## 6 Numerical Results
### 3D Helical Trajectory Tracking
The comparison were performed by the following two quality indexes for the tracking error:  
Figure 10:3  
Figure 10:4  
and the following two indexes for the identification error:  
Figure 10:5  
where \(\tilde{\xi}_{1}\) denotes the mean value of the set of identification error values. Figure 10:6  
The performance index proposed in Equation (61) is given by the Integral of Absolute Error (IAE) to measure the absolute tracking deviation. The second index in Equation (62) is defined by the Integral of Squared Error (ISE) to quantify the absolute squared tracking deviation. The index given in Equation (63) is the standard deviation used to measure the dispersion of the identification error produced by the DNCS scheme. Additionally, the performance index (64) is defined as the root mean square error (RMSE) of the identification error. This is used to measure the amount of error between the set of values of the system and the identifier states. The proposed comparison analysis reports that applying the DNCS controller and the feedback controller solves the trajectory tracking. Tables 3 and 4 show the calculated values for \(J_{IAE}\) and \(J_{ISE}\) (considering \(T=40\) s), with the tracking error data involving the two proposed scenarios. This quantitative comparison validates the argument concerning the advantage of using the proposed DNCS-based controller. The numerical data presented in Tables 3 and 4 demonstrate that the proposed DNCS-based controller yields the smallest tracking error of \(J_{IAE}\) and \(J_{ISE}\); this means that the AUV tracks the reference trajectories with a superior performance. This evaluation intends to highlight the benefit for the application of the suggested controller in contrast with the feedback controller to regulate AUVs when the dynamic model information is incomplete. The information in Tables 5 and 6 shows numerical data of \(J_{STD}\) and \(J_{RMSE}\) for both experiments considering \(T=40\) s. The reported small standard deviation values indicate that the identification error at every step tends to be near the mean value of each set of error values (for each dynamic). In other words, the values of the identification errors have uniform behavior, avoiding abrupt changes or dispersion when they evolve over time. On the other hand, the small numerical values for the RMSE index evidence the fact that each DNCS estimate remains close to its corresponding subsystem trajectory during the simulation time. This validation seems to justify implementation of the proposed DNCScontroller for identifying the closed loop AUV subsystems. **Remark 1**. : _The DNCS algorithm was designed with the aim to provide a low computational complexity solution considering the notion of a lightweight algorithm, which allows an eventual onboard implementation in a small experimental platform. In this sense, the 4 DOF AUV mathematical model is represented by four second-order equations, which are rewritten in their corresponding integrator chain representation. Thus, each second-order system is characterized by two first-order differential equations. The latter aims to avoid matrix operations (nested for loops, which may increment the computational time). Additionally, the right-hand sides of the learning law for the DNN weight matrix \(W(t)\) and the adaptive law for \(b(t)\) were decomposed in simple arithmetic operations. Hence, the closed loop system simulation of the DNCS control algorithm and the complete AUV model is comprised of several simulation steps, but it intends to be computationally efficient since it uses simple calculations. This fact makes it a conservative algorithm without any major impact to system performance._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dynamic** & **RMSE** & **Std Deviation** \\ \hline \(X\) axis & 0.2913 & 0.2846 \\ \(Y\) axis & 0.2558 & 0.2525 \\ \(Z\) axis & 0.1239 & 0.1215 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance index comparison for 3D helical identification.