## 5 Simulation Tests  
In order to validate the foregoing technique based on curve blending for DWMRs, numerical simulation tests are conducted in the context of geometric trajectory planning, kinematic trajectory tracking and dynamic torque computation. The geometric, kinematic and inertial parameters of the DWMR at hand are given in Tables 2 and 3. Moreover, the friction coefficient is \(\beta=2\) Ns/rad, the rated torque of the onboard motors being \(\tau_{r}=20\) Nm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(r\) [m] & \(I\) [m] & \(a\) [m] & \(b\) [m] & \(v_{c}\) [m/s] \\ \hline
0.08 & 0.2 & 0.18 & 0.42 & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Geometric and kinematic parameters.  
Figure 8: Zoom-in of Lam\(\acute{\text{e}}\) curves C\({}_{2}\) and C\({}_{3}\).  
The posture errors are detected as \(e_{\theta}=30^{\circ}\) and \(l_{2}\) = 1.6 m. The trajectory from the current posture of the DWMR to the guide path is obtained by means of different curves: one is a circular arc, and the other is an affine transformed Lame curve. The trajectory curvature and its time-derivative are illustrated in Figures 9 and 10, respectively. In Figure 9, the curvature \(\kappa\) of the circular arc has a positive step change from 0 to 0.16 m\({}^{-1}\) at the starting point, another negative step change from 0.16 to 0 m\({}^{-1}\) at the ending point, and a constant value on the trajectory between these two points. Consequently, one positive and negative impulse occur on the curvature rate of change (RoC) \(\dot{\kappa}\) for the circular arc at these two end-points. On the contrary, in Figure 10, the curvature \(\kappa_{5}\) of the Lame curve has a smooth increasing change from 0 m\({}^{-1}\) at the starting point, to 0.218 m\({}^{-1}\) at the middle point, and then has a smooth decreasing change to 0 m\({}^{-1}\) at the ending point. As a result, the curvature RoC \(\kappa_{5}\) of the Lame curve still remains a smooth curve, changing from an initial finite positive value, going across the 0 value, and reaching a terminal finite negative value.  
For a constant velocity, \(v_{c}\) = 0.5 m/s, the actuated joint rates and their time-derivatives on different trajectories are calculated based on Equations (30) and (31), as shown in Figures 11 and 12, respectively.  
Figure 10: The curvature of the Lame curve and its time-derivative.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(m_{1}\) & 2 kg & \(I_{x}\) & 0.0064 kg\(\cdot\)m\({}^{2}\) & \(I_{y}\) & 0.0032 kg\(\cdot\)m\({}^{2}\) \\ \(m_{2}\) & 2 kg & \(I_{z}\) & 0.0032 kg\(\cdot\)m\({}^{2}\) & \(I_{3}\) & 104 kg\(\cdot\)m\({}^{2}\) \\ \(m_{3}\) & 200 kg & \(H_{4}\) & 0.0192 kg\(\cdot\)m\({}^{2}\) & \(H_{5}\) & 8.8581 kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Inertial parameters.  
Figure 9: The curvature of the circular arc and its time-derivative.  
terminal finite negative value. The maximum amplitude of the motor torque is always kept below the rated torque.  
Moreover, the simulation results reflect the stepwise processing result of the multi-loop vision guidance framework. In the first step, a Lame-curve trajectory is generated by kinematic path tracking, with its shape as shown in Figure 7, and pertinent parameters, as shown in Figure 10. In the second step, the actuated joint rates and angular accelerations are calculated based on the inverse-kinematics model, as shown in Figure 12. In the third step, the required wheel torques are computed based on the inverse-dynamics model, as shown in Figure 14.