## 6 Numerical Results
### 3D Helical Trajectory Tracking  
To verify the performance of the DNCS developed, a more complex scenario is proposed, where a 3D helical reference signal with a radius of 5 m is used. In this case, the desired signals for the X, Y, and Z axes are defined as  
\[ \begin{split}\zeta_{1,r,x}&=5\cos(0.1\pi t),\\ \zeta_{2,x}&=-0.5\pi\sin(0.1\pi t),\end{split}  \tag{57}\]  
Figure 2: Simulation results for trajectory tracking in Z axis. Figure 10: Simulation results for trajectory tracking in yaw dynamic. \[\zeta_{1,y} =5\sin(0.1\pi t) \tag{58}\] \[\zeta_{2,y} =0.5\pi\cos(0.1\pi t),\] \[\zeta_{1,z} =\frac{1}{8}t+1\] (59) \[\zeta_{2,z} =\frac{1}{8}.\]  
In the same way as in Section 6.1, disturbance signals were added to the dynamic model (8) for the 3D helical desired trajectory, which are given as  
\[\tau_{x_{d}}= 15\sin(1.2t)\cos(1.5t), \tag{60}\] \[\tau_{y_{d}}= 10\cos(2t)\sin(0.5t),\] \[\tau_{z_{d}}= 10\cos(t)\sin(3t).\]  
The simulation results for the 3D helical trajectory tracking scenario are shown in Figures 5-8. Figure 5 depicts three-dimensional trajectory tracking with respect to the reference trajectory, which shows the accomplishment of the tracking mission. In this simulation experiment, the yaw angle desired is selected as 0 radians. The initial position for the AUV is selected as \((x(0),y(0),z(0))=(3,-2,2)\). The 3D trajectory begins at a depth of 2 m, reaching up to 6 meters, with the center at the origin and a radius of 5 meters. This kind of trajectory can be useful in 3D reconstruction applications of underwater structures. We can observe the trajectory tracking results for both controllers: DNCS and the feedback controller. Notice that the feedback controller is not able to counteract the effects caused by the lumped disturbances, whereas the DNCS enables the AUV to track the desired 3D trajectory with a small tracking error. Figures 6a, 7a, and 8a illustrate the time evolution of the position of the underwater vehicle in the X, Y, and Z axes. In the three axes are presented the performance of the DNCS; the feedback controller; and the PD controller, which implements a self-tuning process. When the dynamic neural network is added to the feedback controller, performance is significantly increased, reducing the tracking error and enabling the AUV to track the desired reference without abrupt changes. In the case of the PD controller that implements an online parameter adjustment, the deviation with respect to the desired trajectory is reduced; however, this type of controller fails to eliminate oscillations of the system trajectory. Figure 5: Underwater vehicle response in a _3D helical desired trajectory_ scenario by using the DNCS, Feedback and PD Adaptive controllers. Figure 10:0 Simulation results for trajectory tracking of a 3D helical reference signal in Z axis. Figure 10:1 Simulation results for trajectory tracking of a 3D helical reference signal in the X axis. Figure 10:2 Simulation results for trajectory tracking of a 3D helical reference signal in Y axis. The time evolution of the adaptive parameters \(\hat{b}_{x}(t)\), \(\hat{b}_{y}(t)\), and \(\hat{b}_{z}(t)\) is presented in Figures 6b, 7b, and 8b, respectively. These parameters converge from the initial value to a small bounded region in approximately 15 s in all the cases. The importance of the estimation of the adaptive parameters should be recalled because it enables the UAV to track a desired references without the knowledge of any parameter of the dynamic model Equations (17)-(20). In Figures 6b, 7b, and 8b, the estimation of the unknown functions \(f_{x}\), \(f_{y}\) and \(f_{z}\), the so-called lumped disturbances, is presented. After the first 10 s of simulation, one can appreciate that the identifier based on the DNCS estimates in a satisfactory way the unknown dynamics and the external disturbances. Moreover, the identification error converges to the compact residual set around zero given by Equation (48). Additionally, a performance index analysis is conducted to compare both DNCS and feedback controllers.