# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 1 Introduction
This approach results in the algorithms being less complicated from a mathematical point of view and easier to implement. However, information on vehicle dynamics is often less accurate because the algorithm should overcome any model inaccuracies in the control process. Of course, the approach based on the assumption of negligible couplings makes sense if the vehicle is indeed exactly balanced or if it is justified that the couplings have small (negligible) values. However, such conditions are not always met and then the simplified control algorithm may not be effective enough or its performance will be worse. If the asymmetry of the vehicle is so significant that it cannot be ignored, then control strategies are usually proposed using a model with a symmetric inertia matrix but assuming symmetry with respect to the right and left sides. Control schemes designed for such asymmetric underactuated marine vehicles moving horizontally are less common than control algorithms with a diagonal inertia matrix. Various control algorithms are available for the model in which the vehicle is symmetric in its longitudinal plane. For example, an integral backstepping algorithm was proposed in [34] for tracking the trajectory in the presence of ocean current disturbance. The combination of a backstepping technique, cascade analysis, and the Lypunov approach for full state regulation of surface vehicles can be found in [35], and the combination of the Lyapunov direct method with backstepping control in [36,37]. In [38], a control scheme based on the input-output feedback linearization method and usage of the position point of the hand was presented. A trajectory-tracking algorithm for planar under-actuated vehicles that utilize sliding mode control was described in [39]. The design of an improved line of sight (LOS) using an adaptive terminal sliding mode controller was addressed in [40]. In [41], an FUO (finite-time uncertainty observer) was developed in order to enable the separation principle in the algorithm and synthesis of the observer's perspective, making it possible to accurately control the trajectory tracking. Controlling trajectory tracking by developing an accurate control method in a newly defined polar coordinate system moving along a path (PMPCS) was proposed in [42]. The application of neural networks in control strategies was reported, for example, in [43,44]. An adaptive control algorithm with prescribed performance for underactuated surface vehicle trajectory tracking was developed in [45], while, in [46], this method was supported by neural networks. An approach to the tracking control problem based on the prescribed performance function and quantized feedback signals was described in [47]. For a fully asymmetric vehicle, an algorithm based on a combination of backstepping and integral SMC was proposed in [48]. A brief overview of methods to solve the trajectory-tracking problem for vehicle model asymmetry and disturbances is given in Table 1. In the table, the sign \(+\) indicates a positive answer, while \(-\) indicates a negative answer. It may be noted that, in the case of a diagonal inertia matrix (symmetric vehicle), variable disturbance models are used, which should reduce the effect of the environment and modeling inaccuracies on the performance of the controller. However, even when partial asymmetry is assumed, functions representing disturbances are usually introduced in addition. This paper is concerned with using one of the algorithms (in modified form after velocity transformation) to track a desired trajectory of a 3 DOF vehicle model. In order to perform the trajectory-tracking task, a controller expressed in terms of some quasi-velocities (QV) was used. The proposed scheme uses the idea discussed in [38]. This study does not use a combination of different approaches that can make it difficult to use QV, so it was thought appropriate for the considered control task.