## 4 The Improved Self-Collision Detection Technology
### Comparison of the Performance of Self-Collision Detection
#### 5.3.2 Comparison of the Time Required for Approximating Robot  
The time required for the approximation of the mobile manipulator is another key parameter that can be used to improve the performance of self-collision detection. Therefore, to compare the real-time performance, we calculated the required time for the approxi  
Figure 7: Approximating the mobile manipulator robot by a single spherical object. (**a**) The approximation described by the single small spherical object; (**b**) The approximation described by the single large spherical object; (**c**) The approximation described by the method proposed in article [36]; (**d**) The approximation described by the method proposed in this study; (**e**) The approximation result of the mobile manipulator robot using the method proposed in article [36]; (**f**) The approximation result of the mobile manipulator robot using the method proposed in this study.  
mation of all components of the mobile manipulator at different moments in Figure 7e,f. Assumed: the initial position and posture of the mobile platform are \(q_{a}^{int}=(0,0,0^{\circ})\), and the initial angle of each joint of the manipulator robot is \(q_{a}^{int}=(99^{\circ},9^{\circ},-9^{\circ},0^{\circ},81^{\circ},0^{\circ})\), the final position of the mobile platform is the same as the initial position, \(q_{a}^{end}=(0,0,0^{\circ})^{T}\), but the end angle of each joint of the manipulator robot is \(q_{a}^{end}=(180^{\circ},90^{\circ},-90^{\circ},0^{\circ},90^{\circ})\). The total time required for the task is \(t=10\) s. Then, we compared the performance of approximating processes based on the spatial transformation method and the point method used in the approximating process. The time used in solving the center position of all spherical objects is summarized in Table 3. In this table, the approximation object 1 and the approximation object 2 obtained by the VREP software are shown in Figure 7e,f, respectively; \(T_{a}^{i}\) and \(T_{b}^{i}\) indicate the required time for approximating the result \(i\) based on the spatial transformation method in article [36] and the method proposed in this study, respectively; \(T_{c}^{i}\) and \(T_{d}^{i}\), respectively, indicate the required time for solving the spherical objects' center in the result \(i\) based on the spatial transformation method and method proposed in this study, respectively.  
By comparing the spatial transformation method with the point method proposed in this article, the following can be observed:  
(1) Comparing \(T_{a}^{1}\), \(T_{b}^{1}\), \(T_{a}^{2}\) and \(T_{b}^{2}\) in Table 3, the required time based on the critical point method is shorter than the time based on the transformation method in the same position of each component. For example, through the average sampling of 10 points, the key point modeling method is about 47% of the time used based on the spatial transformation method for different approximation robots. Therefore, the key point method can effectively improve the real-time performance of the approximating process and reduce the overall time for self-collision detection.  
(2) Comparing \(T_{c}^{1}\), \(T_{d}^{1}\), \(T_{c}^{2}\) and \(T_{d}^{2}\) in Table 3, the required time to solve the centers of all spherical objects based on the key point method takes less time-consuming than the required time based on the spatial transformation. For example, the required time is shorter by 61% of the time based on the spatial transformation when comparing the average time required for solving the centers of all spherical objects at different moments.  
(3) Comparing \(T_{a}^{1}\), \(T_{c}^{1}\), \(T_{b}^{2}\), and \(T_{d}^{2}\) in Table 3, the improved self-collision detection proposed in this study can effectively reduce the time required for approximating the mobile manipulator robot.