# Robust Input Shapers for Acceleration-Limit Actuators
## 3 Performance Evaluation
### ZVD\({}_{\text{AL}}\) Performance Evaluation  
The residual-deflection reduction performance of ZVD\({}_{\text{AL}}\) and conventional ZVD shapers was compared. The control performance of the residual deflection reduction was evaluated with respect to the completeness of the commands and robustness with a wide range of modeling parameters. Figure 8 depicts the residual deflection reduction performance with respect to the command completeness of the ZVD and ZVD\({}_{\text{AL}}\) shapers as a function of the duration time (\(t_{p}\)) with fixed values of remaining modeling parameters. In region A, illustrated as the short commands that do not reach the set-point velocity level, both shapers exhibit residual deflection. In region B, called the interference commands, large residual deflections were observed. In region C, indicated as the long commands for which the region is not affected by the duration time, the ZVD\({}_{\text{AL}}\) shaper exhibits a much better residual reduction performance than the ZVD shaper. Figure 9 presents a comparison of the residual deflection with the long command for the ZVD\({}_{\text{AL}}\) and ZVD shapers according to the cable length (\(L\)). While the ZVD\({}_{\text{AL}}\) shaper has no effect on the cable length variation with zero residual deflection, the ZVD shaper cannot cope with the cable length variation, as expected. Next, numerical simulations of the residual deflection corresponding to each parameter range of the input shaper were conducted. As a result, a residual deflection graph was provided after applying the variables that are modeled when designing the input shaper to a real system. The robustness against modeling errors in the real system was evaluated \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(L\) & \(\kappa_{a}\) & \(\kappa_{d}\) & \(V_{d}\) & \(t_{p}\) & \(V_{lim}\) \\ \hline
0.8 m & 1 m/s\({}^{2}\) & 1.5 m/s\({}^{2}\) & 0.2 m/s & 3.5 s & 0.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modeling parameter values for ZVD\({}_{\text{AL}}\) and EI\({}_{\text{AL}}\) shapers. using the parameter values specified in Table 1. The acceleration, deceleration, and system frequencies shown in the ramp-shaped input command graph were used. Figure 10 shows a comparison of the residual-deflection reduction performances of the ZVD\({}_{\mathrm{AL}}\) and ZVD shapers regarding \(\kappa_{a}\) and \(\kappa_{d}\) ranges of 0.3\(\sim\)0.5 m/s\({}^{2}\). The residual deflection of the ZVD\({}_{\mathrm{AL}}\) shaper was zero over the entire region, unlike the performance of the ZVD shaper. At \(\kappa_{a}=\kappa_{d}\), which forms the symmetric velocity profile, there is no residual deflection when the effect of acceleration and deceleration is cancelled in the ZVD\({}_{\mathrm{AL}}\) shaper, so both shapers have the same impulse time. Figure 11 presents a comparison of the residual deflection reduction performances of the ZVD\({}_{\mathrm{AL}}\) and ZVD shapers with respect to acceleration (\(\kappa_{a}\)) and \(t_{p}\). The ranges of \(\kappa_{a}\) and \(t_{p}\) were set as 0.3\(\sim\)5 m/s\({}^{2}\) and 2\(\sim\)5 s, respectively. Unlike the ZVD shaper, the ZVD\({}_{\mathrm{AL}}\) shaper exhibited zero residual deflection across the entire evaluation region. Therefore, the ZVD\({}_{\mathrm{AL}}\) shaper can be effectively utilized with a high natural frequency for a short duration for acceleration-limit actuators. Meanwhile, the deflection reduction performance of the ZVD shaper could not be predicted because it produced a periodic deflection magnitude according to \(t_{p}\). Figure 12 presents the robustness of the ZVD\({}_{\mathrm{AL}}\) and ZVD shapers with respect to the acceleration (\(\kappa_{a}\)) and string length (\(L\)), where \(\kappa_{am}=1\) and \(L\) = 0.8 The ZVD\({}_{\mathrm{AL}}\) shaper exhibited a better residual deflection reduction than the ZVD shaper in the entire evaluation region.