## 2 Modeling of Wheel-Legged Robot Systems
### Modeling of the Robot's Slope Movement  
A dynamic amplitude adjustment model of CPG based on force feedback was constructed to solve the problem of motion instability caused by sudden changes in the foot-end force in unstructured terrain. The amplitude is adaptively adjusted according to the deviation between the measured value of the foot-end force sensor and the expected force in this paper, so that the output amplitude of the CPG can be rapidly adjusted to adapt to external disturbances when the foot-end force is abnormal. The CPG amplitude adjustment function is shown in (6). The adjustment angles for the robot's slope movement are shown in Table 2.  
\[A_{i}(t)=A_{i}^{*}-\gamma(F_{foot,i}-F_{ref,i}) \tag{6}\]  
where \(A_{i}(t)\) represents the output amplitude of the i-th CPG unit, and \(\gamma\) is a positive definite adjustment coefficient. For \(F_{foot,i}\), it represents the actual force exerted on the foot of the i-th leg, The term \(F_{ref,i}\) denotes the pre-defined reference force for the i-th leg's foot. When the force on the foot-end is abnormal, the output amplitude of the CPG can be adjusted rapidly to adapt to external disturbances. The robot acquires the terrain slope signal, designated as \(\alpha\), via the torso-mounted pitch direction inclinometer. This signal is subsequently integrated into the feedback phase of the CPG model, denoted as S-G. The torso attitude angle exhibits a linear relationship with the extent of equilibrium position adjustment. Consequently, the vestibular reflexes are modeled as a linear function of slope, as illustrated in (7). \[s_{v}g_{v}=\hat{\alpha} \tag{7}\]  
Let the vestibular feedback term \(s_{v}\) in the reflection information matrix be the actual slope value measured by the inclinometer in the pitch direction. The vestibular reflection coefficient \(g_{v}\) is calculated from (8). \[s_{v}=\alpha \tag{8}\] \[g_{v}=k_{2}/k_{1}*k_{3}\]  
Through iterative adjustment of the slope parameter \(\alpha\) and subsequent numerical fitting, a linear relationship \(\Delta\theta=\ k_{1}\times\alpha(k_{1}=0.23)\) was established between the introduced slope and the equilibrium position adjustment, as depicted in Figure 5.  
A sufficient condition for a wheel-legged robot to achieve slope locomotion with optimal energy efficiency is a linear relationship between the body pitch angle and the slope gradient, expressed as \(\Delta\alpha=k_{2}\times\alpha\), \(k_{2}\). This is a standardized proportional factor used in the slope control of legged robots, quantifying the linear mapping relationship between terrain slope variations and the corresponding adjustment of the torso pitch angle. Through extensive energy-based dynamic modeling, it was derived that when \(k_{2}=\ 0.24\), the metabolically analogous energy consumption can be minimized. Where \(\Delta\alpha\) denotes the body pitch angle adjustment. The relationship between the joint equilibrium position \(\Delta\theta\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Change in Equilibrium Position**} & \multicolumn{2}{c}{**Front Leg**} & \multicolumn{2}{c}{**Back Leg**} \\ \cline{2-5}  & **Hip Joint** & **Knee Joint** & **Hip Joint** & **Knee Joint** \\ \hline Uphill & \(-\Delta\theta\) & \(\Delta\theta\) & \(\Delta\theta\) & \(-\Delta\theta\) \\ \hline Downhill & \(\Delta\theta\) & \(-\Delta\theta\) & \(-\Delta\theta\) & \(\Delta\theta\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Adjustment angles of the robot’s joints on the slope. and \(\Delta\alpha\) is defined in (9), with \(l=0.15\) m representing leg length, \(ll=0.40\) m representing body length, and \(\theta_{0}\) representing the initial equilibrium position. \[\Delta\alpha=\arctan\frac{l[\cos(\theta_{0}-\Delta\theta)-\cos(\theta_{0}+ \Delta\theta)]}{l[\sin(\theta_{0}+\Delta\theta)-\sin(\theta_{0}-\Delta\theta)]+ll} \tag{9}\]  
For \(\Delta\theta\in[0,0.13]\), the solution for \(\Delta\alpha\), as illustrated in Figure 6, approximates a linear dependency \(\Delta\alpha=k_{3}\times\Delta\theta\).