# Energy-Saving Breakthrough in the Point-to-Point Control of a Flexible Manipulator
## 5 Simulation and Experimental Results  
This section presents the results from numerical simulations and experiments conducted to validate the proposed energy-saving trajectory planning method. In these simulations, the numbers of individuals and iterations in the PSO were set to 100 and 200, respectively. The range of optimized parameters was established as follows:  
\[T_{0}\in[0,\ T_{E}/3],\ \theta_{0}\in[0,\ \theta_{E}/4],\ a_{n}\in[-0.4,\ 0.4]\ (n=1,\ 2,\cdots,N), \tag{14}\]  
where the number of terms in the power series (Equation (10)) was \(N=4\). PSO was then employed to minimize the evaluation function (Equation (13)). Initially, the driving conditions were the same as in Figure 3 (\(\theta_{E}=\pi/2\) rad, \(T_{E}=0.8\) s), and the results of the proposed method were compared with those of the previous study [37], as shown in Figure 5. In Figure 5a-e depict the joint angle, angular velocity, angular acceleration, tip displacement of the manipulator, and motor torque, respectively. In the previous study [37], the trajectory of the joint angle was represented by \(\theta_{opt}(t)=\theta_{f}\left(t\right)\), with the number of terms in the power series (Equation (10)) set to \(N=6\). The evaluation function was defined as follows:  
\[\overline{F}_{1}=\max_{t\in S}[|w(t,l)|],\ (S:T_{E}\leq t\leq T_{E}+1\,\mathrm{s}), \ \ \overline{F}_{2}=\int_{0}^{\theta_{E}}\!|\tau|\ d\theta. \tag{15}\]  
Here, \(\overline{F}_{1}\) and \(\overline{F}_{2}\) represent the maximum tip displacement within 1 s after positioning and the operating energy until positioning, respectively. To minimize these two evaluation functions, an optimal trajectory was generated by tuning the coefficients \(a_{n}\) using vector-evaluated PSO [40], a multi-objective optimization method. The range for the coefficients was the same as in Equation (14). The values of the optimized parameters obtained by both methods are provided in Appendix A. As shown in Figure 5d, the residual vibration was effectively suppressed by both methods. A comparison with Figure 3 reveals that both approaches are successful in reducing residual vibrations. The trajectories of angular velocity and acceleration were smoother in the previous method. The trajectory in the proposed method is generally less smooth, with a notable peak in angular velocity at \(t\approx 0.09\) s, attributable to the cycloid function in Equation (8). The cycloid function predominantly influences the trajectory from the start of rotation until \(T_{0}=1.644\ \times 10^{-1}\) s, causing significant negative deflection in the manipulator due to large variations in angular acceleration. This deformation leads to a higher maximum torque in the present method compared to the previous study, although the torque becomes smaller after \(t\approx 0.18\) s.  
The following are the results of the experiments conducted to verify the feasibility of both methods. Figures 6 and 7 show the experimental results of the previous and proposed methods, respectively. The driving conditions were identical to those in Figure 5. As shown in Figures 6 and 7, the simulation and experimental results are consistent, confirming the validity of the flexible manipulator modeling and the feasibility of both methods. The joint angle trajectory in the proposed method is less smooth compared to that in the previous method, which might be interpreted as exciting higher-order vibration modes. However, both methods effectively suppress higher-order vibration modes, significantly reducing residual vibration. Figure 5: Comparison of the simulation results obtained by the proposed method and those obtained by the previous method (\(T_{E}=0.8\) s and \(\theta_{E}=\pi/2\) rad): (**a**) joint angle, (**b**) angular velocity, (**c**) angular acceleration, (**d**) tip displacement, and (**e**) motor torque. Figure 6: Comparison of the simulation and experimental results obtained by the previous method (\(T_{E}=0.8\) s and \(\theta_{E}=\pi/2\) rad): (**a**) joint angle, (**b**) angular velocity, (**c**) tip displacement, and (**d**) motor torque.