# Energy-Saving Breakthrough in the Point-to-Point Control of a Flexible Manipulator
## 4 Trajectory Planning Method Focused on Flexibility Characteristics  
This study addresses the PTP control problem of rotating a manipulator to a target angle \(\theta_{E}\) at time \(T_{E}\). The author's previous study [37] introduced an energy-saving trajectory planning method where the joint angular trajectory of a flexible manipulator is represented by a combination of a cycloid function and a power series. Numerical simulations demonstrated that this method conserves more energy compared to neural networks [32]. However, the physical phenomena discussed in Section 3 suggest that energy savings can also be achieved by significantly deflecting the manipulator in the negative direction immediately following activation. Therefore, the following trajectory planning method is proposed. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{4}{c}{**Value of the Initial Deflection**} \\ \hline \(w(0,l)=-5\) cm & \(w(0,l)=0\) & \(w(0,l)=5\) cm \\ \hline \(5.28\times 10^{-2}\) & \(1.05\times 10^{-1}\) & \(1.60\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effect of initial deflection on driving energy \(E\)[]. Figure 4: Effect of initial deflection on response and torque (\(T_{E}\) = 0.6 s and \(\theta_{E}\) = \(\pi/6\) rad): (**a**) joint angle, (**b**) angular velocity, (**c**) tip displacement, and (**d**) motor torque. The optimized trajectory \(\theta_{opt}(t)\) of the manipulator joint angle is formulated as the sum of a cycloid curve with drive time \(T_{0}\) and target angle \(\theta_{0}\) and a cycloid function with the input represented as a power series:  
\[\theta_{opt}(t)=\theta_{cyc}(t)+\theta_{f}(t), \tag{7}\]  
\[\theta_{cyc}(t)=\theta_{0}\bigg{[}\frac{t}{T_{0}}-\frac{1}{2\pi}\sin\bigg{(} \frac{2\pi t}{T_{0}}\bigg{)}\bigg{]},\ (0\leq t\leq T_{0}), \tag{8}\]  
\[\theta_{f}(t)=(\theta_{E}-\theta_{0})\bigg{\{}u(t)-\frac{\sin[2\pi u(t)]}{2\pi }\bigg{\}}, \tag{9}\]  
\[u(t)=\frac{t}{T_{E}}+(1-T^{2})\sum_{n=1}^{N}a_{n}T^{n-1}, \tag{10}\]  
\[T=-1+\frac{2t}{T_{E}},\ (0\leq t\leq T_{E}). \tag{11}\]  
Here, the ranges of \(\theta_{0}\) and \(T_{0}\) are defined as follows:  
\[0\leq\theta_{0}<\theta_{E},\ \ 0\leq T_{0}<T_{E}. \tag{12}\]  
Equation (8) describes a cycloidal curve with time \(T_{0}\) and angle \(\theta_{0}\), aiming to reduce the driving energy by significantly deflecting the flexible manipulator. Meanwhile, as indicated by Equations (9) and (10), the input \(u(t)\) is given as a power series up to time \(T_{E}\), generating a trajectory toward the target angle \(\theta_{E}\). If \(\theta_{0}=0\), then \(\theta_{opt}=\theta_{f}\), aligning with the trajectory equation from our previous study [37]. The trajectory \(\theta_{opt}(t)\) generated from Equation (7) depends on \(T_{0}\) and \(\theta_{0}\) in Equation (8) and the coefficients \(a_{n}\) of the power series in Equation (10). The procedure for the trajectory planning method for energy-saving and residual vibration suppression is outlined below. Optimization parameters include \(T_{0}\) and \(\theta_{0}\) from Equation (8) and the coefficients \(a_{n}\) from Equation (10), used to generate the joint angle trajectory \(\theta_{opt}(t)\) from Equations (7)-(11). The numerical integration of Equation (3), based on this trajectory, reveals the dynamics of the flexible manipulator. The results from this integration are then applied to deduce the drive torque \(\tau\) via inverse dynamics analysis of Equation (2). To achieve both drive energy minimization and residual vibration suppression, the evaluation function \(F\) is defined as follows:  
\[F=F_{1}+F_{2}=\sum_{i=1}^{I}|\tau_{i}|+\sum_{i=I+1}^{I+I}|\tau_{i}| \tag{13}\]  
where \(\tau_{i}\) is the drive torque per time interval \(\Delta t=2\) ms, and \(I=T_{E}/\Delta t\). Thus, \(F_{1}\) represents the sum of the drive torque during rotation up to time \(T_{E}\), with its minimization signifying drive energy reduction. \(F_{2}\) represents the sum of the torque over 1 s after positioning, set to \(J=1/\Delta t\).