Spatial Path Following for AUVs Using Adaptive Neural Network Controllers

Abstract

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.

1. Introduction

The research of AUV has been a hot topic in recent years with the development of marine robotics. Voluminous literature, for example [1–3], has been presented on the subject of designing path following controllers for AUVs. In order to design an automatic path following control system for AUVs, several problems must be solved. Among them the most difficult and challenging is that AUV’s dynamics are highly nonlinear and the hydrodynamic coefficients of vehicles are difficult to be accurately estimated a priori since the variations of these coefficients with different operating conditions. In addition to vehicle dynamics, Caharija et al. [4] pointed out that the sea currents affect the vehicles significantly which are the lack of actuation in sway. As a universal phenomenon, the single AUV’s spatial path following problem is a basis for formation coordinated control of multiple AUVs [5, 6].

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 [7–9]. And when it comes to system uncertainty, robust control could be used to attenuate disturbances in a relatively suboptimal extent. For example, load disturbance in the modelling of vehicles could be restricted to a satisfied expectation using the robust H_∞ PID controller [10]. Compared with robust control theory, adaptive strategy is an effective method of dealing with optimal control problems in vehicle control systems [11, 12]. Wang et al. [13] designed an adaptive PID controller for the path tracking system. From the simulation results, it was known that the proposed controller could not satisfy the tracking characteristics during automation performance. Based on Lyapunov stability theory and backstepping, Lapierre and Soetanto [14] proposed a path following controller for the motion control system of an AUV. It was demonstrated that the control characteristics of this kind of controller was relied on the accuracy of the hydrodynamic model. Chen and Wang [15] presented an adaptive control law with a parameter projection mechanism to track the desired vehicle longitudinal motion in the presence of tire-road friction coefficient uncertainties and actively injected braking excitation signals. The simulation results demonstrated that the proposed method was valuable for autonomous vehicle systems. A parameter-dependent adaptive H_∞ controller was constructed in [16] to guarantee robust asymptotic stability of the linear parameter-varying systems. And numerical examples were carried out to verify the effective impact on the attenuation in system disturbance. Li et al. [17] found an adaptive controller using a new neural network model, which was effective to improve the control precision by 30% in the case of system with random disturbance. To compensate the uncertainties in robot control systems, the radial basis functional (RBF) neural network was introduced to enhance system stability and transient performance [18].

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 2 formulates the vehicle dynamics for an underactuated AUV in the six-degree-of-freedom (6-DOF) form. Section 3 develops three adaptive neural network controllers to solve the path following problem with uncertain dynamics and external disturbance, such as sea currents. The proposed controllers’ stability is analysed by Lyapunov theory in this section. The simulation results using the proposed controllers are illustrated in Section 4. Finally, Section 5 contains the main conclusions and describes some problems that warrant further investigation.

2. Problem Formulation

2.1. Vehicle Dynamics

The dynamic model of the AUV in the three dimensional space is described in this section. See details in Bian et al. [19]. The vehicle which we studied in this paper measures 7.45×2.32×1.97 m. It is equipped with two main thrusters for propulsion, which are mounted symmetrically about its longitudinal axis in the horizontal plane. A cruciform tail including two different control surfaces is fixed right behind the thrusters to provide an enlarged torque around the transverse axis in the body fixed frame, which is helpful in enhancing the ability of spatial path following control. This vehicle is underactuated for the lack of propellers about its normal axis and transverse axis. The maximum designed speed of the vehicle with respect to the water is 3.08 m/s approximately. An outline of the vehicle with respect to the earth-fixed coordinate and body-fixed coordinate is shown in Figure 1.

Figure 1 

Employed coordinate frame systems.

According to the criteria underwater vehicle motion model in Fossen [20], this 6-DOF model can be described as follows.

Dynamic equation: 𝐌𝝂̇+𝐂(𝝂)𝝂+𝐃(𝝂)𝝂=𝝉. (1)

Kinematic equation: 𝜼̇=𝐉(𝜼)𝝂, (2) where 𝜼=[𝜼₁,𝜼₂]^𝑇, 𝝂=[𝝂₁,𝝂₂]^𝑇, 𝜼₁=[𝑥,𝑦,𝑧]^𝑇, 𝜼₂=[𝜑,𝜃,𝜙]^𝑇, 𝝂₁=[𝑢,𝑣,𝑤]^𝑇, and 𝝂₂=[𝑝,𝑞,𝑟]^𝑇.

The symbols 𝜑, 𝜃, 𝜙, 𝑝, 𝑞, and 𝑟 denote the roll, pitch, and yaw angles and velocities, respectively; 𝑥, 𝑦, 𝑧, 𝑢, 𝑣, and 𝑤 are the surge, sway, and heave displacements and velocities, respectively. The matrix 𝐉(𝜼)=diag{𝐉₁(𝜼₂),𝐉₂(𝜼₂)} is the transformation matrix from the body-fixed coordinated frame to the earth-fixed coordinated frame; 𝐌=𝐌^𝑇>0 is the mass and inertia matrix; 𝐂(𝝂) is the Coriolis and centripetal matrix; 𝐃(𝝂) is the damping matrix; 𝝉 is the control input, including force and moments generated by propellers and hydroplanes.

2.2. Spatial Path Following

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 Γ, the goal is to design some feedback control law which yields the control forces for the vehicle’s thrusters so that its centre of mass would converge asymptotically to a desired path by forcing its speed to track a desired speed assignment. The latter one can be described as below: considering the AUV depicted in Figure 2, where 𝑃 is an arbitrary point on the path and 𝑄 denotes its centre of mass. The objective is to design the controllers which force the vehicle’s position to converge to the desired path by driving the course angle and depth to converge to desired ones.

Figure 2 

Spatial path following control problem.

3. Controller Design

The objective is to realize the path following for AUVs in three dimensions. Consider Healey and Lienard [21], and according to practical operational applications in AUVs, the 6-DOF nonlinear equations of motion can be separated into three lightly interacting subsystems, including diving, steering, and speed control. Our research will focus on dealing with the problem of spatial path following through the three subsystems.

3.1. Speed Control

For control design purposes, the interactions from the other degrees of freedom is neglected; the speed control model could be given by ̇1𝑢=𝑚−𝑋_̇𝑢𝑋_𝑢𝑢𝑢²+𝑋_prop+𝑑_𝑢, (3) where 𝑋_̇𝑢 and 𝑋_𝑢𝑢 are dimensional hydrodynamic coefficients in surge; 𝑋_prop is thrusters’ force; 𝑑_𝑢 represents modelling inaccuracies and external disturbances.

Define 𝑓_𝑢=𝑋_𝑢𝑢𝑢²+𝑑_𝑢. (4)

Then, (4) can be rewritten as ̇1𝑢=𝑚−𝑋_̇𝑢𝑋_prop+𝑓_𝑢, (5) where 𝑋_prop is the forward force generated by the two main thrusters.

To deal with the uncertain terms, a RBF NN is chosen to estimate 𝑓_𝑢 which is described as 𝑓_𝑢=𝑊^∗_𝑢Φ_𝑢+𝜀_𝑢, (6) where 𝑊^∗_𝑢 is an optimized weight estimation of the neural network; Φ_𝑢 is the basis function; 𝜀_𝑢 is its estimation error. An identification diagram of the RBF neural network is shown in Figure 3.

Figure 3 

Diagram of RBF neural network for identification.

Assume 𝑋_prop=−𝑊_𝑢Φ_𝑢−𝑘_𝑢𝑢, (7) where 𝑘_𝑢>0; 𝑊_𝑢 is the weight estimation of the neural network.

Choose a Lyapunov function 𝑉_𝑢=12𝑢²+12𝑊²_𝑢, (8) where 𝑊_𝑢=𝑊_𝑢−𝑊_𝑢.

Equation (8)’s derivative can be calculated as 𝑉̇_𝑢=𝑢𝑢̇+𝑊_𝑢𝑊̇_𝑢=𝑢𝑢̇+𝑊_𝑢𝑊̇_𝑢𝑋=𝑢^𝑢𝑢𝑢²𝑚−𝑋_̇𝑢+𝑋_prop𝑚−𝑋_̇𝑢+𝑑_𝑢𝑚−𝑋_̇𝑢+𝑊_𝑢̇𝑊_𝑢=𝑢𝑚−𝑋_̇𝑢𝑋_𝑢𝑢𝑢²+𝑋_prop+𝑑_𝑢+𝑊_𝑢̇𝑊_𝑢=𝑢𝑚−𝑋_̇𝑢𝑊_𝑢Φ_𝑢+𝜀_𝑢+𝑋_prop+𝑊_𝑢̇𝑊_𝑢. (9)

Combined with (7) and (9), 𝑉̇_𝑢=𝑢𝑚−𝑋_̇𝑢𝑊_𝑢Φ_𝑢+𝜀_𝑢−𝑊_𝑢Φ_𝑢−𝑘_𝑢𝑢+𝑊_𝑢̇𝑊_𝑢=𝑢𝑊_𝑢Φ_𝑢𝑚−𝑋_̇𝑢+𝑢𝜀_𝑢𝑚−𝑋_̇𝑢−𝑘_𝑢𝑢²𝑚−𝑋_̇𝑢+𝑊_𝑢̇𝑊_𝑢. (10)

An adaptive law is designed as follows: 𝑊̇_𝑢=𝑊̇_𝑢=−𝑢Φ_𝑢𝑚−𝑋_̇𝑢+𝜆𝑊_𝑢−𝑊_𝑢0, (11) where 𝜆_𝑢>0; 𝑊_𝑢0 is the initial weight value of the neural network. Then (9) can be changed as 𝑉̇_𝑢=𝑢𝜀_𝑢𝑚−𝑋_̇𝑢−𝑘_𝑢𝑢²𝑚−𝑋_̇𝑢+𝑊_𝑢𝑢Φ_𝑢𝑚−𝑋_̇𝑢+̇𝑊_𝑢=𝑢𝜀_𝑢𝑚−𝑋_̇𝑢−𝑘_𝑢𝑢²𝑚−𝑋_̇𝑢+𝑊_𝑢𝜆_𝑢𝑊_𝑢−𝑊_𝑢0. (12)

Because 𝜀 is small enough, |𝑢|<3.1m/s, 𝑚−𝑋_̇𝑢≫0, and (𝑢/(𝑚−𝑋_̇𝑢))𝜀 can be considered as small positive constant.

In (12), 𝑊_𝑢𝜆_𝑢𝑊_𝑢−𝑊_𝑢0=𝑊_𝑢𝑊_𝑢𝜆_𝑢−𝑊_𝑢𝑊_𝑢0𝜆_𝑢𝜆=−^𝑢2𝑊_𝑢²_𝐹−𝜆2𝑊_𝑢−𝑊_𝑢0²_𝐹+𝜆_𝑢2𝑊_𝑢−𝑊_𝑢0²_𝐹. (13)

Based on Lyapunov stability theorem, it is provided that if one of the inequations in (14) is true, (15) would come true which guarantees the speed error converge to a small neighbourhood zero domain. 𝜆_𝑢2𝑊_𝑢²_𝐹≥𝜆_𝑢2𝑊_𝑢−𝑊_𝑢0²_𝐹,𝜆_𝑢2𝑊_𝑢−𝑊_𝑢0²_𝐹≥𝜆_𝑢2𝑊_𝑢−𝑊_𝑢0²_𝐹,𝑉̇_𝑢≤−𝛾_𝑢𝑉_𝑢+𝜎_𝑢, (14)(15) where 𝛾_𝑢>0; 𝜎_𝑢 is a small positive constant.

3.2. Diving Control

Consider the vehicle dynamics referred in Silvestre and Pascoal [22], and for the sake of diving control design, it is defined as follows: 𝑑𝑧=−𝑢₀(𝜃−sin𝜃)+𝑢₀𝐼−𝑢sin𝜃+𝑤cos𝜃,𝑏=_𝑦−12𝜌𝐿⁵𝑀^′_̇𝑞⁻¹,𝑓_𝑞1=𝑏2𝜌𝐿⁵𝑀^′_𝑞𝑞𝑞𝑞+12𝜌𝐿⁴𝑀^′_𝑢𝑞+1𝑢𝑞2𝜌𝐿³𝑀^′_𝑢𝑢𝑢²,𝑀_𝑇𝑦=12𝜌𝐿³𝑀^′_𝛿𝑠𝑢²𝛿_𝑠. (16)

If the surge speed is 𝑢=𝑢₀, the depth control model can be simplified as 𝑧̇=−𝑢₀̇̇𝜃+𝑑𝑧,𝜃=𝑞,𝑞=𝑓_𝑞+𝑏𝑀_𝑇𝑦,𝑌=𝑧. (17)

To be convenient for controller design, (17) can be rewritten as 𝜍₁̇𝜍₂̇𝜍₃̇𝑌=𝜍₂+𝑑𝜍₁𝜍₃−𝑢₀𝑓_𝑞+−𝑢₀𝑏𝑀_𝑇𝑦𝜍₁=𝑓𝜍₁,𝜍₂𝑓₂𝜍₃𝑓₃−𝑏^′𝑀_𝑇𝑦𝜍₁, (18) where 𝜍=[𝜍₁,𝜍₂,𝜍₃]^𝑇=[𝑧,−𝑢₀𝜃,−𝑢₀𝑞]^𝑇.

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 (18), a more simplified pure-feedback form is shown as follows: 𝜍₁̇𝜍₂̇𝜍₃̇=𝑓𝜍₁,𝜍₂𝑓₂𝜍₃𝑓₃−𝑏^′𝑀_𝑇𝑦. (19)

Step 1. Given a desired depth 𝜍_𝑑1, the depth error is described as 𝑍̇₁=𝜍₁−𝜍_𝑑1=𝑓₁𝜍₁,𝜍₂,𝑢−𝜍̇_𝑑1. (20)

Define a virtual control variable: 𝑣₁=−𝜍_𝑑1+𝐾₁𝑍̇₁. (21)

From 𝜕𝑣₁/𝜕𝜍₂=0, we can know 𝜕𝑓₁𝜍₁,𝜍₂,𝑢+𝑣₁𝜕𝜍₂>𝜀>0. (22)

When we take 𝜍₂ as the input of the virtual control variable, there must exist a 𝜍₂=𝛼^∗₁(𝜍₁,𝑣₁,𝑢) satisfying 𝑓₁𝜍₁,𝛼^∗₁,𝑢+𝑣₁=0. (23)

Referring to the mean value theorem of Lagrange, 𝛾₁  (0<𝛾₁<1) can be found which yields 𝑓₁𝜍₁,𝜍₂,𝑢=𝑓₁𝜍₁,𝛼^∗₁,𝑢+𝛾₁𝜍₂,𝛼^∗₁, (24) where 𝛾₁∶=𝑔₁(𝜍₁,𝜍_2𝛾1,𝑢), 𝜍_2𝛾1=𝛾₁𝜍₂+(1−𝛾₁)𝛼^∗₁.

Combined with (21) and (24), (20) can be rewritten as 𝑍̇₁=−𝐾₁𝑍₁+𝛾₁𝜍₂−𝛼^∗₁. (25)

Then 𝛼^∗₁ can be estimated by RBF neural network as follows: 𝛼^∗₁=𝑊^∗𝑇Φ𝑍₁+𝜀. (26)

Consider 𝑍₂=𝜍₂−𝛼₁ and 𝛼₁=−𝑐₁𝑍₁+𝑊^𝑇Φ(𝑍₁), where 𝑊 is to be the estimation of 𝑊^∗. Then (25) becomes 𝑍̇₁=−𝐾₁𝑍₁+𝛾₁𝑍₂+𝛼₁−𝛼^∗₁=−𝐾₁𝑍₁+𝛾₁𝑍₂−𝑐₁𝑍₁+𝑊^𝑇Φ𝑍₁−𝜀, (27) where 𝑊=𝑊−𝑊^∗.

Choose a Lyapunov function: 𝑉₁=12𝛾₁𝑍²₁+12𝑊^𝑇Γ₁𝑊. (28)

The derivative of (28) can be calculated as 𝑉̇₁=−𝐾₁𝛾₁𝑍²₁+𝑍₁𝑍₂−𝑐₁𝑍²₁−𝛾₁2𝛾²₁𝑍²₁−𝑍₁𝜀+𝑊^𝑇Φ𝑍₁𝑍₁+𝑊^𝑇Γ₁𝑊. (29)

With the adaptive law, 𝑊̇=−Φ𝑍₁𝑍₁−𝜎₁𝑊⋅Γ₁, (30) where 𝜎 is a small positive constant.

Step 2. Consider 𝑍₂=𝜍₂−𝛼₁⟹𝑍̇₂=𝑓₂𝜍₃=𝜍₃. (31)

It is assumed that 𝛼₂=−𝑍₁−𝑐₂𝑍₂+𝛼^∗₁. (32) Choose another Lyapunov function: 𝑉₂=𝑉₁+12𝑍²₂. (33) The derivative of (33) can be calculated as 𝑉̇₂=−𝐾₁𝛾₁𝑍²₁−𝑐₁𝑍²₁−𝜎𝑊²2+𝜎𝑊^∗22+𝜀²4𝑐₁−𝑐₂𝑍²₂+𝑍₂𝑍₃. (34)

Step 3. Define 𝜍₃=𝑍₃−𝛼₂.

It can be calculated as 𝑍̇₃=𝑓_𝑞−𝑏^′𝑀_𝑇𝑦−𝛼̇₂. (35)

And, we can obtain 𝑀^∗_𝑇𝑦=−𝑍₂−𝑐₃𝑍₃−1𝑏^′𝛼̇₂−𝑓_𝑞. (36)

Then the unknown term (1/𝑏^′)𝑓_𝑞 can be estimated by RBF NN, and the ideal optimal control law can be written as 𝑀^∗_𝑇𝑦=−𝑍₂−𝑐₃𝑍₃+𝑊^∗₂Φ𝑍₃−𝛼̇₂𝑏^′+𝜀₂. (37)

Finally, we obtain the actual control input: 𝑀_𝑇𝑦=−𝑍₂−𝑐₃𝑍₃−𝛼̇₂𝑏^′+𝑊₂Φ𝑍₃. (38)

From (35)–(37), it can be derived that 𝑍̇₃=−𝐾₃𝑍₃−𝑏^′−𝑍₂−𝑐₃𝑍₃−𝛼̇₂𝑏^′+𝑊^𝑇₂Φ𝑍₃−𝜀₂. (39)

Consider a Lyapunov function: 𝑉₃=𝑉₂−12𝑏^′𝑍²₃+12𝑊^𝑇₂Γ₂𝑊₂. (40)

We can obtain its derivative 𝑉̇₃=𝑉₂−𝑍₂𝑍₃−𝑐₃𝑍²₃+𝑊₂Φ𝑍₃𝑍₃+𝑊^𝑇₂Γ₂𝑊̇₂, (41) where 𝑊̇₂=Γ₂[−Φ(𝑍₃)𝑍₃−𝜎₃𝑊₂] and 𝜎₃>0.

As what we did in Step 1, (41) can be calculated as 𝑉̇₃<−³_𝑗=1𝑐^∗_𝑗0𝑍²_𝑗−³_𝑗=1𝜎_𝑗𝑊_𝑗2+³_𝑗=1𝜎_𝑗𝑊^∗_𝑗2+³_𝑗=1𝜀_𝑗2, (42) where 𝑐^∗₃₀∶=𝑐₃₀>0.

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.

3.3. Guidance Law and Steering Control

3.3.1. Line-of-Sight Guidance

Referring to Fossen [20] and Oh and Sun [23], we briefly introduce Line-of-Sight guidance law in this section for path following in the horizontal plane and discuss its application for straight lines and circular arcs.

Calculate the angle between the proposed path and the north of earth-fixed coordinate in Figure 4: 𝛼_𝑘∶=arctan2𝜂_𝑘+1−𝜂_𝑘,𝜉_𝑘+1−𝜉_𝑘∈[−𝜋,𝜋]. (43)

Figure 4 

Principle of LOS guidance for straight line.

Considering the cross track error 𝑒_𝛾(𝑡) and a look-ahead distance Δ, the desired course angle for the steering control system can be computed as 𝛾_𝑑=𝛼_𝑘+arctan−𝑒_𝛾Δ=𝛼_𝑘+𝛾_𝑒, (44) where Δ=𝑛𝐿(𝑛=2∼5).

When it comes to the circular arcs in Figure 5, we can obtain the guidance law just as (44) in form.

Figure 5 

Principle of LOS guidance for circular arc.

3.3.2. Steering Controller Design

Referring to the vehicle dynamics in horizontal plane in [24] and considering the advantage of steering controller design, we define 𝑋₁=𝜓,𝑋₂=𝜓̇. (45)

The kinetics model of the AUV in horizontal plane can be simplified as 𝑋̇₁𝑋̇₂𝑌_𝜓=𝑋₂𝑓_𝜓(𝜐)+𝑏_𝜓𝛿_𝑟+𝑑_𝜓𝑋₁, (46) where 𝜐∶=𝑔_𝜓(𝑢,𝑣,𝑟) includes the nonlinear terms with 𝑢, 𝑣, and 𝑟 in the steering equation; 𝑑_𝜓 is the disturbance satisfied with |𝑑_𝜓|≤𝜀₀, 𝜀₀>0.

Consider a heading track error: 𝑒_𝜓=𝑌_𝜓−𝑌_𝜓𝑑, (47) where the desired course angle 𝛾_𝑑=𝑌_𝜓𝑑.

Choose 𝐾∈𝑅 which makes (48) a stabilized system: ℎ(𝑠)=𝑠+𝐾. (48)

Then the derivative of the steering error system can be written as follows: 𝑒_𝜓̇=𝐀𝑒_𝜓+𝐁𝑓_𝜓(𝜐)+𝑏_𝜓𝜏−𝑌̈_𝜓𝑑+𝐊𝑒_𝜓+𝑑_𝜓, (49) where 𝐴=−𝐾 and 𝐵=1.

Then we can find 𝑃>0 and 𝑄≥0 which make the following Lyapunov stable equation solvable: 𝐴^𝑇𝑃+𝑃𝐴=−𝑄. (50)

On the assumption that 𝑓_𝜓(𝜐) and 𝑏_𝜓 are known and 𝑑_𝜓=0, a linear controller can be obtained as (51) using pole-assignment method: 𝑢_𝜓=−𝑏_𝜓⁻¹𝑓_𝜓(𝜐)−𝑌̈_𝜓𝑑+𝐊𝑒_𝜓. (51)

Combined with (46), we can find 𝑒_𝜓̇+𝐊𝑒_𝜓=0. (52)

For 𝐾 is chosen appropriately according to (48), it can be derived that lim_𝑡→∞𝑒_𝜓(𝑡)=0.

In fact, 𝑓_𝜓(𝜐) and 𝑏_𝜓 are uncertain, and 𝑑_𝜓 does exist. To deal with the uncertain terms, a RBF NN is chosen to estimate 𝑓_𝜓(𝜐): 𝑓_𝜓(𝜐)=𝑊^∗Φ(𝜐)+𝜀_𝜓, (53) where 𝑊^∗ is an optimized weight estimation of the neural network; Φ(𝜐) is a vector of Gaussian function Φ(𝜐)=exp(‖𝜐−𝑐₀‖²/𝑏₀²), 𝑐₀∈𝑅 is its centre, and 𝑏₀ is the width of the basis function.

If the weight estimation of neural network 𝑊 is uniformly bounded, a positive constant 𝑤 can be found, which satisfies ‖𝑊‖<𝑤₀.

For 𝜀_𝜓 is the estimation error of the RBF NN, then 𝑓_𝜓(𝜐)−𝑓_𝜓(𝜐)=𝜀_𝜓. (54)

Meanwhile, consider a weight error for the RBF NN: 𝑊=𝑊^∗−𝑊. (55)

Combined with (49), (54), and (55), it can be calculated as 𝑒_𝜓̇=𝐴𝑒_𝜓+𝐵𝑓_𝜓(𝜐)+𝑏̂_𝜓𝛿_𝑟−𝑌̈_𝜓𝑑+𝐾𝑒_𝜓+𝑓_𝜓(𝜐)−𝑓_𝜓(𝜐)+𝑏_𝜓−𝑏̂_𝜓𝛿_𝑟+𝑑_𝜓(𝑡)=𝐴𝑒_𝜓+𝐵𝑓_𝜓(𝜐)+𝑏̂_𝜓𝛿_𝑟−𝑌̇_𝜓𝑑+𝐾𝑒_𝜓+Δ𝑓_𝜓+𝑊𝜙(𝜐)+𝑏̃_𝜓𝛿_𝑟+𝑑_𝜓. (56)

In order to compensate for estimation error and current disturbance as shown in (56), a virtual control input described as (17) is introduced as 𝐵𝜈=−𝜅^𝑇𝑃𝑒_𝜓𝐵^𝑇𝑃𝑒_𝜓+𝜆_𝜓, (57) where 𝜅=𝑃+𝜀_𝜓 and 𝜅≥|Δ𝑓_𝜓|+|𝑑_𝜓|; 𝜆_𝜓 is a small positive constant.

Herein, (56) can be rewritten as follows: 𝑒_𝜓̇=𝐴𝑒_𝜓+𝐵Δ𝑓_𝜓+𝑊𝜙(𝜉)+𝑏̃_𝜓𝜏_𝜓+𝑑_𝜓+𝜈. (58)

Introduce adaptive laws: 𝑊̇=𝛾₁Φ(𝜉)𝐵𝑃𝑒_𝜓,𝑏̇̂_𝜓=𝛾₂𝜏_𝜓𝐵𝑃𝑒_𝜓, (59) where 𝛾₁ and 𝛾₂ are positive constants.

To deal with the problem of stability, a Lyapunov function (60) is chosen to guarantee the proposed adaptive NN controller satisfying that the signals in the steering control system are bounded: 𝑉_𝜓=12𝑃𝑒²_𝜓+12𝛾₁𝑊²+12𝛾₂𝑏̃²_𝜓. (60)

The differentiation of (60) can be calculated as 𝑉̇_𝜓=𝑒_𝜓𝑒_𝜓̇𝑃+1𝛾₁𝑊̇1𝑊+𝛾₂𝑏̃_𝜓𝑏̃_𝜓̇=𝑒_𝜓𝑃𝐵Δ𝑓_𝜓+𝑑_𝜓+𝑊𝜙(𝜐)+𝑏̃_𝜓𝛿_𝑠+1+𝜈𝛾₁𝑊̇1𝑊+𝛾₂𝑏̃_𝜓𝑏̃_𝜓̇. (61)

With a combination of (59) and (61), it can be derived that 𝑉̇_𝜓=𝑒_𝜓𝑃𝐵Δ𝑓_𝜓+𝑑_𝜓+𝜈. (62)

Moreover, because 𝑒^𝑇_𝜓𝑃𝐵Δ𝑓_𝜓+𝑑_𝜓≤𝐵+𝜈_𝑇𝑃𝑒_𝜓Δ𝑓_𝜓+𝑑_𝜓+𝑒^𝑇_𝜓=𝐵𝑃𝐵𝜈_𝑇𝑃𝑒_𝜓Δ𝑓_𝜓+𝑑_𝜓𝐵−𝜅^𝑇𝑃𝑒_𝜓²𝐵^𝑇𝑃𝑒_𝜓+𝜆_𝜓≤𝐵^𝑇𝑃𝑒_𝜓Δ𝑓_𝜓+𝑑_𝜓𝐵−𝜅^𝑇𝑃𝑒_𝜓²𝐵^𝑇𝑃𝑒_𝜓+𝜆_𝜓 (63) and ‖𝐵^𝑇𝑃𝑒_𝜓‖²/(‖𝐵^𝑇𝑃𝑒_𝜓‖+𝜆_𝜓)>‖𝐵^𝑇𝑃𝑒_𝜓‖−𝜆_𝜓 are true, (62) can be calculated as 𝑉̇_𝜓≤𝐵^𝑇𝑄𝑒_𝜓Δ𝑓_𝜓+𝑑_𝜓−𝜅𝐵^𝑇𝑃𝑒_𝜓−𝜆_𝜓. (64) It is known that 𝜅≥|Δ𝑓_𝜓|+|𝑑_𝜓|; then we can obtain 𝑉̇_𝜓≤−𝜅𝜆_𝜓≤0. (65)

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 𝑢_𝜓=−𝑏̂_𝜓⁻¹𝑓_𝜓(𝜐)−𝑌̇_𝜓𝑑+𝐊𝑒_𝜓−𝜈. (66)

4. Simulation Results

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 𝑇=0.1 s (seconds). Considering jacket healthy state detection which is a regular task for offshore platform, a spiral three-dimensional path is programmed to complete the detection job. In order to fulfil this mission successfully, the control objective is going to achieve a high tracking precision with the proposed control method.

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 (𝑥,𝑦,𝑧)=(0m,35m,0.5m) with an unspecified attitude (𝜑,𝜃,𝜓)=(0^∘,0^∘,90^∘). The desired forward speed is 1.8 m/s. The gains and parameters for the adaptive neural network speed controller are 𝜆_𝑢=1.2, 𝑘_𝑢=2.5, and 𝛾_𝑢=1.8, while the ANN steering controller’s initial values are set as follows: 𝑐₁=5, 𝑐₂=12, 𝑐₃=10, 𝑏=3.8386𝑒−005, Γ₁=Γ₂=diag{1.5}, and 𝜎₁=𝜎₃=0.5. The parameters for the diving control system are chosen as 𝐾=2.8, 𝜆_𝜓=0.1, 𝛾₁=2, and 𝛾₂=1.5. And the initial weights of RBF NN for the three subsystems are chosen as zero.

Figures 6–9 show the simulation results for spatial path following between different ocean circumstances. It is shown that the proposed mission under disturbance of current or not could be achieved by the designed adaptive neural network controllers. It is clear that the proposed method is suitable to follow the spatial path with a random position and attitude, which is very practical in jacket healthy state detection mission.

Figure 6 

Spatial path following under different circumstances.

Figure 7 

Position track errors in the horizontal plane.

Figure 8 

Depth track errors in the vertical plane.

Figure 9 

Surge speed errors under different circumstances.

Figure 6 is the response of spatial path following under different circumstances in three-dimensional space, where 𝑆 means the start point and 𝐸 is the end of the mission. We can see that the performances in the two simulations are good in general. It can be seen that the two tracks are identical with the same initial values for controllers’ compensation to the uncertain dynamics and ocean current.

From Figure 7, it is shown that the position track errors during the jacket detection missions are gradually converged to zero. Combined with Figures 7 and 8, we can clearly see that although the overshoots have a little increase at the 67th second and the 54th one, which indicate the presence of ocean current in the process, the errors decrease to zero very quickly using the proposed controllers, which was designed to guarantee the errors in the spatial path following systems to be restricted to a small value gradually. In the meanwhile, when it is compared to the surge speed responses under different circumstances in Figure 9, it is evident that the fluctuant speed is the primary reason responsible for the overshoots in the spatial path following.

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.

5. Conclusions

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.

Acknowledgments

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.