# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 1 Introduction
However, the difference between the proposed controller and those mentioned above is that here it is assumed that the vehicle is not symmetrical about the axes of longitudinal and lateral motion. Moreover, introducing a description of the vehicle dynamics with the use of quasi-velocities is necessary to carry out the task of trajectory tracking because it enables first-order differential equations to be obtained. Such a description causes the inertia matrix to be diagonal and the obtained quasi-accelerations to be independent. The QV adapted here are referred to in [49] as generalized velocity components (GVC), and were applied for mechanical systems. In the past, they were also used to control fully actuated underwater vehicles [50; 51]. Unfortunately, a drawback of the proposed approach to the control task is a failure to show the stability of the entire system. Instead, we show how the dynamics equations used can be transformed to the form considered in the cited work. Selected simulations on 3 DOF models of two vehicles moving horizontally with three trajectories demonstrated the effectiveness of the application. However, in the tests, it turned out that the idea of control is practical for various models of underwater vehicles with different desired trajectories. The contributions of this work can be summarized as follows:  
1. Decomposition of the inertia matrix enabling description of the use of velocity transformation for an underactuated underwater vehicle moving in the horizontal plane, obtaining diagonal equations of motion, and testing of the modified control algorithm. 2. Showing some of the information available from the trajectory-tracking controller when a dynamic description containing quasi-velocities (QV) is used. 3. Proposal of an algorithm for trajectory tracking in horizontal motion when the center of mass lies outside both axes of symmetry of the vehicle. This controller is generalized in the sense that the previously known controllers from [38; 52] are special cases of it and apply only when the center of mass is shifted in the longitudinal direction. In contrast, this scheme also considers shift in the center of mass in the lateral direction. A simulation test of the modified control algorithm was performed for two models of different vehicles and three desired trajectories based on intuitive selection of controller parameters. Compared to [52], this paper is different in the following ways:  
1. First, the algorithm presented is applicable to vehicles whose center of mass lies outside the coordinate axes and is not symmetrical on the longitudinal axis. 2. Second, the theoretical contribution is that the vehicle dynamics model is more realistic and this has been taken into account in the control algorithm. 3. Third, the controller contains more dynamic couplings due to the more complex dynamic model. Therefore, it is possible to assess their influence on the movement of the vehicle along the selected desired trajectory. 4. Fourth, a motion analysis for the cycloid-like trajectory is included in the simulation study, which provides a deeper understanding of the vehicle dynamics using the proposed controller. Instead of a sine trajectory, a sine-cosine trajectory was applied. In addition, quantities that are crucial for coupling evaluation are added. Section 2 describes the marine vehicle model and the transformed equations of motion in terms of the quasi-velocities. Section 3 describes the modified tracking controller. In Section 4, simulations are shown for two different underwater vehicles and three desired trajectories. In Section 5, the conclusions are presented.