Finite-time prescribed performance trajectory tracking control for underactuated autonomous underwater vehicles based on a tan-type barrier Lyapunov function

This paper proposed a finite-time prescribed performance control scheme for underactuated autonomous underwater vehicles (AUVs) based on adaptive neural networks and a tan-type barrier Lyapunov function. Even in the presence of output constraints and environmental disturbances, the AUV can also precisely track the desired trajectory in a finite time. By introducing a tan-type barrier Lyapunov Function (TBLF), the singularity problem in process design is solved and all output errors are guaranteed to satisfy the prescribed performance specifications. Dynamic surface control (DSC) and the minimal learning parameter (MLP) are employed to greatly simplify the complexity of the algorithm and enhance the robustness of the control system, respectively. Lyapunov stability analysis proves that the proposed controller guarantees all signals in the closed-loop system to be uniformly ultimately bounded (UUB), and that the tracking errors converge to a small neighborhood near the origin in a finite time. Finally, the simulation results demonstrate the effectiveness and feasibility of the proposed controller.


I. INTRODUCTION
In recent years, people have increasingly recognized the indispensability of marine resources to the development of human society. As a result, a wide variety of marine equipment has been vigorously developed. Among them, the research and development of underwater vehicles, especially autonomous underwater vehicles (AUVs), have received much attention, and AUVs have important applications in many fields such as surveillance, rescue, ocean mapping, and inspection of underwater structures [1]. To accomplish the above tasks, AUVs are usually maneuvered in three dimensions in a complex ocean environment. Therefore, the research on trajectory tracking control in 3D space has been a research hotspot in the AUV field.
However, it is difficult to design an AUV trajectory tracking controller due to the obvious dynamic model uncertainties and the time-varying unpredictable disturbances generated by the marine environment. Lapierre et al. developed a new type of control law that deals explicitly with vehicle dynamics [2]. Bandara et al. proposed a control method using a vehicle fixed frame adaptive controller and an intrinsic nonlinear PID controller for attitude stabilization in complex-shaped low-speed AUV navigation [3]. Furthermore, the disturbances from the environmental external to the AUV should be considered when designing the controller to improve the control performance. In [4], to address the problem that AUVs are subject to model uncertainty and unknown ocean disturbances, the authors construct an iterative neural update law based on the prediction error, which effectively enables the accurate identification of the unknown dynamics of each vehicle. In [5], the trajectory tracking problem with actuator saturation is solved in the presence of parameter uncertainties.
The above work is discussed for fully driven underwater robots. For practical purposes, the design of the control system for an underwater underactuated vehicle is very complex and cumbersome because there are only three control inputs to control six degrees of freedom. To overcome the underactuated problem of an AUV, Qiao et al. proposed two adaptive fast integral terminal sliding mode control schemes with dual loops [6], and Shojaei et al. used the optically guided approach [7]. Seok et al. developed additional virtual control inputs to solve the underactuated problem [8]. Sliding model control is an effective control strategy for trajectory tracking. In [9], controllers are designed using the concepts of the terminal sliding model to solve the tracking problem of an underactuated AUV. In [10], the author conquered the quantization effect by introduction the bound of quantization error into the switching term of the SMC. In [11], scholars combine adaptive neural networks and dual closed-loop integral sliding mode control to achieve sliding model control of an underactuated AUV with uncertain dynamics.
Disturbance observers are often used to solve disturbances caused by ocean currents, but due to the powerful approximation ability in controlling nonlinear uncertain systems, adaptive neural networks have received much attention from scholars in recent years. There are many types of neural networks such as the radial basic function neural network (RBFNN) and the convolutional neural network (CNN). In paper [12], an improved neural network model based on the Glasius bio-inspired neural network was proposed and used in AUV trajectory. Paper [13] used neural networks and adaptive techniques to estimate and compensate for uncertainty effects in the AUV control system to achieve 3D trajectory tracking control of underactuated AUVs under the effect of parameter uncertainty and external disturbances. Guo et al. introduced a first-order robust exact differentiator, which takes the unknown velocity of the AUV into account, and proposed a nonlinear sliding mode control based on a linear parametric neural network (NSMC-NN) that effectively deals with the unknown dynamics and external environmental disturbances [14]. Elhaki et al. used a multilayer neural network and an adaptive robust controller to guarantee the transient performance of the tracking error at a given maximum overshoot and convergence rate, compensating for the structural and nonstructural uncertainties [15]. However, all neural weights of the NN need to be adjusted, causing an unacceptable learning time and complex calculations, which greatly affects the speed of approximation. To avoid this problem, Miao et al. adopted the Euclidean norm of the neural weights of the NN to approximate the model uncertainties, which is called the MLP algorithm [16].
In order to enhance the control performance, finite-time convergent and prescribed performance are proposed, which are utilized in many fields except underactuated AUVs. In paper [17], a new finite-time adjustable barrier function is introduced, whose design parameters can be dynamically adjusted in real time as the tracking error changes. Considering the safety of an underwater vehicle, we need to constrain its output to ensure that no accidents occur and the AUV can achieve its target as quickly as possible [18,19]. In paper [20], a novel high-order sliding mode (HOSM) controller was designed with asymmetric output constraints. In paper [21], a fixed-time controller for a category of nonlinear systems with output constraints was constructed, which can be considered to be an extension of the finite-time control algorithm. Specifically, the output constraint can be satisfied at any time, and the convergence time is independent of the initial conditions and can be predetermined. When the autonomous underwater vehicle is working in a marine environment with coral reefs, ditches or rocks, the AUV cannot move freely without restriction. If we do not want the AUV to collide with the reef and be damaged when passing through some narrow and cramped environments, we should limit its position error as much as possible. On the other hand, when the angle error is too large, the AUV cannot correct the error quickly due to the mechanical system, which will affect the position error. This requires us to further investigate the underwater robot system with output constraints and a quick response strategy.
Inspired by the above-mentioned works, a finite-time convergent prescribed performance control scheme based on the MLP neural network for an underactuated AUV is proposed. The main contributions of this paper are given in the following points.
(1) Compared with [22] and [23], the TBLF and performance function are developed to guarantee the output errors of the systems to be finite-time convergent and limited within a certain region in a finite time.
(2) In contrast to the existing state observers method [24] and RBFNN algorithm [25], the MLP neural network is designed to approximate the model uncertainties and external disturbances, which guarantees that the computational burden of the algorithm can be drastically reduced and improves the system performance. Furthermore, an adaptive law is introduced to estimate the approximation errors.
(3) Dynamic surface control is used to solve the problem of dimensional explosion in the backstepping method. Different from previous works [26,27], the closed-loop system is finite-time stable and not asymptotically stable. Finite time control can achieve better steady and transient performance and stronger robustness.
The content of this paper is organized as described in the following. Section Ⅱ presents the relevant preparatory knowledge and some theoretical assumptions. Section Ⅲ shows the design of the controller. Section Ⅳ verifies the effectiveness of this scheme by simulation. Finally, Section V makes some concluding remarks.

A. MATHEMATICAL MODELING OF THE AUV
In this paper, an underactuated 5-degree-of-freedom AUV subject to environmental disturbances is studied. Fig.1 shows the frame and state of the AUV, and the kinematic and dynamical model of this AUV is depicted as follows [28]: cos cos sin sin cos sin cos cos sin sin sin cos / cos  ( 1,..., 6) ii d i = are the hydrodynamic coefficients; g ρ is the buoyancy of the AUV; and L GM represents the distance between the center of gravity and the floating center [29].
Remark 1. To simplify the controller design in the next section, the dynamics of the rolling motion are ignored in the AUV model. In practice, a recovery force is generated due to the existence of a metacentric height between the center of gravity and the floating center of the AUV. This allows the AUV to stabilize the rolling oscillation by this restoring force at low speed.
Remark 2. In model (2), there is no control input in the second and third equations, which means that the velocities v and w are not controllable during trajectory tracking. This implies that the AUV can only go through three control inputs to accomplish the trajectory tracking mission. Therefore, the AUV is underactuated. The position and orientation tracking errors between the follower and the target in the inertial reference frame are transformed to the body fixed frame as follows: Differentiating ( Then, the tracking errors can be expressed as: In order to force the AUV to achieve the desired trajectory, we need to impose some restrictions on the tracking error e ρ , e θ and e ψ . That is, the tracking error e ρ , e θ and e ψ need to satisfy the following time-varying limits: , q β ∞ and , r β ∞ represents the maximum error allowed when the system is stabilized. Eq. (8) is also known as the performance function [30].

B. RBFNN APPROXIMATION
 is an unknown smooth nonlinear function and it can be approximated over a compact set m Ω ⊆  with the following RBFNN: where l W ⊆  represents the optimal weight vector, ε is a positive constant representing the approximation error, Ŵ is the estimation of represents the radial basis function vector, the element of which is chosen as the Gaussian function: where m i µ ∈  and i λ ∈  are the center and spread, respectively, and | | ε ε ≤ where ε is an unknown constant.

C. LEMMAS AND ASSUMPTIONS
The following assumptions and theorems will be used in the design. Remark 3. Assumption 1 always stands due to paper [7]. The assumption is appropriate because the velocities in sway and heave will be damped by the hydrodynamic damping forces.

D. TAN-TYPE BARRIER LYAPUNOV FUNCTION
The tan-type barrier Lyapunov function [34] was introduced to constrain the tracking error: Remark 5. According to the formulation of TBLF in (15), we can obtain that: We can realize that when the initial state z satisfies the , then it will always be maintained. Moreover, if there are no constraints on system states, i.e., b β → ∞ , then we can obtain the following equation by using L'Hospital: Therefore, we can directly use the quadratic term to replace the TBLF with no constraints on the dynamic error. Remark 6. There are many types of barrier Lyapunov functions (BLF) such as the log-type BLF [17,22] and the tan-type BLF [35]. Eq. (15) should be rewritten to Therefore, the log-type BLF cannot be used as a universal BLF without constraints. Certainly, the tan-type BLF can be used to deal with both constrained and unconstrained situations, but the tan-type BLF is quite complex for underactuated AUV control.

E. CONTROL OBJECTIVE
For a smooth reference trajectory ( , , ) d d d x y z , the control target of this paper is to design TBLF-based control input signals , u q τ τ and r τ to make the AUV follow the target trajectory successfully. In the presence of environmental disturbances, the tracking errors of the AUV achieve the prescribed performance and eventually converge to an arbitrary small neighborhood of the origin. Finally, the adaptive neural network is used to estimate and approximate unknown environmental disturbances from wind, wave and ocean currents.

III. MAIN RESULTS
In this part, a controller based on the TBLF backstepping method with combined DSC and RBFNN techniques is designed for the trajectory tracking problem of an underactuated AUV. The design process is divided into three parts: the design of the virtual control law, the design of the actual control input, and the elimination of environmental disturbances.

A. SURGE MOTION CONTROLLER
The following TBLF candidate should be considered: Differentiating ue V , we have the following: where csc( ) 1 / sin( ) =   . By differentiating e ρ in (5) The following error coordinate transformation can be defined as follows: The following virtual control law u α should be designed: where the design parameters According to (16)(17)(18), ue V  is simplified as The time derivative of 2 ue V is given by: are positive design parameters.

B. PITCH MOTION CONTROLLER
The following TBLF candidate should be considered: 2 The following error coordinate transformation should be defined: The following virtual control law q α should be defined: The virtual control law is filtered using a first-order filter below: design parameters.

C. YAW MOTION CONTROLLER
The following TBLF candidate should be considered:

IV. SIMULATION RESULTS
In order to verify that the proposed control scheme is effective, simulation results of the developed controller in this paper are compared to the existing literature [22]   show the trajectory tracking result, which indicates that the AUV can successfully track the target. Figs.5-7 shows tracking errors and the prescribed performance constraints. Compared with the tradition backstepping method and BLF tracking trajectory method, the proposed controller converges quickly and has better control precision. Fig.8 shows the velocities of the AUV in u , q and r . Fig.9 illustrates that the three controllable inputs u τ , q τ and r τ are continuous and smooth. Fig.10 shows the adaptive parameters ˆu  , ˆq  and ˆr  .

V. CONCLUSIONS
In this paper, a finite-time convergent control strategy based on the tan-type barrier Lyapunov function backstepping method, combining DSC technique and MLP algorithm, is proposed for the trajectory tracking problem of an underactuated autonomous underwater vehicle. The proposed controller ensures that all signals of the closed-loop system are bounded, and the AUV can converge to the desired trajectory in a finite time. The DSC technique improves the efficiency of the controller by simplifying the calculation process, while the MLP also enhances the robustness against the model uncertainties. Then, an adaptive law can effectively estimate NN errors and environmental disturbances to guarantee the accuracy of system. Finally, simulation results show that the proposed controller is effective.