## 2 Dynamic Models and Problem Formulation
\(\dot{m}_{\mathrm{a}}\) can be calculated by [16]:  
\[\dot{m}_{\mathrm{a}}=(p_{\mathrm{f}}-p_{\mathrm{a}})\frac{p_{\mathrm{v}}A_{ \mathrm{f}}D_{\mathrm{f}}}{32 RT_{\mathrm{s}}L_{\mathrm{f}}\mu}, \tag{3}\]  
where \(p_{\mathrm{f}}\) is the average air pressure in the transmission line, \(p_{\mathrm{v}}\) is the measured air pressure at the work port of the control valve, \(L_{\mathrm{f}}\), \(A_{\mathrm{f}}\), and \(D_{\mathrm{f}}\) denote the length, cross section area, and inner diameter of the transmission line, respectively, and \(\mu\) is the dynamic viscosity of air. As shown in Figure 2, the long transmission line is assumed to be an intermediate chamber connected between the control valve and the actuator in series, thus, the following equation is formulated to represent the pressure dynamic of the line [16; 35]:  
\[\dot{p}_{\mathrm{f}}=\frac{RT_{\mathrm{s}}}{V_{\mathrm{f}}}(\dot{m}_{\mathrm{ v}}-\dot{m}_{\mathrm{a}}), \tag{4}\]  
Figure 2: Schematic of the pneumatic servo system used to drive the needle insertion motion. where \(V_{I}\) is the volume of the transmission line, and \(\dot{m}_{\rm v}\) is the mass flow rate through the valve, which can be calculated by [35]:  
\[\dot{m}_{\rm v}=A(u)K_{\rm q}(p_{\rm u},p_{\rm d},T_{\rm u})=\left\{ \begin{array} []{ll}A(u)C_{\rm d}C_{1}\frac{p_{\rm u}}{\sqrt{T_{\rm u}}},&\frac{p_{\rm d}}{p_ {\rm u}}\leq p_{\rm r},\\ A(u)C_{\rm d}C_{1}\frac{p_{\rm u}}{\sqrt{T_{\rm u}}}\sqrt{1-\left(\frac{p_{\rm d }}{p_{\rm u}}-p_{\rm r}\right)^{2}},&\frac{p_{\rm d}}{p_{\rm u}}>p_{\rm r}\\ \end{array} \right., \tag{5}\]  
where \(A(u)\) is the effective valve orifice area, \(u\) is the control valve's control input, \(C_{\rm d}\) is the discharge coefficient, \(C_{1}\) is a constant with a value of 0.0404, \(p_{\rm u}\) and \(p_{\rm d}\) are the upstream pressure and the downstream pressure, respectively, \(T_{\rm u}\) is the upstream temperature of air, and \(p_{\rm r}\) is the critical pressure ratio. Therefore, \(\dot{m}_{\rm v}=A(u)K_{\rm q}(p_{\rm s},p_{\rm v},T_{\rm s})\) when the line is charging, and \(\dot{m}_{\rm v}=A(u)K_{\rm q}(p_{\rm v},p_{0},T_{\rm s})\) when the line is discharging. Choose the state vectors \(x=[x_{1},x_{2},x_{3},x_{4}]^{\rm T}\) as \(x_{1}=x\), \(x_{2}=\dot{x}\), \(x_{3}=p_{\rm a}\) and \(x_{4}=p_{l}\), thus the system dynamics can be expressed in a state-space form as:  
\[\left\{ \begin{array}{ll}\dot{x}_{1}=x_{2}\\ m\dot{x}_{2}=A_{\rm a}x_{3}-A_{\rm b}p_{\rm b}-p_{0}A_{\rm r}-bx_{2}-A_{\rm f}S _{\rm f}(x_{2})-d_{\rm f}\\ \dot{x}_{3}=\frac{RT_{\rm s}}{V_{\rm s}}\dot{m}_{\rm a}-\frac{A_{\rm a}x_{2}}{ V_{\rm a}}x_{3}\\ \dot{x}_{4}=\frac{RT_{\rm s}}{V_{\rm l}}(m_{\rm v}-m_{\rm a})\\ \end{array} \right.. \tag{6}\]  
The control objective is to synthesize a control input \(u\) for the system Equation (6) such that \(x\) tracks the desired trajectory \(x_{\rm d}\) with a guaranteed transient and final tracking accuracy. Instead of using the pressure sensor, a globally stable pressure observer is developed to estimate the pressure in chamber A. The parametric uncertainties due to unknown \(b\) and \(A_{\rm f}\) and unknown external disturbances will be explicitly considered in this paper. **Assumption 1:**_The desired trajectory \(x_{d}\) is at least second-order differentiable, and \(x_{d}\), \(\dot{x}_{\rm d}\), and \(\ddot{x}_{\rm d}\) are bounded. Then, there exists a compact set \(\Omega_{0}=\left\{\left[x_{\rm d},\dot{x}_{\rm d},\ddot{x}_{\rm d}\right]^{ \rm T}:x_{\rm d}^{2}+\dot{x}_{\rm d}^{2}+\ddot{x}_{\rm d}^{2}\leq B_{0}\right\}\) such that \(\left[x_{\rm d},\dot{x}_{\rm d},\ddot{x}_{\rm d}\right]^{\rm T}\in\Omega_{0}\), where \(B_{0}\) is a positive constant._  
**Assumption 2:**_The extent of parametric uncertainties and external disturbances can be predicted and given by \(b_{\rm min}\leq b\leq b_{\rm max}\), \(F_{\rm{fmin}}\leq F_{\rm f}\leq F_{\rm{fmax}}\), and \(d_{\rm min}\leq d_{\rm f}\leq d_{\rm max}\)._