## 5 Optimization Result
### Optimization Results
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(t_{f}\) (s) & \multicolumn{2}{c}{**Mean (u)**} & \multicolumn{2}{c}{**Standard Deviation (\(\sigma\))**} & \multicolumn{2}{c}{**Coefficient of Variation (=\(\sigma\)/u)**} \\ \hline \multirow{2}{*}{**Case 1**} & \multirow{2}{*}{2.0} & Initial & \(1.693\times 10^{-2}\) & \(1.845\times 10^{-2}\) & \(108.9\%\) \\  & & Optimal & \(4.522\times 10^{-2}\) & \(5.921\times 10^{-5}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 2**} & \multirow{2}{*}{3.0} & Initial & \(9.163\times 10^{-3}\) & \(7.069\times 10^{-3}\) & \(77.1\%\) \\  & & Optimal & \(2.452\times 10^{-2}\) & \(5.624\times 10^{-6}\) & \(0.0\%\) \\ \hline \multirow{2}{*}{**Case 3**} & \multirow{2}{*}{4.0} & Initial & \(6.678\times 10^{-3}\) & \(4.496\times 10^{-3}\) & \(67.3\%\) \\  & & Optimal & \(1.172\times 10^{-2}\) & \(1.563\times 10^{-5}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 4**} & \multirow{2}{*}{5.0} & Initial & \(5.162\times 10^{-3}\) & \(3.369\times 10^{-3}\) & \(65.3\%\) \\  & & Optimal & \(6.347\times 10^{-3}\) & \(7.260\times 10^{-6}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 5**} & \multirow{2}{*}{6.0} & Initial & \(4.056\times 10^{-3}\) & \(2.719\times 10^{-3}\) & \(67.0\%\) \\  & & Optimal & \(3.837\times 10^{-3}\) & \(2.385\times 10^{-6}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 6**} & \multirow{2}{*}{7.0} & Initial & \(3.357\times 10^{-3}\) & \(2.290\times 10^{-3}\) & \(68.2\%\) \\  & & Optimal & \(2.331\times 10^{-3}\) & \(1.599\times 10^{-6}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 7**} & \multirow{2}{*}{8.0} & Initial & \(2.872\times 10^{-3}\) & \(1.982\times 10^{-3}\) & \(69.0\%\) \\  & & Optimal & \(1.743\times 10^{-3}\) & \(1.727\times 10^{-6}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 8**} & \multirow{2}{*}{9.0} & Initial & \(2.514\times 10^{-3}\) & \(1.749\times 10^{-3}\) & \(69.6\%\) \\  & & Optimal & \(1.435\times 10^{-3}\) & \(1.224\times 10^{-6}\) & \(0.1\%\) \\ \hline \multirow{2}{*}{**Case 9**} & \multirow{2}{*}{10.0} & Initial & \(2.238\times 10^{-3}\) & \(1.567\times 10^{-3}\) & \(70.0\%\) \\  & & Optimal & \(1.209\times 10^{-3}\) & \(6.390\times 10^{-7}\) & \(0.1\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Acceleration factors for the initial and the optimal motion. Figure 2 shows the average value of the acceleration factors according to cycle time and indicates that performing optimization while reducing cycle time can result in a motion profile having a higher acceleration factor. Figure 3 represents an objective function according to the number of iterations when optimization is performed by setting cycle time \(t_{f}=2\), and Figure 4 shows an acceleration factor of each joint according to number of iterations. It can be confirmed that the objective function converges to the global minimum of 0, and accordingly, the acceleration factors converge to the same value. Figure 4: Acceleration factors at each optimization iterations (case 1). Figure 3: Objective function value at each optimization iterations (case 1). Figure 2: Mean acceleration factor \((\mu)\) according to the cycle time \((t_{f})\). Figures 5 and 6 represent the initial and the optimal motion profiles, respectively, for the case of \(t_{f}=2\) s. Each joint begins to move at the initial angle and stops at the final angle, which are given in Table 4, at the end of the cycle time \(t_{f}\). In the initial motion, the speed is uniform except for the initial and final acceleration/deceleration of the cycle time, while in the optimal motion, the speed fluctuates significantly. In particular, for optimal motion, it can be seen that the speed of joint 1 and torque of joint 5 reach the lower limit in 0.6 s and remain within the bound in Table 3 for the rest of the time (Figure 6). Figure 5: Joint angle, speed and torque profiles for the initial motion (Case 1).