## 4 Experiments with a Group of Low-Cost UUVs  
### Experimental Setup  
The direct graph of three UUVs is shown in Figure 4. The distributed adaptive neural network control and the collision and obstacle avoidance have been tested and implemented in UUV-Simulator [23], which is based on ROS/Gazebo. The objective of the proposed experiments is to coordinate the low-cost underwater drones by simultaneously controlling some of their local information, and introducing a complete operating scenario in hazardous environments. We consider a leader-follower scheme composed of three identical underwater drones. The model that was used for the simulation is a BlueROV-1, actuated in surge, sway, heave, and yaw via a 6 thruster set configuration (the motion in roll and pitch has been neglected as both DoFs are passive and minimally affect the others). According to [30], the main inertial and hydrodynamic parameters of BlueROV-1 are given by  
\[M=M_{RB}+M_{A}=\left[\begin{array}{ccc}12.81&0&0\\ 0&20.01&0\\ 0&0&0.12\end{array}\right],\hskip 14.226378 ptD=\left[\begin{array}{ccc}-28.27&0&0\\ 0&-134.74&0\\ 0&0&-0.07\end{array}\right] \tag{31}\]  
then,  
\[B=\left[\begin{array}{c}0_{3\times 3}\\ M_{i}^{-1}\end{array}\right]=\left(\begin{array}{ccc}0&0&0\\ 0&0&0\\ 0&0&0\\ 0.0781&0&0\\ 0&0.0050&0\\ 0&0&8.3333\end{array}\right) \tag{32}\]and the mass of BlueROV-1 is \(m=7.31\) kg, the length is 483 mm, the width is 330 mm, and the height is 267 mm. The unmodeled dynamics \(f_{i}\) and disturbances \(w_{i}\) are considered randomly as  
\[f_{i}\left(x_{i}\right)=\left[\begin{array}{c}-0.01u_{i}v_{i}^{2}\\ -0.02u_{i}v_{i}\\ -0.01r_{i}u_{i}\end{array}\right]\text{ and }w_{i}=\left[\begin{array}{c}-0.1 \sin\left(0.5t\right)\cos\left(0.3t\right)\\ -0.1\sin\left(0.2t\right)\cos\left(0.5t\right)\\ -0.1\sin\left(0.1t\right)\cos\left(0.6t\right)\end{array}\right] \tag{33}\]  
The initial states of three BlueROV-1 are given by \(\mathbf{x}_{1}=[6,0,\
icefrac{{\pi}}{{2}},0,0,0]^{\top}\), \(\mathbf{x}_{2}=[-6,1,0,0,0,0]^{\top}\), \(\mathbf{x}_{3}=[0,3,\
icefrac{{\pi}}{{2}},0,0,0]^{\top}\). The desired formation is given by \(\delta_{1}=[5\sqrt{3},0,0,0,0]^{\top}\), \(\delta_{2}=[0,5,0,0,0,0]^{\top}\), \(\delta_{3}=[0,-5,0,0,0,0]^{\top}\). The distance of a narrow area is chosen as \(h=8\) m, \(\Delta=2\) m.  
The number of neural network neurons for each UUV has adjusted to 10 neurons. The number of input values for neural network is 6. The neural network activation function is chosen as a log-sigmoid of the form \(\phi(\mathbf{x}_{i})=1/(1+e^{-\kappa t})\), where \(\kappa\) is a positive constant (\(\kappa=2\) in this work).  
From Equations (10), (15), and (32), a solution \(P\) of the Riccati equation with \(Q=\text{diag}\{1,1,1,1,1\}\) is  
\[P=\begin{pmatrix}5.1595&0&0&12.8100&0&0\\ 0&6.4047&0&0&20.0100&0\\ 0&0&1.1136&0&0&0.1200\\ 12.8100&0&0&66.0926&0&0\\ 0&20.0100&0&128.1578&0\\ 0&0&0.1200&0&0&0.1336\end{pmatrix} \tag{34}\]  
where \(P\) is a positive-definite matrix, then,  
\[K=-B^{\top}P=\begin{pmatrix}-1.0000&0&-5.1595&0&0\\ 0&-1.0000&0&-6.4047&0\\ 0&0&-1.0000&0&-1.1136\end{pmatrix} \tag{35}\]  
The leader UUV assumes that it has been controlled to follow the trajectory with the way-points \((0,6)\), \((0,12)\), \((30,12)\), \((30,6)\). The position of the obstacle is \((15,9)\) and \(D_{out}=7\) m, \(d_{in}=4\) m.