# Nonlinear Trajectory Tracking Controller for Underwater Vehicles with Shifted Center of Mass Model
## 1 Introduction  
Researching and developing underwater vehicles has attracted the attention of many scientists and researchers. Underwater research is a major challenge because the marine environment is complex and variable; thus, autonomous underwater vehicles (AUVs) can replace humans in performing many difficult and dangerous marine operations. Among this group of vehicles, underactuated marine vehicles are of interest as they have fewer actuators than degrees of freedom. An important reason for investigating underactuated systems is that fully actuated autonomous vehicles usually generate considerable costs and may be impractical given their weight, reliability, complexity, and efficiency. Of the various tasks that can be accomplished with these systems, one is tracking the desired trajectory. Since general vehicle models have a complex mathematical form, due to the need to take into account both nonlinearities and disturbances from the environment, e.g., [1, 2], various simplifications of these models are therefore used to design simplified control schemes. These difficulties make it very common to use reduced models of marine vehicles so that control strategies are simplified. The basic model of a marine vehicle has six DOF and includes inertial couplings. Relevant trajectory tracking schemes can be found, for example, in [3, 4, 5]. In [3], the backstepping method and Lyapunov function were used, but the simulation example included a diagonal mass matrix. The strategy in [4] was based on artificial neural networks (ANNs) with an event-triggered control approach. The control scheme in [5] consisted of the model predictive control (MPC) method and an extended active observer. It is also sometimes assumed that the mutual couplings between variables can be ignored because they have small values, making it easier to design a control system. Such a simplification leads to the users obtaining a model with six DOF but with a diagonal inertia matrix. Examples of approaches based on the above assumption are shown in [6] (backstepping, Lyapunovfunction, fuzzy observer), in [7] (combination of adaptive fuzzy control with other methods), in [8] (backstepping and sliding mode control (SMC)), in [9; 10] (neural network), and in [11; 12] (backstepping and Lyapunov function). Making the model simpler is mathematically beneficial, but it moves the model further away from the real object. In such a situation, experimental studies on a real vehicle are convincing, e.g., [13]. Another known simplification is to reduce the vehicle model to five DOF so that it contains a diagonal mass matrix. Such a model can be found in the book by [PERSON] and [PERSON] [1]. There are a number of trajectory tracking algorithms for a vehicle, as described. Schemes connecting ANNs to other controllers are given in [14] (with SMC), in [15] (with optimal control), in [16] (with prescribed performance), in [17] (with Lyapunov approach), and in [18] (backstepping and event-triggered approach). Other control methods for this class of models are as follows: the dynamic surface control (DSC) technique [19], the integral terminal SMC [20], a combination of a single critic network, adaptive dynamic programming (ADP), and backstepping [21], backstepping with an integral SMC scheme using bioinspired neurodynamics [22], backstepping with a Lyapunov function [23], a bioinspired model [24], backstepping, a Lyapunov function, and active disturbance rejection control (ADRC) [25], a PID regulator with SMC [26], SMC and optimal control [27], a method based on error and velocity transformation techniques enabling the design of a practical prescribed-time controller [28] and a controller that allowed the design of the desired error dynamics combining state variables in the body's fixed coordinate and world coordinate [29]. Thus, it can be seen that the five DOF model is also used in real vehicles.