## 2 System Description  
As Figure 1 shows, the pneumatic constant force actuator mainly included a tilt sensor (KELAG Kabel variante kas90), a cylinder (SMC MGPM20-30Z), a force sensor (LH Z05A-200N), a base, a proportional pressure regulator (Festo VPPE-3-1-1/8-10-010-E1) and an electromagnetic valve (Parker A05 PS25X-1s). Among them, the base was connected with the robot by bolts, and the force sensor had its thread for connecting the workpiece. The pneumatic system control diagram is shown in Figure 2. The force sensor fed back the contact force F to the electrical signal U/ in real time, and the tilt sensor was used for obtaining the attitude signal U/ of the actuator. The signals were transmitted to the  
Figure 1: Compliant constant-force actuator structure diagram. host computer through the data acquisition (DAQ) board. Then, the host computer ran the controller and adjusted the control voltage U\(c\) of the proportional pressure regulator, making the output force F meet the set value. It also controlled the on and off state of the electromagnetic valve to change the direction of the force. As a result, the system had force perception and force control capabilities. The pneumatic system has nonlinear factors that are difficult to quantify [13]. To understand the characteristics of the system intuitively and to facilitate the following simulations, a linear approximate model of the system needed to be obtained first. The mathematical model G(s) of the system represents the corresponding relationship between the control voltage of the proportional pressure regulator U\(c\)(s) and the output force of the actuator F(s), including three links: the control voltage U\(c\)(s) to the output pressure of the proportional pressure regulator Pu(s); the Pu(s) to the input pressure of the cylinder action chamber Pd(s); and the Pd(s) to the output force F(s). According to the device structure above, it is divided into three parts to calculate the mathematical model. The first part is the proportional pressure regulator. The schematic diagram of its internal structure is shown in the Figure 3. The following equation is the force balance equation of the proportional pressure regulator spool:  
\[P_{pv}A_{1}+P_{dp}A_{2}+mg-P_{dp}A_{1}-P_{sv}A_{2}-c\dot{x}_{p}-k_{f}(x_{v}+x_ {0})-F_{c}=0 \tag{1}\]  
where \(P_{pv}\) is the air pressure in the pilot cavity; \(P_{dp}\) is the output pressure of the proportional pressure regulator; \(P_{sv}\) is the supply pressure of the proportional pressure regulator; \(A_{1}\) is the effective area of the diaphragm; \(A_{2}\) is the cross-sectional area of the valve chamber; \(m\) is the self-weight of the valve spool; \(x_{v}\) is the displacement of the valve spool; \(x_{0}\) is the initial deformation of the spring; \(c\) is the viscous damping coefficient; \(k_{f}\) is the equivalent spring stiffness; and \(F_{c}\) is the Coulomb friction. Ignoring the influence of the Coulomb friction, we can write Equation (1) in incremental form and perform a Laplace transform:  
\[\left(cs+k_{f}\right)\Delta x_{v}(s)-\Delta P_{pv}(s)A_{1}+\Delta P_{dp}(s)(A_ {1}-A_{2})=0 \tag{2}\]  
According to the properties of the proportional pressure regulator, the spool displacement \(x_{v}\) and the air pressure \(P_{pv}\) of the pilot cavity can be approximated as proportional to the control voltage \(U_{c}\). Figure 2: Pneumatic system control diagram. Then, we can obtain the following relationship:  
\[\frac{\Delta P_{u}(s)}{\Delta U_{c}(s)}=\frac{K_{1}cs+K_{1}k_{f}-K_{2}A_{1}}{A_{1} -A_{2}}=k_{1}s+k_{2} \tag{4}\]  
where \(K_{n}\), \(k_{n}\) (\(n\) = 1, 2, 3... ) are all undetermined coefficients. The second part is the gas pipeline. Assuming that the trachea is circular and the airflow movement is laminar, according to [PERSON] theory [28], we have  
\[\Delta q_{m}(s)=K_{3}(\Delta P_{u}(s)-\Delta P_{d}(s)) \tag{5}\]  
where \(q_{m}\) is the mass flow rate. The gas enters the cylinder through the proportional pressure regulator and the gas pipe.