# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 1 Introduction  
Non-holonomic systems, such as autonomous underwater vehicles (AUVs) among others, are difficult to control because of their underactuated nature. In general, underwater vehicle models are strongly nonlinear, contain mechanical couplings, and are described by 6 DOF (degrees of freedom), and the equations of motion depend on many factors, as shown in, for example, [1, 2]. Moreover, as is known from the literature, many control algorithms have been developed for fully actuated marine vehicles. One of the important issues for control is tracking a desired trajectory. In the case of underactuated marine vehicles, building a tracking controller is a significant challenge. Although there are algorithms for models with 6 DOF, for example, [3, 4], there are many controllers for models that are simplified from a mechanical point of view. One possible simplification is to use a diagonal inertia matrix as in [5, 6]. Another increasingly common simplification is to design the controller for a 5 DOF model with a diagonal inertia matrix, as in [7, 8, 9]. One of the significant problems that make it difficult to control a marine vehicle is its asymmetry. This occurs when the center of mass does not coincide with the geometric center. This results in additional inertial forces that must be reduced during the controller operation. This problem becomes even more important when the vehicle is underactuated because not all control signals are then available. For this reason, the control algorithm should include additional components to reduce unwanted effects due to the presence of asymmetry. This means that, in the simplest model, when the inertia matrix is diagonal, the control algorithm must guarantee tracking of the desired trajectory with imprecise knowledge of the model parameters. Because this work addresses the issue of tracking control for underactuated 3 DOF underwater vehicles moving in the horizontal plane, it is worth reviewing publications on this type of object. Designing controllers for underactuated vehicles is a complex process when the dynamics model is complicated, hence the need to simplify the model. There are many trajectory-tracking strategies described in the literature where the inertia matrix is diagonal. They concern either underwater vehicles in planar motion or surface vehicles. Some control schemes use the Lyapunov theory, as shown in [10,11]. Another solution to the trajectory-tracking problem is to use sliding mode control (SMC) methods [12,13]. In addition, the prescribed performance method has been applied as a control strategy [14,15]. Other important solutions to the trajectory-tracking problem are control schemes based on neural networks (NN), e.g., in [16], or fuzzy logic [17]. There are also many other algorithms for trajectory tracking, such as terminal sliding mode control (TSMC) [18], output feedback control [19] and the linear algebra approach [20]. In order to achieve good performance of control algorithms, combinations of different methods are often proposed, e.g., the backstepping and Lyapunov methods [21], backstepping and SMC (plus Lyapunov stability theory) [22,23], NN and SMC [24,25], NN and SMC plus backstepping [26,27]), backstepping and neural network plus low-frequency techniques [28], event-triggered tracking control with prescribed performance using radial basis function NN (RBFNN) [29], and strategy based on virtual control points and RBFNN [30]. A control scheme which is a combination of an extended state observer (ESO) and a super-twisting second-order sliding mode controller was proposed in [31]. [PERSON] et al. [32] developed a tracking control scheme based on a combination of prescribed performance and the Lyapunov logarithmic barrier function. Event-triggered composite learning using an NN controller was considered in [33]. Taking into account the inertia matrix, it can be seen that most control methods are designed for a vehicle described by a simplified model with a diagonal matrix.