## 3 Controller Design
### Proof of Stability
The interaction between needle and soft tissue is very complex and may decrease the needle insertion precision in practical. Since the current research focused on addressing the issue of long transmission line and realizing high accuracy control of a pneumatic actuator with a pressure observer, modeling of needle-tissue interaction and further improving needle insertion precision would be the subject of the next phase of this research. The process of parameter and disturbance estimations is shown in Figure 7. It can be seen that the estimates converged quickly, and the tracking error improvement could be achieved within several seconds. Comparisons between the observed chamber pressure and the measured one are shown in Figure 8. As can be seen, the estimated chamber pressure was close to its actual value, which indicates the effectiveness of the proposed pressure observer. However, the estimation error during the charging process was slightly bigger than the one during the discharging process. This might be due to the fact that charging process was close to adiabatic and the discharging process was close to isothermal. \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Value** & **Unit** \\ \hline \(m\) & 0.32 & kg \\ \(A_{\text{a}}\) & \(7.854\times 10^{-5}\) & m\({}^{2}\) \\ \(A_{\text{b}}\) & \(6.597\times 10^{-4}\) & m\({}^{2}\) \\ \(A_{\text{r}}\) & \(1.257\times 10^{-5}\) & m\({}^{2}\) \\ \(R\) & 287 & N m/(kg K) \\ \(T_{\text{s}}\) & 300 & K \\ \(p_{\text{s}}\) & \(6\times 10^{5}\) & Pa \\ \(V_{\text{a}0}\) & \(6\times 10^{-7}\) & m\({}^{3}\) \\ \(L_{\text{c}}\) & 0.125 & m \\ \(V_{l}\) & \(1.257\times 10^{-4}\) & m\({}^{3}\) \\ \(C_{\text{d}}\) & \(1.099-0.1075\times\frac{p_{\text{u}}}{p_{\text{d}}}\) & \\ \(p_{\text{r}}\) & 0.29 & \\ \(A_{l}\) & \(1.257\times 10^{-5}\) & m\({}^{2}\) \\ \(D_{l}\) & 0.004 & m \\ \(L_{l}\) & 10 & m \\ \(\mu\) & \(1.79\times 10^{-5}\) & N s/m\({}^{2}\) \\ \(p_{0}\) & \(1\times 10^{5}\) & Pa \\ \(\hat{b}(0),b_{\text{min}},b_{\text{max}}\) & 9, 0, 15 & N s/m \\ \(\hat{F}_{l}(0),\hat{F}_{\text{min}},\hat{F}_{\text{max}}\) & 6, 0, 20 & N \\ \(\hat{d}_{l}(0),d_{\text{min}},d_{\text{max}}\) & 0, \(-10,10\) & N \\ \(k_{1}\), \(\tau_{1}\) & 40, 0.5 & \\ \(k_{2}\), \(h_{1}\), \(\eta_{1}\), \(\tau_{2}\) & 50, 40, 0.4, 0.75 & \\ \(k_{3}\), \(\tau_{3}\) & 50, 1.2 & \\ \(k_{4}\) & 90 & \\ \(\gamma_{1},\gamma_{2},\gamma_{3}\) & 100, 10, 10 & \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the system and the proposed controller. Figure 4: Tracking response of the same control algorithm as the proposed controller but without using long transmission line compensation. Figure 5: Tracking response of the proposed controller for three different types of trajectories. Figure 6: Tracking errors of the proposed controller for three different types of trajectories. Figure 7: Parameter and disturbance estimations of the proposed controller when the reference is a 0.5 Hz sinusoidal trajectory. Experiments were also conducted to verify the performance robustness of the proposed controller. To simulate a sudden disturbance acting on the system, a big step signal was added to the output of the position sensor at \(t\) = 22 s, and removed 10 s later. Figure 9 shows the control accuracy of the proposed controller in this scenario for tracking a sinusoidal trajectory with a frequency of 0.5 Hz and amplitude of 62.5 mm. It can be seen that the system performance did not deteriorate except the transient spikes when the sudden disturbance was added or removed.