# Adaptive Robust Admittance Control of Robots Using Duality Principle-Based Impedance Selection
## 5 Simulations  
To verify the proposed methodology, simulations are implemented on a two-DOF planar parallel robot with the mechanism diagram in Figure 2, and the related parameters in Table 1, where \(A_{i},i=0,1,\cdots,4\) are joint points and \(A_{0}\) denotes the interaction point. Driven by the motors located at \(A_{1}\) and \(A_{2}\), the robot can move cooperatively with the user in the workspace of the robot. Along with the advantages of the parallel mechanism in comparison with the series mechanism, the parallel robot is with a complex model.  
Figure 1: The indices on error convergence.  
In the simulation, the friction torque is \(F(\hat{x})=0.2\hat{x}+0.4*\text{sign}(\hat{x})\) and the human-robot system requires following the reference trajectory  
\[x_{d}=\bigg{[}\begin{array}{c}0.075+0.1\sin(\pi/3)\\ 0.55+0.1\cos(\pi/3)\end{array}\bigg{]}. \tag{43}\]  
In the environmental impedance estimation, choose \(\beta_{i}=1,k_{d1}=1,k_{d2}=0.1\), \(d_{d1}=1\), and \(d_{d2}=0.1\). In Figure 3, it can be seen that the estimated stiffness and damping are very close to their actual values \(k_{ei}=0.8\), and \(d_{ei}=0.5\) after \(10\,s\), which shows the effectiveness of the designed projection-based adaptive estimation for the environmental impedance. Based on the estimated stiffness, we choose \(\alpha_{1}=10\) and \(\varsigma_{i}=0.707\) for (21) and (22) to construct the desired robot impedance presented in Figure 4.  
For the proposed adaptive robust admittance controller in (30), choose the following control parameters: \(\bar{M}=0.02I\), \(K=3I\), \(K_{1}=15I\), and \(\mu=0.01\). The proposed controller in Figure 5 with the adaptive gain in Figure 6 makes the tracking error \(e_{r}\) in Figure 7 for the admittance trajectory is very close to zero after \(20\) seconds, which guarantees the impedance error \(e_{im}\) being very close to zero ultimately and the achievement of the desired impedance dynamics. Figure 8 presents the desired trajectory tracking error \(e_{d}\) and the force tracking error \(f_{e}-f_{d}\), from which we can see that the designed admittance control develops a dynamic balance between the tracking error and the interactive force. Based on the above analysis, one can conclude the effectiveness of the proposed environmental impedance estimation and adaptive robust admittance control for robot-environment interaction.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Link** & **Mass (kg)** & **Length (m)** & **Inertial Moment (kg-m\({}^{2}\))** \\ \hline \(A_{2}A_{4}\) & \(m_{2}\ =\ 4.859\) & \(l_{2}\ =\ 0.5\) & \(l_{2}\ =\ 0.128\) \\ \(A_{4}A_{0}\) & \(m_{4}\ =\ 4.053\) & \(l_{4}\ =\ 0.5\) & \(l_{4}\ =\ 0.093\) \\ \(A_{0}A_{3}\) & \(m_{3}\ =\ 1.423\) & \(l_{3}\ =\ 0.5\) & \(l_{3}\ =\ 0.0327\) \\ \(A_{3}A_{1}\) & \(m_{1}\ =\ 4.859\) & \(l_{1}\ =\ 0.5\) & \(l_{1}\ =\ 0.128\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the parallel robot.  
Figure 2: The structure diagram of the considered robot.  
Figure 4: The desired robot stiffness and damping.  
Figure 5: The proposed adaptive robust admittance controller in (30).  
Figure 3: The estimation performance for the environmental stiffness and damping.  
Figure 6: The adaptive parameters for the adaptive robust control term (35).  
Figure 7: The tracking performance for the admittance trajectory and impedance error.