# Deflection Reduction Shaping Commands with Asymmetric First-Order Actuators
## 4 Experimental Results
### Experimental Assessment  
The control performances of the UMZV\({}_{\rm DL}\), UMZV\({}_{\rm DRF}\), and UMZV\({}_{\rm DRFO}\) shapers are experimentally verified from the aspects of transient and residual deflections and the robustness to modeling uncertainties. For a specific case, all three input shapers are presented with the magnitude difference of the UMZV\({}_{\rm DRFO}\) shaper from the others as shown in Table 2 even if the UMZV\({}_{\rm DL}\) shaper is utilized for the initial condition during numerical optimization. In addition, Figure 15 shows the first-order velocity commands applied to the mini-bridge crane and resulting in the time responses shown in Figure 16, where the UMZV\({}_{\rm DRF}\) and UMZV\({}_{\rm DRFO}\) shapers present a control performance improvement of approximately 25% compared to that of the UMZV\({}_{\rm DL}\) shaper.  
Figure 17 shows the sensitivity curves with respect to the system modeling uncertainty \(L/L_{m}\). The theoretical analysis accurately predicts the experimental results as expected. The UMZV\({}_{\rm DRF}\) and UMZV\({}_{\rm DRFO}\) shapers show better robustness against modeling errors than that of the UMZV\({}_{\rm DL}\) shaper that has an unpredictable robustness at \(L/L_{m}=1\). This implies that the UMZV\({}_{\rm DL}\) shaper cannot be coped with the nonlinearity of actuators. Once again, the UMZV\({}_{\rm DRF}\) and UMZV\({}_{\rm DRFO}\) shapers illustrate a comparable control performance in the experimental robustness evaluation. This result shows the high applicability of UMZV\({}_{\rm DRF}\) shapers for control systems with micro  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & \multicolumn{6}{c}{**Input Shaper**} \\ \hline UMZV\({}_{\rm DL}\) & \(\left[\begin{array}{cccc}1&-1&0.6&-0.6&1&-1\\ 0&0.1740&0.5356&3&3.3616&3.5356\\ 1&-1&0.6&-0.6&1&-1\\ 0&0.1935&0.5331&3&3.3437&3.5380\\ \end{array}\right]\) \\ UMZV\({}_{\rm DRFO}\) & \(\left[\begin{array}{cccc}1&-1&0.5478&-0.5478&1&-1\\ 0&0.1390&0.5304&3&3.3870&3.5252\\ \end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Typical input shapers for start and stop motions.  
Figure 15: Experimental velocity commands.  
Figure 18 shows the robustness with respect to the time constant uncertainty of actuators. The UMZVDL shaper shows a consistent residual deflection performance that means no counteraction to the modeling error of the actuator. However, the UMZVDRF and UMZVDRFO shapers present an experimentally predictable deflection reduction performance unlike the UMZVDL shaper.  
Lastly, the transient limiting performance is evaluated with respect to the time constant and switching-on time. Figures 19 and 20 show the maximum deflection magnitude during the period of transient responses at \(D_{film}=0.6\). As shown in Figure 19, the UMZVDRFO shaper shows a slightly better maximum transient deflection magnitude than that of the others. From the aspect of the usability of input shapers, the UMZVDRF shaper presents a physical transient maximum deflection performance similar to that of the UMZVDRFO shaper. Figure 20 shows the transient maximum deflections with regard to switch-on time \(t_{p}\). The UMZVDRF shaper illustrates a comparable control performance of a less than 10% discrepancy comparing to that of the UMZVDRFO shaper. However, the UMZVDL shaper shows 20% error in the maximum transient deflection limit compared to that of the UMZVDRF.  
Figure 16: Experimental time responses.  
Figure 17: Experimental sensitivity to L/Lm.  
Figure 19: Transient deflection to \(t_{p}\).  
Figure 20: Transient deflection to \(t_{p}\).