## 4 Simulation Results and Analysis
\begin{bmatrix}0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\end{bmatrix} \]  
The controller from [33] is given as (53)-(58). \[\tau=\mathcal{G}(q)\hat{p}-K_{d}\dot{q}-K_{p}e-K_{s}diag\{\mathcal{X}_{1}, \mathcal{X}_{2}\}z \tag{53}\]  
\[\mathcal{X}_{i}=\frac{\zeta_{i}^{2}}{\zeta_{i}^{2}-q_{i}^{2}}+\frac{\zeta_{i} ^{2}q_{i}z_{i}}{\left(\zeta_{i}^{2}-q_{i}^{2}\right)^{2}},\;i=1,2 \tag{54}\]  
\[z_{i}=e_{i}+\int_{0}^{t}\eta_{i}(\Theta)d\Theta,\;i=1,2 \tag{55}\]  
\[\eta_{i}=-\beta_{i}z_{i},\;i=1,2 \tag{56}\]  
\[\dot{\hat{p}}=-\Psi\mathcal{G}^{T}(q)\dot{q} \tag{57}\]  
\[\mathcal{G}(q)= \begin{bmatrix}\cos(q_{1}+q_{2})&\cos(q_{2})\\ \cos(q_{1}+q_{2})&0\end{bmatrix}  \tag{58}\]  
where \(\hat{p}=[\hat{p}_{1},\hat{p}_{2}]^{T}\), \(z_{1}(0)=e_{1}(0)\) and \(z_{2}(0)=e_{2}(0)\). \(\zeta_{1}\) and \(\zeta_{2}\) are the specific constraint (designed upper bound) of the 1 st and 2 nd joint respectively, therefore, \(\zeta_{1}=\varepsilon_{1}=50^{\text{o}}\) and \(\zeta_{2}=\varepsilon_{2}=60^{\text{o}}\). The parameters in (53)-(58) are given as follows:  
\[K_{d}= \begin{bmatrix}60&0\\ 0&20\end{bmatrix} ,\ K_{p}= \begin{bmatrix}24&0\\ 0&24\end{bmatrix} ,\ K_{s}= \begin{bmatrix}20&0\\ 0&20\end{bmatrix} \]  
\[\beta_{1}=30,\ \beta_{2}=30,\ \Psi= \begin{bmatrix}0.1&0\\ 0&0.1\end{bmatrix} \]  
The controller from [43] is given as (59)-(61). \[\tau=K_{D}^{-1}\hat{M}\left[K_{p}\dot{e}-K_{I}e+K_{D}\ddot{q}_{r}+\dot{\beta}s -K_{D}\hat{\Gamma}-K_{D}\Lambda s-K_{D}k_{s}sgn(\dot{s})\right] \tag{59}\]  
\[\dot{s}+\beta s=K_{P}e+K_{I}\int_{0}^{t}e+K_{D}\dot{e} \tag{60}\]  
\[\dot{k}_{s}=\Gamma\big{|}\dot{s}\big{|} \tag{61}\]  
where \(\dot{s}(0)=[0,0]^{T}\), \(k_{s}(0)=0\), \(\Gamma=0.1\), \(\hat{\Gamma}\) is the known part of lumped uncertainty \(\Gamma\). In this simulation, we let \(\hat{\Gamma}=0.7\Gamma\). The parameters \(K_{P}\), \(K_{D}\), \(K_{I}\) and \(\beta\) are given as follows. \[K_{D}= \begin{bmatrix}20&0\\ 0&20\end{bmatrix} ,\ K_{P}= \begin{bmatrix}10&0\\ 0&10\end{bmatrix} ,\ K_{I}= \begin{bmatrix}0.01&0\\ 0&0.01\end{bmatrix} ,\ \dot{\beta}= \begin{bmatrix}0.1&0\\ 0&0.1\end{bmatrix} \]  
The controller from [44] is given as (62)-(65). \[ \begin{cases}\dot{D}=z+Y\dot{q}\\ \dot{z}=-Y\hat{M}^{-1}z+Y\hat{M}^{-1}\left(-\tau-Y\dot{q}-\overline{D}\right) \end{cases}  \tag{62}\]  
\[\tau=\hat{M}\big{(}\ddot{q}_{r}-\Lambda\dot{q}+\Lambda\dot{q}_{r}\big{)}- \overline{D}-K_{D}sat(s)-\hat{D} \tag{63}\]  
\[sat(s_{i})= \begin{cases}sgn(s_{i}),\ |s_{i}|\geq\sigma\\ s_{i}/\sigma,|s_{i}|<\sigma\end{cases}  \tag{64}\]  
where \(\sigma=0.1\), \(sat(s)=[s_{1},s_{2}]^{T}\). \(z(0)=[0,0]^{T}\) and \(s=\dot{e}+\Lambda e\). \(\overline{D}\) is known part of \(\hat{M}^{-1}\Gamma\), \(\overline{D}=0.7\hat{M}^{-1}\Gamma\) in the simulation. The parameters \(Y\), \(\Lambda\) and \(K_{D}\) are given as follows. \[K_{D}= \begin{bmatrix}40&0\\ 0&16\end{bmatrix} ,\ Y= \begin{bmatrix}\frac{1}{0.16}&0\\ 0&\frac{1}{0.106}\end{bmatrix} ,\ \Lambda= \begin{bmatrix}40&0\\ 0&20\end{bmatrix} \]  
Furthermore, two cases are considered to run the simulation. In the 1 st case, the parameters of the dynamic model are fully known, and there is no external disturbance and no friction. In the 2 nd case, the parametric uncertainty is considered and the unknown external disturbance and frictions are applied on the dynamics model. **Remark 19**. : _The selections of parameters for the proposed controller, the controller from [26] and the controller from [33] are all carried out in case 1. In other words, all the parameters for the three different controllers are fine-tuned to have a good performance in case 1. In case 2, all the selected parameters of the three controllers remain unchanged to test the robustness to the lumped uncertainty._  
**Case 1. In the absence of system uncertainty, external disturbance and friction.