## 4 Slope Climbing and Trenching Capability
### Slope Climbing Ability Study  
In mobile robot design, the stability of the robot's large slope motion is a key factor in measuring the success or failure of mobile robot design. It is an essential indicator of the robot's ability to move longitudinally or laterally on a slope without tipping over, and it is closely related to the robot's ability to smoothly traverse a rough terrain. Every concave and convex cell on the ground can be treated as a slope, either significant or minor. Therefore, the anti-tip performance of a mobile robot on a slope can be simplified as stability of its travel on a rough road [23]. Slopes can be classified into two types based on the orientation of the mobile robot's movement on the sloping terrain--positive slope and side slope terrain. Positive slope terrain refers to the situation where the robot moves along the slope direction of the slope, while side slope terrain refers to the condition where the robot moves along the slope direction of the vertical slope [24]. The size of the robot's climbing ability is related to the road adhesion coefficient and the size of its structure. The height of the center of mass should be adjusted by altering the angle between the front and rear swing arms. This adjustment ensures that any variations in the slope  
Figure 9: Schematic diagram of the robot crossing trench. angle of the slope will not cause the robot to tip over. The slope angle \(\gamma\) is a function of \(\theta_{1},\ \theta_{2},\ \text{and}\ \theta_{3}\). As shown in Figure 10, the robot climbs the slope longitudinally. To avoid tipping, the robot must meet the following condition, where \(G_{c}\) represents the robot's gravity. \[G_{c}\text{cos}\gamma_{1}(C_{x}+L_{2}\text{cos}\theta_{2})>G_{c}\text{sin} \gamma_{1}\left(C_{y}+R\right) \tag{21}\]  
\[\gamma_{1}<\text{arctan}\left(\frac{C_{x}+L_{2}\text{cos}\theta_{2}}{C_{y}+R}\right) \tag{22}\]  
During the climbing process, the robot has the interaction of the air resistance \(F_{1}\), and the traction force \(F\) of the swing arm tracked robot, in addition to the gravitational force, the support force, to complete the climbing process of the swing arm tracked robot [25]. Let the coefficient of friction between the robot track and the slope be \(\mu_{1}\), and the conditions for climbing the slope without slipping are as follows. \[F-F_{1}-\mu_{1}G_{c}\text{cos}\gamma_{2}-G_{c}\text{sin}\gamma_{2}\geq 0 \tag{23}\]  
The speed of the robot is relatively slow, and the air resistance of the swing arm tracked robot is also very low when moving against the wind, which is generally negligible. Therefore, the above equation can be simplified at this point:  
\[F-\mu_{1}G_{c}\text{cos}\gamma_{2}-G_{c}\text{sin}\gamma_{2}\geq 0 \tag{24}\]  
\[\gamma_{2}\leq\text{arcsin}\frac{F}{G_{c}\sqrt{{\mu_{1}}^{2}+1}}-\text{arctan }\frac{1}{\mu_{1}} \tag{25}\]  
In summary, the maximum slope angle that a robot can climb longitudinally without tipping or slipping is:  
\[\gamma_{max}=\text{min}\ \left(\gamma_{1},\gamma_{2}\right) \tag{26}\]  
Realistic terrain slopes are not all single slopes, and sometimes the terrain may be shaped as an inverted figure--the \"Chinese eight\". In such cases, the width of the flat part at the bottom may be smaller than the width of the main body of the robot. If the robot is allowed to move directly in its original state, the outermost sides of the track and wheels may encounter the inverted figure, a \"Chinese eight\" slope, making it difficult for the swing-arm tracked robot to continue moving. For the robot to move effectively, it needs to move completely to one side of the slope, where the track and wheel sections are in full contact with the slope, as shown in Figure 11.  
Figure 10: Robot climbing slopes longitudinally. The robot moves laterally on the slope, and the maximum slope angle at which the robot does not tip over is obtained. The condition for lateral movement on a slope without tipping is as follows.