## 4 Control Design  
Based on the estimates from the SMO, the main controller is constructed with the Lyapunov function in order to guarantee prescribed performance in all error components, which is different from the authors' previous work. The control diagram is shown in Figure 2.  
The control signal is proposed as follows:  
\[ \begin{array}{l}\mathbf{u}=J_{a}^{-T}(\mathbf{q})(V(\dot{q},\dot{\hat{q}})+G(\mathbf{q} )+M(\mathbf{q})\hat{\mathbf{d}}+\mathbf{u}_{1})\\ \mathbf{u}_{1}=-M(\mathbf{q})T^{-1}(\mathbf{q})\mathbf{I}(\mathbf{q},\dot{\hat{q}})\hat{q}-M(\mathbf{q} )T^{-1}(\mathbf{q})T^{-1}\mathbf{u}_{2}+M(\mathbf{q})T^{-1}\ddot{\mathbf{r}}_{d}\end{array}  \tag{14}\]  
Figure 2: Control schematics. where \(r_{d}\) denotes the reference trajectory of the end-effector \(r=(x,y,\phi)\), \(u_{2}\) is an auxiliary control signal to be designed later, and \(T\) is the task coordinate transformation matrix, which denotes the following relation:  
\[\varepsilon=Te \tag{15}\]  
where \(\varepsilon=\left[\varepsilon_{n},\varepsilon_{t},\varepsilon_{\phi}\right]^{T}\) is a new presentation of the original tracking error \(e=r_{d}-r\), which includes the normal, tangential, and orientation tracking error components, respectively. The transformation matrix \(T\) is calculated based on the slope angle \(\theta\) of the desired contour as shown in Figure 3.  
In the case that linear contour is adopted, i.e., \(T\) is constant, from (14) and (15), the error dynamics is derived as  
\[\ddot{\varepsilon}=\mathbf{u}_{2}+\mathbf{f}_{d} \tag{16}\]  
where \(f_{d}\) is the lumped error components including velocity and disturbance estimation errors of the SMO, which is bounded, i.e., \(\|f_{d}\|\leq\Delta_{f},\Delta_{f}>0\). The output constraint is pre-defined as follows:  
\[ \begin{array}{l}-b_{n}(t)\leq\varepsilon_{n}\leq b_{n}(t)\\ -b_{t}(t)\leq\varepsilon_{t}\leq b_{t}(t)\\ -b_{\phi}(t)\leq\varepsilon_{\phi}\leq b_{\phi}(t)\\ \end{array}  \tag{17}\]  
where \(b_{i}(t)\) are predefined limits. To maintain these prescribed performances during operation, the auxiliary control signal \(\mathbf{u}_{2}\) is designed by  
\[u_{2i}=\dot{a}_{i}-\frac{\varepsilon_{i}}{b_{i}(t)^{2}-\varepsilon_{i}^{2}}+k _{2i}z_{i}\quad(i=n,t,\phi) \tag{18}\]  
Figure 3: Illustration of the error transformation mechanism [4], Reprinted/adapted with permission from Ref. [4]. 2021, Elsevier. where \(z_{i}=\alpha_{i}-\dot{\epsilon}_{i}\), \(\alpha_{i}=-(k_{1i}+\lambda_{i}(t))\epsilon_{i}\), \(\lambda_{i}(t)=\{\dot{b}_{i}(t)^{2}/b_{i}(t)^{2}+\beta\}\), \(k_{1i}\) and \(k_{2i}\) are control gains, and \(\beta\) is a positive constant. For the system (1), the proposed control algorithm (14), (18) together with the SMO (4), (12) guarantees prescribed performance on both contouring, tangential, and orientation error components under the effects of lumped uncertainties and disturbances. Lyapunov functions are defined as follows:  
\[V_{i}=\frac{1}{2}\ln\Biggl{(}\frac{b_{i}^{2}(t)}{\dot{b}_{i}^{2}(t)-\dot{ \epsilon}_{i}^{2}}\Biggr{)}+\frac{1}{2}{z_{i}}^{2}\ \ (i=n,t,\phi) \tag{19}\]  
Taking derivatives of them and substituting (16) and (18) into them, one obtains  
\[ \begin{array}{l}\dot{V}_{i}=\frac{z_{i}}{b_{i}^{2}(t)-\dot{ \epsilon}_{i}^{2}}\left(\dot{\epsilon}_{i}-\epsilon_{i}\frac{\dot{b}_{i}(t)}{ \dot{b}_{i}(t)}\right)+z_{i}\dot{z}_{i}\\ =\frac{\dot{\epsilon}_{i}}{\dot{b}_{i}^{2}(t)-\dot{\epsilon}_{i}^{2}}\left( \dot{\alpha}_{i}-\epsilon_{i}\frac{\dot{b}_{i}(t)}{\dot{b}_{i}(t)}\right)- \frac{\dot{\epsilon}_{i}\dot{z}_{i}}{\dot{b}_{i}^{2}(t)-\dot{\epsilon}_{i}^{2 }}+z_{i}(\dot{\alpha}_{i}-u_{2i}-f_{di})\\ \leq-\frac{k_{1i}\epsilon_{i}^{2}}{\dot{b}_{i}^{2}(t)-\dot{\epsilon}_{i}^{2}} -k_{2i}z_{i}^{2}+z_{i}f_{di}\\ \leq-k_{1i}\ln\left(\frac{\dot{b}_{i}^{2}(t)}{\dot{b}_{i}^{2}(t)-\dot{ \epsilon}_{i}^{2}}\right)-\left(k_{2i}-\frac{1}{2}\right)z_{i}^{2}+\frac{1}{2 }f_{di}\\ \leq-\lambda_{i}V_{i}+\Delta_{i}\end{array}  \tag{20}\]  
where \(\lambda_{i}=2\text{min}(k_{1i},k_{2i}-0.5)\) and \(\Delta_{i}=0.5\Delta_{f}^{2}\).