# Cross-Coupled Contouring Control of Multi-DOF Robotic Manipulator
## 3 Stability Analysis
Therefore, the Lyapunov function \(V\) in Equation (16) is a positive definite function, i.e., \(V\geq 0\). Differentiating Equation (16) with respect to time, we have:  
\[\dot{V}=\left[ \begin{array}{cc}e_{q}&\dot{e}_{q}\end{array} \right]^{T} \left[ \begin{array}{cc}\overline{K}_{p}+\overline{K}_{d}&M\\ M&M\end{array} \right]\left[ \begin{array}{c}\dot{e}_{q}\\ \ddot{e}_{q}\end{array} \right]+\frac{1}{2}\left[ \begin{array}{cc}e_{q}&\dot {e}_{q}\end{array} \right]^{T}\left[ \begin{array}{cc}0&\dot{M}\\ \dot{M}&\dot{M}\end{array} \right]\left[ \begin{array}{c}e_{q}\\ \dot{e}_{q}\end{array} \right] \tag{19}\]  
Substituting \(M\ddot{e}_{q}\) from Equation (14) into Equation (19) and using properties P2 and P3 yields  
\[\dot{V}=-\dot{e}_{q}^{T}\left(K_{dd}+C_{0}-M\right)\dot{e}_{q}-e_{q}^{T}K_{pp} e_{q}+e_{q}^{T}\left(\overline{K}_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0} \right)\dot{e}_{q}+\left(e_{q}+\dot{e}_{q}\right)^{T}\rho \tag{20}\]  
From Equation (15), it is easy to prove that  
\[e_{q}^{T}\left(\overline{K}_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0} \right)\dot{e}_{q}\leq\frac{\lambda_{M}\left(\overline{K}_{p}-K_{pp}+ \overline{K}_{d}-K_{dd}+C^{T}-C_{0}\right)}{2}\left(\dot{e}_{q}^{T}e_{q}+\dot {e}_{q}^{T}\dot{e}_{q}\right) \tag{21}\]  
From property P4, it can be proven that parameter \(\rho\) is bounded with \(\rho_{b}\) (\(>0\)) as the upper bound. It means \(\rho\leq\ ||\,M\,||\,||\,\ddot{q}_{d}\,||\,+\,||\,C_{0}\,||\,||\,\dot{q}_{d}\,||\,+\,|| \,G\left(q\right)\,||\,+\,||\,G\left(q\right)\,||\,+\,||\,F\left(q,\dot{q} \right)\,||\,=\rho_{b}\). Therefore, the following inequality is hold:  
\[\left(e_{q}+\dot{e}_{q}\right)^{T}\rho\leq\left(\,||\,e_{q}\,||\,+\,||\,\dot{e}_{ q}\,||\,\right)\rho_{b} \tag{22}\]  
Substituting Equations (21) and (22) into Equation (20) yields  
\[ \begin{array}{ll}\dot{V}&=-\dot{e}_{q}^{T}\left(K_{dd}+C_{0}-M-\frac{\lambda _{M}(\overline{K}_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0})}{2}I\right) \dot{e}_{q}\\ &-e_{q}^{T}\left(K_{pp}-\frac{\lambda_{M}(\overline{K}_{p}-K_{pp}+\overline{K} _{d}-K_{dd}+C^{T}-C_{0})}{2}\right)e_{q}+\left(\,||\,e_{q}\,||\,+\,||\,\dot{e}_ {q}\,||\,\right)\rho_{b}\end{array}  \tag{23}\]  
According to Equation (15), it can be seen that  
\[\left\{ \begin{array}{ll}\gamma_{e}&=\lambda_{m}\left(K_{pp}-\frac{\lambda_ {M}(\overline{K}_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0})}{2}I\right) >0\\ \gamma_{\dot{e}}&=\lambda_{m}\left(K_{dd}+C_{0}-M-\frac{\lambda_{M}(\overline {K}_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0})}{2}I\right)>0\end{array} \right. \tag{24}\]  
Applying Equation (24) into Equation (23) yields  
\[\dot{V}\leq-\gamma_{\dot{e}}\,||\,\dot{e}_{q}\,||\,^{2}+\rho_{b}\,||\,\dot{e} _{q}\,||\,-\gamma_{e}\,||\,e_{q}\,||\,^{2}+\rho_{b}\,||\,e_{q}\,||\, \tag{25}\]  
For all positive parameters, the following inequalities hold:  
\[\left\{ \begin{array}{ll}-\gamma_{e}\,||\,e_{q}\,||\,^{2}+\rho_{b}\,||\,e_{q }\,||\,\leq-\frac{\gamma_{\dot{e}}}{4}\,||\,e_{q}\,||\,^{2}+\frac{(\rho_{b})^{ 2}}{\gamma_{e}}\\ -\gamma_{\dot{e}}\,||\,\dot{e}_{q}\,||\,^{2}+\rho_{b}\,||\,\dot{e}_{q}\,||\, \leq-\frac{\gamma_{\dot{e}}}{4}\,||\,\dot{e}_{q}\,||\,^{2}+\frac{(\rho_{b})^{ 2}}{\gamma_{\dot{e}}}\end{array} \right. \tag{26}\]  
Substituting Equation (26) into Equation (25) yields  
\[\dot{V}\leq-\frac{\gamma_{\dot{e}}}{4}\,||\,\dot{e}_{q}\,||\,^{2}-\frac{\gamma _{e}}{4}\,||\,e_{q}\,||\,^{2}+\left(\frac{\gamma_{e}+\gamma_{\dot{e}}}{\gamma_ {e}\gamma_{\dot{e}}}\right)(\rho_{b})^{2} \tag{27}\]  
From Equation (27), it immediately follows that  
\[\dot{V}<0\,,\,\,\forall\quad\left\{ \begin{array}{ll}\,||\,e_{q}\,||\,>\, \delta_{e}=\frac{2\rho_{b}}{\gamma_{e}}\sqrt{\left(\frac{\gamma_{e}+\gamma_{ \dot{e}}}{\gamma_{\dot{e}}}\right)}\\ \,||\,\dot{e}_{q}\,||\,>\,\delta_{\dot{e}}=\frac{2\rho_{b}}{\gamma_{\dot{e}}} \sqrt{\left(\frac{\gamma_{e}+\gamma_{\dot{e}}}{\gamma_{e}}\right)}\end{array}