## 1 Introduction  
The control of marine vehicles is an area where active research is being conducted due to the various theoretical challenges and significant practical applications of these vehicles. Most marine vehicles are underactuated, which means they have more degrees of freedom to control than the number of independent control inputs. Particularly interesting and challenging among the control problems studied for marine vehicles are trajectory tracking and path tracking. The latter issue is particularly relevant for mobile vehicles and robots, e.g., [1; 2; 3]. In [1], the predictive Stanley controller was applied to path tracking. The authors of [2] has presented a robust path tracking algorithm under the vehicle-infrastructure cooperative system (i-VICS) based on Kalman filter. Another article [3] proposed a navigation method modeled on the use of bookmarks as in the memory of living beings to reach a destination using key points. In general, an underwater vehicle is modeled with 6 degrees of freedom (DOF) and control schemes are designed for such a model, as can be found in [4; 5; 6]. However, in some works, the dynamics model is reduced to 5 DOF and only the controller is designed for it. Such an approach is presented, e.g., in [7; 8]. However, this paper only addresses the problem of tracking the desired trajectory for a 3 DOF vehicle model in horizontal motion. In the simplest case, the inertia matrix is assumed to be diagonal. There are various control schemes that have been successfully applied to this class of vehicles. One can point to works that use sliding mode control (SMC), such as [9; 10]. Another group of controllers is based on Lyapunov method [11; 12]. Control strategies based on the backstepping approach are also widely used [13; 14; 15]. This technique is often combined with other methods, such as SMC [16; 17], direct Lyapunov method [18; 19], averaging approach [20], dynamic surface control method [21], nonlinear model predictive control (NMPC) [22], neural network (NN) and SMC [23; 24]. Another group of methods uses NN, such as in [25]. Such approaches are also a combination of different algorithms, such as with SMC [26] or with fuzzy logic [27]. Fuzzy logic-based control schemes are also available for solving trajectory tracking problems, for example [28; 29]. Many control algorithms based on other methods can be found, such as event-triggered control (ETC) with fuzzy system approximation [30] or with NN [31] as well as proportional integral derivative sliding mode control (PID-SMC) [32] or with prescribed performance [33]. Models with a diagonal inertia matrix allow simplification of the controller, but from the point of view of dynamics they often deviate from the real vehicle. For this reason, some control schemes are designed based on a model with a symmetric inertia matrix. Many approaches to the issue under consideration can be identified here. The NN has been used, for example, in [34; 35], although additional methods such as backstepping in [36] or backstepping and SMC in [37] are sometimes applied. Various approaches can be cited here, such as those based on the backstepping method [38], SMC [39] terminal sliding mode control [40], prescribed performance [41] or input-output linearization [42]. For this type of model, control strategies that are a combination of different methods are also applied, such as backstepping or the Lypunov approach [43]. Some control algorithms are based on the transformation of variables. In such cases, the control scheme is designed in new variables. One possible solution is to introduce coordinate transformation as in [14; 29] for a model with a diagonal inertia matrix. The coordinate transformation is also used for models with a non-diagonal inertia matrix as in [41; 42; 44]. The proposed control scheme for tracking the trajectory of 3 DOF underactuated underwater vehicles that move in the horizontal plane is based on a velocity transformation instead of a coordinate transformation.