## 2 Modeling of Wheel-Legged Robot Systems
### Mechanical Design of Wheel-Legged Robots
\begin{array}{l}\left[\begin{array}{l}\dot{x}_{i}\\ \dot{y}_{i}\end{array}\right]=\left[\begin{array}{ll}\alpha(\mu-r_{i}^{2})&- \omega_{i}\\ \omega_{i}&\alpha(\mu-r_{i}^{2})\end{array}\right]\left[\begin{array}{l}x_{i} \\ y_{i}\end{array}\right]+\Delta_{i}(t)+\sum\limits_{j=1}^{4}R(\theta_{i}^{j}) \left[\begin{array}{l}x_{j}\\ y_{j}\end{array}\right]\\ \dot{r_{i}^{2}}=\left(x_{i}-u_{1}\right)^{2}+\left(y_{i}-u_{2}\right)^{2}\\ \omega_{i}=\frac{\omega_{st}}{e^{-\omega_{i}}+1}+\frac{\omega_{sw}}{e^{\alpha _{ij}}+1}\\ \omega_{st}=\frac{\beta}{1-\beta}\omega_{sw}\\ \theta_{hi}=x_{i}\\ \theta_{ki}=\begin{cases}-\text{sgn}(\varphi)\frac{A_{k}}{A_{h}}y_{i},&y_{i} \leq 0\\ 0,&y_{i}>0\end{cases}\end{array} \right. \tag{2}\]  
In order to break through the traditional limitation of only considering the convergence of the rhythm, the augmented and broadened Lyapunov functions of the CPG state and the body motion state were constructed to evaluate the stability of the control system, as shown in (3). Specifically, \(V\) serves as the total cost function for motion optimization, balancing \begin{table}
\begin{tabular}{l l l l} \hline
**Symbol** & **Definition** & **Symbol** & **Definition** \\ \hline \(x_{i},y_{i}\) & State variables of the i-th oscillator & \(w_{st},w_{sw}\) & Natural frequencies in stance and swing phases, respectively \\ \hline \(r_{i}^{2}\) & Euclidean distance from the i-th joint to the reference point \(\mu_{1}\), \(\mu_{2}\). & \(\theta_{hij}\) & Hip joint angle of leg i, directly equal to \(x_{i}\) \\ \hline \multirow{4}{*}{a = 10} & Convergence rate constant & \multirow{4}{*}{\(\theta_{kj}\)} & Knee joint angle of leg i, A\({}_{k}\), A\({}_{h}\) serves as amplitude gains for hip and knee oscillators \\  & Sensitivity: larger a accelerates & & \\ \cline{1-1}  & stabilization but may induce overshoot; & \(\theta_{kj}\) & \\ \cline{1-1}  & smaller a slows response, risking & & \\ \cline{1-1}  & instability under rapid terrain changes. & & \\ \hline \multirow{4}{*}{\(\mu\) = 0.25} & Desired squared limit-cycle amplitude & \multirow{4}{*}{\(\Delta_{i}(t)\)} & Additional phase correction term \\  & Sensitivity: higher \(\mu\) increases stride & & \\ \cline{1-1}  & motion amplitude, enhancing obstacle & & \\ \cline{1-1}  & crossing but raising energy costs; lower & & \\ \cline{1-1}  & \(\mu\) conserves energy but may reduce & & \\ \cline{1-1}  & terrain adaptability. & & \\ \hline \multirow{4}{*}{\(\omega_{i}\)} & Adaptive weighting factor for & \multirow{4}{*}{\(R(\theta_{ji})\)} & Motion mapping function that \\ \cline{1-1}  & hip-knee coordination & & \\ \cline{1-1}  & Larger \(\omega_{i}\) strengthens hip-driven & & \\ \cline{1-1}  & coordination, improving stability on & & \\ \cline{1-1}  & slopes; smaller \(\omega_{i}\) increases knee & & \\ \cline{1-1}  & autonomy, useful for fine terrain & & \\ \cline{1-1}  & adjustment but risking phase mismatch. & & \\ \hline \multirow{4}{*}{\(\beta\) = 0.5} & Decay coefficient for & \multirow{4}{*}{\(R(\theta_{ji})\)} & \multirow{4}{*}{Foot-ground contact signal} \\  & stance-swing transition & & \\ \cline{1-1}  & Sensitivity: higher \(\beta\) accelerates phase & & \\ \cline{1-1}  & transitions, enabling faster gait & & \\ \cline{1-1}  & adaptation but potentially causing & & \\ \cline{1-1}  & abrupt motions; lower \(\beta\) ensures smooth & & \\ \cline{1-1}  & transitions but may delay & & \\ \cline{1-1}  & obstacle response. & & \\ \hline \end{tabular}
\end{table} 
Table 1: Symbol definitions for locomotion control equations. position tracking errors. \(x_{i}\) denotes the actual position of the i-th joint, while \(x_{i}^{*}\) indicates the desired target position. \(\beta_{ij}\) as inter-joint coupling weights and \(R\left(\theta_{ji}\right)\) is compared as the motion mapping output with the reference value \(R\left(\theta_{ji}\right)^{*}\).