# Energy-Saving Breakthrough in the Point-to-Point Control of a Flexible Manipulator
## 5 Simulation and Experimental Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\theta_{E}\) **[rad]** & \(T_{E}\) **[s]** & **Cyc** & \begin{tabular}{c} **Previous** \\ **Method** \\ \end{tabular} & \begin{tabular}{c} **Proposed** \\ **Method** \\ \end{tabular} \\ \hline \(\pi/2\) & 0.8 & \begin{tabular}{c} 5.65 \(\times\) 10\({}^{-1}\) \\ (5.59 \(\times\) 10\({}^{-1}\)) \\ \end{tabular} & \begin{tabular}{c} 2.98 \(\times\) 10\({}^{-1}\) \\ (2.88 \(\times\) 10\({}^{-1}\)) \\ \end{tabular} & \begin{tabular}{c} 2.34 \(\times\) 10\({}^{-1}\) \\ (2.06 \(\times\) 10\({}^{-1}\)) \\ \end{tabular} \\ \(\pi/4\) & 0.7 & \begin{tabular}{c} 1.98 \(\times\) 10\({}^{-1}\) \\ (1.84 \(\times\) 10\({}^{-1}\)) \\ \end{tabular} & \begin{tabular}{c} 9.82 \(\times\) 10\({}^{-2}\) \\ (8.57 \(\times\) 10\({}^{-2}\)) \\ \end{tabular} & \begin{tabular}{c} 8.06 \(\times\) 10\({}^{-2}\) \\ (6.94 \(\times\) 10\({}^{-2}\)) \\ \end{tabular} \\ \(\pi/6\) & 0.6 & \begin{tabular}{c} 1.18 \(\times\) 10\({}^{-1}\) \\ (1.05 \(\times\) 10\({}^{-1}\)) \\ \end{tabular} & \
\begin{tabular}{c} 6.58 \(\times\) 10\({}^{-2}\) \\ (5.52 \(\times\) 10\({}^{-2}\)) \\ \end{tabular} &
\begin{tabular}{c} 5.43 \(\times\) 10\({}^{-2}\) \\ (4.85 \(\times\) 10\({}^{-2}\)) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of driving energy \(E\) [J]. Figure 9: Comparison of the simulation and experimental results obtained by the proposed method (\(T_{E}=0.6\) s and \(\theta_{E}=\pi/6\) rad): (**a**) joint angle, (**b**) angular velocity, (**c**) tip displacement, and (**d**) motor torque. illustrated in Figure 11, the proposed method can generate an optimal trajectory whereby the angular acceleration and the elastic restoring force counteract each other, enabling energy-efficient operation by utilizing the flexibility characteristics of the manipulator. Finally, the efficacy of the proposed trajectory planning method was validated. In a previous study [37], the amplitudes of residual vibration and driving energy were used as the two evaluated values, with trajectories for energy-saving residual vibration suppression derived from multi-objective optimization. In contrast, the present study generates trajectories through the minimization of the sum of the drive torques from the start of operation to the positioning time \(T_{E}+1\) s. In this case, driving energy was not evaluated directly. Hence, to reaffirm the validity of the proposed method's evaluation function, a comparison with simulation results from the previous study [37] was conducted. The following parameters were employed in Equations (2) and (3) [37]:  
\[\alpha_{1}=2.383\times 10^{-2}\ [\mathrm{kgm}^{2}],\ \alpha_{2}=9.261\times 10^{-2 }\ [\mathrm{kgm}],\ c=3.091\times 10^{-2}\ [\mathrm{Nms}/\mathrm{rad}], \tag{16}\]  
\(\varsigma=9.963\times 10^{-3}\ [-],\ \beta_{1}=2.555\times 10^{-1}\ [\mathrm{m}],\ \beta_{2}=2.614\times 10^{-1}\ [-]\,\ \omega=10.43\ [\mathrm{rad}/\mathrm{s}]\)  
Table 3 presents a comparison of driving energies under the three driving conditions. The coefficient values obtained by the proposed method and a comparative diagram of the time history data from both methods, demonstrating the effectiveness in suppressing residual vibration, are provided in Appendix B. Table 3 indicates that the proposed method achieves a significant reduction in driving energy compared to the previous study. Thus, it can be concluded that minimizing the evaluation function in Equation (13) is effective for both suppressing residual vibration and minimizing drive energy. Figure 11: Schematic of the offset relationship between acceleration and elastic restoring force due to deformation. Figure 10: Time histories of torque components (\(T_{E}=0.8\) s and \(\theta_{E}=\pi/2\) rad): (**a**) proposed method and (**b**) previous method.