## 5 Case Study
\[\theta_{2}(t)=-4.142+4.142t+0.84t^{2}-0.061t^{3}\;3\leq t\leq 8 \tag{42}\]  
The planning function of the \(PQ\) segment is shown in Equation (43) below. \[\theta_{3}(t)=-76.069+31.115t-2.531t^{2}+0.08t^{3}\;8\leq t\leq 13 \tag{43}\]  
The planning function of the \(QN\) segment is shown in Equation (44) below. \[\theta_{4}(t)=125.677-15.442t+1.05t^{2}-0.012t^{3}\;13\leq t\leq 19 \tag{44}\]  
The planning function of the \(ND\) segment is shown in Equation (45) below. \[\theta_{5}(t)=1057.847-162.627t+8.796t^{2}-0.148t^{3}\;19\leq t\leq 25 \tag{45}\]  
Validate whether the motion planning functions meet the requirements of angular displacements and angular velocities at the initial target points, as shown in Table 3. Validation results show the planning functions meet the requirements. Validate the continuities of the motion planning functions at the connecting points between every two adjacent planning segments, as shown in Table 4. The validation results show that angular displacements, angular velocities and angular accelerations at connecting points are all continuous. According to the motion planning functions above, the motion planning curves can be obtained, shown in Figure 12. As can be seen from Figure 12a, the angular displacement curve is very smooth and meets the angular displacement requirements at the initial target points \(A\), \(B\), \(C\), \(D\). As can be seen from Figure 12b, the angular velocity curves are also smooth, the velocity fluctuation is gentle, and meets the angular velocity requirements at the initial target points \(A\), \(B\), \(C\), \(D\). As can be seen from Figure 12c, the angular acceleration curves are continuous at the connecting points \(M\), \(P\), \(Q\), \(N\), and the whole acceleration curves does not fluctuate much. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Target Point** & **Angular Displacement** & **Angular Velocity** & **Time** \\  & **\(\theta\) (\({}^{\circ}\))** & **\(\omega\) (\({}^{\circ}\)/s)** & **t (s)** \\ \hline \(A\) & 0 & 0 & 0 \\ \(M\) & 14.21 & 7.548 & 3 \\ \(P\) & 51.747 & 5.952 & 8 \\ \(Q\) & 76.147 & 5.805 & 13 \\ \(N\) & 129.443 & 11.53 & 19 \\ \(D\) & 180 & 0 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constraint conditions at newly-built targets points. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Initial Target** & **Planning** & **Angular Displacement** & **Angular Velocity** & **Time** \\
**Point** & **Function** & **\(\theta\) (\({}^{\circ}\))** & **\(\omega\) (\({}^{\circ}\)/s)** & **t (s)** \\ \hline \(A\) & \(\theta_{1}(t)\) & \(\theta_{1}(0)=0\) & \(\dot{\theta}_{1}(0)=0\) & 0 \\ \(B\) & \(\theta_{2}(t)\) & \(\theta_{2}(5)=30\) & \(\dot{\theta}_{2}(5)=8\) & 5 \\ \(C\) & \(\theta_{4}(t)\) & \(\theta_{4}(15)=90\) & \(\dot{\theta}_{4}(15)=8\) & 15 \\ \(D\) & \(\theta_{5}(t)\) & \(\theta_{5}(25)=180\) & \(\dot{\theta}_{5}(25)=0\) & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rotational requirement validation at the initial target points. The motion planning curves once more verify that the motion planning functions can not only meet the constrained requirements of angular displacements and angular velocities at the initial target points but also guarantee the continuity of angular acceleration and angular velocity fluctuates gently. It proves that the proposed method can be applied well to joint motion planning with velocity constraints, and avoid the problems of discontinuous angular acceleration in cubic polynomial planning and large fluctuation of angular velocity in the quintic polynomial.