## 4 Simulation Results and Analysis
\(Lin\big{(}\hat{q}_{1}\big{)}=\left\{\mathfrak{z}_{2,1}^{(1)},\ \mathfrak{z}_{2,2}^{(1)},\ldots, \mathfrak{z}_{2,8}^{(1)}\right\}=\)\(\{-0.175,\ -0.138,\ -0.097,\ -0.058,\ -0.019,\ 0.019,\ 0.058,\ 0.097,\ 0.138,\ 0.175\}\times\)\(10^{-3}\). \(Lin(e_{2})=\left\{\mathfrak{z}_{1,1}^{(2)},\ \mathfrak{z}_{1,2}^{(2)},\ldots, \mathfrak{z}_{1,8}^{(2)}\right\}=\)\(\{-0.0017,\ -0.0014,\ -0.001,\ -0.0006,\ -0.0002,\ 0.0002,\ 0.0006,\ 0.001,\ 0.0014,\ 0.0017\}\). \(Lin\big{(}\hat{q}_{2}\big{)}=\left\{\mathfrak{z}_{2,1}^{(2)},\ \mathfrak{z}_{2,2}^{(2)},\ldots, \mathfrak{z}_{2,8}^{(2)}\right\}=\)\(\{-0.175,\ -0.138,\ -0.097,\ -0.058,\ -0.019,\ 0.019,\ 0.058,\ 0.097,\ 0.138,\ 0.175\}\times\)\(10^{-3}\). The action candidates for each fuzzy rule are: \(\mathfrak{U}_{i,\boldsymbol{n}}=\{0,22,44,\ 66,\ 88,\ 111,\ 133,\ 1 55,\ 177,\ 200\}\) for all \(i=1,2\) and \(\boldsymbol{n}_{1}=1,2,\ldots,\ 64\). Therefore, the amount of action candidates for each fuzzy rule is \(P=10\). The discount factor is \(\alpha_{1}=\alpha_{2}=0.01\) and the learning rate is \(\Gamma_{1}=\Gamma_{2}=0.2\). The threshold of obtaining rewards is \(\sigma_{1}=\sigma_{2}=0.01\). To show the superiority, four existing controllers, Refs. [26; 33; 43; 44] are compared with the proposed controller. The controller from [26] is given as (44)-(52)  
\[\boldsymbol{\tau}=\boldsymbol{\tau_{1}}+\boldsymbol{\tau_{2}} \tag{44}\]  
\[\boldsymbol{\tau_{1}}=\hat{\boldsymbol{M}}(\boldsymbol{q})\big{(}\tilde{ \boldsymbol{q}}_{\boldsymbol{r}}-K_{\boldsymbol{V}}\hat{\boldsymbol{e}}-K_{ \boldsymbol{P}}\boldsymbol{e}\big{)}+\hat{\boldsymbol{C}}\big{(}\boldsymbol{q}, \hat{\boldsymbol{q}}\big{)}\hat{\boldsymbol{q}}+\hat{\boldsymbol{G}}( \boldsymbol{q}) \tag{45}\]  
\[\boldsymbol{\tau_{2}}=-\hat{\boldsymbol{M}}(\boldsymbol{q})\hat{\boldsymbol{f}}+ \hat{\boldsymbol{M}}(\boldsymbol{q})\boldsymbol{u}_{\boldsymbol{r}} \tag{46}\]  
\[\boldsymbol{u}_{\boldsymbol{r}}= \begin{bmatrix}-\hat{\xi}\tan\ln(a_{1}p/\rho_{1}) \\ -\hat{\xi}\tan\ln(a_{2}p/\rho_{1})\end{bmatrix}  \tag{47}\]  
\[\hat{\boldsymbol{f}}=\hat{\boldsymbol{W}}^{\boldsymbol{T}}\boldsymbol{\sigma} \Big{(}\hat{\boldsymbol{V}}^{\boldsymbol{T}}\boldsymbol{X}\Big{)} \tag{48}\]  
\[\hat{\xi}=a_{1}\boldsymbol{p}\boldsymbol{tanh}\bigg{(}\frac{a_{1}p}{\rho_{1}} \bigg{)}+a_{1}\boldsymbol{p}\boldsymbol{tanh}\bigg{(}\frac{a_{2}p}{\rho_{1}} \bigg{)}-K_{\xi}^{\xi} \tag{49}\]\[\hat{\hat{W}}=\Big{(}\mathbf{\sigma}-\mathbf{\sigma}^{\prime}\hat{V}^{T}X\Big{)}x^{T}PB-Y_{W }\hat{W} \tag{50}\]  
\[\hat{\hat{V}}=Xx^{T}PB\hat{W}^{T}\mathbf{\sigma}^{\prime}-Y_{V}\hat{V} \tag{51}\]  
\[p=1+||X||+||\hat{V}||\cdot||X||+||\hat{W}||\cdot||X|| \tag{52}\]  
where \(\mathbf{\sigma}=[\sigma_{1},\sigma_{2},\ldots,\sigma_{m}]^{T}\) is the vector of hidden neurons with the activation function \(\sigma_{i}(s)=1/(1+e^{-s})\). \(\mathbf{\sigma}^{\prime}\) is the vector of partial derivative of \(\sigma\) such that \(\mathbf{\sigma}^{\prime}=\partial\mathbf{\sigma}(s)/\partial s\). \(x=[e_{1},e_{2},\dot{q}_{1},\dot{q}_{2}]^{T}\) and \(X=[q_{r1},q_{r2},q_{1},q_{2},\dot{q}_{1},\dot{q}_{2},\ddot{q}_{1},\ddot{q}_{2 }]^{T}\). \(P\) satisfies \(PA+A^{T}P=-Q\) with \(A= \begin{bmatrix}0&I\\ -K_{P}&-K_{V}\end{bmatrix} \). \([a_{1},a_{2}]=x^{T}PB\). In this simulation, we let the controller from [26] to fully know the system parameters such that \(\hat{\mathbf{M}}(\mathbf{q})=\mathbf{M}(\mathbf{q})\), \(\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}})=\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\) and \(\hat{\mathbf{G}}(\mathbf{q})=\mathbf{G}(\mathbf{q})\). The parameters in (44)-(52) are given as:  
\[K_{V}= \begin{bmatrix}300&0\\ 0&300\end{bmatrix} ,\;K_{P}= \begin{bmatrix}200&0\\ 0&200\end{bmatrix} \]  
\[B= \begin{bmatrix}0&0\\ 0&0\\ 1&0\\ 0&1\end{bmatrix} ,\;Q= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \]  
\(\rho_{1}=0.01\), \(K=0.005\), \(Y_{W}=0.15\), \(Y_{V}=0.15\), \(\hat{\xi}(0)=0.01\)  
\[\hat{W}(0)= \begin{bmatrix}-0.1&-0.1\\ -0.1&-0.1\\ -0.1&-0.1\\ -0.1&-0.1\\ -0.1&-0.1\end{bmatrix} \]  
\[\hat{V}(0)=