## 5 Numerical Evaluations  
Figure 2 depicts the numerical results related to the tracking result for the reference trajectory for the left-hip articulation of the BRD using the selected PID and the adaptive controllers. These results confirm that both controllers show similar tracking outcomes. A high-quality tracking of the reference trajectories is attained after 0.2 s for both controllers. The equivalence in the tracking is no longer valid, considering that the proposed adaptive controller tracks the reference without exhibiting fast oscillations as the regular PID enforces. PID cannot provide an efficient, robust tracking of the references, highlighting the suggested controller's benefits. The three articulations validate this fact on the left side of the biped robot. Consequently, the tracking error converges to a characterized invariant set, which characterizes this study's main result. Figure 2a-c detail the evolution of angular  
displacements for the hip, knee and ankle, respectively, including the comparison of PID and the proposed adaptive controller. These individual comparisons confirm that the adaptive controller tracks each reference with smaller oscillations around the reference trajectories. Figure 3 depicts the evolution of the tracking deviations generated with the adaptive and the regular PID controllers. The comparison validates that the selected PID controller produces oscillations with larger amplitudes around the zero value when the time increases. These oscillations confirm that despite the fact that the PID controller may work robustly concerning external perturbations and uncertain modeling sections, there is no efficient tracking of the reference (measured in terms of the tracking errors). At the same time, implementing the adaptive gains improves the tracking quality. Even when the differences between the tracking errors produced by the PID and the adaptive controller are insignificant, their values are relevant in the BDR performing the gait cycle process. Figure 4 shows the temporal evolution of the Euclidean norm for the tracking deviation produced with the implementation of both controllers, the PID, and the proposed adaptive form. These norms are presented simultaneously here to evaluate the adaptive control quality. The comparison highlights that the suggested controller drives the tracking error toward the origin, implementing both adaptive and PID forms. The more significant norm produced by the PID controller also proves that the adaptive form induces better tracking of the reference trajectory, which yields a better gait cycle developed by the suggested biped robotic device. This figure also shows that the adaptive control form produces a minor mean square error, significantly improving traditional control forms such as the PID. Figure 2: Position tracking for each joint, comparing the proposed adaptive controller (red) scheme with a regular PID controller (green) for the hip (**a**), knee (**b**) and ankle (**c**). Figures 5 and 6 compare the torque evolution measured at all three joints in a particular leg of the biped robot. This comparison exhibits similar temporal evolutions for both controllers (for both PID and adaptive structures) for the three articulations. The individual torque obtained for each articulation is larger if the PID controller is compared with the adaptive form. Notice that such comparison confirms that the adaptive form enhances the tracking in both senses, the better tracking quality and the corresponding smaller energy (Figure 6). Figure 6 also highlights the high energy consumed by the PID controller, while the adaptive controller is significantly smaller in comparison. The temporal evolution of the control norm motivates the inclusion of the adaptive controller for mobile robots, such as the biped structure considered in this study, with the linear actuators as driving elements. Figure 4: Tracking error signals norm.