## 3 Simulation and Experimental Validation  
### Numerical Validation on a Robotic System  
To validate the previous results, the robotic arm analyzed in [23] and depicted in Figure 3 is considered.  
The dynamics of this system is given by the following:  
\[u(t)=D(\theta(t))\dot{\theta}(t)+h(\theta(t),\dot{\theta}(t))+C(\theta) \tag{10}\]  
where  
\[D(\theta(t))=\left[\begin{array}{cc}d_{11}(\theta(t))&d_{12}(\theta(t))\\ d_{21}(\theta(t))&d_{22}(\theta(t))\end{array}\right], \tag{11}\]  
\[h(\theta(t),\dot{\theta}(t))=\left[\begin{array}{c}h_{1}(\theta(t),\dot{ \theta}(t))\\ h_{2}(\theta(t),\dot{\theta}(t))\end{array}\right], \tag{12}\]  
\[c(\theta(t))=\left[\begin{array}{c}c_{1}(\theta(t))\\ c_{2}(\theta(t))\end{array}\right],\;u(t)=\left[\begin{array}{c}u_{1}(t)\\ u_{2}(t)\end{array}\right], \tag{13}\]  
and \(d_{11}(\theta(t))=\frac{1}{3}m_{1}l^{2}+\frac{4}{3}m_{2}l^{2}+m_{2}\cos(\theta _{2}(t))l^{2}\), \(d_{12}(\theta(t))=\frac{1}{3}m_{2}l^{2}+\frac{1}{2}m_{2}l^{2}\cos(\theta_{2}(t ))\), \(d_{21}(\theta(t))=\frac{1}{3}m_{2}l^{2}+\frac{1}{2}m_{2}l^{2}\cos(\theta_{2}(t ))\), \(d_{21}(\theta(t))=\frac{1}{3}m_{2}l^{2}+\frac{1}{2}m_{2}l^{2}\cos(\theta_{2}(t ))\), \(d_{22}(\theta(t))=\frac{1}{3}m_{2}l^{2}\), \(c_{1}(\theta(t))=\frac{1}{2}m_{2}l\delta l\cos(\theta_{1}(t))+\frac{1}{2}m_{2} gl\cos(\theta_{1}(t)+\theta_{2}(t))+m_{2}gl\cos(\theta_{1}(t))\), \(c_{2}(\theta(t))=\frac{1}{3}m_{2}gl\cos(\theta_{1}(t)+\theta_{2}(t))\), \(h_{1}(\theta(t))=\frac{1}{3}m_{2}\sin(\theta_{2}(t))l^{2}\theta_{1}(t)\dot{ \theta}_{2}\), \(h_{2}(\theta(t),\dot{\theta}(t))=\frac{1}{3}m_{2}\sin(\theta_{2}(t))l^{2}\dot{ \theta}_{1}(t)^{2}\). where \(\theta_{1}(t)\) is the angle (in radians) between the inner link and the horizontal axis, \(\theta_{2}(t)\) represents the angle (in radians) between the outer link and the inner link, \(x_{2}(t)=\dot{\theta}_{1}\) is the angular velocity of the inner link and \(x_{4}(t)=\dot{\theta}_{2}\) is the angular velocity of the outer link.  
The rigid links have masses of \(m_{1}=0.4\) kg and \(m_{2}=0.3\) kg, while lengths \(l=l_{1}=l_{2}=0.5\) m. The input torques applied to the system are denoted as \(u_{1}(t)\) and \(u_{2}(t)\), both measured in \(N\cdot m\). The gravitational acceleration is \(g=9.81\) m/s\({}^{2}\).  
Also consider that the control signals \(u_{1}(t)\) and \(u_{2}(t)\) are generated by two mistuned PID controllers of the form (2), with parameters \(K_{P}=100\), \(K_{I}=10\), \(K_{D}=5\), and \(N=300\) for both PIDs.  
In the following examples, the robotic arm starts at rest from the lower vertical position, with initial conditions \(\theta_{1}(0)=\frac{3\pi}{2}\), \(\dot{\theta}_{1}(0)=0\), \(\theta_{2}(0)=0\), and \(\dot{\theta}_{2}(0)=0\). Then, both the typical PID controller and the proposed controller are applied to move the robotic arm to different setpoints during 15 s, illustrating the distinct behaviors induced by these controllers.  
Figure 3: Schematic representation of the robotic arm modeled in MATLAB/Simulink using the Multibody toolbox.  
**Remark 4**.: _Although the behavior achieved by the proposed control scheme can approximate that of the classical scheme shown in Figure 1, the classical scheme requires much more precise tuning of the PID controllers._