## 4 Simulation
### Tracking Control Performances and Cost  
In this subsection, the control performances of the ADPFTNs in (19) are further analyzed from the perspective of control cost quantitatively and compared with the backstepping control scheme in [11] (denoted as Backstepping) and the learning-based robust control scheme in [50] (denoted as LBRC). For a fair comparison, the same cost functions are defined as \(\mathbf{V}_{p}=\int_{0}^{\infty}(\mathbf{e}_{1}^{\intercal}\mathbf{Q}_{1}\mathbf{e}_{1}+\mathbf{u}^ {\intercal}\mathbf{R}_{1}\mathbf{u})\) and \(\mathbf{V}_{a}=\int_{0}^{\infty}(\mathbf{e}_{2}^{\intercal}\mathbf{Q}_{2}\mathbf{e}_{2}+\mathbf{ \tau}^{\intercal}\mathbf{R}_{2}\mathbf{\tau})\), in which \(\mathbf{V}_{p}\) and \(\mathbf{V}_{a}\) represent the system of position and attitude control cost, respectively. For a convenient comparison of the position subsystem control cost, the gravity necessary for the three schemes was ignored in the calculation of the control cost \(\mathbf{V}_{p}\).  
Figure 8 shows the time response of the tracking errors for the ADPFTNs, Backstepping, and LBRC, respectively. As shown in Figure 8, the stabilization accuracy of the ADPFTN was \(10^{-5}\) orders of magnitude higher than the \(10^{-4}\) orders of magnitude of Backstepping and LBRC in the same convergence time. In the convergence step, Backstepping and LBRC had steeper convergence trends and faster transient responses, while the ADPFTNs has a smooth convergence tendency. It is worth noting that the steep convergence trends and fast transient responses led to higher control cost. Figure 9 illustrates the control cost of the three control schemes. It can be observed that the ADPFTNs were capable of reducing the control cost while guaranteeing better control performances. In 0-10 s, the tracking errors lied in the convergence process, and it could be obtained that the ADPFTNs had lower control cost than Backstepping and LBRC, which demonstrates the superior performance of the ADPFTNs in reducing the control cost. After 10 s, the tracking error converged to a neighborhood of the origin, and there was no major difference in the control cost of the three control schemes. Compared to Backstepping, the total control cost of the position subsystem was reduced by 5.5% and the total control cost of the attitude subsystem was reduced by 38.9%. In addition, the control inputs of the position system were larger than those of the attitude system, which accordingly led to a larger control cost of the position system. The 3D trajectory tracking results are illustrated in Figure 5. It can be observed that all three control schemes can successfully track the desired trajectory. Further, it can be seen that the red solid line had a more-stable tracking result in comparison with the blue dotted line (Backstepping in [11]) and the black dashed line (LBRC in [50]). Figure 10 shows the time response of control inputs in the frame \(\xi_{E}\). As shown in Figure 9, the initial control inputs of the ADPFTNs were the smallest among the three schemes. Smaller control inputs had a smaller control cost, which corresponds to the results in Figure 9. This further illustrates the superiority of ADPFTNs in balancing control cost.  
Figure 8: Time response of tracking errors under different schemes.  
Figure 11: Time response of tracking errors of ADPFTNs in Simscape.  
Figure 10: Time responses of control inputs under different schemes.  
Figure 9: The position control cost and attitude control cost under different schemes.