# Nonlinear Trajectory Tracking Controller for Underwater Vehicles with Shifted Center of Mass Model
## 1 Introduction
However, the verification problem is similar to that for control based on six DOF models, namely, if experimental tests are not feasible for any reason, then the effectiveness of the control strategy for the simplified model with five DOF should be validated in the case of a shift in the center of mass. An effort to solve this problem has just been made in this work. This paper proposes a tracking control for a five DOF underwater vehicle model with couplings due to a shift in the center of mass. First, the symmetric inertia matrix is decomposed to obtain the diagonal form expressed in inertial quasi-velocities (IQVs). Only then is a control strategy designed that combines backstepping and SMC. Controllers that use IQVs are applicable to tracking trajectories set in three-dimensional space for marine vehicles, but only when all input signals (for fully actuated systems) are available. In contrast, for underactuated vehicles, designing control algorithms using IQVs is challenging. The novelty of the proposed approach is that, in the event of the lack of even a single input signal, control using IQVs is not comparable to algorithms for a fully forced system. An additional advantage of control algorithms using IQVs is the ability to obtain insight into vehicle dynamics when changing dynamic parameters and the desired trajectory. Such information may be useful for preparing the correct operating conditions or modifying model parameters at the initial step of vehicle design. The proposed approach has been verified on a model representation of realistic vehicle parameters via numerical simulations. It turned out to be effective in the task of trajectory tracking, confirming the usefulness of using IQVs for control purposes. The simulation tests included one vehicle and two scenarios. The article's contributions are as follows:  
1. Obtaining a five DOF underactuated marine vehicle model with coupling by diagonalization of the inertia matrix. This model is a generalization of the usually considered models. This is an extension of commonly used control schemes for marine vehicles with a diagonal inertia matrix to an algorithm suitable for vehicles with a lack of symmetry. 2. The design of a trajectory tracking control scheme for a vehicle moving in three-dimensional space that uses IQVs in traditional control methods. An important feature of the approach offered is the ability to examine the effect of shifting the mass center of the vehicles on the controller's performance to see some of the consequences of simplifying a model containing dynamic couplings to one that ignores those couplings. 3. Simulation verification of the proposed strategy assuming different operating conditions regarding trajectories and couplings. A model of a real vehicle was adopted and, in contrast to some of the known work, the technical possibilities of the actuators were taken into account in order to provide operating conditions close to real ones. It is not difficult to see that in the literature, trajectory tracking of five DOF vehicles by means of an experiment is rarely found because various problems associated with it have to be solved [29,30]. Much more often, work ends with simulation verification and the vehicle model is described by a diagonal inertia matrix. The proposed approach seems particularly useful at the design stage or in deciding whether to carry out an eventual real-world experiment because it indicates what difficulties may arise from using the controller if the center of mass is not identically located to the geometric center. The article is organized as follows. Section 2 describes the problem under consideration. Section 3 presents the theorem concerning the proposed strategy. Section 4 contains simulation results confirming the effectiveness of the controller. Section 5 concludes the paper. The theoretical supplements can be found in Appendixes A and B.