This paper addresses the problem of trajectory tracking of underactuated surface vessels (USVs) in the presence of thruster failure. Multilayer neural networks (MNNs) are employed to estimate the unknown model parameters and external disturbances. To design a fault-tolerant controller without a fault detection scheme, we use the Nussbaum gain technique. We introduce an additional control to resolve the difficulty arising from having fewer inputs than degrees-of-freedom. Further, an approach angle is proposed to track both a straight and curved path. Stability analysis and simulations are performed to demonstrate the effectiveness of the proposed scheme.

As land resources are depleted, interest in ocean exploration has increased. There are many marine vehicles designed to explore the ocean. Among these, unmanned marine vehicles (UMVs) such as autonomous underwater vehicles (AUVs) and unmanned surface vessels are useful because they can reduce the cost and the risk of personal injury [

An important problem in the control of UMVs is failure occurrence on the actuators [

Motivated by these observations, a neural network-based fault-tolerant control is presented for underactuated surface vessels (USVs). The proposed algorithm can be applied to AUVs because the equations of motion for surface vessels are the same as those of AUVs in the horizontal plane. For a practical perspective, we assume that the bow and stern are not symmetric and the model parameters for the hydrodynamic terms are unknown. To estimate the unknown model parameters and external disturbances, we employ multilayer neural networks (MNNs) [

The main contributions of our paper are as follows. (1) The fault-tolerant controller is designed without the fault detection scheme. (2) For underactuated surface vessels, we develop the additional control input and analyze the stability including the sway dynamics. (3) The approach angle which is composed only of position information of USVs is introduced. With the help of a novel approach angle, we can track any trajectory including both a straight- and curved-line path. (4) For a practical application, the model parameters for the hydrodynamic terms and external disturbances are assumed to be unknown. These uncertainties are estimated by MNNs.

Throughout this paper, the following notations are used: (1)

The kinematics and dynamics of USVs are described as follows [

Unlike other works, we assume that the parameters

To address the problem stated in Remark

When faults occur in the control vector

The control objective is to design a fault-tolerant controller such that it tracks the reference trajectory

The desired velocities

In this paper, MNN is used to estimate the unknown function

The Taylor series expansion of

The unknown term

See the appendix.

In this section, we design the neural network-based fault-tolerant control system. To avoid the difficulty of detecting the faults, the controller is proposed using the Nussbaum gain technique. The following lemma is used to design the neural network-based fault-tolerant control law.

Let

We now design the neural network-based fault-tolerant control law using the dynamic surface design approach.

Define the following errors:

In this paper, we use an approach angle

Using (

In this step, we use the NNs to estimate uncertain terms

Define the following errors:

The threshold

Using (

See the appendix.

From (

Consider the USV model (

See the appendix.

In this section, we simulate a neural network-based fault-tolerant controller to verify the performance of the proposed control system. The parameters of the underactuated surface vessel are as indicated in [

Figure

Simulation results. (a) Trajectory tracking result. (b) Control inputs (solid:

This paper presented a neural network-based fault-tolerant controller for an underactuated USV with faults, model uncertainties, and external disturbances. MNNs were employed to estimate the highly nonlinear uncertain terms, and approach angle was implemented to track any reference trajectory including both straight- and curved-line paths. To avoid the difficulty of detecting the faults, the neural network-based fault-tolerant controller was designed using the Nussbaum gain technique. From the simulation results, it was demonstrated that the proposed control system has good tracking performance in the presence of the faults, model uncertainties, and external disturbances. In the future, the actuator saturation problem will be addressed and evaluate the performance through real experiments.

Since

Since

We prove Theorem

Since

The choice of control gains has some suggestions as follows: (i) increasing

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A1041216).