## 4 Simulation and Experimental Results  
### Simulation Analysis  
To verify the effectiveness of the proposed intelligent control method using machine learning based on RBFNNs, the two-link robot manipulator model of FFSGR shown in Figure 6 was utilized. Based on Reference [10], the dynamic equation for the two-link robot manipulators with joint friction and external disturbance can be described as follows:  
\[M(q)=\left[ \begin{array}{cc}M_{11}&M_{12}\\ M_{21}&M_{22}\end{array} \right]\!,\ C(q)=\left[ \begin{array}{cc}C_{11}&C_{1 2}\\ C_{21}&C_{22}\end{array} \right]\!,\ G(q)=\left[ \begin{array}{c}G_{1}\\ G_{2}\end{array} \right]\!,\ F\Big{(}\dot{q}\Big{)}=\left[ \begin{array}{c}F_{ 1}\\ F_{2}\end{array} \right]\!,\ \tau_{d}=\left[ \begin{array}{c}\tau_{1}\\ \tau_{2}\end{array} \right] \tag{47}\]  
and  
\[\left\{ \begin{array}{c}M_{11}=(m_{1}+m_{2})l_{1}^{2}+m_{2}l_{2}^{2}+2m_{2}l_ {1}l_{2}\cos(q_{2})\\ M_{22}=m_{2}l_{2}^{2}\\ M_{12}=M_{21}=m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}\cos(q_{2})\end{array} \right., \left\{ \begin{array}{c}C_{11}=-m_{2}l_{1}l_{2}\sin(q_{2})\dot{q}_{2}\\ C_{12}=-m_{2}l_{1}l_{2}\sin(q_{2})\Big{(}\dot{q}_{1}+\dot{q}_{2}\Big{)}\\ C_{21}=m_{2}l_{1}l_{2}\sin(q_{2})\dot{q}_{1},C_{22}=0\\ G_{2}=m_{2}l_{2}g\cos(q_{1}+q_{2})\end{array} \right., \left\{ \begin{array}{c}F_{1}=0.2\,\mathrm{sign}\Big{(}\dot{q}_{1}\Big{)} \\ F_{2}=0.2\,\mathrm{sign}\Big{(}\dot{q}_{2}\Big{)}\end{array} \right.^{,}\left\{ \begin{array}{c}\tau_{1}=0.5\sin(t)\\ \tau_{2}=0.5\cos(t)\end{array} \right. \tag{48}\]  
here \(m_{1}\) and \(m_{2}\) are the masses of joint 1 and joint 2, respectively; \(l_{1}\) and \(l_{2}\) are the length of joint 1 and joint 2, respectively; and \(g\) is the acceleration of gravity. The parameters of the considered two-link robot manipulators are given as:  
\[m_{1}=3\text{kg},\ m_{2}=2\text{kg},\ l_{1}=0.8\text{m},\ l_{2}=1\text{m},\ g=1 0\text{m}/\text{s}^{2} \tag{49}\]  
The chosen dead zone parameters are:  
\[d^{+}=13,\ d^{-}=-15,\ m(u_{C})=u_{C}-d^{+},\ n(u_{C})=u_{C}+d^{-} \tag{50}\]  
The parameter values used in the adaptive control system are:  
\[c=diag\Big{[} \begin{array}{cc}6&6\end{array} \Big{]},\ K_{g}=diag\Big{[} \begin{array}{cc}30&30\end{array} \Big{]},\ K_{s}=diag\Big{[} \begin{array}{ cc}1&1\end{array} \Big{]} \tag{51}\]  
The parameter values used in the RBFNNs are:  
\[S=diag\Big{[} \begin{array}{cc}400&400\end{array} \Big{]},\ T=diag\Big{[} \begin{array}{cc}500&500\end{array} \Big{]},\ b_{i}=100,\ c_{i}=\left[ \begin{array}{cccccc}-50&-20&-10&0&10&20&40\\ -50&-20&-10&0&10&20&40\end{array} \right] \tag{52}\]  
where the initial values of the weights are all 20. The control objective was to control the joint angles of the two-link robot manipulators to follow the desired trajectories. The initial positions of the joints and the initial velocities of the joints were 0, while the desired position trajectories of the two-link robot manipulators were chosen as \(q=\Big{[} \begin{array}{cc}2\sin(0.2\pi t)&\cos(0.2\pi t)\end{array} \Big{]} ^{T}\) (in radians). As illustrated in Figure 7, due to the existence of dead zone dynamics, there was a large error between the actual output of the joint driver and the ideal output of the controller. The existence of this error significantly affected the performance of the controller, so it was necessary to identify and compensate for the dead zone characteristics. Figure 6: Two-link robotic manipulator model of free form surface grinding robots(FFSGRs). As shown in Figure 8, the designed identification method based on RBFNNs can converge in limited time, and the dead zone characteristics can be accurately identified. To eliminate the influence of a dead zone on system performance, the identification results should be added to the controller as a compensation term. The position tracking results are shown in Figure 9. Due to the existence of a dead zone, the control method without compensation has an obvious steady-state error.