## 5 Numerical Evaluations
Figure 3: Tracking error signal comparison between the proposed adaptive controller (red) scheme and the regular PID controller (green) for the hip (**a**), knee (**b**) and ankle (**c**). Figures 7 and 8 show the comparison of the torque enforced by both controllers, adaptive and PID forms. Notice that these reference torques are different due to the backstepping formulation using either the PID or the adaptive control forms, which limits the ability to obtain a unified comparison of all three trajectories. The numerical outcomes correspond to the tracking of the reference PID and adaptive torques for hip, knee, and ankle articulations of the biped robot. In all these comparisons, the adaptive controller performs better in the tracking of the reference trajectories. The reference tracking for the requested torques attains a similar time (0.12 s), which is smaller than the needed time for the position tracking. The equivalent quality tracking for torques in the PID case is not as good as in the case of the adaptive control form. Considering that the adaptive control structure forces a better tracking of the reference torque, the better tracking of the reference angular motion is even more justified. PID cannot provide an efficient tracking of the reference torques, highlighting the benefits of the suggested controller. This fact is confirmed for all three articulations in the biped robot structure corresponding to the calculated torques. The tracking errors for both torques produced by adaptive and PID converge to a particular invariant set that is part of this study's main result for such variables. The individual comparisons of torques confirm that the adaptive controller tracks each reference torque with smaller oscillations around  
Figure 5: Tracking control signal comparing the proposed adaptive controller (red) scheme with the regular PID controller (green) for the hip (**a**), knee (**b**) and ankle (**c**). Figure 6: Control signals norm  
the reference torques. Notice also that tracking both torques at each articulation seems to correspond to the tracking of the reference position for each joint. This fact confirms the interplay between the linear actuator performance and the articulation motion in the biped robot joint. Figure 9 depicts the comparison of the torque's deviation from their corresponding reference torques. This comparison shows a similar control tracking performance for both  
Figure 8: Comparison of the control evolution for the tracking problem based on the application of the PID controller (green) for the hip (**a**), knee (**b**) and ankle (**c**). Figure 7: Comparison of the control evolution for the tracking problem based on the application of the adaptive controller (red) for the hip (**a**), knee (**b**) and ankle (**c**). controllers (PID and adaptive forms) with similar quality. Nevertheless, the comparison seems to offer a similar quality to the torque tracking, but the improved position tracking exhibited by the adaptive form justifies its application for enforcing a more efficient gait cycle for biped robots. Figure 10 illustrates the temporal evolution of the Euclidean norm for the torque tracking error enforced with the pair of studied controllers (PID and adaptive structures). These norms are shown simultaneously in the same figure. Such comparison may highlight that the proposed controller drives the torque tracking error toward the origin by implementing both adaptive and PID forms. Nonetheless, both norms shown in this figure also prove that the adaptive form induces a good tracking of torque as the adaptive form does, but the adaptive formulation yields a better gait cycle developed by the suggested biped robotic device. Figure 11 renders the square of voltage needed to drive the tracking of torque as well as the position. The temporal evolution of the voltage proves that the adaptive form requires  
Figure 10: Norm of the tracking errors with the application of the regular PID and the adaptive controllers.