## 3 Controller Design
### Proof of Stability
 \tag{32}\]  
It has been proved in [36] that with the projection mapping, the following properties hold:  
\[\overline{\dot{b}}\left[\mathrm{Proj}_{\dot{b}}(-\frac{\gamma_{1}}{m}x_{2}s_{ 1})-(-\frac{\gamma_{1}}{m}x_{2}s_{1})\right]\leq 0. \tag{33a}\]  
\[\overline{F}_{\rm f}\left[\mathrm{Proj}_{\dot{F}_{\rm f}}(-\frac{\gamma_{2}}{ m}S_{\rm f}(x_{2})s_{1})-(-\frac{\gamma_{2}}{m}S_{\rm f}(x_{2})s_{1})\right]\leq 0. \tag{33b}\]  
\[\overline{\dot{d}}_{\rm f}\left[\mathrm{Proj}_{\dot{d}_{\rm f}}(-\frac{\gamma _{3}}{m}s_{1})-(-\frac{\gamma_{3}}{m}s_{1})\right]\leq 0. \tag{33c}\]  
According to Assumption 1-2, there exists a known function \(h_{1}\) satisfies:  
\[h_{1}(t)\geq\frac{1}{m}\left[|b_{\rm max}-b_{\rm min}||x_{2}|+|F_{\rm max}-F_ {\rm min}||S_{\rm f}(x_{2})\right|+|d_{\rm max}-d_{\rm min}|\right]+\overline{ A}_{\rm a}\left|\rho_{1}\right|, \tag{34}\]  
where \(\rho_{1}\) is the bound of \(\overline{x}_{3}\) as proven in the above step. From the smoothed sliding mode control theory, the following inequality holds. \[s_{1}\left\{-\frac{h_{1}^{2}}{4\eta_{1}}s_{1}-\frac{1}{m}\overline{\dot{b}}x_ {2}+\overline{F}_{\rm f}S_{\rm f}(\dot{x})+\overline{\dot{d}}_{\rm f}\right]+ \overline{A}_{\rm a}\overline{x}_{3}\right\}\leq\eta_{1}. \tag{35}\]  
Substituting Equation (34) into Equation (32) yields  
\[\dot{V}_{1}\leq-k_{1}e_{1}^{2}-k_{2}s_{1}^{2}-k_{3}s_{2}^{2}-k_{4}s_{3}^{2}+e_ {1}s_{1}+e_{1}z_{1}+\overline{A}_{\rm a}(s_{1}s_{2}+s_{1}z_{2})+\frac{RT_{s}a }{V_{\rm a}}(s_{3}+z_{3})+\eta_{1}. \tag{36}\]  
The following inequalities can be obtained by using Young's inequality. \[ \begin{array}{rl}e_{1}s_{1}\leq\frac{e_{1}^{2}}{2}+\frac{s_{1}^{2}}{2},e_{1 }z_{1}\leq\frac{e_{1}^{2}}{2}+\frac{z_{2}^{2}}{2},s_{1}s_{2}\leq\frac{s_{1}^{2 }}{2}+\frac{s_{2}^{2}}{2}\\ s_{1}z_{2}\leq\frac{s_{1}^{2}}{2}+\frac{z_{2}^{2}}{2},s_{2}s_{3}\leq\frac{s_{2}^ {2}}{2}+\frac{s_{2}^{2}}{2},s_{2}s_{3}\leq\frac{s_{2}^{2}}{2}+\frac{z_{2}^{2} }{2}\end{array}  \tag{37}\]  
Substituting Equation (37) into Equation (36) gives:  
\[ \begin{array}{rl}\dot{V}_{1}\leq&-(k_{1}-1)e_{1}^{2}-(k_{2}-\frac{1}{2}- \overline{A}_{\rm a})s_{1}^{2}-(k_{3}-\frac{\overline{A}_{\rm a}}{2}-\frac{ RT_{s}a}{V_{\rm a}})s_{2}^{2}-(k_{4}-\frac{RT_{s}a}{2V_{\rm a}})s_{3}^{2}+\\ &\frac{1}{2}z_{1}^{2}+\frac{\overline{A}_{\rm a}}{2}z_{2}^{2}+\frac{RT_{s}a}{2V _{\rm a}}s_{2}^{2}+\eta_{1}\end{array}  \tag{38}\]  
Following the similar analysis as done in [28], the following inequalities hold. \[\dot{z}_{1}+\frac{z_{1}}{\tau_{1}}\leq\left|\dot{z}_{1}+\frac{z_{1}}{\tau_{1}} \right|\leq\xi_{1},\quad\dot{z}_{2}+\frac{z_{2}}{\tau_{2}}\leq\left|\dot{z}_ {2}+\frac{z_{2}}{\tau_{2}}\right|\leq\xi_{2},\quad\dot{z}_{3}+\frac{z_{3}}{ \tau_{3}}\leq\left|\dot{z}_{3}+\frac{z_{3}}{\tau_{3}}\right|\leq\xi_{3}, \tag{39}\]  
The same control algorithm as the proposed controller, but in which the long transmission line was characterized as a part of the cylinder's dead volume and the chamber pressure was directly measured, was first tested for tracking a sinusoidal trajectory and a smooth square trajectory. As shown in Figure 4, the performance was poor for practical application, which indicates the need for an advanced method to address the issue of long transmission line. The performance of the proposed controller was tested on three types of reference trajectories: A 0.5 Hz sinusoidal signal, a smooth square signal, and a 4 s periodic signal. Figure 5 shows the response of the system, and the tracking errors are presented in Figure 6. The maximum absolute tracking errors (\(\max_{l}\{|e_{1}|\}\)) were about 2.51 mm, 2.68 mm, and 1.81 mm, while the final steady-state tracking errors (\(\max_{t\geq 5}\{|e_{1}|\}\)) were about 0.96 mm, 1.05 mm, and 1.39 mm. As seen, the proposed controller could significantly improve the system performance, which indicates the effectiveness of the proposed long transmission line compensation method. However, it should be noted that these needle insertion experiments were performed in a gel phantom, whose resisting force behavior was simpler than real soft tissue.