## 5 Case Study  
In order to verify the rationality and effectiveness of the proposed methodology, the following study case is employed. The usage of the proposed methodology is further described in detail during the study case analysis. Case: In order to accomplish a task, joint 1 of the industrial robot needs to carry out velocity-constrained PTP motion. As shown in Figure 11a, joint 1 turns from start target point \(A\) to intermediate target points \(B\), \(C\) and to end target point \(D\) in turn. The velocities at the intermediate target points \(B\) and \(C\) are specified. The rotational requirements at each target point are shown in Table 1. Using the proposed planning of modified cubic Hermiteinterpolation presented above, carry out velocity-constrained PTP motion planning for joint 1. There are four initial knots \(A\), \(B\), \(C\), and \(D\) to be interpolated. As seen in Figure 11b, newly-built knot \(M\) needs to be inserted between initial \(A\) and initial \(B\), and \(N\) between \(C\) and \(D\), and \(P\), \(Q\) between \(B\) and \(D\), according to the proposed method. Therefore, the initial knots \(A\), \(B\), \(C\), \(D\) are replaced by the newly-built knots \(A\), \(M\), \(P\), \(Q\), \(N\), and \(D\). The initial time knots become the newly-built time knots correspondingly. In order to be unified with the previous formula, the initial time knot is denoted as \(x\), and the newly-built time knots is denoted as \(t\). According to formula (39), the newly-built time knots are arranged from the initial time knots \(x=0\), \(5\), \(15\), \(25\) (s) and the newly-built time knots are \(t=0\), \(3\), \(8\), \(13\), \(19\), \(25\) (s) respectively with proper rounding. We need to find out the angular displacements and angular velocities at the newly-built knots \(M\), \(P\), \(Q\) and \(N\) from the initial knots \(A\), \(B\), \(C\), and \(D\). According to formula (30), \(31\), \(32\), \(33\), and \(34\), we can obtain Equation (40) to solve out the angular displacements and angular velocities of the rebuilt knots, \(\theta_{M}\), \(\theta_{P}\), \(\theta_{Q}\), \(\theta_{N}\), \(\omega_{M}\), \(\omega_{P}\), \(\omega_{Q}\), \(\omega_{N}\). \[\left[ \begin{array}{cccccccc}-0.4&-0.225&0&0&2&0.375&0&0\\ 0.3&0&-0.3&0&0.5&2&0.5&0\\ 0&0.327&-0.1&-0.227&0&0.546&2&0.455\\ 0&0&0.25&0&0&0.5&2\\ 0.648&0.352&0&0&0.72&-0.48&0&0\\ -0.288&0.288&0&0&-0.12&-0.32&0&0\\ 0&0&0.741&0.259&0&0&0.889&-0.444\\ 0&0&-0.222&0.222&0&0&0&-0.333\\ \end{array} \right]\left[ \begin{array}{c}\theta_{M}\\ \theta_{P}\\ \theta_{Q}\\ \omega_{N}\\ \omega_{P}\\ \omega_{Q}\\ \omega_{N}\\ \end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 45\\ 30\\ \omega_{Q}\\ \omega_{N}\\ \end{array} \right] \tag{40}\]  
The values of \(\theta_{M}\), \(\theta_{P}\), \(\theta_{Q}\), \(\theta_{N}\), \(\omega_{M}\), \(\omega_{P}\), \(\omega_{Q}\), \(\omega_{N}\) can be obtained by solving the linear Equations. In addition to the previously calculated amount of time, the information of the newly-built knots for joint motion planning can be listed in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Target Point** & **Angular Displacement** & **Angular Velocity** & **Time** \\  & **\(\theta\) (\({}^{\circ}\))** & **\(\omega\) (\({}^{\circ}\)/s)** & **\(x\) (s)** \\ \hline \(A\) & 0 & 0 & 0 \\ \(B\) & 30 & 8 & 5 \\ \(C\) & 90 & 8 & 15 \\ \(D\) & 180 & 0 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Constraint requirements for target points of joint 1. Figure 11: Point-to-point joint motion planning with constrained velocity: (**a**) Initial knots; (**b**) Newly-built knots. According to the motion conditions at newly-built target points in Table 2, the piecewise motion planning of joint 1 can be carried out by formula (18). The planning function of \(AM\) segment is shown in Equation (41) below. \[\theta_{1}(t)=2.221t^{2}-0.214t^{3}\;0\leq t\leq 3 \tag{41}\]  
The planning function of the \(MP\) segment is shown in Equation (42) below.