## 3 Energy Consumption Analysis  
The energy consumption of a robot moving on undulating ground differs significantly from that of a robot moving on flat ground. It is crucial to consider not only the energy consumption due to the robot's horizontal movement but also the energy expenditure associated with its vertical movement. Robot movement on undulating ground can be approximated as movement on an inclined surface with varying angles, as illustrated in Figure 2(a). In Figure 2(a), \(\theta\) represents the inclination angle of the inclined plane, \(\alpha\) represents the inclination angle of the robot's traveling direction, \(F_{f}\) is the friction force acting on the robot, G stands for the gravitational force, and F represents the traction force required for the robot to move at a constant speed. Figure 2(b) illustrates a force analysis in cross-section based on the robot's traveling angle. Where \(N_{i}\) and \(N_{i+1}\) represent the information of two grid cells in the map.  
It is assumed that the robot moves at a constant speed throughout the entire coverage operation on an inclined surface. There are two primary aspects of energy consumption that need to be considered during the coverage process: the length of the traveling path and the elevation difference along the path.  
Let the mass of the robot be m (kg), the acceleration of gravity be g, and the coefficient of friction be \(\mu\). The traction force F on the robot on the uphill and downhill slopes are  
\[F =mg\mu\cos\alpha+mg\sin\alpha \tag{1}\] \[F =mg\mu\cos\alpha-mg\sin\alpha \tag{2}\]  
Figure 3: Robot force analysis.  
Figure 2: Modeling process.  
Taking the movement of a robot in a raster map as an example, the relationship between path length and energy consumption can be expressed as follows:  
\[E_{1}=F\cos\alpha S \tag{3}\]  
\[S=\sqrt{\left(x_{i+1}-x_{i}\right)^{2}+\left(y_{i+1}-y_{i}\right)^{2}} \tag{4}\]  
\[\alpha=\tan^{-1}\frac{\left|h_{i+1}-h_{i}\right|}{S} \tag{5}\]  
where \(E_{1}\) is the energy consumption generated by the length of the traveling path(J), \(S\) is the distance between the two grids (m), \(x_{i}\), \(x_{+1i}\), \(y_{i}\), \(y_{i+1}\) is the coordinate value of the grid, and \(h_{i}\), \(h_{i+1}\) is the surface height of the grid (m).  
The relationship between path elevation difference and energy consumption is:  
\[E_{2}=F\sin\alpha|h_{i+1}-h_{i}| \tag{6}\]  
The energy consumption of the robot traveling on an inclined surface is obtained by bringing in the traction force \(F\) calculated above:  
\[E_{1}=\left\{\begin{matrix}mg\cos\alpha S(\mu\cos\alpha+\sin\alpha)\\ mg\cos\alpha S(\mu\cos\alpha-\sin\alpha)\end{matrix}\right. \tag{7}\]  
\[E_{2}=\left\{\begin{matrix}mg\sin\alpha|h_{i+1}-h_{i}|\left(\mu\cos\alpha+\sin \alpha\right)\\ mg\sin\alpha|h_{i+1}-h_{i}|\left(\mu\cos\alpha-\sin\alpha\right)\end{matrix}\right. \tag{8}\]  
Analyzing the aforementioned energy consumption equation, it can be concluded that the larger the inclination angle of the inclined plane, the greater the energy consumption of the robot during a full coverage operation. Furthermore, when covering an inclined surface, it is advisable to minimize the elevation difference along the path, i.e., to cover along the direction that aligns most closely with the horizontal line on the inclined surface, as illustrated in Figure 4.