## 3 Analysis of Movement over a Step  
### Strategy  
The following three steps are necessary for a simple tracked vehicle to traverse a step:  
1. The front end of the vehicle climbs onto the step. 2. The rear end of the vehicle climbs onto the step. 3. The vehicle lands on the ground without turning over forward. We first analyze the condition for Step 1. For simplicity, Figure 5 shows a force diagram for the condition when the vehicle is stationary at the time the front end starts to move up along the step wall. A and B represent the contact points between the track and the ground and the step wall, respectively. \(N_{\text{A}}\) and \(N_{\text{B}}\) are normal forces applied to A and B; \(f_{\text{A}}\) and \(f_{\text{B}}\) are the static friction applied to A and B; and \(x\) and \(l\) are the distances between B and the COG and between A and B, respectively. Figure 4: CADs of “_Dyjob_”: (**a**) the top view and (**b**) the right-hand side view at the dashed cross section in (**a**). The rubber tracks are omitted. Since the horizontal and vertical forces are balanced in Figure 5, \(f_{\rm A}=N_{\rm B}\) and \(N_{\rm A}+f_{\rm B}=mg\) (where \(g\) is the acceleration of gravity). Considering moment equilibrium about B, \(N_{\rm A}l-mgx=0\) holds, assuming that the vehicle is low enough. Therefore, we have the following:  
\[N_{\rm A}=\frac{mgx}{l} \tag{1}\]  
and:  
\[f_{\rm B}=mg\Big{(}1-\frac{x}{l}\Big{)}. \tag{2}\]  
The conditions for the vehicle not to slip at A and B are \(f_{\rm A}<\mu_{A}N_{\rm A}\) and \(f_{\rm B}<\mu_{B}N_{\rm B}\), respectively, where \(\mu_{A}\) and \(\mu_{B}\) are the coefficients of static friction at A and B. Therefore, the larger value of \(x\) in Equations (1) and (2) allows the conditions to be easily met. This means the backward-positioned COG can prevent the track from slipping on the ground and wall, thus providing large propulsion needed to achieve Step 1. Since \(\mu_{A}\) and \(\mu_{B}\) depend on the surfaces of the terrain and step wall, respectively, the conditions \(f_{\rm A}<\mu_{A}N_{\rm A}\) and \(f_{\rm B}<\mu_{B}N_{\rm B}\) are variable according to the environment. Some works estimate contact models between the vehicle and the ground for autonomous off-road UGVs [36; 37; 38]. For our teleoperated vehicle, the operator judges whether the vehicle is slipping through the onboard camera and manually controls the acceleration and the movable mass position to achieve Step 1. We will analyze Step 2 in Section 3.2. Regarding Step 3, the forward-positioned COG causes the front end to land on the ground at a steep angle, as shown in Figure 6a. In addition, since the COG is close to the contact point P, the vehicle easily turns over. The backward-positioned COG can avoid the risk of turning over, as shown in Figure 6b. Figure 5: Force diagram when the vehicle stops at the time the front end starts to move up along the step wall. Figure 6: High (**a**) and low (**b**) risks of turning over with the adjustable COG position when the vehicle is landing on the ground. Geometric Analysis of Effect of Adjustment of COG Position on Rear End of the Vehicle Climbing a Step  
Here, we analyze the geometric boundary condition needed to achieve Step 2 after finishing Step 1, as shown in Figure 7. Here, the Points O, P, and Q are the center of the vehicle and the centers of the front and rear track pulleys, respectively; \(R\) and \(h\) represent the radius of the track pulley and the height of the lug, respectively; and \(x\) and \(y\) represent, respectively, the forward and upward distances between O and COG. The time when the COG is located directly above the edge of the step, as shown in Figure 6, is the boundary condition required for the vehicle to achieve Step 2. The height of a step that the vehicle can climb is given by:  
\[H=A-B, \tag{3}\]  
where \(A\) and \(B\) are the heights of the COG from the ground and the step tread, respectively.