# Nonlinear Trajectory Tracking Controller for Underwater Vehicles with Shifted Center of Mass Model
## 4 Numerical Simulations
### Analysis of Results  
The control scheme was tested for a five DOF model used in ROPOS underwater vehicle marine research. The aim was to verify the effect of dynamic coupling resulting from the displacement of the center of mass relative to the geometric center. It was also taken as an assumption that verification should apply to different trajectories (linear and complex) and be performed taking into account the real technical possibilities and not only the performance of the control algorithms. Position error convergence times of 50 to 100 s were obtained for the proposed algorithm. However, it must be taken into account that the initial position points were 2 to 6 m. Such distances, taking into account the weight of the vehicle (2268 kg) and the drag of movement in the water, significantly increase the time to reach the final positions in a steady-state motion. The convergence time depends on the initial distance points from the desired trajectory, the inertial parameters of the vehicle, and the assumed motion disturbance functions. Control algorithms for the ROPOS vehicle were tested, for example, in [37, 40]. This work also took into account the limitations of the thrusters. From [37], the error convergence was achieved after about 50 s, but for an initial point less than 5 cm away from the reference trajectory. In [40], the error convergence was achieved after 25 s, but the starting distances from the desired trajectory were an order of magnitude shorter than assumed in this paper. From the analysis of the chosen signals, the following was concluded:  
1. The performance of the QV controller was correct (i.e., it ensured that the task was completed in a finite period of time) for two trajectories and for selected slow, time-varying disturbances. 2. Using the QV algorithm, it is possible to control the vehicle for both weak couplings and stronger ones resulting from the lack of full symmetry of the vehicle. 3. The simulation test based on QV provides an insight into the impact of dynamic changes in the vehicle model resulting from a shift in the center of mass and makes it possible to determine whether the control strategy in this case is effective for different trajectories. The abovementioned observations were confirmed by an analysis of the selected indexes. This analysis shows that tracking a complex trajectory needs more kinetic energy than tracking a linear trajectory. Remarks on the comparison of the QV algorithm with another algorithm of the same type (based on a similar control strategy and denoted as CL):  
1. The QV algorithm provided acceptable performance for the assumed linear and complex trajectories, as well as for specified center-of-mass displacements. 2. This scheme can be used for comparative tests with other controls designed for trajectory tracking containing both diagonal and symmetric inertia matrix models because, after reducing the elements containing couplings, a controller for systems without couplings will be obtained. This can provide information on the effects of shifting the center of mass. 3. The QV controller allows one to determine the reduced kinetic energy of each quasi-velocity and therefore includes the coupling energy or deformation of linear velocities arising from the presence of couplings. No similar information is available with the CL controller or it will be incomplete, as in the case of kinetic energy. 4. The CL algorithm, i.e., the one in which the inertia matrix is diagonal, is able to track a complex trajectory if the dynamic couplings are weak. However, the tracking accuracy is not sufficient for the linear position. In addition, due to inaccurate information on the model parameters, higher gain values are required for the sliding surfaces and for the kinematic controller. The selection of controller parameters may not be as easy as it is for the QV algorithm, which can be explained by the lack of inclusion of dynamic couplings in the control process. 5. The CL algorithm tested included an option for robustness to dynamic modeling errors.