# Spatial Data-Based Automatic and Quantitative Approach in Analyzing Maintenance Reachability
## 3 Materials and Methods
### Intersection Detection-Based Spatial Relationship Judgment between Arm and Product
#### 3.3.3 Intersection Detection Efficiency Test  
Considering the difference in computer performance, studying the intersection detection efficiency is necessary. In this study, efficiency is expressed in computational time. The size of the initial solution set of the arm pose and the intersection detection method are the main factors affecting efficiency. By reasonably adjusting the division value n in the discretization process of the arm's maximum solution set, the different initial solution set of the arm pose that includes different quantity solutions can be obtained. Six objects, including one sphere, one cylinder, one capsule, two OBBs, and one triangular patch, are created for the efficiency test. Eight initial solution sets of the arm pose are obtained under the division values from \(3^{\circ}\) to \(10^{\circ}\). Each initial solution set of the arm pose detected the intersection with these six objects three times (Table A1 in the Appendix A), meanwhile, the run time and DOR should also be recorded (Tables A2 and A3). Furthermore, by processing the data, the statistical parameters of the DOR under different division values can be obtained (Table A4). Based on the original test data above, the test results can be obtained in Figure 10.  
According to the figures above, the following analyses can be obtained. 1. As the division value increases, the quantity of initial solutions rapidly decreases. 2. The run time of the detection between line segments and different model objects and the quantity of initial solutions is linear. 3. Therefore, as the division value increases, the run time decreases rapidly. 4. The variation coefficients of the DOR obtained by different objects under different division values are below 2.5%. Little change was exhibited, as a larger division value can be chosen to solve the DOR. Figure 10: Results of the Test. Five objects were used to test Scene 2, as shown in Figure 11b, which contains one sphere, one cylinder, and three OBBs. Table 4 shows the necessary geometric characteristic data of these objects, and the meanings are similar to those in Scene 1. Nineteen objects were tested in Scene 4, as shown in Figure 11d, which contains six spheres, four cylinders, three capsules, four OBBs, one irregular object represented by 10 triangular patches, and one irregular object represented by 12 triangular patches. Table 6 shows the necessary geometric characteristic data of these objects. For irregular objects represented by triangular patches, position parameter is the coordinates for the vertices and the direction parameter is the vertexes connection order. For other objects, the parameter meanings are similar to those in Scene 1 and 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Objects to Be**} & \multicolumn{3}{c}{**Necessary Geometric Characteristic Data (mm)**} & **Hand** \\ \cline{2-4}  & **Position Class** & **Range Class** & **Direction Class** & **Coordinates (mm)** \\ \hline Object 1 & \multirow{2}{*}{(\(-\)306.84, 33.05, 0)} & \multirow{2}{*}{148.65} & \multirow{2}{*}{—} & \\ (Sphere) & & & & \\ Object 2 & \multirow{2}{*}{(\(-\)128.13, 463.47, 23.52)} & \multirow{2}{*}{102.64} & \multirow{2}{*}{—} & \\ (Cylinder) & & & & \\ (\(-\)145.23, 335.65, 176.75) & & & & \\ (Object 3 & \multirow{2}{*}{(240.31, 423.67, 73.945)} & \multirow{2}{*}{[124.49, 82.90, 73.95]} & \multirow{2}{*}{v [\(-\)0.0529, 0.1547, \(-\)0.9865]} & \multirow{2}{*}{x = \(-\)14.76} \\ (OBB) & & & & \\ Object 4 & \multirow{2}{*}{(493.56, 115.57, 18.635)} & \multirow{2}{*}{[67.06, 63.16, 82.47]} & \multirow{2}{*}{v [\(-\)0.7581, \(-\)0.5197, \(-\)0.3939]} & \multirow{2}{*}{} \\ (OBB) & & & & \\ (OBB) & & & & \\ \hline Object 5 & \multirow{2}{*}{(28.5, \(-\)331.99, 77.8)} & \multirow{2}{*}{[70.58, 143.56, 77.80]} & \multirow{2}{*}{v [\(-\)0.2627, \(-\)0.8654, \(-\)0.4266]} & \multirow{2}{*}{x = \(-\)14.76} \\ (OBB) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Necessary Geometric Characteristic Data of the Object in Scene 2.