## 4 Computer Simulation Results
### Scenario 1  
Considering Scenario 1, one presents the computer simulation results of the proposed 3D path plan algorithm (Algorithm 3). One obtains the path length as \(L=2439\) in meters.  
Regarding Scenario 1, Figure 2 depicts the path formed utilizing Algorithm 3. Obstacles are depicted with spheres, and the constructed smooth path is illustrated as blue line segments. The start position is marked with a red asterisk, and the destination position is marked with a black asterisk. There are several obstacles close to the destination, which makes a safe path planning more challenging. The formed path is safe for a vehicle with safety margin \(r\) meters.  
Regarding Scenario 1, let \([n_{1},n_{2},\ldots,n_{end}]\) stand for the simulation nodes along the path formed using Algorithm 3. Let \(Angle(n_{j})\), where \(j\in\{1,2,3,\ldots,end-2\}\) stands for the angle formed by two adjacent line segments, \(Line(n_{j},n_{j+1})\) and \(Line(n_{j+1},n_{j+2})\), in the path. Figure 3 presents \(Angle(n_{j})\) with respect to \(j\), where \(j\in\{1,2,3,\ldots,end-2\}\). \(A_{max,z}\) is marked with a red dotted line. Note that \(Angle(n_{j})\leq A_{max,z}\), where \(j\in\{1,2,3,\ldots,end-2\}\).  
Regarding Scenario 1, let \(Dist(n_{j})\), where \(j\in\{1,2,3,...,end-2\}\) stands for the relative distance between \(n_{j}\) and its closest obstacle boundary. Figure 4 presents \(Dist(n_{j})\) with respect to \(j\), where \(j\in\{1,2,3,...,end-2\}\). The safety radius \(r\) is marked with a red dotted line. Note that \(Dist(n_{j})\geq r\), where the safety radius is set as \(r=3\) m.  
Figure 3: Regarding Scenario 1, this plot shows \(Angle(n_{j})\) with respect to \(j\), where \(j\in\{1,2,...,end-2\}\). \(A_{max,z}\) is marked with a red dotted line.  
Figure 2: Regarding Scenario 1, this plot shows the path formed utilizing Algorithm 3. Obstacles are depicted with spheres. The constructed smooth path is illustrated as blue line segments. The start position is marked with a red asterisk, and the destination position is marked with a black asterisk.  
Comparison Algorithms Through Monte Carlo Simulations  
For comparison, we run the 3D RRT-star algorithm [21] considering the turn rate limits, (6) and (7). In the 3D RRT-star algorithm in [21], the steer function is used to spawn a new tree node from these two nodes: a randomly generated node and an existing tree node which is nearest to the randomly generated node. In the 3D RRT-star algorithm [21], the steer function is set considering the turn angle limits, (6) and (7). This implies that a new tree node is generated by considering the turn angle limits. In the 3D RRT-star algorithm [21], two nodes are _neighbors_, as their relative distance is shorter than \(r_{s}\) in (8).  
For a fair comparison, 100 Monte Carlo (MC) simulations are applied as we run the proposed plan algorithm \([\)_Proposed_\(]\), and the 3D RRT-star \([\)_RRT_\(]\). In total, 1000 tree nodes are generated in the 3D RRT-star algorithm in [21]. In the case where a safe path to the destination is not found using 1000 tree nodes, we keep re-running the associated MC simulation until a safe path to the destination is found.  
In Table 1, \(simTime\) (in seconds) stands for the average simulation time of running a single MC simulation utilizing MATLAB. \(meanLen\) stands for the mean of path length for all MC simulations. \(varLen\) stands for the variance of path length for all MC simulations. Note that the proposed plan algorithm outperforms \([\)_RRT_\(]\) considering both time efficiency and path length. Five hours are spent to find a path using \([\)_RRT_\(]\) in a single MC simulation. However, \([\)_RRT_\(]\) cannot succeed in finding a path within five hours. Thus, Table 1 shows - for \([\)_RRT_\(]\).