# Numerical Test of Several Controllers for Underactuated Underwater Vehicles
## 3 Short Description of Selected Algorithms
### Selected Trajectory Tracking Methods  
Since there are many strategies for controlling underactuated underwater vehicles that move horizontally, some selection had to be made based on assumptions. Assumptions for control schemes selection and test conditions. The method of choosing a control scheme for preliminary tests (1.)-(3.) and the test conditions (4.)-(6.) is as follows:  
1. the controller is related to the vehicle mathematical model taken into consideration, i.e., (1)-(6) whereas the testing algorithm refers to the dynamic model (7)-(9);
2. the controller was verified at least on simulation using the vehicle parameters given in the original work;
3. the control algorithm provides complete information necessary for simulation verification;
4. in this preliminary test, the linear and sine desired trajectories and vehicles ROPOS and Kambara are used only;
5. the disturbances and other inaccuracies are not taken into consideration in the first test, but in the additional test (done only for algorithms which gave acceptable results in the first part of the test) the disturbances occur;
6. only a simple method for selecting control gains is allowed, i.e., the trial and error method; the use of additional searching algorithms is unacceptable. Comment. The above assumptions were based on the rule that if simulation studies of an algorithm do not produce acceptable results, they also do not guarantee their practical application. On the other hand, selecting the control algorithms for the test was guided by the fact that the source work shows that no additional methods of optimal selection of gain coefficients are necessary, even in the case when these parameters are numerous. Taking into account the above assumptions and remarks, five algorithms were chosen for the test. In this preliminary study, neural networks, fuzzy control methods were omitted due to the need to use different mathematical approaches. Similarly the more complex strategy described in [29] (despite the significant advantages) has not been investigated. From a practical point of view, it would be better to examine the usefulness of more control algorithms. However, only five were used for the preliminary tests to show their performance for selected vehicles with different dynamics. The first controller is suitable for a hovercraft and it serves as a tester of each vehicle's dynamics. Here, it is applied to investigate dynamics of the vehicle, e.g., possibility of control, necessary control inputs (limitation of the force and torque), body velocities. It belongs to the Lyapunov-based methods which are easy for interpretation. The second algorithm is a combination of backstepping and SMC. Therefore, it reflects the benefits of both methods. The third one represents the SMC approach which has repeatedly proved effective in controlling vehicles. The fourth and the fifth selected control schemes belong to the TSMC which is considered an efficient tool for trajectory tracking. Selection of control algorithms was guided by the compliance of the dynamic model given in the source literature with the model adopted in this work. After the initial test, the strategy described in [4] proved promising to achieve this goal. Other selected control strategies fulfill the assumptions made above. Controller 1 (Lyapunov method-based controller--Lyapunov MBC) comes from [4]. The control purpose is to have a fixed point in the body frame \(\delta\in R^{2}\) which does not necessarily have to be the center of mass, to track a desired trajectory \(\mathbf{p}_{d}\).