## 2 Topographic Features and Modeling
### Modeling of Hilly Farmland  
Environment map modeling is an important part of full-coverage path planning, i.e., a certain mathematical model is employed to represent the robot's working environment [17]. Three-dimensional map modeling constitutes the first step towards achieving three-dimensional-coverage path planning. The surface is generally represented by measuring the elevation of discrete points in the area, known as DEM (Digital Elevation Model),  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Region**} & \multicolumn{2}{c}{**Arable Land**} & \multicolumn{2}{c}{**Sloping Cultivated Land**} \\ \cline{2-5}  & **Area (MH)** & **Percentage (\%)** & **Area (MH)** & **Percentage (\%)** \\ \hline Eastern Region & 3695.59 & 28.4 & 38.67 & 6.4 \\ Central Region & 5611.89 & 43.2 & 104.00 & 17.1 \\ Western Region & 3696.45 & 28.4 & 464.00 & 76.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Distribution of arable land in China.  
and interpolation methods such as kriging are utilized to estimate the elevation of any point within the area [18]. However, this interpolation technique can only provide elevation information at specified points. Conversely, the analytical model derived from DEM data can continuously describe the terrain surface in 3D space, facilitating terrain characterization as well as slope and curvature computations, which are crucial for evaluating different decomposition and coverage models. In addition, polynomial interpolation is a fitting method that employs a polynomial expansion to fit all observations within a small analysis area encompassing several analysis grids. Polynomials have been widely used for curve or surface fitting. However, when a function needs to be approximated over a large interval, the method of approximating with polynomials may no longer be applicable. For three-dimensional terrain, areas with numerous slopes and gullies necessitate more accurate polynomial models, and the use of low-order polynomials necessitates the avoidance of overfitting. To circumvent these issues with polynomial models, a spline interpolation model can be utilized, which is a special function defined through polynomial segmentation. Spline interpolation typically offers better accuracy than polynomial interpolation, as it can achieve similar accuracy with a lower degree of polynomials. In this paper, we will employ the spline method to interpolate grid-like DEM data.  
Compared with farmland in plain areas, the topography of farmland in hilly areas is more undulating, and a 2D raster map fails to accurately reflect the real environment of farmland in such areas. Therefore, this paper adopts a three-dimensional raster map as the map model for farmland, where the value of each raster in the map represents the height of the terrain surface at its corresponding point.  
The map is constructed as follows. Firstly, DEM (Digital Elevation Model) data for the area are obtained by traversing the entire area, as depicted in Figure 1a. Subsequently, a series of data processing steps, including interpolation and rasterization, are performed using GIS (Geographic Information System) with software version 10.8. Finally, the processed raster data are imported into MATLAB to generate the raster map model of the hilly farmland, as illustrated in Figure 1b. The specific process of map modeling is shown in Figure 2.  
Figure 1: Map data processing. (**a**) DEM data; (**b**) raster ap.