# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 3 Controller in Terms of Quasi-Velocities
### Advantages and Limitation of Using the Proposed Tracking Controller  
The benefits of using the QV obtained from the decomposition of the inertia matrix for control purposes can be summarized in a few points:  
* It can be used to control fully asymmetric vehicles, making the dynamics model more realistic than a model with a diagonal inertia matrix. Simplifying the dynamics model still allows the algorithm to work, but loses some information about the effects of dynamic coupling. * The resulting equations employed in the algorithm are decoupled because of the diagonalization of the inertia matrix instead of calculating the inverse inertia matrix. The equations of motion are in diagonal form. From the dynamics equations, it can be determined which values of mass correspond to the quasi-accelerations when couplings are included. * Insights into vehicle dynamics with the controller are available. * Each quasi-velocity is controlled separately. The time history provides information about the deformation of the velocity during vehicle movement. * Control signals are related to the vehicle dynamics because they are scaled by \(s_{f}\) and the set of parameters is included in the algorithm. * Compared to other algorithms, additional information hidden in the inertia matrix can be obtained after the velocity transformation from the modified controller. _Comment_. In terms of the scope of applicability, control algorithms can be divided as follows:  
* suitable for fully asymmetrical vehicle models (about both axes of symmetry), e.g., the proposed QV controller;
* suitable for vehicle models with partial asymmetry (about one axis), e.g., [38; 39; 41; 43; 45; 46];
* suitable for fully symmetrical vehicle models (with diagonal inertia matrix), e.g., [14; 17; 18; 21; 22; 23; 24; 40]. Much more commonly, asymmetry is understood as a shift of the center of mass in the x direction). For such a simplified model, the proposed algorithm is also suitable for trajectory tracking and the effect of coupling during motion will be detectable. Thus, both functions of the algorithm (trajectory tracking and detection of coupling effects) are still satisfied. Many control schemes deal with tracking the desired trajectory only when the inertia matrix is diagonal. In this extreme case, the algorithm containing QV will only fulfill the task of control without detecting the effect of couplings on the movement of the vehicle since the inertia matrix is diagonal. At the same time, this form will be simplified compared to the general form. Control methods developed for a dynamics model with a diagonal inertia matrix cannot be compared with the proposed algorithm since it is suitable for estimating the effect of dynamic couplings. By assuming a diagonal inertia matrix, the control algorithm will compensate for any inaccuracies due to inertia forces. The QV-based control scheme allows tracking the desired trajectory under the full asymmetry of the vehicle model while studying the effect of dynamic coupling (i.e., assuming the center of mass is located outside the x- and y-axes). The symmetric inertia matrix plays a key role in the proposed approach. The proposed method is limited to horizontal vehicle movement only. With the current state of knowledge, it cannot be extended to a three-dimensional space. The difficulty is that it is not clear how, in the case of the considered algorithm, to extend the definition of the hand point to the three-dimensional system in such a way as to also preserve the idea of control by means of transformation of variables. In addition, the proposed approach is based on the strategy described in [38], which, in turn, requires restrictive assumptions (e.g., planar motion of the vehicle, linear damping coefficients, only the constant ocean current is taken into account and other disturbances are ignored). For this reason, more complex disturbances are also neglected in the proposed method.