## 2 Materials and Methods
### Passability Model  
To find the optimal route through the forest, it is necessary to know: parameters of the vehicle, parameters of the trees, start and end points of a route, and impassable areas. The most important parameters of a vehicle are vehicle width (\(VW\)), length, turning radius, and tolerance (\(T\))--see Figure 5.  
Figure 4: Dependence of a forest structure determination.  
\(T\) determines the minimum distance of the vehicle from a trunk to pass between two trees safely. To simplify the passability model, \(T\) also replaces the effect of other vehicle parameters (length, turning radius, etc.). In turn, tree parameters refer to those characteristics that are key to finding the optimal route in the forest. In our case, those parameters are stem simulated coordinates, mean tree spacing (\(MTS\)), mean \(DBH\), mean riding corridor (\(MRC\)), and a \(VW\). We can express the relationship between \(MTS\), \(MDBH\), and \(MRC\) (see Figure 6) using Formula (4):  
\[MRC=MTS-MDBH. \tag{4}\]  
If we do not know the exact coordinates of each tree (which is the usual situation), we can generate the simulated forest structure from average values--see the procedure above. We use the process of random tree deployment to determine the probability of crossing the forest.  
The start and end points of a route can be substituted by an initial and final area in a forest region. It is usually not possible to go between these points or areas directly, and we must maneuver between tree stems. Impassable objects (steep slopes, rocks, lakes, rivers, burning forest, etc.) can be obtained from GIS databases or using aerial or satellite images. For the purposes of modeling vehicle mobility in the above-described forest, the impassable areas of the simulated burning polygons were chosen, though other objects (obstacles) were not considered. To search the optimal vehicle route, the following algorithms were used (see Figure 7): Voronoi graph and Delaunay triangulation, Dijkstra algorithm, and optimization of the fractional line.  
Figure 7 shows the positions of individual trees (Vertex M)--blue points. The closest two trees to the given tree create the Delaunay triangle--Figure 7b, and the most secure route sections (Voronoi edges) are intersected in the Voronoi nods--blue lines in Figure 7c. The Dijkstra algorithm was used to find the shortest routes from the nod of the graph given to all other nods--see also [23,24]. Using Figure 7c, we can simulate a forest path (see Figure 7d). Trees that were obstacles are marked in red, and Voronoi nods (pale blue points) are connected with Voronoi edges (dark blue lines). All Voronoi edges are rated by weights. These weights may represent the distances or time for which a vehicle passes through the Voronoi edges. In the event that we search for the shortest route from point 1 to point 15, the condition of the minimum sum of the Voronoi edges (weighing) is to be compliant with  
Figure 5: Fire and tested vehicles. (**a**) Fire vehicle Tatra 815 \(4\times 4\), Length/Width/Height: 7950/2550/300 mm. (**b**) Tested terrain vehicle Tatra 815 \(8\times 8\), Length/Width/Height: 8950/2550/3300 mm.  
Figure 6: The relationship between particular tree spacing, \(DBH\), and riding corridor.  
the shortest route--red line in Figure 7d passing through Voronoi nods 1-3-7-6-11-12-14-15, since the sum of route segment values (3.0 m + 3.3 m + 0.8 m + 3.8 m + 1.2 m + 2.1 m + 1.6 m) = 15.8 m is the smallest (shortest) compared to all possible routes connecting points 1 and 15.