**5. Simulation studies**

Another important property of the proposed planner takes place in the RRT\* algorithm where the steering factor in the original RRT\*, which is 'step', will be automatically replaced by the sampling radius. As stated before, the samples will be created randomly from the perimeter of the current Poisson disks. As a result, the highest distance between any two samples exactly

be calculated as the total Euclidean distance between all members (nodes) of the optimum path

*i i*

*i i*

\* \*\* 1

\* \*\* 1

 ww+

This upper bound merely depends on the size of the final graph/tree structure. On the other hand, reducing the total number of samples (*n*), will reduce the number of samples in any solution. Therefore, it can be concluded that using a Poisson-disk distribution as the sampling domain will improve the cost of the final solution and maintains the asymptotic optimality property of the proposed algorithm. **Figure 5** illustrates the graph construction phase for the

1 || *n*

*i*

=

 ww+

where *n*\* is the number of nodes in the optimal path resulted from the algorithm. Considering

1 || *n*

*i*

=

( ) \*

( ) \*

*c* w

*c* w

(n). As stated before, the cost of the final optimum solution can

\* <*r <sup>s</sup>*(*n*), now it is possible to find an upper bound for the path of the

= - å <sup>P</sup> (3)

= - å <sup>P</sup> (4)

equals the sampling radius rs

32 Autonomous Vehicle

the fact that *ωi*+1

optimal solution:

PRM\* planner.

which can be calculated as follows:

\* - *<sup>ω</sup><sup>i</sup>*

**Figure 5.** The connection strategy in the proposed algorithm.

According to the properties of the proposed algorithm, three different situations have been designed for simulation studies. **Figure 7** depicts an instance of the solutions in these scenarios. The proposed method is able to plan difficult motions for the vehicle such as following or takeover. As can be seen in **Figure 7**, when there are no other vehicles, the planner restricts the sampling domain to the current lane and other parts of the road are not included in the sampling procedure. When there is another car in front but takeover is not an option, the generated rapidly exploring random tree just follows the front vehicle with a proper distance. Finally, when takeover is possible, the sampling domain will be expanded to include suitable space.

**Figure 7.** Simulation results in different scenarios. The initial and final positions of the vehicle are shown by yellow and green, respectively.

The average results of the performance of the proposed planner as well as the performances of RRT and RRT\* planners are shown in **Table 1**.


**Table 1.** Simulation average results for the proposed algorithm, RRT and RRT\* planners.

As can be seen, the proposed planner provides optimal solutions with surprisingly smaller set of samples. The runtime of the planner is also less than other planners. **Figure 8** shows the variations of performance variables for 1000 iterations of each planner.

**Figure 8.** Variation of the results over 1000 iterations for the proposed algorithm (green), RRT (red) and RRT\* (blue) in terms of the number of nodes, optimality (%) and runtime (s).
