## 2 Dynamic Models and Problem Formulation  
As shown in Figure 1, a 5-DOF (degree of freedom) MRI compatible robot for abdominal and thoracic punctures was built in our laboratory. The robot's mechanism design was developed such that all motions were decoupled and actuated by pneumatic cylinders. With the help of the robot, the physician could manipulate the needle remotely without moving the patient out of the MRI scanners. A schematic of the pneumatic servo system driving the needle insertion motion is depicted in Figure 2. The pneumatic cylinder (customized product from XMC Corp., Ningbo, China), which had a 150 mm stroke and 10 mm diameter bore, was made of nonmagnetic material to ensure MRI compatibility. While Chamber A was controlled by a proportional directional control valve (FESTO MPYE-5-M5-010-B), a tank was used to maintain the pressure of Chamber B at a specified constant level for safety reasons. The control valve was connected to the cylinder via a 10 m transmission line since it had to be placed away from the MRI scanner. A pressure observer was needed to estimate the pressure of Chamber A, since direct measurement was expensive for the lack of MRI-compatible pressure sensors. The main purpose of this paper was to find a way to deal with the issue of the long transmission line and realize high accuracy control of a pneumatic cylinder with pressure observation. For the purpose of comparison, the pressures at the two ends of the long transmission line, as well as the supply pressure were measured by three pressure sensors (FESTO SDET-22T-D10-G14-I-M12). The piston position was obtained with an optical position sensor (Micronor MR328). The controller was programmed in Simulink on a dSPACE DS1103 control system with a 1 ms sampling period. Figure 1: Virtual prototype of a 5-DOF (degree of freedom) magnetic resonance imaging (MRI) compatible robot for abdominal and thoracic punctures. As shown in [34], the motion of the piston-rod-needle assembly can be expressed as:  
\[m\ddot{x}=p_{\mathrm{a}}A_{\mathrm{a}}-p_{\mathrm{b}}A_{\mathrm{b}}-p_{0}A_{ \mathrm{r}}-b\dot{x}-F_{\mathrm{f}}S_{\mathrm{f}}(\dot{x})-d_{\mathrm{f}}, \tag{1}\]  
where \(x\), \(\dot{x}\), and \(\ddot{x}\) are the piston position, velocity, and acceleration, respectively, \(m\) is the normal mass of the piston-rod-needle assembly, \(p_{\mathrm{a}}\) and \(p_{\mathrm{b}}\) denotes the absolute pressures of actuator chambers, \(p_{0}\) is the ambience pressure, \(A_{\mathrm{a}}\) and \(A_{\mathrm{b}}\) are the cross section areas of piston chambers, \(A_{\mathrm{r}}\) is the cross section area of the rod, \(b\) is the total load and cylinder viscous friction coefficient, \(F_{\mathrm{f}}\) is the unknown friction coefficient, \(S_{\mathrm{f}}(\dot{x})\) is a continuous function, which is always chosen as \(S_{\mathrm{f}}(\dot{x})=\frac{2}{\pi}\mathrm{arctan}(900\dot{x})\), \(F_{\mathrm{f}}S_{\mathrm{f}}(\dot{x})\) is utilized as the smooth approximation for the usual static discontinuous Coulomb friction force, and \(d_{\mathrm{f}}\) represents the unmodeled dynamics and external disturbances. With the assumption that the discharging and charging processes are both isothermal, the pressure dynamic in the actuator chamber can be modeled as [16; 35]:  
\[\dot{p}_{\mathrm{a}}=\frac{RT_{\mathrm{s}}}{V_{\mathrm{a}}}\dot{m}_{\mathrm{a }}-\frac{A_{\mathrm{a}}p_{\mathrm{a}}}{V_{\mathrm{a}}}\dot{x},\quad V_{ \mathrm{a}}=V_{\mathrm{a}0}+A_{\mathrm{a}}(\frac{L_{\mathrm{c}}}{2}+x), \tag{2}\]  
where \(R\) is the gas constant, \(T_{s}\) is the ambient temperature, \(V_{\mathrm{a}}\) is the volume of the cylinder chamber A, \(\dot{m}_{\mathrm{a}}\) is the mass flow entering or exiting the chamber A, \(V_{\mathrm{a}0}\) is the dead volume of the cylinder chamber A, and \(L_{\mathrm{c}}\) is stroke of the actuator.