# A Sampling-Based Unfixed Orientation Search Method for Dual Manipulator Cooperative Manufacturing
## 4 Minimum Distance Prediction
### Simulation of Dual Manipulator Avoidance  
Similar to single manipulator avoidance, 5 cases will be used to simulate the operation of the SbUOSM algorithm in a dual manipulator avoidance scenario. Case 1: the EEF of manipulator A moves from \(\mathbf{P}_{A}^{start}=(400,-450,\,500)\) towards \(\mathbf{P}_{A}^{end}=(-400,-450,\,500)\) while manipulator A moves from \(\mathbf{P}_{B}^{start}=(-400,-500,\,500)\) towards \(\mathbf{P}_{B}^{end}=(400,-500,\,500)\). Case 2: the EEF of manipulator A moves from \(\mathbf{P}_{A}^{start}=(400,-450,\,500)\) towards \(\mathbf{P}_{A}^{end}=(-400,-450,\,500)\) while manipulator A moves from \(\mathbf{P}_{B}^{start}=(-400,-450,\,500)\) towards \(\mathbf{P}_{B}^{end}=(400,-450,\,500)\). Case 3: the EEF of manipulator A moves from \(\mathbf{P}_{A}^{start}=(400,-450,\,500)\) towards \(\mathbf{P}_{A}^{end}=(-400,-450,\,500)\) while manipulator A moves from \(\mathbf{P}_{B}^{start}=(-400,-400,\,500)\) towards \(\mathbf{P}_{B}^{end}=(400,-400,\,500)\). Case 4: the EEF of manipulator A moves from \(\mathbf{P}_{A}^{start}=(400,-400,\,500)\) towards \(\mathbf{P}_{A}^{end}=(-400,-500,\,500)\) while manipulator A moves from \(\mathbf{P}_{B}^{start}=(-400,-400,\,500)\) towards \(\mathbf{P}_{B}^{end}=(400,-500,\,500)\). Case 5: the EEF of manipulator A moves from \(\mathbf{P}_{A}^{start}=(400,-450,\,400)\) towards \(\mathbf{P}_{A}^{end}=(-400,-450,\,600)\) while manipulator A moves from \(\mathbf{P}_{B}^{start}=(-400,-450,\,400)\) towards \(\mathbf{P}_{B}^{end}=(400,-450,\,600)\). Similarly, when we refer to Case 2 as the benchmark case, based on the fact that both manipulators have relative motions in the dual manipulator avoidance, Cases 2 and 4 in Figure 8 are fused into Case 5 in Figure 13, which is used to show how this paper's proposal operates when adding the displacement of the initial path in the \(Z\)-axis direction of the WCS; In Figure 13, Cases 1-3 investigate the effect of the relative distance between two manipulators on the planning results of the algorithm, similar to cases 1, 5 and 6 in single manipulator avoidance; Case 4 intersects the paths of the two manipulators in the X-Y plane of the world coordinate system, similar to Case 3 in Figure 8, aiming to reveal the specific simulation results of the algorithm for displacement increments in the \(Y\)-axis direction in the WCS. As the effect of changing the end-effector pose vector on the planning results has already been shown in the section on single-arm avoidance (Figure 9), this variable will not be modeled in this chapter. Figures 14-16 focus on the difference between the results of the three different algorithms. The analogous results for dual manipulator avoidance are similar to those for single manipulator avoidance. SbUOSM can always produce smoother paths. The results of the map search algorithm are always the most space-complex. Except for Figure 16, both SbUOSM and SbOSMS perform avoidance only in the \(X\)-\(Y\) plane of the world coordinate system. In Figure 16, the path points of SbUOMS are all dis-tributed in the X-Z plane, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** \\ \hline SbUOSM & 6876.7/95 & 6825.9/93 & 7703.3/95 & 7045.5/96 & 6405.9/88 & 8892.3/121 \\ A* in C-space & 45,992.6/145 & 38,291.4/131 & 48,498.9/151 & 38,153.8/131 & 42,050.7/138 & 46,764.1/142 \\ SbOSMS & 13,477/95 & 13,313.9/93 & 13,148.3/97 & 13,340.8/97 & 12,391/89 & 17,043.8/112 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary table of algorithm execution times/total planning steps for the three algorithms under the six cases (Unit \(\upmu\)s/step). mainly because the dimensions of the end links are more distributed in the X-Y plane than in the X-Z plane. Figure 13: Comparison of dual manipulators avoidance simulation results for five initial path cases. As the simulation results are 3D path curves, the results are presented in a comprehensive and extensive visualization with four views: axonometric, top, front and side views.