## 3 Problem Specification  
The AUV has a single-integrator dynamics model that is expressed as follows:  
\[\dot{x}_{i}=u_{i},i=1,\ldots\ldots,n\] (3a) or in the form of matrix \[\dot{\mathbf{X}}=\mathbf{U}\] (3b) where \[\mathbf{X}=\left[\mathbf{x}_{1}^{T},\ldots,\mathbf{x}_{\mathbf{n}}^{T}\right]^{T},\mathbf{U}=\left[ \mathbf{u}_{1}^{T},\ldots,\mathbf{u}_{n}^{T}\right]^{T},\]  
in which \(\mathbf{x}_{i}(\mathbf{t})\in\mathbb{R}^{m}\) is the state of the \(i\)th AUV, \(\mathbf{u}_{i}(\mathbf{t})\in\mathbb{R}^{m}\) are the control inputs of the \(i\)th AUV, \(\mathbf{X}\in\mathbb{R}^{nm}\) are the aggregate states of all AUVs, and \(\mathbf{U}\in\mathbb{R}^{mm}\) are the control inputs of all AUVs.  
Figure 1 depicts multiple AUVs that are distributed in the sea, the biggest AUV can provide power and all the AUVs can exchange data. In this figure, different colors represent the changes of water depth. MAUV systems can generally reside on the seabed for a long time. When marine geological disasters occur, an AUV will sense the occurrence of the disasters and automatically identify possible disaster sites to evaluate the overall environment. When the system senses an unusual change in the environment, the AUVs will cooperate, and in this phase, they sense the nearby AUVs through the sensors that they carry and determine the location of the collection through negotiation.  
In this study, the consensus problem involves the design of a distributed control law \(\mathbf{u}_{i}(\mathbf{t})\) that depends on the information exchange topology such that the states of all AUVs converge to the same value, i.e., \(\left\|\mathbf{x}_{i}(\mathbf{t})-\mathbf{x}_{j}(\mathbf{t})\right\|\to\mathbf{0}\). Furthermore, it is guaranteed that obstacles along the AUV trajectory can be avoided.  
Figure 2 depicts an example scene of the consensus problem with five AUVs and one obstacle. Three zones are established: the collision, the diagnostic, and the reaction zones, which are defined as follows.  
Figure 1: MAUV multi-water area search scenario.  
Collision zone for the _j_th obstacle: \(\Lambda_{j}\triangleq\left\{\mathbf{x}|\mathbf{x}\in\mathbb{R}^{m},\left\|\mathbf{x}-\mathbf{O}_{ \mathbf{bj}}\right\|\leq\mathbf{r}_{\mathbf{j}}\right\}\). The AUV absolutely cannot enter the interval and each obstacle is solid.  
Diagnostic zone for the _j_th obstacle: \(\Psi_{j}\triangleq\left\{\mathbf{x}|\mathbf{x}\in\mathbb{R}^{m},\left\|\mathbf{x}-\mathbf{O}_{ \mathbf{bj}}\right\|\leq\mathbf{R}_{\mathbf{j}}\right\}\). This is the range within which the AUV can detect only one obstacle. Outside this area is the AUV safe area, within which the AUV navigates according to coordinated commands.  
Reaction zone for the _j_th obstacle: \(\Gamma_{j}\triangleq\left\{\mathbf{x}|\mathbf{x}\in\mathbb{R}^{m},\mathbf{r}_{j}<\left\| \mathbf{x}-\mathbf{O}_{\mathbf{bj}}\right\|\leq\mathbf{R}_{\mathbf{j}}\right\}\). In this area, the AUV can sense and avoid obstacles.  
Where \(\mathbf{r}_{\mathbf{j}}\) means the radius of the obstacle, \(\mathbf{R}_{\mathbf{j}}\) means the range that the AUV can detect the obstacle.  
Accordingly, the entire safety area can be represented as \(\Theta=\left(\cup_{j}\Lambda_{j}\right)^{\mathbf{c}}\) and the entire outside diagnostic zone can be represented as \(\Pi=\left(\cup_{j}\Psi_{j}\right)^{\mathbf{c}}\). The symbol \(\cup\) and superscript \(\mathbf{c}\) indicate the union and complement of sets, respectively.  
The following three assumptions are included in this study:  
1. All obstacles can be modeled as spheroidal objects.
2. \(\Psi_{j}\cap\Psi_{\mathbf{k}}=\varnothing,\mathbf{j}\
eq\mathbf{k}\).
3. It is assumed that the topology of information exchange between AUVs is unconnected.  
According to A2, the diagnostic areas of multiple obstacles are completely independent. This assumption precludes the inability of the AUV to determine which obstacle to avoid after entering the crossover area. Thus, each AUV will encounter only one obstacle at a given time.