## 5 Simulation Results and Discussions  
Simulations are run to demonstrate the effectiveness of the proposed tracking control algorithms for AUVs. The proposed integrated localization and control algorithm is applied in following a reference path. Such applications can be seen in sea mapping scenarios where only waypoints in the concerned ocean region are given. The AUV is positioned at the initial point \([100,100,300]\) m and is controlled to reach the destination \([600,600,300]\) m along the following curved reference path  
\[\begin{cases}x_{\text{d}}(t)=600-500\cos(\pi t/1200),\\ y_{\text{d}}(t)=100+500\sin(\pi t/1200),\end{cases}\]  
for \(t\in[0,600]\) s. The three surface buoys are positioned at \([0,0,0]\) m, \([500,800,0]\) m, and \([1000,0,0]\) m, respectively. The isogradient SSP parameters in (7) are set as:\(a=-0.05\) and \(b=1540\) m/s. In this region, ocean currents exist. The average speed is predicted to be 0.5 m/s and the angle is predicted to be \(\pi/6\). However, there might be prediction biased errors. In the simulations, we will first validate the performance of the proposed integrated localization and tracking control algorithm with zero prediction biased errors, i.e., \(\varepsilon_{\rm{p}}=0\) and \(\varepsilon_{\beta}=0\). Then, to illustrate the inherent robustness (to a certain degree) of the feedback due to acoustic localization, we also consider non-zero prediction biased errors with \(\varepsilon_{\rm{p}}=0.15\) m/s and \(\varepsilon_{\beta}=\pi/20\) rad. Note that the biased errors are not considered explicitly in the algorithm design, and are only implemented for simulation purposes to show the inherent robustness due to the feedback from the acoustic localization. Moreover, there exist uncertainties with the predicted values following the normal distribution with covariance \(\Sigma_{b}=diag([0.1\,{\rm m}^{2}/{\rm s}^{2}\,0.01\ {\rm rad}^{2}])\), which means low prediction accuracy in the ocean current. It should be noted that AUVs with any design may fail at those worst predictions of the sea. The uncertain acoustic measurement is with covariance \(\Sigma=diag([0.1\ {\rm m}^{2}\ 0.1\ {\rm m}^{2}\ 0.01\ {\rm rad}^{2}])\). The AUV depth is 300 m, and the buoy depths are all 0 m. Controller parameters are set as follows: prediction horizon \(N_{\rm{p}}=10\); weight parameter \(\mathbf{Q}_{f}=\mathbf{Q}=diag([100\ 100])\) and \(\mathbf{R}=diag([1\ 180/\pi])\); sampling time \(T_{\rm{s}}^{\rm{s}}=0.5\) s and \(T_{\rm{s}}^{\rm{t}}=1\) s, which means the system feedback states are corrected by the acoustic localization results every two control time steps, i.e., \(n=2\) in Algorithm 1. Physical system constraints are: \(u_{\rm{max}}=-u_{\rm{min}}=2\) m/s, \(\xi_{\rm{max}}=-\xi_{\rm{min}}=86\) N, and \(\delta_{\rm{max}}=-\delta_{\rm{min}}=13.6\pi/180\) rad. The REMUS AUV system parameter values are referred to [25]. Note that the practical parameter values might be different from the above set values. For adverse environmental situations, the algorithm might not be applicable. All the algorithms are implemented in MATLAB 2016b [29] with solver Cplex [32] on a platform with Intel(R) Core(TM) i3-7100 CPU @ 3.70 GHz.