## 4 Test Case: A Two-Link Flexible Robot Arm
### Experimental Application  
#### 4.3.1 Application of the Controller of Test Case 1  
The controller proposed in Section 4.2.1 is applied to the experimental two-link flexible robot arm shown in Figure 2. An impulse-like torque disturbance is applied through the DC motor, the experimental responses of the open-loop and closed-loop system with \(\mathbf{k}\) are shown in Figure 7 through the relative angle of the second link \(\theta_{2}^{\prime}=\theta_{2}-\theta_{1}\). The modal parameters of the open-loop and closed-loop system are identified through the \(modalfit\) function and compared with the expected numerical counterpart in Table 4, where \(f_{n}\) denotes the natural frequency and \(\zeta_{n}\) the related damping ratio. Additionally, the expected and experimental inertances for the open-loop and closed-loop system are shown in Figure 8. The fulfillment of the assignment task is evidenced by the agreement between the numerical data and the experimental measurements. Figure 6: Test case 2: Nyquist curve of \(L(j\omega)\). Figure 7: Test case 1: experimental impulse responses, relative angle of the second-link. The agreement between the numerical expectations and the actual experimental results corroborates the controller robustness and its effectiveness to handle perturbations. Besides the linearization and the unavoidable, although small, errors on the measured receptances, the system is affected by the warp of the wire of encoder B that slightly perturbs the system mass distribution during the motion and also the equilibrium configuration. This effect is evident in Figure 9 where a train of pulses with different amplitudes is applied to obtain different nonlinear effects due to the geometrical non-linearity of the system (see Equation (13)). The motion of the wire causes slightly different equilibrium configurations once the response settles. Nonetheless, the controller ensures stable, fast, and repeatable vibration control, thus proving robustness. If no adequate robustness in obtained, the nominally stabilizing controller might result unstable or, at least, remarkably downgrade the actual performances. The robustness of the controller synthesized through the proposed method, i.e., the one with gains \(\mathbf{k}\), is even more evident if it is compared with the non-robust controller tuned through \(\mathbf{k_{0}}\). The experimental impulsive response of \(\mathbf{k_{0}}\) is shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Open-Loop** & \multicolumn{2}{c}{**Mode 1**} & \multicolumn{2}{c}{**Mode 2**} \\ \hline
**Modal Parameter** & **Numerical** & **Experimental** & **Numerical** & **Experimental** \\ \hline \(f_{n}[Hz]\) & 1.15 & 1.29 & 6.08 & 6.16 \\ \(\xi_{n}\) & 0.166 & 0.161 & 0.027 & 0.025 \\ \hline \multicolumn{4}{c}{**Closed-Loop**} \\ \hline \(f_{n}[Hz]\) & 2.84 & 3.08 & 6.13 & 6.14 \\ \(\xi_{n}\) & 0.770 & 0.749 & 0.130 & 0.190 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Test case 1: numerical and experimental modal parameters. Figure 8: Test case 1: expected and experimental inertances. The numerical and experimental responses of the open-loop and closed-loop systems with \(\mathbf{k}\) and \(\mathbf{k_{0}}\) are compared in Figure 11. These provide a satisfying agreement for the open-loop and closed-loop system with \(\mathbf{k}\), while in case of \(\mathbf{k_{0}}\) the experimental results do not match with the expected numerical results. It is evident that improving robustness is mandatory to enable for practical applications with adequate and reliable performances. Last, Figure 12 shows the application of the controller to a step-tracking problem, that represents a typical pick-and-place task [37; 38]. A sequence of step references to move the end-effector to the target value \(x_{ee}=0.1\) m is applied. The state-feedback controller is supported by a feedforward controller to compensate for gravity and friction and ensure no (or negligible) steady state error.