## 3 Experimental Results  
In previous sections, the following aspects were presented: (a) theoretical ones supporting the algorithms running in the ECU-RBP and the DT-RBP, and (b) the hardware and software resources required to complete the test. The P3-DX is a differential drive robot developed by MobileRobots Inc. It is made of an aluminium body (44 \(\times\) 38 \(\times\) 22 cm) with 16.5 cm diameter drive wheels. The two DC motors use 38.3:1 gear ratio and contain 500-tick encoders. The differential drive platform is non-holonomic and can rotate in place moving both wheels, or it can swing around a stationery wheel in a circle of 32 cm radius. A rear caster is included for balancing the robot. On flat floor, the P3-DX can move at speeds of 1.6 m/s. At slower speeds it can carry payloads up to 23 kg. In addition to motor encoders, the P3 DX base includes eight ultrasonic transducer (range-finding sonar) sensors arranged to provide 180-degree forward coverage [41]. As a case study, the tracking of two different trajectories for the P3-DX is proposed. The closed-loop solution comprising the ECU-RBP and the DT-RBP (i.e., virtual robot), is initially evaluated following the configuration shown in Figure 1 (right side). Subsequently, as indicated in Figure 1 (left side), the same electronic control unit (ECU-RBP) is applied to the physical robotic platform. In both configurations, the ECU-RBP and DT-RBP are connected to a host laptop, enabling testing and result evaluation through the Simulink graphical interface. Nevertheless, each RBP platform can execute its respective functions, control (ECU-RBP) and modeling (DT-RBP), independently in stand-alone mode. The experimental setup corresponding to this configuration is illustrated in Figure 6 and the real work scenario in the Engineering School at University of Alcala is shown in Figure 7.  
In the first experiment, an eight-form trajectory reference is defined, starting from the coordinates (0, 0, 0) to the final target (0, 2, \(\pi\)). Figure 8a shows the results obtained from the DT (red line) and the robot (yellow line) with respect to the reference (blue line). Linear and angular velocities registered from the real (Robot) and virtual (DT) mobiles are depicted in Figure 8b. The highest difference is accumulated at the end of the trajectory. To gain deeper insight into the advantages and limitations of the proposed experiment, the tracking errors shown in Figure 9 are analyzed. These include the position (\(e_{x}\), \(e_{y}\)) and orientation (\(e_{\theta}\)) errors of the robot (red line) and the DT (blue line) along the trajectory. Compared to Figure 8, an increase in pose errors is observed during transitions in reference signals. However, the overall errors remain within acceptable margins, under 5 cm in X and Y, and under 0.05 rads in \(\theta\). Figure 5: Block diagram to validate by simulation the global control solution for nonlinear trajectory tracking of the robotic unit. Figure 8: (**a**). Comparison of reference and followed trajectories, by DT-RBP (red line) and P3-DX (yellow line) (**b**). Linear and angular velocities obtained from the real robot and its DT. Figure 6: Set-up for obtaining experimental results working with the same electronic controller ECU-RBP applied to the virtual process (ECU-DT) or the real one (P3-DX robot). Figure 7: Real work scenario (laboratories corridor) for validating nonlinear trajectory tests with the P3-DX robotic platform remotely controlled by the ECU-RBP. A longer trajectory, including a simulated disturbance at the plant input has also been tested. As shown in Figure 10, the trajectory starts at the door of the laboratory L03 (red dot) with a initial pose \((0,0,\pi/2)\) and ends at the door of the OL5 (green dot) with a pose \((3.80,6.80,\pi/2)\), following the path represented in blue color. The measures of the corridor are also included. The simulated disturbance, shown in Figure 11, is added to the angular velocity plant input to evaluate the behaviour of the control system.