## 6 Comparison and Discussion  
In order to compare with the proposed planning methodology, the most commonly used PTP joint planning method, cubic polynomial and quintic polynomial, are applied to the same case study above as well. The results of the study and analysis below reflect there are more or fewer deficiencies for these two methods to be used for joint planning with constrained velocity. The detailed comparison and analysis process are as follows. Figure 13 shows the angular displacement curves obtained by the proposed methodology, cubic polynomial, and quintic polynomial respectively. As can be seen from Figure 13a,b, the angular displacement curve obtained by the proposed method is similar to the curves obtained by the cubic polynomial. Both of the curves are smooth and the fluctuation is very small. In Figure 13c, the angular displacement curves obtained by quintic polynomial fluctuate a little bit. In fact, a curve with a higher order is more likely to fluctuate than a curve with a lower order. Therefore, we should try to avoid using higher-order functions for planning. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Connecting** \\ **Point** \\ \end{tabular} & \begin{tabular}{c} **Planning** \\ **Function** \\ \end{tabular} & \begin{tabular}{c} **Angular** \\ **Displacement** \(\theta\) \\ **(C\({}^{\circ}\)) \\ \end{tabular} & \begin{tabular}{c} **Angular Velocity** \\ **\(\omega\) (C\({}^{\circ}\)/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Angular** \\ **Acceleration** \\ \(\varepsilon\) (C\({}^{\circ}\)/s\({}^{2}\)) \\ \end{tabular} &
\begin{tabular}{c} **Time \(t\) (s)** \\ \end{tabular} \\ \hline \multirow{3}{*}{\(M\)} & \(\theta_{1}(t)\) & \(\theta_{1}(3)=14.21\) & \(\hat{\theta}_{1}(3)=7.55\) & \(\hat{\theta}_{1}(3)=0.59\) & \multirow{3}{*}{3} \\  & \(\theta_{2}(t)\) & \(\theta_{2}(3)=14.21\) & \(\hat{\theta}_{2}(3)=7.55\) & \(\hat{\theta}_{2}(3)=0.59\) & \multirow{3}{*}{8} \\  & \(\theta_{2}(t)\) & \(\theta_{2}(8)=51.75\) & \(\hat{\theta}_{2}(8)=5.96\) & \(\hat{\theta}_{2}(8)=-1.23\) & \multirow{3}{*}{8} \\  & \(\theta_{3}(t)\) & \(\theta_{3}(8)=51.75\) & \(\hat{\theta}_{3}(8)=5.96\) & \(\hat{\theta}_{3}(8)=-1.23\) & \\ \cline{1-1}  & \(\theta_{3}(t)\) & \(\theta_{3}(13)=76.15\) & \(\theta_{3}(13)=5.81\) & \(\hat{\theta}_{3}(13)=1.17\) & \\ \cline{1-1}  & \(\theta_{4}(t)\) & \(\theta_{4}(13)=76.15\) & \(\theta_{4}(13)=5.81\) & \(\hat{\theta}_{4}(13)=1.17\) & \\ \cline{1-1}  & \(\theta_{4}(t)\) & \(\theta_{4}(19)=129.44\) & \(\hat{\theta}_{4}(19)=11.53\) & \(\hat{\theta}_{4}(19)=0.74\) & \\ \cline{1-1}  & \(\theta_{5}(t)\) & \(\theta_{5}(19)=129.44\) & \(\hat{\theta}_{5}(19)=11.53\) & \(\hat{\theta}_{5}(19)=0.74\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Continuity validation at the connecting points. Figure 12: Motion planning curves by proposed method. (**a**) Angular displacement planning curves; (**b**) Angular velocity planning curves; (**c**) Angular acceleration planning curves. Figure 14 is the angular velocity curves obtained by these three methods. In Figure 14a, the velocity curves obtained by the proposed method are smooth, the velocity fluctuation is gentle, and the variation trend is roughly similar to that in Figure 14b by the cubic polynomial. In Figure 14b, the overall velocity waveform by a cubic polynomial is simple and appropriate. However, there are turning points at junctions \(B\) and \(C\), which will lead to discontinuous angular acceleration at the connection points. In Figure 14c, the overall fluctuation of the velocity curves by a quintic polynomial is obviously large. In particular, because of inappropriate preset angular acceleration at point \(C\), the velocity near point \(C\) fluctuates back and forth sharply, which is not conducive to the control of joint rotation speed. This shows that quintic polynomial planning is likely to cause angular velocity to fluctuate too much. Moreover, the higher the order of the polynomial is, the higher the velocity peak will be, which makes greater volatility.