## 4 Analysis  
### Analysis of the Rocker Arm in a Retracted State during Step Crossing  
By determining the relationship between the center of mass and the angle of rotation, the theoretical height for the robot climbing the step can be analyzed [29]. Considering Figure 4d, it can be concluded that the robot can climb the step smoothly only when the center of gravity is higher than the edge of the step. Therefore, further analysis of this part was performed to establish a coordinate system with the axis of the rear crawler wheel of the robot as the coordinate origin. The mathematical model of the climbable obstacle height of the robot can be obtained from the geometric relationship of Figure 5:  
\[H=R+X_{G}\sin\eta+Y_{G}\cos\eta-\frac{Y_{G}+R}{\cos\mu} \tag{9}\]  
where \(R\) is the radius of the large wheel of the robot, \(H\) is the height of the step, \(X_{G}\) is the coordinate of the mass center on the \(X\)-axis, and \(Y_{G}\) is the coordinate of the mass center on the \(Y\)-axis. Figure 4: The process of crossing a step. (**a**) the start of the climbing process, and the rocker arm starts to touch the step; (**b**) the robot is ready to climb, and the rocker arm is already on the step; (**c**) crossing over; (**d**) the critical point of climbing—at this point, the center of gravity of the robot and the contact point are on the same vertical line; (**e**) the end of the climb, at which point the robot has climbed the step completely. It can be seen from Figure 4a that \(\eta=0\) at this time. Combined with Equation (7), the coordinate formula of robot centroid can be obtained:  
\[ \begin{array}{c}X_{G}=\frac{G_{1}L_{2}+G_{2}L_{1}}{G_{1}+G_{2}}+\frac{G_{2}L_{ 4}}{G_{1}+G_{2}}\cos\gamma\\ Y_{G}=\frac{G_{1}\eta}{G_{1}+G_{2}}+\frac{G_{2}L_{4}}{G_{1}+G_{2}}\sin\gamma \end{array}  \tag{10}\]  
where \(\gamma\) is the swing angle of the rocker arm, and \(\eta\) is the pitch angle of the robot. The design parameters of the robot are: \(G=38\) kg; \(m_{1}=30\) kg; \(m_{2}=8\) kg; \(L_{1}=510\) mm; \(L_{2}=255\) mm; \(L_{3}=300\) mm; \(L_{4}=115\) mm; \(h_{2}=70\) mm; \(R=100\) mm; \(r=30\) mm. To determine the relationship between the height \(H\) of the step and the angle \(\gamma\), \(\eta\), first, the first and second derivatives of \(\gamma\) and \(\eta\) in Equation (9) must be determined. When the robot climbs the step, \(\gamma\) is in the range (0, 2\(\pi\)), and \(\eta\) is in the range (0, \(\pi/2\)). Therefore, when the first-order derivative is zero and the second-order derivative is less than zero, \(H\) is the largest. That is, when the rocker arm is horizontal (\(\eta+\gamma=2\pi\)), the robot reaches the maximum height \(H_{max}\) of climbing the step. Finally, the parameters of the robot are substituted into Equations (9) and (10). The mathematical model related to \(H\) is solved using the simulation software optimization toolbox and is simulated to obtain the relationship between the height \(H\) of the robot across the step and the swing angle \(\gamma\) of the rocker arm, as well as the pitch angle \(\eta\) of the robot body. In Figure 6, \(\gamma\) is the rocker arm swing angle, and \(\eta\) is the pitch angle. From the simulation results, one can learn that when \(\gamma=5.4605\) rad and \(\eta=0.8227\) rad, the maximum height when the robot climbs the step is \(H_{max}=187.74\) mm. That is, \(\eta+\gamma=6.2832\approx 2\pi\). When the robot crosses the barrier, it first needs the rocker arm to be able to touch the end of the step, and then the rocker arm and the main body need to be in a straight line during the crossing process. The robot's center of gravity should exceed the contact point between the robot and the step so that the crossing can be carried out smoothly. It can be seen from the figure that when the pitch angle is 0, the swing angle of the rocker arm has no effect on the obstacle height of the robot; when the swing angle of the rocker arm is 0, a change of the pitch angle will result in a change of the obstacle height of the robot.