## 1 Introduction  
In recent years, autonomous underwater vehicles (AUVs) have been gaining much attention because of their usefulness for various operations. In terms of the availability of input signals, vehicles can be divided into fully actuated and underactuated. Therefore, the control methods in each of these cases are usually different. Furthermore, among many problems, trajectory tracking and path following are of particular interest. The problem of modeling real underwater or surface vehicles is difficult because the system itself as well as the physical phenomena occurring in it and the environmental disturbances can be taken into account only approximately. In addition, the equations of motion are strongly nonlinear, and many physical phenomena are only partially taken into account. For this reason, there are various methods for modeling dynamics of the system and analyzing its properties. Some of these methods may even allow a more accurate analysis of the vehicle behavior in motion. The issues related to marine vehicle modeling have been considered many times, e.g., in [1; 2; 3; 4; 5]. An important issue examined in the literature is the control of underactuated marine vehicles exposed to wind, waves and ocean currents, as shown in [6]. Control in the presence of irregular waves was addressed in [7]. When a marine vehicle may encounter obstacles while performing a control task, it is necessary to solve the collision avoidance problem. This was done, for example, in [8; 9; 10]. If the variable ballast has to be taken into account, then it is necessary to modify the equations of motion accordingly as in [11]. Automatic control of a single unmanned surface vehicle pushing a floating payload was analyzed in [12]. Such a problem is difficult to solve because the payload being manipulated is insufficiently controlled and its open-loop dynamics are inherently unstable. Problems related to marine vehicle motion are indicated here by way of example to indicate the various difficulties encountered in building a dynamic model and control strategy. However, since the purpose of this paper is to try to apply a modified control algorithm to a basic model of a vehicle in horizontal motion, the above problems were omittedin this paper. They are very important at the stage of modeling and construction of the control scheme, but they are not crucial for the topic of this work. Here, the aim is to show that using a velocity transformation for the given vehicle model it is possible not only to track the desired trajectory, but at the same time to have an insight into the vehicle behavior for the model with different parameters and for tracking different trajectories. On the other hand, the mentioned previous problems should be investigated in the future. This paper is limited to the problem of trajectory tracking of a 3-DOF underactuated underwater vehicle moving in the horizontal plane. At present, there are many control methods into this control problem of underactuated AUV. There exist various controllers that have been used for this class of vehicles. If the inertia matrix in the dynamical model is a diagonal one, then the methods representing [PERSON] approach [13; 14; 15], sliding mode control (SMC) [16; 17; 18], terminal sliding mode control (TSMC) [19; 20], the proportional-integral-derivative sliding mode control (PID-SMC) [21], backstepping method [22; 23; 24] or backstepping and [PERSON]'s methods [25; 26], backstepping and SMC [27], backstepping and a composite system method [28], feedback control, backstepping and averaging approach [29], SMC and backstepping based on a neurodynamical model [30; 31], and output-feedback represented in [32] are used. Another group of strategies is based on neural networks (NN), e.g., in [33; 34]. However, these methods are often a combination of different algorithms, e.g., [35] (NN, SMC), [36; 37] (NN, backstepping, SMC) or [38] (an event-triggered adaptive neural fault-tolerant control scheme).