*3.3. Comparative Experiments in a Virtual Picking Environment*

The motion-planning experiment of the Franka manipulator was initially realized through Robotics toolbox, which proved that the TO-RRT algorithm was feasible in the motion of the manipulator. MoveIt! was used in this section to build a virtual picking environment and to conduct comparative experiments on different algorithms in this environment. The experimental parameters are shown in Table 5.

During the experiment, the maximum search time was 10 min, the maximum number of failed searches was 10,000, and the search domain was {*x*, *y*, *z*| − 1 < *x* < 1, −1 < *y* < 1, −1 < *z* < 1}(m). Due to the large number of sampling points generated, the global search time of the RRT algorithm was 243.322451 s. Compared with the RRT algorithm, the search time of the biased-RRT algorithm was only 3.720342 s. However, affected by the nature of obstacles and the probability threshold, the collision-free path generated by the biased-RRT algorithm was less smooth. In contrast, since the NC-RRT algorithm controlled the sampling interval, its trajectory was the smoothest among all the algorithms. Compared with the previous algorithms, the TO-RRT search time and path length were only 0.074915 s and 0.63548128 m, respectively, due to the generation of smaller random trees. The simulation results are shown in Table 6 and Figure 14.

**Figure 13.** The shortest distance from the center of the sphere to the origin of the coordinate system of adjacent links of the manipulator. (**a**) Obstacle with coordinates (25,55,48); (**b**) Obstacle with coordinates (25,53,47); (**c**) Obstacle with coordinates (25,51,46); (**d**) Obstacle with coordinates (25,49,45). *CiCi*+<sup>1</sup> represents the distance between the line segment between coordinate system *i* and coordinate system *i* + 1 and the center of the sphere.



**Table 6.** Experimental data using MoveIt!.


**Figure 14.** The use of MoveIt! with: the RRT algorithm (**a**–**c**); the biased-RRT algorithm with a target offset probability of 50% (**d**–**f**); the TO-RRT algorithm (**g**–**i**); the RRT-BCR algorithm (**j**–**l**); and the NC-RRT algorithm (**m**–**o**).
