# Deflection Reduction Shaping Commands with Asymmetric First-Order Actuators
## 3 Numerical Results  
The control performances of the proposed input shapers are evaluated in comparison to previously presented input shapers [18; 19]. For numerical evaluation, the parameters are selected as \(\tau_{a}=0.03\) s, \(\tau_{d}=0.01\) s, \(L=0.8\) m, \(V_{d}=0.2\) m/s, \(t_{p}=3\) s, \(D_{lim}=0.3\). and \(D_{film}=0.6\). To initially start the parameter optimization, the UMZV DL shapers are used for the solution search of the UMZVDRF and UMZVDRF shapers during the process of start and stop motions. Because of the rise time, the effect of time-delayed commands is demonstrated by evaluating the solution range and maintaining the same values of the remaining analysis constants as shown in Figure 5. In the case of a long time constant, the shaped commands can be collapsed resulting in control performance degradation indicated as a range of no feasible solution. It is shown that the long length of the suspension cable can allow for a long time constant. Therefore, the proposed shaping technique should be utilized with a validation process of command profile completeness. In addition, the residual maximum deflection is evaluated to illustrate the effect of the suspension cable length \(L\) as shown in Figure 6. The UMZVDRF shaper and UMZVDRF shaper are superior to the UMZVDL shaper except for at a cable length \(L=1\) m. This result means that the first-order actuator dynamics should be considered for better control performance in the application of the input shaping technique. To assess the internal command interference resulting from rising time delays of first-order actuators, the residual deflection is evaluated as a function of the switch-on time \(t_{p}\) as shown in Figure 7. Even if negative commands are generated in the short and medium command regions, the UMZVDRF and UMZVDRF shapers show better deflection reduction performance than the UMZVDL shaper. In the long command region, the UMZVDRF shaper shows the best control performance among the shapers with the selected parameters. Note that the switch-on time \(t_{p}\) should be longer than the final impulse time of an input shaper to preserve the designed velocity command profile, although the performance degradation is small in the interference region. For the evaluation of residual oscillation as shown in Figure 8, the UMZV\({}_{\rm DL}\) shapers produce only a low oscillation magnitude at \(\tau_{a}=\tau_{b}\). However, the UMZV\({}_{\rm DRF}\) and UMZV\({}_{\rm DRFO}\) shapers present much better residual deflection performance than that of the UMZV\({}_{\rm DL}\) shaper within the entire evaluation range. In the difference plot, the UMZV\({}_{\rm DRFO}\) shaper shows slightly better control performance than that of the UMZV\({}_{\rm DRF}\) shaper owing to the approximation effect in the region of relatively large time constants. Figure 6: Residual deflection to \(L\). Figure 7: Residual deflection to \(t_{p}\). Figure 9 and 10 show the accuracy of assigned transient deflection limits in the case of \(D_{film}=[0.6,0.8]\), which is the deflection percentage [60%, 80%] with respect to the maximum deflection determined by the UMZV\({}_{\text{DRF}}\) shaped velocity command as shown in Figure 3a with \(A_{3}=1\) and \(D_{film}=1\). The transient deflection limiting performance of the UMZV\({}_{\text{DRF}}\) shapers is degraded by the 10% discrepancy from that of the UMZV\({}_{\text{DRFO}}\) shapers because of the negligence of the exponential term. On the other hand, the UMZV\({}_{\text{DRFO}}\) shapers show accurate transient control performance as expected. The numerical results obtained with respect to \(D_{film}=1\) mean there is no deflection limit while considering the time constants. The transient deflection limits are controlled by maintaining the consistent deflection magnitudes in the evaluation region. Based on the numerical analysis, the transient deflection magnitude can be slightly varied for \(\tau_{a}\) and \(\tau_{d}\) with regard to the deflection percentage. Figure 8: Residual deflection to \(\tau_{a}\) and \(\tau_{d}\).