The aim of this paper is to develop a fuzzy iterative sliding mode control (FISMC) scheme for special autonomous underwater vehicles (AUVs) on three-dimensional (3D) path following. In this paper, the characteristics of the AUV are considered, which include a large scale, large inertia, and high speed. The FISMC controller designs iterative sliding mode surfaces by using a hyperbolic tangent function to keep the system with fast convergence and robust performance. At the same time, system uncertainties and environmental disturbances are taken into account. The control algorithm introduces fuzzy control to optimize the control parameters online to enhance the adaptability of the system and inhibit the chattering of the actuators. The performance of the proposed FISMC is demonstrated with numerical simulations.
In the last decades, AUVs have attracted more and more attention for their multifunctions and practicability in many fields. They have been used in exploring resources, graphic mapping, underwater pipelines inspection, seabed surface reconstruction, and so on [
Before choosing a reliable control scheme, it is essential to comprehend the features of the AUV. Firstly, a traditional AUV is an underactuated and second-order nonholonomic system [
Various researches have been carried out for AUVs’ control, such as adaptive control, suboptimal control, backstepping control, genetic control, neural network control, and fuzzy control. Different control algorithms have their advantages and disadvantages. For the adaptive control [
As sliding mode control is based on the sliding mode surfaces, the stability and rapidity are evident if the Lyapunov equation is right established. Hence, sliding mode control becomes popular. Furthermore, sliding mode control is with strong robustness against model uncertainties and external environmental disturbances [
In this paper, based on the advantages of each control algorithm introduced above and then doing further optimizing, a novel fuzzy iterative sliding mode control scheme is presented which introduces a hyperbolic tangent function in multiple sliding mode surfaces and control laws and uses fuzzy logic to tune the parameters automatically. Due to the boundedness of the hyperbolic tangent function, the controller will also be bounded. It is concluded that the stability, the accuracy, and the rapidity of the system can be in favorable control. In this way, FISMC is more suitable for engineering applications.
The rest of this paper is organized as follows: Section
The purpose of modeling for the AUV is to establish kinematic and kinetic equations. The kinematic model deals with geometrical aspects of motion, while the kinetic model focuses on the active and passive forces and moments about the vehicle [
As shown in Figure
Definition of coordinate frames.
The AUV has neutrally buoyant and hydrodynamic restoring forces which are large enough to neglect the roll motion in 3D space; the model of the AUV can be described as following the kinetic and kinematic model.
The detailed descriptions of the model and parameters in formulas (
The AUV in this paper is characteristic with a large scale, large inertia, and high speed. With these special features, controller designing of this AUV is much more difficult and of higher requirements in terms of stability, accuracy, and rapidity.
For example, in order to drive the vehicle and converge to a predefined path, there are so many parameters by using the PID controller, such as input variable, error of present and desired heading angle, and output variable and control-rudder angle. In addition, the parameters in the PID controller are sensitive to the AUV’s initial position, initial attitude, environmental disturbances, and so on. The drawback of the PID controller applied for this AUV will be illustrated in detail by simulation results in Section
The 3D path following problem deals with the design of control law which could control this kind of special AUV with a large scale, large inertia, and high speed to reach a desired path stably, accurately, and rapidly. The control objectives are as follows:
Deriving a control law so that the position error
In this section, the two-layered control framework is established for 3D path following of the AUV. As presented in Figure
Control framework of 3D path following.
The input of the control system is the desired path
In this subsection, the LOS guidance law for path following is presented. The 3D motion of the AUV can be divided into motions in two different planes which are the horizontal plane and vertical plane, respectively.
Control objectives of path following need to meet the following two requirements [ Decreasing the offset distance between the AUV’s position and the desired position to zero Aligning the direction between the AUV’s velocity vector and the desired path’s tangential vector
As shown in Figure
Path following in 3D space.
The line-of-sight (LOS) guidance angle in the horizontal plane named azimuth angle
Analyzing Figure
As
Similarly, the elevation angle
As previously mentioned, the PID control scheme lacks robustness and could not keep the accuracy while guaranteeing the rapidity. To solve this problem, a fuzzy iterative sliding mode controller based on the LOS guidance law is designed in this subsection. More than one of the sliding mode nonlinear surfaces settle the contradiction between static performance and dynamic performance of the control system which mainly refer to errors of the steady state and time of approaching the steady state, respectively. In this way the controller can ensure the robustness and the adaptability, despite of system uncertainties and environmental unknown disturbances. Next, application of fuzzy logic heuristic knowledge further enhances the self-adaptability which makes it more convenient to tune the parameters of the controller automatically.
Consider the nonlinear scalar system with zero order defined as The equation of state is Sign for the gain of output
the incremental feedback control law is chosen as
Based on the proposed conditions and control law above, the system
Choose the following Lyapunov function candidate:
Taking the derivative of formula (
By using formulas (
Therefore, it can be concluded that the system
According to the features of the AUV in this paper, the revolving speed of the propeller is fixed. Aiming at the actuators of the AUV, yaw rudder, and horizontal rudder, the controllers used for the horizontal plane and vertical plane are designed, respectively.
In the horizontal plane, iterative sliding mode control surfaces are given as follows:
When applying to the controller with defined surfaces (
Expanding formula (
Analyzing formula (
Due to the surface
Substituting formula (
Next, analyzing formulas (
On one hand, the signs of velocities
Go on analyzing formulas (
Considering the case of steering yaw rudder in the horizontal plane, the values of
Based on formula (
As
As the values of
Obviously,
Because of the characteristic of the AUV which voyages forward at a fixed speed, that is to say,
Based on Theorem
Due to the uniform monotonicity of
Therefore, the errors of path following for the AUV asymptotically convergence to zero.
In the vertical plane, iterative sliding mode control surfaces and control law are given as follows:
When applying to the controller with defined surfaces and control law (
The proving process can be in a similar method and it is omitted.
Based on ISMC, the FISMC focuses on the self-adaptability of control parameters which guarantee the AUV with environmental disturbance rejection and inhibition of chattering for the rudder.
Analyzing the control laws
The fuzzy logic rules for the parameters
Fuzzy logic rules for
| | |||||||
---|---|---|---|---|---|---|---|---|
NB | NM | NS | Z | PS | PM | PB | ||
| NB | NB | NB | NB | NB | NB | NB | NB |
NM | NM | NM | NM | NM | NM | NM | NM | |
NS | PS | PS | PS | PB | PS | PS | PS | |
Z | PS | PM | PS | Z | PS | PM | PS | |
PS | PS | PM | PS | PB | PS | PM | PS | |
PM | NM | NM | NM | NM | NM | NM | NM | |
PB | NB | NB | NB | NB | NB | NB | NB |
Fuzzy logic rules for
| | |||||||
---|---|---|---|---|---|---|---|---|
NB | NM | NS | Z | PS | PM | PB | ||
| NB | NB | NB | NB | NB | NB | NB | NB |
NM | NM | NM | NM | NM | NM | NM | NM | |
NS | PS | PS | PS | PB | PS | PS | PS | |
Z | PS | PM | PS | Z | PS | PM | PS | |
PS | PS | PM | PS | PB | PS | PM | PS | |
PM | NM | NM | NM | NM | NM | NM | NM | |
PB | NB | NB | NB | NB | NB | NB | NB |
The fuzzy subsets are divided into traditional types which are NB, NM, NS, ZE, PS, PM, and PB, respectively, namely, negative big, negative medium, negative small, zero, positive small, positive medium, and positive big. The input scaling factors of
Membership function.
Numerical simulation researches are performed to test the performance of the proposed control schemes in this paper through a special AUV whose key parameters are partly presented in Table
Key parameters of the AUV.
Parameter name | Parameter signal | Value |
---|---|---|
Mass | | |
Length | | |
Breadth | | 6 m |
Height | | 6 m |
Displacement | | |
Speed | | |
In the simulations, the delay time between the control signal and the actuators is considered according to the large-inertia feature of the AUV. The rudder control signals lag behind the yaw rudder and the horizontal rudder with 100 milliseconds, and the sample time in numerical simulation is 1 second.
It should be noted that the AUV mentioned in this paper has two water tanks, and the mass described in Table PID parameters: ISMC parameters: FISMC parameters:
Figures
Paths in 3D space.
Paths in the horizontal plane and vertical plane.
Yaw rudder.
Horizontal rudder.
Heading angles and pith angles.
Sliding mode surfaces.
Sliding mode surfaces.
In Figure
More details can be noticed in Figure
Focusing on the status of steering rudders from Figures
The changes of the desired angles
Finally, regarding Figure
In this part, simulation is carried out under the environmental disturbance which is added from multiple directions in 3D space. Referring to formula (
The time-varying disturbances which are an 11-ton force at maximum occur in the process of rising stage, 100 ~ 200 s, and steady state, 1200 ~ 1300 s. Compared with ideal conditions, simulation under environmental disturbances achieves better testing of three different controllers.
Glancing at Figure
Paths in 3D space.
Paths in the horizontal plane and vertical plane.
Making a comparison with Figures
It can be basically estimated that FISMC provides the best performance when applying for path following of the AUV with large inertia. But in many cases, the AUV cannot start at zero initial conditions. For example, the initial heading angle of the AUV may not be zero absolutely. This comes to a question: If the special AUV is with the different initial heading angle,
Paths in the horizontal plane with PID controller.
Paths in the horizontal plane with ISMC.
Paths in the horizontal plane controlled by ISMC and FISMC.
Figure
As Figure
And making further improvement, the simulation used FISMC are carried out to be in contrast with ISMC. In Figure
In general, applying fuzzy iterative sliding mode control to the special AUV with a large scale, large inertia, and high speed, the performance can be guaranteed. Particularly when considering environmental disturbances and different initial state, the advantages of FISMC are evidently demonstrated.
This paper presented a novel fuzzy iterative sliding mode control scheme for the mentioned AUV. On the basis of previous research, the modeling of the AUV is introduced firstly, along with a problem formulation. Guidance laws for path following are then proposed applying the line-of-sight scheme. In this way, iterative sliding mode controller can be designed. Meanwhile, considering the self-adaptability of the control parameters, fuzzy logic is added on ISMC and then to form the FISMC. Finally, the robustness and self-adaptability of FISMC are verified through representative simulations. The AUV is able to complete path following while it is under environmental disturbances and various initial states.
Focusing on the results, the FISMC method can be applied to an underactuated vehicle with large-scale, large-inertia, and high-speed characters, in 2D and 3D path following, such as autopilot of oil tankers and cargo ships.
Partial data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.