## 2 The Problem Statement  
The aim of the work is to develop a theory of using groups of underwater and wave gliders in relation to the applied problems of monitoring and patrolling, including patrolling areas with underwater potential hazardous objects (UPHOs) [28; 33; 34]. The combination of two heterogeneous vehicles makes it possible to receive and transmit data from the underwater glider in near real time modes, as well as to quickly correct the position of the underwater glider(s) and change the mission parameters. Our research is based on the consideration of the following glider models: the SHADOW underwater glider and a wave glider developed by SMTU and Oceanos JSC, as well as the possibility of their complex application [35; 36; 37]. Underwater and wave glider models are shown in the Figure 1.  
To achieve the primary objective of the study, namely the interaction of actual vehicles in a group, it is necessary to solve the following problems (subject to the availability of operating low-power broadcast stations, communication equipment and MRC navigation equipment with a certain technical specification):  
* Ensuring a sufficient distance between objects, which allows maintaining an operational communication channel;
* Setting up the intra-group logic for vehicles' action planning in case of unforeseen situations;
* Maintaining the group while the group is moving. A key factor for the successful completion of the mission is the availability of high-quality communication between the vehicles (both hydroacoustic and radio/satellite communication). As an example of hydroacoustic communication channel implementation, we consider the parameters of a widely used acoustic modem from Evologic (30/60 underwater acoustic modem) and USBL transceivers. The operating range of such a system is approximately 1 km. Generally, the acoustic modem is installed in the underwater part of the wave glider. In this case, an important and necessary condition for fulfilling the mission is not only a high degree of complexity of the control system (both a separate device and a group of devices), but also the design features of the vehicles associated with certain glider design schemes. A case, when a group of gliders is assigned the task to monitor environment state changes in the area of a given position (similar tasks are the UPHO monitoring tasks) is considered. Let there be \(N\) number of robots in some space of global coordinates \(M\) with the coordinates (1):  
Figure 1: SHADOW Underwater Glider and Wave Glider from SMTU and Oceanos JSC. (**a**) Underwater Glider SHADOW; (**b**) Wave glider by SMTU. \[P_{group}(i)=[x(i)_{group},y(i)_{group}] \tag{1}\]  
The set of points \(P_{group}(i)\) composes an \(H\) formation; its center is defined as the center of mass of the figure defined by the points \(x(i)_{group},y(i)_{group}\). The task for the group is to move to the given target points \(T_{p}=[T_{p1},T_{p2},T_{p3},T_{p4}]\), retaining the \(H\) formation. We consider the formation as the spatial structure of the vehicles in a group. In order to coordinate an interaction, the elements of the group should have an information communication channel, but its range has a physical limitation. In this case, it is necessary to create such a type of formation that prevents the vehicles from moving away from each other by more than the radius \(R\), which is the maximum allowable range of the optimal communication system channel. The formalized statement of the problem is shown in the Figure 2:  
In Figure 2, \(UG_{1}\), \(UG_{2}\), \(WG\) are positions of two underwater and one wave glider, respectively, describing the function of the formation structure \(H_{group}(UG_{1},UG_{2},WG)\). \(D_{1}\), \(D_{2}\), \(D_{3}\) are distances between \(UG_{1}\), \(UG_{2}\), \(WG\) accordingly. \(\Phi\) is an area in which it is required to maintain the formation of the group. This area is based on the intersection of three circles \(\Phi=R_{UG_{1}}\cap R_{UG_{2}}\cap R_{WG}\).