## 4 Robot Stability Analysis
### Stability Analysis of Unilateral Unfolding of Transformed Wheels  
According to the previous analysis, \(\varepsilon_{min}\) should be greater than 0. The larger \(\varepsilon_{min}\) is, the more stable the robot is. When crossing a larger unilateral obstacle, the transformed wheel is passively transformed at the moment of contacting the obstacle, at this time the transformed wheel A moves in the wheeled mode, and the transformed wheel B moves in the wheel-legged mode. The state of crossing the obstacle is shown in Figure 15, where one side of the transformed wheel is subjected to a larger load acting on the edge, and the height of the center of gravity changes, causing the body to tilt, and the robot may move with an unstable motion [25; 26; 27].  
The height of the transformed wheel centers on the upper and lower sides are \(H_{a}\) and \(H_{b}\), and \(\Delta\delta\) is the cross-roll angle of the body. \(\Delta\theta\) is the adjustment angle of the robot relative to the slope, and \(a_{s}\) is the equivalent slope of the ramp. The following equations are derived:  
\[\left\{\begin{array}{l}\Delta\delta=a_{s}-\Delta\theta\\ a_{s}=\arcsin\frac{\Delta t\cos\Delta\delta}{a}\\ \Delta\theta=\arctan\frac{H_{b}-H_{b}}{a}\end{array}\right. \tag{20}\]  
The wheel center height \(H_{a}\) is varied with the side wheel spread radius \(r_{uu}\), \(H_{b}\) is varied with the lower side wheel spread radius \(r_{ww}\), and the robot's body cross-roll angle is  
\[\Delta\delta=a_{s}-\arctan\frac{H(r_{wb})-H(r_{wa})}{a} \tag{21}\]  
Figure 14: Analysis of robot slope motion based on the stable cone principle. (**a**) Lateral movement; (**b**) longitudinal movement.  
The following equations are derived:  
\[\left\{\begin{array}{l}\varepsilon_{1}=\arctan\frac{\mu}{H(T_{uu})}-\Delta \delta\\ \varepsilon_{2}=\arctan\frac{\mu}{H(T_{uu})}+\Delta\delta\end{array}\right. \tag{22}\]  
When \(w\) = 1 rad/s and \(r_{uu}\) = 75 mm, the fluctuation of the robot's body traverse angle during obstacle crossing is solved through the stability criterion. Figure 16a shows the relationship between the height of the obstacle or the inclined surface and the effective radius of the transformation wheel. Figure 16b shows the relationship between the maximum traverse angle of the robot, the effective radius of the transformation wheel, and the stability criterion.  
The following conclusions were obtained.  
1. When the terrain height difference is small, the excessive wheel diameter difference affects the significant increase of the fuselage cross-roll angle, the stability criterion decreases, and the fuselage tilts backward.
2. When the terrain height difference is greater than 100 mm, the fluctuation of the fuselage cross-roll angle decreases with the increase of the wheel diameter difference. The cross-roll angle decreases by more than 25% in the wheel-leg maximum spread mode and the wheeled mode.
3. Due to the small change in wheel diameter of the lower transformed wheels, the effect of fuselage adjustment affected by the radius difference cannot offset the decrease in the stability angle caused by the increase in its center of gravity. There is a phenomenon where the stability angle first decreases and then increases as the effective radius of the wheels becomes larger.  
Figure 15: Stable cone model for unilateral expansion of transformed wheels. (**a**) Vertical obstacle; (**b**) sloping terrain.  
When the terrain height difference is 185 mm, the stability angle appears less than 0, and the fuselage tilts, increasing the possibility of tipping.