## 1 Introduction
Fuzzy logic approach, due to its simple control structure, is also employed to the solution of the trajectory tracking problems as, for example, in [39; 40]. One can also find other trajectory tracking methods, e.g., prescribed performance [41; 42], model predictive control (MPC) [43], dynamic surface control (DSC) method [44], bounded feedback [45], dynamic inversion method [46], linear algebra approach [47], viability control [48]. Models with a diagonal inertia matrix, on the one hand, indeed simplify the theoretical considerations, but on the other hand, are less of a representation of the actual vehicle. Therefore, algorithms are designed for models with a non-diagonal inertia matrix. Because in this paper an attempt was made to use the quasi-velocities (QV) for a planar vehicle trajectory tracking, therefore we are interested in control algorithms appropriate when dynamics models are described by an inertia matrix containing non-diagonal elements. In relevant control strategies of this type, for example, neural networks [49; 50; 51; 52], backstepping method [53; 54], sliding mode control [55], input-output linearization [56], an improved line-of-sight (LOS) using terminal sliding mode controller [57], combination of backstepping technique, cascade analysis and Lypunov approach [58], combination of FUO-based observations (finite-time uncertainty observer) and cascade analysis [59], combination of Lyapunov direct method, backstepping control method, and disturbance observer [60] are applied. The QVs considered in this work are known as the generalized velocity components originally introduced for mechanical systems in [61]. In order to obtain some information, which is inaccessible in classical control algorithms as well as to demonstrate some of the system's properties, quasi-velocities-based controllers were introduced. However, they were also used to fully actuated underwater vehicles control [62; 63]. For vehicles with incomplete input signals, the desired trajectory tracking is a big challenge and to the best of the author's knowledge, no effective solution to this problem has been developed. In this work, an attempt was made to describe vehicle dynamics in the sense of the QV and then to apply a known control algorithm. Since this is a preliminary test, methods requiring a combination of different techniques (also including neural networks) have been omitted because they require additional knowledge. Therefore, it was decided to use the input-output linearization method described in [56]. Simulation results show effectiveness of the proposed approach and give a preliminary answer to the question about the applicability of the QV for underactuated underwater vehicles trajectory tracking control. The originality of the proposed work can be stated as follows:1. A proposal for solving the trajectory tracking problem for an underactuated underwater vehicle moving horizontally described by equations resulting from the transformation of the inertia matrix (expressed in terms of the QV);
2. Description with the quasi-velocities was used to detect properties of the vehicle model that are not available directly using the classical equations of motion (indication of some information that can be obtained based on the proposed vehicle dynamic equations);
3. Controller based on transformed dynamic equations and expressed in QV;
4. Simulation verification of the proposed control algorithm for 3-DOF planar models of 2 vehicles with different dynamics and for 2 different trajectories (this issue is important because it happens that the algorithms effectively working in the original literature do not work after changing the parameters of the model or the desired trajectory as demonstrated in [64]). Effects related to the properties of the equations are also presented. This work is organized in the following way. In Section 2, the problem formulation is presented. In Section 3, equations of motion in terms of the quasi-velocities and the selected tracking control algorithm are described.