## 4 Experiment
### Verification of the Lower-Level Controller for Velocity Tracking  
In this section, velocity tracking experiments were conducted to validate the proposed lower-level controller based on current-velocity mappings. Additionally, we conducted  
Figure 5: Block diagram of the lower-level controller.  
Figure 6: Experiment platform.  
During the experiments, we set different target trajectories, including a simple L-shaped trajectory and a more complex circle-shaped trajectory, in order to evaluate the tracking performance of the proposed control strategy.  
Figure 9 depicts the results of the mobile robot tracking the L-shaped trajectory. The initial position and heading angle of the robot are [0, 0, 0]. From Figure 9a, it can be observed that the controller proposed in this study demonstrates better tracking performance compared to the comparative controller (LMPC + PID). The position tracking errors are shown in Figure 9b. Figure 9c,d illustrate the angular velocities of the left and right hydraulic motors during the motion of the robot. It can be observed that when adopting our control method, the velocity fluctuations are smaller, thereby resulting in better motion stability for the mobile robot.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameters** & **Proposed Controller** & **LMPC + PID** \\ \hline Upper-level & NMPC & LMPC \\ Sample time \(T\) & 0.1 & 0.1 \\ Prediction horizon \(N_{p}\) & 10 & 10 \\ Control horizon \(N_{m}\) & 10 & 10 \\ Weight \(Q\) & \(diag([1,1,0.1])\) & \(diag([1,1,0.1])\) \\ Weight \(W\) & \(diag([0.5,0.5])\) & \(diag([0.5,0.5])\) \\ Linear velocity constraints \([v_{\min},v_{\max}]\) (m/s) & \([-1.8,\,1.8]\) & \([-1.8,\,1.8]\) \\ Angular velocity constraints \([\omega_{\min},\omega_{\max}]\) & \([-\pi/4,\pi/4]\) & \([-\pi/4,\pi/4]\) \\ (rad/s) & \(\text{Mappings}\) & \(\text{PID}\) \\ \(\text{Mappings}\) & Equation (7) & / \\ Proportional coefficient \(k_{i,p}\) & 4 & 10 \\ Integral coefficient \(k_{i,j}\) & 0.4 & 0.5 \\ Derivative coefficient \(k_{i,d}\) & 0.5 & 1 \\ Dead zone of valve \(Dz_{value,i}\) (A) & \([-0.2,\,0.2]\) & \([-0.2,\,0.2]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Controller settings of the trajectory tracking experiment.  
Figure 9: Tracking result of the L-shaped trajectory: (**a**) trajectory; (**b**) errors; (**c**) left-side velocity; (**d**) right-side velocity.  
Figure 10 presents the results of the mobile robot tracking the circle-shaped trajectory. The initial position and heading angle of the robot are [100, 0, 1.57]. As expected, we obtained results which are similar to the tracking of the L-shaped trajectory, even with the more complex trajectory. The position tracking errors are shown in Figure 10b. Similarly, Figure 10c,d demonstrate that our controller leads to smaller velocity fluctuations, thereby providing the mobile robot with better motion stability. The experimental results indicate that the hierarchical control strategy proposed in this study exhibits superior trajectory tracking control performance.  
The tracking errors of different controllers are shown in Table 3. Due to the initial position of the robot being quite distant from the starting point of the L-shaped trajectory, error calculation was commenced after 20 s. The experimental results show that our proposed hierarchical control scheme exhibits better trajectory tracking control performance, whether it is following an L-shaped trajectory or a circular trajectory.