. \tag{8}\]  
According to the assumption that the gas temperature in the cylinder cavity is the same as the ambient temperature and the air source temperature, which are also constant, it can be expressed by \(T_{0}\). Then, Equation (8) becomes  
\[\left\{ \begin{array}{l}q_{m_{1}}=\frac{1}{RT_{0}}\left(p_{1}\frac{dV_{1}}{ dt}+\frac{V_{1}}{k}\frac{dp_{1}}{dt}\right)\\ q_{m_{2}}=\frac{1}{RT_{0}}\left(p_{2}\frac{dV_{2}}{dt}+\frac{V_{2}}{k}\frac{dp _{2}}{dt}\right)\end{array} \right. \tag{9}\]  
The pressure dynamic equation of the two cylinder cavities is  
\[\left\{ \begin{array}{l}\frac{dp_{1}}{dt}=\frac{kRT_{0}}{V_{1}}q_{m_{1}}- \frac{kp_{1}}{V_{1}}\frac{dV_{1}}{dt}\\ \frac{dp_{2}}{dt}=\frac{kRT_{0}}{V_{2}}q_{m_{1}}-\frac{kp_{2}}{V_{2}}\frac{ dV_{2}}{dt}\end{array} \right. \tag{10}\]  
In the pneumatic servo system of the robot grinding and polishing device, the grinding force is controlled by the cylinder, and the spatial position motion is completed by the robot. When the robot positioning error is very small, the grinding wheel is always in contact with the workpiece, so the displacement of the piston rod is very small.