## 1 Introduction
The controller uses the inertial quasi-velocity (IQV) originally described for mechanical systems in [45]. Because of the difficulties associated with the velocity transformation (since there is additional dynamic coupling due to the coupling between these velocities), controllers incorporating the IQV are used in systems with fully actuated input signals, as can be found in [46; 47] (for serial manipulators) or [48; 49] (for fully actuated marine vehicles). This means that for vehicles with incomplete input signals, developing control algorithms in terms of IQV is still a difficult problem. The present work is an attempt to fill this research shortfall. An important advantage of controllers based on IQV is the availability of information about the impact of vehicle dynamics during the control task. This means that, in addition to trajectory tracking, it is possible to gain insight into the dynamics of the system, and thus study the behavior of the vehicle with the controller depending on the assumed desired trajectory, as well as changes in the model parameters. An example of such studies for a system with full forcing is given in [49]. Thus, the novelty of the work is also due to the fact that IQV cannot be easily applied to mechanical systems with underactuation. Simulation results show effectiveness of the proposed control scheme and give an answer to the question of the usability of using QV for trajectory tracking control of underactuated underwater vehicles. Paper [50] dealt with an algorithm based on a velocity transformation but also a coordinate transformation, while this one uses only the first transformation. The dynamic models are different in the two works and the disturbances are defined differently. The algorithm expressed in IQV [50] was only a slight modification of the original controller while here the control scheme is based on a combination of IQV, backstepping and SMC, i.e., it is grounded on a different control idea. The main contributions of this study are:  
1. Development of a trajectory tracking controller in terms of IQV for asymmetric underactuated vehicles moving in the horizontal plane. 2. Extension of IQV concept for tracking controller for underactuated vehicles with 3 DOF. 3. Compared to existing controllers employing a combination of baskstepping and SMC methods, the proposed algorithm uses a velocity transformation resulting in a vector of new variables that includes the dynamic parameters of the vehicle model. 4. Indication of the mathematical conditions for the implementation of such an algorithm and its verification based on simulation studies performed for two vehicles with different dynamics and for two different desired trajectories. _Comment._ Verification of the control algorithm for models with different parameters, as well as the use of two trajectories and an intuitive method of selecting controller gains seems reasonable, as it can happen that such changes affect the performance of the controller, as shown, for example, in [51]. The present work is organized as follows. Section 2 presents the formulation of the problem. Section 3 describes the equations of motion in terms of quasi-velocities. Section 4 provides a theoretical analysis of the proposed control scheme. Section 5 presents numerical results for two different underwater vehicles and two trajectories to demonstrate the effectiveness of the controller. Section 6 discusses the obtained results in comparison with other works on the issue under consideration. Finally, Section 7 presents conclusions.