The spatial path following control problem of autonomous underwater vehicles (AUVs) is addressed in this paper. In order to realize AUVs’ spatial path following control under systemic variations and ocean current, three adaptive neural network controllers which are based on the Lyapunov stability theorem are introduced to estimate uncertain parameters of the vehicle’s model and unknown current disturbances. These controllers are designed to guarantee that all the error states in the path following system are asymptotically stable. Simulation results demonstrated that the proposed controller was effective in reducing the path following error and was robust against the disturbances caused by vehicle's uncertainty and ocean currents.

The research of AUV has been a hot topic in recent years with the development of marine robotics. Voluminous literature, for example [

Hereinabove, it is shown that modelling inaccuracy is primary difficult to achieve oriented results. The data driven fault diagnosis and process monitoring methods based on input and output data could be an effective way in real-time implementation where the physical model is hard to obtain [_{∞} PID controller [_{∞} controller was constructed in [

In this paper, we propose an adaptive neural network control method for spatial path following control of an AUV. Three lightly interacting subsystems are introduced to fulfil this mission. RBF neural network (NN) is introduced to estimate unknown terms including inaccuracies of the vehicle. Adaptive laws are chosen to guarantee optimal estimation of the weight of NN to make the approximation more accurate. The control performance of the closed-loop systems are guaranteed by appropriately choosing the design parameters. Based on the Lyapunov stability theorem, the proposed controllers are designed to guarantee all the error states in the subcontrol systems which are asymptotically stable.

The paper is organized as follows. Section

The dynamic model of the AUV in the three dimensional space is described in this section. See details in Bian et al. [

Employed coordinate frame systems.

According to the criteria underwater vehicle motion model in Fossen [

Dynamic equation:

Kinematic equation:

The symbols

Spatial path following problems of AUV can be solved by a dynamic task and a geometric task, whose objectives are to make the vehicle sail at an expected speed and move to the proposed three-dimensional path.

The former process of path following problem can be briefly stated as follows. Given a spatial path

Spatial path following control problem.

The objective is to realize the path following for AUVs in three dimensions. Consider Healey and Lienard [

For control design purposes, the interactions from the other degrees of freedom is neglected; the speed control model could be given by

Define

Then, (

To deal with the uncertain terms, a RBF NN is chosen to estimate

Diagram of RBF neural network for identification.

Assume

Choose a Lyapunov function

Equation (

Combined with (

An adaptive law is designed as follows:

Because

In (

Based on Lyapunov stability theorem, it is provided that if one of the inequations in (

Consider the vehicle dynamics referred in Silvestre and Pascoal [

If the surge speed is

To be convenient for controller design, (

In this section the backstepping techniques are adopted based on iterative methodology, where a virtual control input is introduced to ensure that the diving error can be converged to zero. And based on the Lyapunov stability theorem, an adaptive neural network controller is designed to guarantee that all the error states in the diving control system are asymptotically stable.

From (

Given a desired depth

Define a virtual control variable:

From

When we take

Referring to the mean value theorem of Lagrange,

Combined with (

Then

Consider

Choose a Lyapunov function:

The derivative of (

With the adaptive law,

Consider

It is assumed that

Define

It can be calculated as

And, we can obtain

Then the unknown term

Finally, we obtain the actual control input:

From (

Consider a Lyapunov function:

We can obtain its derivative

As what we did in Step

Consider the Lyapunov stability theorem, it can be concluded that all the signals in the diving control system are bounded. Furthermore, the output tracking error of the system will converge to a small neighbourhood zero domain by appropriately choosing control parameters.

Referring to Fossen [

Calculate the angle between the proposed path and the north of earth-fixed coordinate in Figure

Principle of LOS guidance for straight line.

Considering the cross track error

When it comes to the circular arcs in Figure

Principle of LOS guidance for circular arc.

Referring to the vehicle dynamics in horizontal plane in [

The kinetics model of the AUV in horizontal plane can be simplified as

Consider a heading track error:

Choose

Then the derivative of the steering error system can be written as follows:

Then we can find

On the assumption that

Combined with (

For

In fact,

If the weight estimation of neural network

For

Meanwhile, consider a weight error for the RBF NN:

Combined with (

In order to compensate for estimation error and current disturbance as shown in (

Herein, (

Introduce adaptive laws:

To deal with the problem of stability, a Lyapunov function (

The differentiation of (

With a combination of (

Moreover, because

Similar to the derivation of diving controller, it can be concluded that all the signals in the steering control system are bounded. Furthermore, the output tracking error of the system will converge to a small neighbourhood zero domain by appropriately choosing control parameters.

Finally, the control input can be given by

In order to validate the proposed controller, it is assessed in the C/C++ simulation environment with a full nonlinear model for the designed vehicle. It is assumed that the states of the system are updated with a period of

Two simulations are carried out to demonstrate the advantage of the proposed method, including path following conditions without sea current and undersea current, where the unvarying current is set to be heading east with 0.25 m/s. The vehicle is initially rest at a random position

Figures

Spatial path following under different circumstances.

Position track errors in the horizontal plane.

Depth track errors in the vertical plane.

Surge speed errors under different circumstances.

Figure

From Figure

The simulation results obtained illustrate that the proposed methodology is effective and reduces the path following errors. Moreover, it is relatively simple to apply this proposed control in simulation.

The objective of this paper was to accurately follow a given path in the presence of systemic variations and ocean current. On one hand, three lightly interacting subsystems, including diving, steering, and speed control, were proposed to simplify the controller design for the spatial path following with 6-DOF nonlinear equations. On the other hand, those three controllers were designed to guarantee that all the error states in the spatial path following system were asymptotically stable by using adaptive neural network method. The simulation results illustrated that the proposed methodology was effective and attenuated the path following error under current. Future work will address the problems of path following under more common spatial curves. The problem of external disturbance about varying sea currents also warrants further research.

This work was supported by the Fundamental Research Funds for the Central Universities, under Grant HEUCF041330; the Fundamental Research Funds for the Central Universities of key laboratory’s open key; the National Natural Science Foundation of China, under Grant 51309067/E091002. And the authors also would like to thank the editor and three reviewers for their helpful comments.