# Numerical Test of Several Controllers for Underactuated Underwater Vehicles
## 1 Introduction  
The control of autonomous underwater vehicles (AUVs) is a complex problem mainly due to the nonlinear dynamics, uncertainty in model parameters, and external disturbances. The AUVs are applied to inspections, surveillance, maintenance, and security within the maritime industry. Due to the availability of forcing signals, these vehicles can be divided into fully actuated and underactuated. In this work, the problem of the application of known methods of trajectory tracking for underactuated underwater vehicles which move in the horizontal plane is considered. There exist many control schemes that have been applied to this class of vehicles. Various types of control strategies which are suitable for underactuated vehicles may be recalled here. One method is the Lapunov-based approach which has proven to be useful for the trajectory tracking of marine vehicles [1; 2; 3] or hovercrafts [4]. A lot of work has been done to solve the problem of trajectory tracking for underactuated AUVs, namely using the backstepping method [5; 6; 7; 8] or backstepping and Lyapunov's direct methods [9]. The backstepping techniques were employed to realize the trajectory tracking of underactuated ASVs many times. In the traditional approach, however, a serious disadvantage is the computational complexity. Despite this difficulty, this type of technique can be found in [10]. Combinations of this approach with other methods are also available, e.g., in [11] (backstepping and sliding mode control (SMC) method), [12] (averaging and backstepping), ref. [13] (backstepping and integral SMC). Another approach is the output-feedback [14; 15], sometimes together with backstepping [14]. Sliding mode control (SMC) is also an effective nonlinear control technique which is characterized by low sensitivity to systematic uncertainty and good disturbances rejection. There are various kinds of this strategy [16; 17]. However, some authors suggest modifications of the SMC method to achieve greater precision in tracking trajectories or other benefits [18; 19]. Sometimes the terminal sliding mode control (TSMC) method is considered better than the conventional sliding mode control technique (taking into account the finite-time convergence and high steady state tracking precision) and is proposed, e.g., in [18]. The proportional-integral-derivative sliding mode control (PID-SMC) [19] guarantees global convergence and boundedness of all tracking errors in thefinite time. Quite different groups of methods but with significant application are strategies based on neural network (NN) learning, e.g., in [20; 21; 22; 23; 24]. These methods also include a controller that is a combination of different algorithms, e.g., [23; 25] (NN,SMC). Fuzzy logic approach, due to its simple control structure, easy, and cost-effective design, is also employed to the application of underactuated AUVs control. The following works may be mentioned here [26; 27]. A review of applications is given in [28]. Another strategy consists of a combination of artificial intelligence, learning with other techniques as it was shown in [29]. Less often, however, other trajectory tracking methods are used as, e.g., prescribed performance [30; 31], model predictive control (MPC) [32; 33], input-output linearization [34], linear algebra approach [35], and modified dynamic inversion [36]. Valuable works comparing known control algorithms can be found in the literature, e.g., [37]. Since such works have a defined goal, consequently also the research has a limited scope. Of course, for the purposes of initial design verification, known algorithms dedicated to horizontally moving vehicle's underwater vehicle control can be applied. The aim of the work was to show some preliminary numerical tests of selected known control algorithms that perform the task of tracking two set trajectory types for arbitrarily adopted two underactuated underwater vehicle models.