## 2 Modeling of Wheel-Legged Robot Systems
### Coordinated CPG and Dynamic Balance Control for Legged Robots  
Reference joint trajectories \((\theta_{\text{ref}},\dot{\theta}_{\text{ref}},\ddot{\theta}_{\text{ref}})\) are generated by the CPG to establish gait rhythms, and the CPG's oscillator amplitude and frequency support real-time modulation. Because the CPG outputs contain purely kinematic information, the dynamic layer maps  
\begin{table}
\begin{tabular}{l l l} \hline
**Symbol** & **Meaning** & **Test Value** \\ \hline \(\theta\) & Joint angle & - \\ \(A\) & CPG oscillation amplitude & \(12^{\circ}\) \\ \(T\) & CPG oscillation period & 2 s \\ \(\mathbf{\varphi}\) & Phase matrix of hip joint relative to oscillator & Diagonal gait of [0, \(\pi\), 0, \(-\pi\)] \\ \(\varphi_{hk}\) & Hip-knee joint phase difference & \(\pi/2\) \\ \(F_{k}\) & Knee joint waveform mode & 1: Half-wave mode \\ \(\theta_{0}\) & Initial position & \(30^{\circ}\) \\ \(n\) & Number of legs & 4 \\ \(A_{s}\) & Oscillation amplitude of gap terrain & \(40^{\circ}\) \\  & stemp-reflection & \\ \(T_{s}\) & Oscillation period of gap terrain & \\ \(T_{s}\) & stop-reflection & 0.53 s \\ \(t_{s}\) & Equivalent time within the rhythmic & - \\  & movement cycle & \\ \(F_{s}\) & Stomp-reflection response switch & 0.1 \\ \(\varphi_{s}\) & Equivalent delay phase of ideal stomp time & 0.36\(\pi\) \\ \hline \end{tabular}
\end{table}
Table 3: Definition and value of stomp-reflection parameters.  
them to feasible forces based on physical constraints (13) and calculates the ground reaction forces \(f_{i}\) through whole-body dynamics equations. In (13), \(\mathbf{\tilde{v}}\) denotes the center of mass acceleration, and \(\dot{\omega}\) denotes the angular acceleration. \(I_{s}\) denotes the moment of inertia of the robot's torso about its center of mass, and \(\mathbf{P}_{g,i}\) is the moment arm from the point of force application to the center of mass.  
\[\begin{array}{c}m\mathbf{\tilde{v}}=mg+\sum\limits_{i=1}^{n}f_{i}\\ l_{s}\dot{\omega}=\sum\limits_{i=1}^{n}P_{g,i}\times f_{i}\end{array} \tag{13}\]  
While the CPG provides idealized kinematic references, the zero moment point (ZMP) (14) introduces physical feasibility constraints--bridging the gap between gait planning and real-world stability [23; 24]. In (14), \(z_{i}\) is the z-axis coordinate of the i-th foot-end contact point; \(f_{i}^{y}\) represents the y-axis component of the ground reaction force on the i-th foot end; \(x_{i}\) denotes the x-axis coordinate of the i-th foot-end contact point; and \(f_{i}^{z}\) is the z-axis component of the ground reaction force on the i-th foot end.  
\[P_{z}=\frac{\sum\limits_{i=1}^{n}\left(z_{i}f_{i}^{y}-x_{i}f_{i}^{z}\right)}{ \sum\limits_{i=1}^{n}f_{i}^{z}} \tag{14}\]  
Combine the optimized foot-end forces \(f_{i}\) with the CPG trajectories and substitute them into the inverse dynamics model (15), where \(M(\theta)\) is the inertia matrix, \(C(\theta,\dot{\theta})\) is the Coriolis/centrifugal force matrix, the gravity vector, and \(J^{T}(\theta)\) is the Jacobian transpose. Directly output the joint driving torque \(\tau\) to complete the closed-loop mapping from the robot's whole-body balance requirements to the joint execution commands.  
\[\tau\approx M(\theta)\ddot{\theta}_{\text{ref}}+C(\theta,\dot{\theta})\dot{ \theta}_{\text{ref}}+G(\theta)+J^{T}(\theta)f \tag{15}\]  
MATLAB was combined with CoppeliaSim to verify the aforementioned robot gait control scheme involving CPG, ZMP, and inverse dynamics. The simulation results were visualized in MATLAB, as shown in Figure 7, and demonstrate that the scheme satisfies both stability and dynamic constraints: the ZMP trajectory strictly lies within the support polygon (meeting core gait stability needs), and the CoM trajectory aligns coordinately with the ZMP trajectory, thereby verifying the ability of the scheme to be implemented.  
Figure 7: Gait stability verification with ZMP, CoM, and support polygon.