# Energy-Saving Breakthrough in the Point-to-Point Control of a Flexible Manipulator
## 2 Single-Link Flexible Manipulator
### Equations of Motion  
A schematic of the experimental setup is illustrated in Figure 2, where \(\theta\) is the joint angle, \(a\) represents the radius of the motor hub securing one end of the flexible manipulator, and \(w\) is the displacement of the manipulator; \(u\) is the displacement in the \(x\)-axis direction owing to significant deflection [7; 34; 38]. From a theoretical analysis based on finite deformation theory, the system's equation of motion is given by the following equations [34]:  
\[\alpha_{1}\ddot{\theta}+\alpha_{2}\ddot{W}+c\dot{\theta}=\tau, \tag{2}\]  
\[\ddot{W}+2c\omega\dot{W}+\omega^{2}W+\beta_{1}\ddot{\theta}+\beta_{2}\dot{ \theta}^{2}W=0. \tag{3}\]  
Equations (2) and (3) correspond to the rotational motion and vibration of the flexible manipulator, respectively. Here, \(W\) is the amplitude of the first-order vibration mode, \(\tau\) is the driving torque, and \(\alpha_{i}\) and \(\beta_{i}\) are the coefficients of the equations of motion. The model incorporates the viscous damping coefficient \(\zeta\) and the viscous friction coefficient \(c\) to account for the damping and friction effects observed in the experiments. The trajectory planning method, discussed in Section 4, falls under feedforward vibration control and requires an accurate mathematical model. Therefore, the values of the coefficients \(\zeta\), \(c\), \(\alpha_{i}\), and \(\beta_{i}\) and the natural frequency \(\omega\) were determined through parameter identification experiments.  
Figure 1: Photograph of the experimental setup.  
For parameter identification [8; 34], cycloidal motion was implemented to rotate the manipulator on an actual machine, as described by Equation (4):  
\[\theta_{cyc}(t)=\theta_{E}\bigg{[}\frac{t}{T_{E}}-\frac{1}{2\pi}\sin\bigg{(} \frac{2\pi t}{T_{E}}\bigg{)}\bigg{]},\ (0\leq t\leq T_{E}), \tag{4}\]  
where \(t\) represents time, \(T_{E}\) is the driving time, and \(\theta_{E}\) is the target angle of the manipulator. Parameters were optimized to ensure that the numerical simulation results aligned with the experimental data on the deflection and drive torques of the flexible manipulator. The identified values are as follows:  
\[\alpha_{1}=1.663\times 10^{-2}\ [\mathrm{kgm}^{2}],\ \alpha_{2}=6.310\times 10^{-2} \ [\mathrm{kgm}],\ c=3.205\times 10^{-2}\ [\mathrm{Nms}/\mathrm{rad}],\] \[\varsigma=8.238\times 10^{-3}\ [-],\ \beta_{1}=2.353\times 10^{-1}\ [ \mathrm{m}],\ \beta_{2}=3.311\times 10^{-1}\ [-],\ \omega=12.52\ [\mathrm{rad}/\mathrm{s}]\ \bigg{\}}. \tag{5}\]  
Figure 3 presents a time-series comparison of simulation and experimental results. Figure 3a-d show the joint angle, angular velocity, manipulator displacement, and motor torque, respectively. This comparison confirms the validity of the parameter identification, as indicated by the close agreement between the simulation and experimental findings. Notably, cycloidal motion generates a large-amplitude residual vibration, as shown in Figure 3c.  
Figure 3: Comparison of the simulation and experimental results obtained using cycloidal motion (\(T_{E}=0.8\ \mathrm{s}\) and \(\theta_{E}=\pi/2\ \mathrm{rad}\)): (**a**) joint angle, (**b**) angular velocity, (**c**) tip displacement, and (**d**) motor torque.  
Figure 2: Schematic of the flexible manipulator.