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This paper addresses the problem of composite curve path following for an underactuated autonomous underwater vehicle by utilizing an adaptive integral line-of-sight (AILOS) guidance and nonlinear iterative sliding mode (NISM) controller. First, the composite curve path is parametrized by a common scalar variable in a continuous way. Then, the kinematics error of an underactuated vehicle is described based on the nonprojection Frenet–Serret frame with a virtual point, which can be eliminated by the virtual point control and AILOS guidance. Meanwhile, the subpath switching algorithm is studied to realize the global path following for the composite curve path. Besides, the NISM controller is cascaded with the AILOS guidance law, and the cascade structure proved to be globally

In recent years, much research has been done in the field of path following for autonomous underwater vehicles (AUVs). In most of the existing literature, the desired geometric path is a single curve [

It is known that the shape and the properties of the composite curve path have a great inﬂuence on the path following control. Generally, the composite curve path is obtained by a path planning algorithm with two steps. Firstly, utilizing a path search algorithm, the given order of waypoints is obtained based on certain optimization objectives. Then, considering the kinematics constraints of the vehicle, multiple curves (such as straight lines, circular arcs, spiral lines, and polynomial curves) can be used to connect all the waypoints to generate a flyable path [

As the expressions of curve segments vary, and there is no unified parametrization for the composite curve path, they will be inconvenient to calculate and expand. Hence, for better tracking of the composite curve path, it is necessary to choose an appropriate path description method. To solve this problem, all the curve segments are parametrized in a continuous way in this paper. Besides, each curve segment is parametrized by a common scalar variable with the same interval. However, it is usually difficult for the composite curve path to satisfy parametric continuity between the curve segments. Hence, to realize the global path following of the composite curve path, the problem of subpath switching [

After the composite curve path has been designed and parametrized, an efficient path following system is proposed. For underactuated AUVs, which have no independent control input in the sway and heaven direction, the line-of-sight (LOS) guidance principle is a very suitable and efficient solution for path following. When combined with guidance, the position and heading can be controlled simultaneously just by the heading control. Hence, utilizing LOS guidance, this has been no longer an underactuated problem in terms of the variables to be controlled [

Generally, the LOS guidance laws are intuitive as they are formulated at a kinematics level without using the vehicle parameters. Through a cascaded system approach [

In this paper, a new composite curve path following controller is proposed for an underactuated AUV, based on nonlinear iterative sliding mode (NISM) controller [

To eliminate the tracking error, a cascade structure is established based on AILOS guidance law and a NISM controller, which proved to be UGAS (uniformly globally asymptotically stable).

Based on the AILOS guidance law and the subpath switching algorithm, the global path following of the composite curve path in the kinematics layer is realized with the unknown ocean currents.

The NISM control is proposed in the dynamics layer, which has the characteristics of fast convergence and strong antidisturbance ability. In addition, the strictly bounded nonlinear hyperbolic tangent function is used to avoid excessive control input caused by the discontinuity at each connection point between two subpaths. Besides, the incremental feedback control law is designed, which is independent of hydrodynamic parameters.

The remainder of this paper is organized as follows. Section

The composite curve path can be defined as a set of curves connected in a specific order. Each curve segment is regarded as a subpath. The parametrization and the continuity of the composite curve path are discussed in this section. Besides, the parametric description is proposed for typical curves.

In complex and limited marine environment, the composite paths can meet the demands of varied tasks flexibly by adjusting the type of subpaths. However, as the expressions of subpaths may vary, they will be inconvenient to calculate and hard to expand for the path following of composite curve path.

To handle this problem, the path parametrization method is adopted to describe the composite curve path. Many parametrization methods can be used to describe a path, which may be continuous, discrete, or even hybrid [

For a continuous parametrization, the composite curve path

The path is then simplified by the set

With a continuous parametrization, for any given parameters of the composite curve path, the position of the corresponding point can be determined uniquely.

Regular curves are desired for the subpaths, which means that such paths never degenerate into a point nor do they have corners. Specifically, these curves include both straight lines and circles [

The first derivative of

The path-tangential angle (or course angle of the path) is computed as

The curvature of the path can be calculated as

The angular speed of the path can be described as

The smoothness of the desired path has an essential impact on the motion control of the underactuated vehicle. Moreover, two notions can be used to describe the path smoothness, namely, the geometric continuity (GC) and the parametric continuity (PC) [

GC is denoted by

Similarly, PC is denoted by

Compared with GC, PC is a stricter form of continuity which imposes constraints on how the parameter propagates along the path. Moreover, PC is a measure of smoothness for parametrizations.

From (

Some common composite curve paths can be used as examples (see Table

Common composite curve paths.

Path | GC | Comments |
---|---|---|

Piecewise linear path | Generated directly through waypoints but is not suitable for path following of underactuated AUV | |

Circular smoothing | Generated by approximating methods, and curvature between straight and circular segments is discontinuous | |

Dubins path | Generated by interpolating methods, and curvature between straight and circular segments is discontinuous | |

Clothoid smoothing | Generated by approximating methods, linear varying curvature with an increased computational cost | |

Fermat’s spiral smoothing | Generated by interpolating methods, curvature-continuous paths with a very low computational cost compared to clothoid smoothing [ |

Based on [

Furthermore, the parametrization of circular arcs can be expressed as

To avoid singularity, the parametrization of Fermat’s spiral can be described as

For the mirrored curve of Fermat’s spiral, the following parametrization is proposed:

REMUS vehicles are low-cost AUVs designed by the Woods Hole Oceanographic Institution serving in a range of oceanographic applications, such as surveying and mapping. The vehicles are torpedo-shaped and underactuated without lateral thrust; a propeller and fins are used for steering and diving. Besides, the mathematical model of REMUS vehicles has been well researched, which can be used in motion control simulation of underactuated AUVs [

This section describes the kinematics and dynamics expressions of the REMUS vehicles and problem formulation of the composite curve path following.

Considering the influences of currents, the kinematics model in the horizontal plane can be expressed in terms of the relative surge and sway velocities [

As depicted in Figure

The kinematics description of AUVS in the horizontal plane.

The relative surge and sway velocity can be defined as

Besides, the relative resultant velocity can be expressed as

The REMUS vehicle considered in this paper is based on the following assumptions.

The vehicle has two axial planes of symmetry, top-bottom and port-starboard symmetry, respectively.

The vehicle center of gravity is the same as the vehicle center of buoyancy, and the origin of the vehicle body-fixed coordinate system is located at the vehicle center of buoyancy.

Neglecting the motions in heave, roll, and pitch directions, the 3-DOF dynamics model of REMUS vehicle in the horizontal plane can be simplified as

Parameters of the REMUS AUV.

Parameter | Value |
---|---|

As depicted in Figure

The kinematics description of path following in horizontal plane.

The tracking error expressed in

Differentiating (

Generally, the problem of the composite curve path following for underactuated AUV can be formulated as follows:

Given a constant thrust

To eliminate the tracking error under the influence of unknown static currents, a new path following controller is proposed with cascade structure, as shown in Figure

The cascade structures with a current observer.

First, the designed composite curve path is parametrized by a common scalar variable in a continuous way, and a point

The path-tangential speed of the virtual point can be used as a control input [

Substituting (

Simultaneously, adaptive ILOS guidance law is used to derive a desired heading angle to eliminate the cross-track error. As illustrated in Figure

As the heading error can be expressed as

Substituting (

Then, (

where

To eliminate the influence of the currents,

As the currents considered in this paper are much smaller than the speed of the vehicle, it is easy to conclude that

The first derivative of the kinematics error can be described as

Assume that

By substituting

However, currents are generally difficult to measure. To eliminate the influence of the unknown currents, the current observer is designed to predict the currents. The currents can be described as

The current observer can be designed as [

Assume that the currents are changing slowly relative to time, which means

The current observer (

Consider the following Lyapunov function candidate:

When

The time derivative of

Substituting of (

Consequently, the equilibrium point

The path-tangential speed

It can be seen that

Besides, as each curve segment with different characteristics is parametrized with the same parameter interval, the length of the curve segments will have a significant impact on the value of

The parameter

When

Besides, an additional path switching mechanism can be introduced to achieve better tracking effect at path switching, especially for the piecewise linear path with

The subpath switching algorithm of the composite curve path.

Generally, for composite curve path following, each subpath switching can be regarded as the beginning of tracking a new path initialized by the current state of the AUV.

To obtain the desired heading angle given by the guidance law, the heading control system is designed to make the heading error converge to zero.

A non-model-based NISM control algorithm is utilized to realize the heading control. The nonlinear sliding surfaces are designed as

To stabilize

Different from the standard NISM controller [

According to (

Hence,

Hence,

Assume that the sliding mode surfaces (

The Lyapunov Function Candidate (LFC)

According to (

According to (

Hence, it can be derived that

Depending on the properties of the hyperbolic function and Lyapunov stability theory, it can be concluded that

Consider the following cascade system:

System (

Besides, the theorems proposed by [

Cascaded system (

Assumption on

where

Assumption on the interconnection: the function

where

Assumption on

In addition to the assumptions in Theorem

For path following under the disturbance of unknown static currents, the stability of cascade system can be proved in two steps. First, without considering the currents, the cascade system described in Theorem

Without considering the currents, the guidance system expressed as (

The cascade can be described as

To prove Theorem

The LFC is selected as

Besides, the interconnection term satisfies

As

Considering the prediction error of observer, the cascade structure can be described as

Assume that the currents are changing slowly relative to time and the current observer is designed as (

The nominal system (

To verify the effectiveness of the control system for the composite curve path following, three simulation cases are carried out. Case 1 is implemented to verify the performance of the proposed path following controller for the composite curve path without currents. Besides, the PID control and the conventional sliding mode control are introduced as the heading controller to be compared with the improved NISM controller. Based on Case 1, Case 2 is carried out to prove the effectiveness of the additional subpath switching algorithm. In Case 3, unknown static currents are introduced to test the anti-interference ability of the controller based on the current observer.

The composite curve path is designed to contain G0, G1, and G2 continuity at the same time and is composed of seven subpaths, including straight lines, circular arcs, and Fermat’s spirals. The parameters of the composite curve path and the controller are shown in Tables

Parameters of the composite curve path.

No. | Subpath type | Parameter | Value |
---|---|---|---|

1 | Straight line | ||

2 | Straight line | ||

3 | Circular arc | ||

_{100} | |||

_{0} | |||

4 | Straight line | ||

154.9566 | |||

5\6 | Fermat’s spiral | ||

0.1329 | |||

1 | |||

7 | Straight line | ||

237.7993 | |||

Parameters of the guidance and control system.

Parameter | Value |
---|---|

0 | |

For path following with no currents,

The planned composite curve path and the actual trajectory.

As shown in Figure

The tracking error and the heading control error without currents.

The comparison of the heading control effect after changing the model parameter.

The degree of the GC has a high impact on the control of path following. Comparatively,

The discontinuity of the connection point between two curve segments will cause excessive rudder speed which is not practical and may cause damage to the rudder system. Strictly bounded nonlinear hyperbolic tangent function is used in the NISM control, and the parameter

The rudder control of the NISM controller.

From Case 1, it is known that the tracking trajectory deviates from the desired path obviously at the switching point due to the sudden change of the heading angle for

As shown in Figures

The planned composite curve path and the actual trajectory.

The parameter value of the virtual point.

For path following with static currents,

From Figure

The tracking error under constant currents.

The estimate of constant currents and the compensation coefficient.

This paper addresses the problems of composite curve path following for an underactuated AUV in the horizontal plane. The global path following of the composite curve path is realized by the virtual point control and the subpath switching algorithm after the parametrization of the composite curve and description of the kinematics error. Besides, the cascade structure composed of AILOS guidance and the improved NISM control proved to be UGAS under the influence of constant currents. Future work will expand the application of the composite curve path following to three-dimensional space.

Partial data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China (Grant no. 52001132) and Special Funds for Basic Scientific Research in Central Universities of China (Grant no. 2662020GXQD003).