## 5 Simulation and Numerical Calculation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Numerical Value** \\ \hline Surface roughness & 1.2 \\ Robot speed/(m/s) & 1 \\ Obstacle height/(mm) & 300 \\ Static Coefficient & 0.3 \\ Dynamic Coefficient & 0.25 \\ Gully width/mm & 300 \\ Ramp angle/(\({}^{\circ}\)) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Crossing step simulation parameters. Figure 13: Numerical solution for robot climbing stairs. (**a**) Low step-down climbing height and effective radius curve. (**b**) Climbing height and effective radius curve under high steps. (**c**) Robot over-height surface. The simulation of the robot climbing a step with a height of 300 mm is shown in Figure 15. The robot first approaches the step at a steady speed. Then, the rear swing arm rotates clockwise to raise the robot's body, and the front swing arm rotates clockwise to contact the step. As the robot advances, the rear swing arm rotates clockwise, and the robot lifts until it successfully crosses the high step. Overall, the simulation demonstrates the robot's ability to climb steps with ease and accuracy. The use of swing arms and the coordinated motion of the robot's body allowed it to elevate itself and cross the step effectively. This proves the effectiveness of our gait planning. Future work can focus on enhancing the robot's ability to climb higher steps and improving its stability during the climbing process. Substitute the parameters of the robot into Equations (21)-(26). When the rubber timing belt drives on a smoother road surface (without considering the influence of track spurs on the size of adhesion force), the adhesion coefficient f can be taken as 0.7, or, when the rubber timing belt drives on the road surface with higher friction, the adhesion coefficient can be taken as 1.2. According to the calculation, the maximum slope angle of the robot in longitudinal climbing is \(\gamma=35^{\circ}\) for an attachment coefficient of 0.7 and \(\gamma=50.2^{\circ}\) for an attachment coefficient of 1.2. Figure 14: Robot speed curve. Figure 15: Step climbing (300 mm). (**a**) Climbing high steps stage 1. (**b**) Climbing high steps stage 2. (**c**) Climbing high steps stage 3. (**d**) Climbing high steps stage 4. (**e**) Climbing high steps stage 5. (**f**) Climbing high steps stage 6. Substitute the parameters of the robot into Equations (27)-(31). According to the calculation, the maximum slope angle of the robot in longitudinal climbing is \(\gamma=35^{\circ}\) for an attachment coefficient of 0.7 and \(\gamma=50.2^{\circ}\) for an attachment coefficient of 1.2. Establish the slope climbing simulation model of the mobile robot. The process of the robot's ascent up a slope of \(30^{\circ}\) has been simulated, and the simulation findings are elaborated in Figure 16. The initial scenario in Figure 16a shows the robot moving along steadily at a consistent speed prior to reaching the slope. During this stage, the robot's swing arm is held in a contracted state. Moving on to the subsequent stage in Figure 16b, the robot is depicted in an open swing arm state, negotiating the \(30^{\circ}\) slope. Additionally, the trajectory of the robot while it climbs up the slope is illustrated in Figure 17. When observing Figure 17, we can see that, from 0 to 0.66 s on the time axis, the robot was driving on a flat surface until it reached the slope position. At 0.66 s, the robot started to climb the stairs, and at 1.09 s, the robot had fully entered the slope and continued to travel on it. Substitute the robot's parameters into Equation (20). According to the calculation, when \(\theta_{1}=-12.42^{\circ}\) and \(\theta_{2}=192.42^{\circ}\), the maximum width of the transverse trench \(L_{d}=388.05\) m. Now, establish a simulation model of the robot that can cross the trench.