## 4 Simulation Results  
The presented analytical results are evaluated via simulation. The goal of the simulation is to demonstrate the performance of the proposed control approaches for a scenario where a robot manipulator is subjected to exogenous disturbance torques, which are caused by vibration and payload variation. For this purpose, two signals are generated and combined to represent the disturbance torques due to vibration and payload variation. These signals are formed for the proposed approaches to match the properties and assumptions for each approach. In order to represent this scenario, a two-link robot manipulator is considered with the physical parameters in Table 1.  
The two-link robot manipulator model, the disturbance torques, and the proposed controllers are simulated in MATLAB-SIMULINK as shown in Figure 1. The function that is used to integrate the states is ode45 (i.e., the solver order of the integration is 45) and the sample time that is used to solve the integration is 0.53 ms.  
The dynamics of the two-link robot manipulator can be expressed using (1) as  
\[\begin{bmatrix}M_{11}&M_{12}\\ M_{21}&M_{22}\end{bmatrix}\begin{bmatrix}\ddot{q}_{1}\\ \ddot{q}_{2}\end{bmatrix}+\begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}+\begin{bmatrix}G_{1}\\ G_{2}\end{bmatrix}=\begin{bmatrix}\tau_{1}\\ \tau_{2}\end{bmatrix}-\begin{bmatrix}\tau_{v1}\\ \tau_{v2}\end{bmatrix}-\begin{bmatrix}\tau_{1}\\ \tau_{2}\end{bmatrix} \tag{37}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Symbol** & **Description** & **Value** & **Unit** \\ \hline \(m_{1}\) & Mass of link 1 & 0.5 & kg \\ \(m_{2}\) & Mass of link 2 & 0.4 & kg \\ \(l_{1}\) & Length of link 1 & 0.6 & m \\ \(l_{2}\) & Length of link 2 & 0.5 & m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical parameters of the two-link manipulator.  
Figure 1: Scheme of the simulated robot and the first controller.  
where  
\[M_{11} =(m_{1}+m_{2})l_{1}^{2}+m_{2}l_{2}^{2}+2m_{2}l_{1}l_{2}cos(q_{1})\] \[M_{12} =m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}cos(q_{2})\] \[M_{21} =m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}cos(q_{2})\] \[M_{22} =m_{2}l_{2}^{2}\] \[V_{1} =-m_{2}l_{1}l_{2}(2q_{1}q_{2}+{q_{2}}^{2})sin(q_{2})\] \[V_{2} =m_{2}l_{1}l_{2}q_{1}^{2}sin(q_{2})\] \[G_{1} =(m_{1}+m_{2})gl_{1}cos(q_{1})+m_{2}gl_{2}cos(q_{1}+q_{2})\] \[G_{2} =m_{2}gl_{2}cos(q_{1}+q_{2})\]  
and \(g=9.807\) m/s\({}^{2}\) is the Earth gravity constant. As we will show subsequently, \(\tau_{v}\) and \(\tau_{l}\) are created for each of the proposed approaches according to their assumptions.  
The desired trajectories for both joints are assumed to be as follows:  
\[\begin{bmatrix}q_{d1}\\ q_{d2}\end{bmatrix}=\begin{bmatrix}114.95^{\circ}\ sin(1.5t)\ e^{-0.03t}\\ 85.94^{\circ}\ cos(2t)\ e^{-0.03t}\end{bmatrix}\]  
with the following initial conditions:  
\[\begin{bmatrix}q_{d1}(0)\\ q_{d2}(0)\end{bmatrix}=\begin{bmatrix}0^{\circ}\\ 85.94^{\circ}\end{bmatrix}.\]  
The proposed controllers are compared with a standard PD controller in order to demonstrate their performance over recently proposed controllers. The input torque of the PD controller is designed as  
\[\tau=K_{p}\Delta q+K_{d}\Delta q \tag{38}\]  
where \(K_{p}\) and \(K_{d}\in\Re^{n\times n}\) denote the proportional and derivative diagonal gain matrices, respectively.  
The proposed controllers are tested in terms of rejecting the uncertain disturbance torques due to vibration and payload variation. Beyond the restriction on the disturbance properties, which are considered while formulating the disturbances, the simulation shows that the purpose of each controller can be adequately achieved. The control approaches in Sections 3.2 and 3.3 are investigated in Sections 4.1 and 4.2, respectively. Moreover, quantitative analysis of the results is provided in Section 4.3.