# Barrier Lyapunov Function-Based Adaptive Control of an Uncertain Hovercraft with Position and Velocity Constraints

Authors: Mingyu Fu; Taiqi Wang; Chenglong Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper considers the problem of constrained path following control for an underactuated hovercraft subject to parametric uncertainties and external disturbances. A four-degree-of-freedom hovercraft model with unknown curve-fitted coefficients is first rewritten into a parameterized form. By introducing a barrier Lyapunov function into the line-of-sight guidance, the specific transient tracking performance in terms of position error is guaranteed. A novel constrained yaw rate controller is proposed to ensure time-varying yaw rate constraint satisfaction, in which the yaw rate barrier is required to vary with the speed of the hovercraft. Moreover, a command filter is incorporated into the control design to generate the desired virtual controls and its time derivatives. Theoretical analyses show that, under the proposed controller, the position tracking error constraints and the yaw rate constraint can be strictly guaranteed. Finally, numerical simulations illustrate the effectiveness and advantages of the proposed control scheme.


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## Body

## 1. Introduction

As a high-performance amphibious marine craft, a hovercraft utilizes a flexible skirt system around its periphery such that the hull is totally supported by a pressurized air cushion. The hovercraft has attracted increasing attention in both military and civil fields because of its superior high speed and amphibious characteristics [1]. Due to its complex wave-making resistance and skirt drag caused by the cushion system, the dynamics of a hovercraft are very uncertain, nonlinear, and coupled [2]. Moreover, a hovercraft is underactuated because the actuators are equipped for surge and yaw motion only [3]. In addition, a hovercraft is essentially hovering over the water surface, which causes less water friction drag than conventional displacement ships. Therefore, a hovercraft can slip considerably and undergo great heeling during fast turning, requiring the yaw rate of a hovercraft to be regulated within specific safe ranges during maneuvering. These requirements make controller design of a hovercraft a challenging task.Path following, which requires a hovercraft to follow a geometric path that is time independent, is one of the typical control scenarios for a marine surface vessel (MSV). In [4], a global path following controller was designed for MSVs based on a cascaded approach, and the stability was proved by using the linear time-varying theory. Reference [5] utilized a backstepping technique to develop a nonlinear path following controller for MSVs, in which the control design was based on feedback dominance instead of feedback linearization. Reference [6, 7] presented a model predictive control scheme with line-of-sight (LOS) guidance to improve the path following performance. Reference [8] proposed an adaptive path following controller to estimate the ocean currents, while a new integral LOS guidance law was obtained based on adaptive control. Neural network (NN) control is usually used to deal with the uncertain nonlinear system [9], which has also been widely introduced into path following control to cope with the uncertain dynamics of surface vessels. In [10, 11], the NN control approach together with the integral LOS guidance was proposed for underactuated surface vessels with parameter uncertainties. Reference [12] developed a saturated path following controller for surface vessels, and the uncertainties and disturbances were approximated by using NN.One of the greatest challenges for hovercraft control is the inexact dynamics, which are due to the intricate interactions between the cushion system and the water surface; these interactions lead to an unclear hydrodynamic structure and parametric uncertainties for control design. An exceedingly simplified hovercraft model was derived in [13–15], in which the hydrodynamic damping coefficients were assumed to be zero. The same hovercraft model was adopted in [16, 17]. However, because the hydrodynamic force and moment are not included in this model, the proposed model-based controllers might not achieve the desired control objective in practical applications. Furthermore, a curve-fitted hovercraft model was obtained in [1] by replacing the complex hydrodynamic and aerodynamic force and moment functions with curve-fitted approximations; by neglecting the heave and pitch motion, a four-degree-of-freedom (DOF) control-oriented hovercraft model was obtained for control design. However, the effects of parametric uncertainties and external disturbances were not considered in [1]. The uncertainties and disturbances always exist in a practical control system, and some works have investigated the feedback control schemes for different form uncertain nonlinear systems, such as triangular form systems [18, 19] and strict-feedback systems [20]. In addition, the adaptive control method is regarded as a powerful method to cope with such system uncertainties [21]. For example, [22] presented a backstepping-based adaptive control scheme to estimate the unknown parameters, and the tuning-function based approach was proposed to avoid the overparametrization in backstepping design. The adaptive backstepping technique has been widely applied in path following control of vessels [23–25]. To improve the tracking performance of a hovercraft under these system uncertainties, the above 4-DOF hovercraft dynamics [1] are rewritten into a parameterized form in this paper, and the adaptive control method is adopted to estimate the uncertain parameters and external disturbances.LOS guidance, which has been widely explored for surface vessels due to its simplicity and small computational burden, is an effective guidance algorithm for path following control. In [26, 27], it was shown that the equilibrium point of the proportional LOS guidance law is uniformly and exponentially semi-globally stable. To compensate for the vehicle’s constant sideslip due to environmental disturbances, an integral LOS (ILOS) approach was proposed in [28, 29] by adding an additive integral action into the conventional LOS guidance. In [10, 30], an adaptive LOS (ALOS)-based controller was developed with a parameter adaption law to eliminate the effects of system uncertainties. In [31], the authors discussed the drawbacks of the ILOS and ALOS. A modified extended state observer- (ESO-) based LOS guidance approach was proposed to identify the time-varying sideslip angle. Reference [32] presented a compound LOS guidance by combining the time delay control and ESO technique to estimate the time-varying ocean currents. Note that all the aforementioned LOS guidance algorithms only regulate the steady state of position tracking errors to be zero, while the transient tracking performance cannot be guaranteed. However, it is important to ensure the prescribed transient tracking performance of a hovercraft in practice. Moreover, the yaw rate of a hovercraft should be limited to below the stability boundary for safe navigation [33]. If the maximum yaw rate of a hovercraft exceeds the corresponding stability boundary for a period of time, the hovercraft will become unstable, especially at high speeds, which can even lead to a capsizing hazard. As shown in [33], the stability boundary of the yaw rate varies with the surge speed of a hovercraft. Thus, the yaw rate constraint should also be time varying because the desired speed command for a hovercraft can change in practice. To the author’s knowledge, the problem of tracking error-constrained path following control for an uncertain hovercraft with a time-varying yaw rate constraint has rarely been considered.To guarantee the state or output constraints of the system, some control methods have been included in model predictive control [34], nonovershooting control [35], and prescribed performance control [36]. However, in these methods, the state constraints remain difficult to guarantee in practical applications. More recently, a barrier Lyapunov function (BLF) has been proposed for nonlinear systems to ensure the state and output constraints [37, 38], which has been applied to strict-feedback systems with output constraints [39], pure-feedback systems [40], switched systems [41], and practical applications for hypersonic flight vehicles [42] or missile guidance [43]. In addition to the conventional logarithmic function form, a modified tan-type BLF was proposed in [44], which is a general method for systems with state constraints because it works even if the state constraints are removed. In [45], the authors utilized the tan-type BLF to design an error-constrained LOS path following controller for a 3-DOF conventional surface vessel. To break the static constraint limitations in the abovementioned works, a time-varying output constraint was handled in [46] by using a time-varying BLF, which allowed the output constraints to be both time varying and asymmetric. Moreover, to facilitate the backstepping design for the attitude subsystem of a hovercraft, a command filter was introduced to avoid the analytical computation of the virtual control laws [47].Motivated by the above considerations, a BLF-based adaptive path following control scheme is developed for a 4-DOF underactuated hovercraft subject to parametric uncertainties and external disturbances. The main contributions of this paper can be summarized as follows:( 1 ) A novel position-constrained LOS guidance algorithm is proposed by introducing BLF into the classical LOS guidance design procedure. The proposed LOS guidance can guarantee the prescribed tracking performance in terms of position tracking errors.( 2 ) By virtue of the time-varying BLF, an attitude controller is proposed to ensure the yaw rate time-varying constraints of a hovercraft. The yaw rate of a hovercraft will not exceed the stability boundary, which has practical significance for the safe navigation of a hovercraft.( 3 ) A command filter is integrated into the control design to generate the amplitude-constrained virtual control laws, which avoid the analytic computation of the virtual control law derivative. In addition, a novel auxiliary system is developed to compensate for the filtering error.The remainder of this paper is organized as follows. The problem formulation is introduced in Section2. Section 3 is devoted to the LOS guidance, altitude subsystem, and velocity subsystem control design. Numerical simulation examples are presented in Section 4. Section 5 concludes the paper.


## 2. Problem Formulation and Preliminaries

### 2.1. Preliminaries

Notation. Throughout this paper, ·T denotes the transpose of a matrix ·; · represents the absolute value of a scalar; · represents the Euclidean norm of a vector; ·∧ denotes the estimate of ·; the estimation error is defined as ·~=·-·∧; and R+ denotes the set of nonnegative real numbers.Lemma 1 (see [44]).
For anyδ>0 and η∈R, the following inequality always holds: (1)0≤η-ηtanh⁡ηδ≤0.2785δLemma 2 (see [37]).
For any positive constantkb, positive integer p, and z∈R satisfying z<kb, there exists (2)log⁡kb2pkb2p-z2p<z2pkb2p-z2pDefinition 3 (see [39]).
A barrier Lyapunov function is a scalar functionVx defined with respect to the system x˙=fx on an open region D containing the origin. It is continuous, is positive definite, has continuous first-order partial derivatives at every point of D, has the property Vx→∞ as x approaches the boundary of D, and satisfies Vx≤b∀t≥0 along the solution of x˙=fx for x0∈D and some positive constant b.


### 2.2. Dynamic Model of a Hovercraft

The typical configuration of the hovercraft is shown in Figure1, in which two sets of ducted air propellers are mounted at the stern and a pair of rudders is mounted behind the duct flaps to create turning moments.Figure 1
The hovercraft model in the body-fixed frame.For motion control of the hovercraft, we first define the earth-fixed frameOExEyEzE and the body-fixed frame Obxbybzb. The origin of the body-fixed frame is located at the center of gravity (CG). By neglecting the pitch and heave motion and making some basic assumptions, the 4-DOF curve-fitted hovercraft dynamics are derived in [1] and are formulated as follows:(3)u˙=vr+Fxa+Fxh+Fxn+FxTmv˙=-ur+Fya+Fyh+Fyc+Fynmr˙=Mza+Mzh+Mzc+Mzn+MzδIzp˙=Mxa+Mxh+Mxc+Mxn+MxGIxwhere m is the mass of the hovercraft; Ix and Iz are the moments of inertia of the hovercraft about the xb axis and zb axis, respectively; u and v are the surge and sway linear velocities, respectively; and p and r represent the roll and yaw angular velocities, respectively. The right side of (3) consists of various forces and moments acting on the hovercraft, in which the suffix a represents the aerodynamic force, h denotes the hydrodynamic force, c denotes the cushion force, n denotes the air momentum force, T denotes the thrust, δ denotes the rudder moment, and G denotes the restoring moment during heeling of the hovercraft. The curve-fitted aerodynamic forces and moments in (3) are written as(4)Fxa=0.5ρaVa2SxaCxa0+Cxaδrδr+Cxaββ+Cxaβ˙β˙Fya=0.5ρaVa2SyaCya0+Cyaδrδr+Cyaββ+Cyaβ˙β˙Mza=0.5ρaVa2SzalaCmza0+Cmzaδrδr+Cmzaββ+Cmzaβ˙β˙Mxa=FyazaIn addition, the curve-fitted hydrodynamic forces and moments in (3) are written as(5)Fxh=0.5ρwU2ShCxh0+Cxhϕϕ+CxhββFyh=0.5ρwU2ShCyh0+Cyhϕϕ+CyhββMzh=0.5ρwU2ShlcCmzhϕϕ+CmzhββMxh=Fyhzgwhere Va denotes the relative wind speed. Sxa, Sya, and Sza are the projection areas of the hovercraft along the three axes. The average length of the hovercraft is denoted by la, the vertical distance from the center of the lateral area is denoted by za, the velocity of the hovercraft is denoted by U=u2+v2, the lateral area of the skirts in contact with the water surface is denoted by Sh, the cushion length is denoted by lc, and the height of CG is defined as zg. δr, ϕ, and β represent the rudder angle, heeling angle, and sideslip angle of the hovercraft, respectively. C∗# represents the curve-fitted coefficient. The expressions of other component forces and moments can be found in [1] (Chapter 6).In this paper, we assume thatSxa, Sya, Sza, Sh, and C∗# in (4) and (5) are unknown constants. Thus, (4) and (5) can be parameterized into the following form:(6)Fxam=ϑu,1Tφu,1Fyam=ϑv,1Tφv,1MzaIz=ϑr,1Tφr,1MxaIx=ϑp,1Tφp,1,Fxhm=ϑu,2Tφu,2Fyhm=ϑv,2Tφv,2MzhIz=ϑr,2Tφr,2MxhIx=ϑp,2Tφp,2By substituting (6) and other component forces and moments into (3), a 4-DOF parameterized hovercraft model can be given as(7)u˙=vr+fu+ϑu,1Tφu,1+ϑu,2Tφu,2+τu+duv˙=-ur+fv+ϑv,1Tφv,1+ϑv,2Tφv,2+dvr˙=fr+ϑr,1Tφr,1+ϑr,2Tφr,2+τr+drp˙=fp+ϑp,1Tφp,1+ϑp,2Tφp,2+dpwhere the control inputs in (7) are τu=FxT/m and τr=Mzδ/Iz. The dynamics fu, fv, fp, and fr are the integration of other known component forces and moments in (3), which are defined as fu=Fxn/m, fv=Fyc+Fyn/m, fr=Mzc+Mzn/Iz, and fp=Mxc+Mxn+MxG/Ix. In addition, du, dv, dr, and dp in (7) denote the bounded disturbances, including the unmodeled dynamics and external disturbances induced by wind and waves. See Appendix B for the detailed expressions of the parameters in (7).The desired parameterized path is defined byXp=xpω,ypω, where ω∈R denotes the path variable. A path-fixed reference frame is defined with its origin at Xp. The path-tangential angle is calculated by (8)γpω=atan⁡2yp′ω,xp′ωwhere xp′ω=∂xp/∂ω and yp′ω=∂yp/∂ω. For a hovercraft located at X=x,y, the geometry of LOS guidance is illustrated in Figure 2.Figure 2
The LOS guidance geometry.The along-track errorξst and the cross-track error ξet in the path-fixed frame are written as(9)ξsξe=cos⁡γp-sin⁡γpsin⁡γpcos⁡γpTx-xpy-ypAccording to the kinematic equations of surface vessels given in [48], the 4-DOF kinematics of the hovercraft is expressed as(10)x˙y˙ψ˙ϕ˙=cos⁡ψ-sin⁡ψcos⁡ϕ00sin⁡ψcos⁡ψcos⁡ϕ0000cos⁡ϕ00001uvrpwhere x and y are the coordinates of the hovercraft’s CG in the earth-fixed frame. ψ is the heading angle of the hovercraft, and ϕ is the heeling angle of the hovercraft.The time derivative of the tracking errors in (9) is written as(11)ξ˙sξ˙e=γ˙p-sin⁡γpx-xp+cos⁡γpy-yp︸ξe-cos⁡γpx-xp-sin⁡γpy-yp︸-ξs+cos⁡γpx˙-x˙p+sin⁡γpy˙-y˙p-sin⁡γpx˙-x˙p+cos⁡γpy˙-y˙pSubstituting (8) and (10) into (11) yields(12)ξ˙sξ˙e=ucos⁡ψ-γp-v-sin⁡ψ-γp-ω˙xp′2+yp′2usin⁡ψ-γp+v-cos⁡ψ-γp+γ˙pξe-γ˙pξsSetting β=atan⁡2v-,u, (12) can finally be written as(13)ξ˙sξ˙e=U-cos⁡ψ-γp+β-ω˙xp′2+yp′2+γ˙pξeU-sin⁡ψ-γp+β-γ˙pξswhere v-=vcos⁡ϕ is the projection of the lateral velocity onto the horizontal plane. U-=u2+v-2 is the hovercraft speed. If the roll motion is not considered, we obtain cos⁡ϕ=1. The obtained equation (13) is similar to the previous results [8, 31] for 3-DOF surface vessels.Suppose thatks, ke, and kue are positive constant constraints and krt is the symmetric time-varying yaw rate constraint, where the time derivatives of ks, ke, kue, and krt are bounded. The control objective of this study is to design a constrained path following controller for a hovercraft in the presence of parametric uncertainties and external disturbances that can guarantee the prescribed tracking performance: ξs<ks and ξe<ke. The yaw rate satisfies the constraint rt<krt, and the surge velocity satisfies the constraint ue<kue for all t>0, where ue=u-ud.Remark 4.
In the previous path following literature [8, 10, 29, 31], the proposed LOS guidance algorithms only regulate the steady-state tracking errors to be zero or within the neighborhood of zero, while the transient tracking performances are not considered. Reference [45] also proposed a LOS guidance to limit the position tracking errors. However, only the output constraints of the vessel were guaranteed; the yaw rate and velocity constraints were not guaranteed. In this paper, both the state and the output constraints of vessel system can be strictly guaranteed by the proposed control scheme. In addition, the yaw rate of a hovercraft is of particular concern to sailors during maneuver. Because the hovercraft usually hovers over the water with a high forward velocity, a large yaw rate or an excessive rudder angle can easily influence the navigation safety of a hovercraft as compared to a conventional displacement ship. Reference [33] shows that the stability boundary of the yaw rate varies with the surge speed of the hovercraft, which leads to the conversion of our design task into a time-varying state constrained problem.To facilitate the control design, we make the following assumptions.Assumption 5.
The disturbancesdut, dvt, drt, and dpt are bounded. There exist unknown positive constants Du, Dv, Dr, and Dp such that du≤Du, dv≤Dv, dr≤Dr. and dp≤Dp hold for all t≥0.Assumption 6.
The initial tracking errorsξs0, ξe0, and ue0 satisfy the prescribed constraints ξs0<ks, ξe0<ke, and ue0<kue, and the yaw rate tracking error satisfies re0<kre0.Assumption 7.
For anyks>0 and ke>0, the first and second time derivatives of the desired path xp,yp are both bounded.Assumption 8.
Due to underactuated configuration of the hovercraft, a constraint on sway dynamics without control input is required that the sway velocity of the hovercraft is passive bounded.


## 2.1. Preliminaries

Notation. Throughout this paper, ·T denotes the transpose of a matrix ·; · represents the absolute value of a scalar; · represents the Euclidean norm of a vector; ·∧ denotes the estimate of ·; the estimation error is defined as ·~=·-·∧; and R+ denotes the set of nonnegative real numbers.Lemma 1 (see [44]).
For anyδ>0 and η∈R, the following inequality always holds: (1)0≤η-ηtanh⁡ηδ≤0.2785δLemma 2 (see [37]).
For any positive constantkb, positive integer p, and z∈R satisfying z<kb, there exists (2)log⁡kb2pkb2p-z2p<z2pkb2p-z2pDefinition 3 (see [39]).
A barrier Lyapunov function is a scalar functionVx defined with respect to the system x˙=fx on an open region D containing the origin. It is continuous, is positive definite, has continuous first-order partial derivatives at every point of D, has the property Vx→∞ as x approaches the boundary of D, and satisfies Vx≤b∀t≥0 along the solution of x˙=fx for x0∈D and some positive constant b.


## 2.2. Dynamic Model of a Hovercraft

The typical configuration of the hovercraft is shown in Figure1, in which two sets of ducted air propellers are mounted at the stern and a pair of rudders is mounted behind the duct flaps to create turning moments.Figure 1
The hovercraft model in the body-fixed frame.For motion control of the hovercraft, we first define the earth-fixed frameOExEyEzE and the body-fixed frame Obxbybzb. The origin of the body-fixed frame is located at the center of gravity (CG). By neglecting the pitch and heave motion and making some basic assumptions, the 4-DOF curve-fitted hovercraft dynamics are derived in [1] and are formulated as follows:(3)u˙=vr+Fxa+Fxh+Fxn+FxTmv˙=-ur+Fya+Fyh+Fyc+Fynmr˙=Mza+Mzh+Mzc+Mzn+MzδIzp˙=Mxa+Mxh+Mxc+Mxn+MxGIxwhere m is the mass of the hovercraft; Ix and Iz are the moments of inertia of the hovercraft about the xb axis and zb axis, respectively; u and v are the surge and sway linear velocities, respectively; and p and r represent the roll and yaw angular velocities, respectively. The right side of (3) consists of various forces and moments acting on the hovercraft, in which the suffix a represents the aerodynamic force, h denotes the hydrodynamic force, c denotes the cushion force, n denotes the air momentum force, T denotes the thrust, δ denotes the rudder moment, and G denotes the restoring moment during heeling of the hovercraft. The curve-fitted aerodynamic forces and moments in (3) are written as(4)Fxa=0.5ρaVa2SxaCxa0+Cxaδrδr+Cxaββ+Cxaβ˙β˙Fya=0.5ρaVa2SyaCya0+Cyaδrδr+Cyaββ+Cyaβ˙β˙Mza=0.5ρaVa2SzalaCmza0+Cmzaδrδr+Cmzaββ+Cmzaβ˙β˙Mxa=FyazaIn addition, the curve-fitted hydrodynamic forces and moments in (3) are written as(5)Fxh=0.5ρwU2ShCxh0+Cxhϕϕ+CxhββFyh=0.5ρwU2ShCyh0+Cyhϕϕ+CyhββMzh=0.5ρwU2ShlcCmzhϕϕ+CmzhββMxh=Fyhzgwhere Va denotes the relative wind speed. Sxa, Sya, and Sza are the projection areas of the hovercraft along the three axes. The average length of the hovercraft is denoted by la, the vertical distance from the center of the lateral area is denoted by za, the velocity of the hovercraft is denoted by U=u2+v2, the lateral area of the skirts in contact with the water surface is denoted by Sh, the cushion length is denoted by lc, and the height of CG is defined as zg. δr, ϕ, and β represent the rudder angle, heeling angle, and sideslip angle of the hovercraft, respectively. C∗# represents the curve-fitted coefficient. The expressions of other component forces and moments can be found in [1] (Chapter 6).In this paper, we assume thatSxa, Sya, Sza, Sh, and C∗# in (4) and (5) are unknown constants. Thus, (4) and (5) can be parameterized into the following form:(6)Fxam=ϑu,1Tφu,1Fyam=ϑv,1Tφv,1MzaIz=ϑr,1Tφr,1MxaIx=ϑp,1Tφp,1,Fxhm=ϑu,2Tφu,2Fyhm=ϑv,2Tφv,2MzhIz=ϑr,2Tφr,2MxhIx=ϑp,2Tφp,2By substituting (6) and other component forces and moments into (3), a 4-DOF parameterized hovercraft model can be given as(7)u˙=vr+fu+ϑu,1Tφu,1+ϑu,2Tφu,2+τu+duv˙=-ur+fv+ϑv,1Tφv,1+ϑv,2Tφv,2+dvr˙=fr+ϑr,1Tφr,1+ϑr,2Tφr,2+τr+drp˙=fp+ϑp,1Tφp,1+ϑp,2Tφp,2+dpwhere the control inputs in (7) are τu=FxT/m and τr=Mzδ/Iz. The dynamics fu, fv, fp, and fr are the integration of other known component forces and moments in (3), which are defined as fu=Fxn/m, fv=Fyc+Fyn/m, fr=Mzc+Mzn/Iz, and fp=Mxc+Mxn+MxG/Ix. In addition, du, dv, dr, and dp in (7) denote the bounded disturbances, including the unmodeled dynamics and external disturbances induced by wind and waves. See Appendix B for the detailed expressions of the parameters in (7).The desired parameterized path is defined byXp=xpω,ypω, where ω∈R denotes the path variable. A path-fixed reference frame is defined with its origin at Xp. The path-tangential angle is calculated by (8)γpω=atan⁡2yp′ω,xp′ωwhere xp′ω=∂xp/∂ω and yp′ω=∂yp/∂ω. For a hovercraft located at X=x,y, the geometry of LOS guidance is illustrated in Figure 2.Figure 2
The LOS guidance geometry.The along-track errorξst and the cross-track error ξet in the path-fixed frame are written as(9)ξsξe=cos⁡γp-sin⁡γpsin⁡γpcos⁡γpTx-xpy-ypAccording to the kinematic equations of surface vessels given in [48], the 4-DOF kinematics of the hovercraft is expressed as(10)x˙y˙ψ˙ϕ˙=cos⁡ψ-sin⁡ψcos⁡ϕ00sin⁡ψcos⁡ψcos⁡ϕ0000cos⁡ϕ00001uvrpwhere x and y are the coordinates of the hovercraft’s CG in the earth-fixed frame. ψ is the heading angle of the hovercraft, and ϕ is the heeling angle of the hovercraft.The time derivative of the tracking errors in (9) is written as(11)ξ˙sξ˙e=γ˙p-sin⁡γpx-xp+cos⁡γpy-yp︸ξe-cos⁡γpx-xp-sin⁡γpy-yp︸-ξs+cos⁡γpx˙-x˙p+sin⁡γpy˙-y˙p-sin⁡γpx˙-x˙p+cos⁡γpy˙-y˙pSubstituting (8) and (10) into (11) yields(12)ξ˙sξ˙e=ucos⁡ψ-γp-v-sin⁡ψ-γp-ω˙xp′2+yp′2usin⁡ψ-γp+v-cos⁡ψ-γp+γ˙pξe-γ˙pξsSetting β=atan⁡2v-,u, (12) can finally be written as(13)ξ˙sξ˙e=U-cos⁡ψ-γp+β-ω˙xp′2+yp′2+γ˙pξeU-sin⁡ψ-γp+β-γ˙pξswhere v-=vcos⁡ϕ is the projection of the lateral velocity onto the horizontal plane. U-=u2+v-2 is the hovercraft speed. If the roll motion is not considered, we obtain cos⁡ϕ=1. The obtained equation (13) is similar to the previous results [8, 31] for 3-DOF surface vessels.Suppose thatks, ke, and kue are positive constant constraints and krt is the symmetric time-varying yaw rate constraint, where the time derivatives of ks, ke, kue, and krt are bounded. The control objective of this study is to design a constrained path following controller for a hovercraft in the presence of parametric uncertainties and external disturbances that can guarantee the prescribed tracking performance: ξs<ks and ξe<ke. The yaw rate satisfies the constraint rt<krt, and the surge velocity satisfies the constraint ue<kue for all t>0, where ue=u-ud.Remark 4.
In the previous path following literature [8, 10, 29, 31], the proposed LOS guidance algorithms only regulate the steady-state tracking errors to be zero or within the neighborhood of zero, while the transient tracking performances are not considered. Reference [45] also proposed a LOS guidance to limit the position tracking errors. However, only the output constraints of the vessel were guaranteed; the yaw rate and velocity constraints were not guaranteed. In this paper, both the state and the output constraints of vessel system can be strictly guaranteed by the proposed control scheme. In addition, the yaw rate of a hovercraft is of particular concern to sailors during maneuver. Because the hovercraft usually hovers over the water with a high forward velocity, a large yaw rate or an excessive rudder angle can easily influence the navigation safety of a hovercraft as compared to a conventional displacement ship. Reference [33] shows that the stability boundary of the yaw rate varies with the surge speed of the hovercraft, which leads to the conversion of our design task into a time-varying state constrained problem.To facilitate the control design, we make the following assumptions.Assumption 5.
The disturbancesdut, dvt, drt, and dpt are bounded. There exist unknown positive constants Du, Dv, Dr, and Dp such that du≤Du, dv≤Dv, dr≤Dr. and dp≤Dp hold for all t≥0.Assumption 6.
The initial tracking errorsξs0, ξe0, and ue0 satisfy the prescribed constraints ξs0<ks, ξe0<ke, and ue0<kue, and the yaw rate tracking error satisfies re0<kre0.Assumption 7.
For anyks>0 and ke>0, the first and second time derivatives of the desired path xp,yp are both bounded.Assumption 8.
Due to underactuated configuration of the hovercraft, a constraint on sway dynamics without control input is required that the sway velocity of the hovercraft is passive bounded.


## 3. Control System Design

### 3.1. Position-Constrained LOS Guidance

In this subsection, the barrier Lyapunov function will be introduced into the LOS guidance to guarantee the position constraints. The Lyapunov function is defined as follows:(14)V1=12log⁡ks2ks2-ξs2+12log⁡ke2ke2-ξe2The time derivative of V1 is calculated as(15)V˙1=ϑsξ˙s+ϑeξ˙ewhere ϑs=ξs/ks2-ξs2 and ϑe=ξe/ke2-ξe2.Substituting (13) into (15), we obtain (16)V˙1=ϑsU-cos⁡ψ-γp+β-ω˙xp′2+yp′2+γ˙pξe+ϑeU-sin⁡ψ-γp+β-γ˙pξs=ϑsU-cos⁡ψ-γp+β-ω˙xp′2+yp′2+ϑsγ˙pξe1-ks2-ξs2ke2-ξe2+ϑeU-sin⁡ψ-γp+βTaking the time derivative of γp yields(17)γ˙p=ω˙yp′′xp′-xp′′yp′xp′2+yp′2Invoking (17) in (16), we have(18)V˙1=ϑsU-cos⁡ψ-γp+β-ω˙σ+ϑeU-sin⁡ψ-γp+βwith(19)σ=xp′2+yp′2-ξeyp′′xp′-xp′′yp′xp′2+yp′21-ks2-ξs2ke2-ξe2Defining the tracking error as (20)ψe=ψ-ψdand substituting (20) into (18) yield(21)V˙1=ϑsU-cos⁡ψ-γp+β-ω˙σ+ϑeudsin⁡ψd-γp+v-cos⁡ψd-γpcos⁡ψe+uesin⁡ψ-γp+udcos⁡ψd-γp-v-sin⁡ψd-γpsin⁡ψe=ϑsU-cos⁡ψ-γp+β-ω˙σ+ϑeU-dsin⁡ψd-γp+β-dcos⁡ψe+uesin⁡ψ-γp+U-dcos⁡ψd-γp+β-dsin⁡ψewith U-d=ud2+v-2 and β-d=atan⁡2v-,ud.From (21), we can obtain the update law for ω:(22)ω˙=U-cos⁡ψ-γp+β+k1ξsσNote that σ is the denominator of (22); to avoid σ=0, we can take σ=σ0sgn⁡σ when σ<σ0, where σ0 is a small positive constant.The desired heading angle of the hovercraft is chosen to be(23)ψd=γp-β-d+atan⁡2-ξe,k2where k1,k2>0 are positive design constants and k2 is the look-ahead distance.By invoking (22) and (23), (21) becomes(24)V˙1=-k1ξs2ks2-ξs2+ϑe-cos⁡ψeU-dξe+sin⁡ψeU-dk2k22+ξe2+uesin⁡ψ-γp=-k1ξs2ks2-ξs2-U-dk22+ξe2ξe2ke2-ξe2+ϑeψeχ+ϑeuesin⁡ψ-γp≤-ς1V1+ϑeψeχ+ϑeuesin⁡ψ-γpwhere (25)χ=-U-dξek22+ξe2cos⁡ψe-1ψe+k2U-dk22+ξe2sin⁡ψeψewith ς1=min⁡k1,U-d/k22+ξe2.Remark 9.
In (25), we have ξe/k22+ξe2≤1 and k2/k22+ξe2≤1. By using L’Hopital’s rule, we obtain limψe→0⁡cos⁡ψe-1/ψe=0 and limψe→0⁡sin⁡ψe/ψe=1. Thus, singularity will not occur in (25).


### 3.2. Attitude Control

In this subsection, the attitude kinematic controller for the hovercraft will be designed to stabilize the yaw angle tracking errorψe. Subsequently, the yaw rate controller will be developed via BLF-based adaptive control, while the time-varying constraint on the yaw rate, r<krt, will be strictly guaranteed.Suppose that the virtual yaw rateαr satisfies(26)αr<kαrt<krtwhere kαrt:R+→R+and krt:R+→R+. The yaw rate tracking error is defined as re=r-αr. If a yaw rate controller exists that can ensure the inequality(27)re<kret≤krt-kαrtwhere kret:R+→R+ and then the time-varying constraint krt on yaw rate rt can be guaranteed.Remark 10.
If the designed yaw rate controller can makere<kret, then we have r-αr<krt-kαrt and r-αr<krt-kαrt. Using (26), we obtain r<krt-kαrt-αr<krt.However, one problem is how to ensure the inequalityαr<kαrt in (26) because there is no effective mechanism to guarantee it. Motivated by [49], the command filter is employed to constrain the magnitude of the virtual yaw rate by the time-varying constraint kαrt. The command filter is shown in Figure 3.Figure 3
Configuration of the command filter.Here, the nominal virtual yaw rateαr,c is filtered to provide the magnitude and bandwidth limited virtual yaw rate αr and its derivative α˙r. In addition, an auxiliary system will be designed to compensate for the estimation error Δαr=αr-αr,c.Step 1 (kinematics controller).
Choose the following candidate Lyapunov functionV2:(28)V2=12ψe2Taking the time derivative of (28), we obtain (29)V˙2=ψerecos⁡ϕ+Δαrcos⁡ϕ+αr,ccos⁡ϕ-ψ˙dTo avoid the complex derivative computations of ψd included in the nominal virtual control law αr,c, a second-order tracking-differentiator [50, 51] will be used for the estimation of ψ˙d, which is described as(30)v˙1=v2v˙2=-R2a1v1-ψd+a2v2Rwhere R, a1, and a2 are positive design constants; ψd is the input signal of (30); v1 and v2 denote the respective estimations of ψd and ψ˙d. Let v2=ψ˙d,c and define the estimation error as(31)ε=ψ˙d,c-ψ˙dAccording to Theorem 2 presented in [50], we have limR→+∞⁡ε=0, which means that ε can be sufficiently small by choosing a large R. Hence, it can be concluded that there exists arbitrarily small positive constant εM such that ε≤εM.As such, the nominal virtual control lawαr,c for (29) is given by(32)αr,c=-k3ψe+ψ˙d,c-ϑeχ+c1ζr-c12/2ψecos⁡ϕwhere k3>1/2 and c1>0 denote the design constants. ϕ≠±π/2. ζr is the state of the auxiliary system and compensates for the constraint effect Δαr. The auxiliary system is designed as(33)ζ˙r=-k4ζr-f1ζr+γ1Δαrcos⁡ϕ,ζr>ρ10,ζr≤ρ1where k4>1 and γ1>0. ρ1 is a small positive design constant that satisfies the performance requirement. f1 is given by(34)f1=ψeΔαrcos⁡ϕ+1/2γ12Δαrcos⁡ϕ2ζr2Remark 11.
Note that, in (33), ζr will exponentially converge onto the small set ζr≤ρ1 if Δαr=0. If the state ζr in (33) satisfies ζr>ρ1, then saturation effects occur again and the initial value of the auxiliary system should be properly reset to ensure that ζr can respond to the case Δαr≠0.Consider the candidate Lyapunov function(35)V3=V2+12ζr2Invoking (32) and (33), the time derivative of V3 is given by(36)V˙3=ψere+Δαrcos⁡ϕ-k3ψe+ψ˙d,c-ψ˙d-ϑeχ+c1ζr-c122ψe+ζr-k4ζr-f1ζr+γ1Δαrcos⁡ϕ=ψerecos⁡ϕ+ψeΔαrcos⁡ϕ+ψeε-ϑeψeχ+c1ψeζr-k3ψe2-k4ζr2-ψeΔαrcos⁡ϕ-12γ12Δαrcos⁡ϕ2-c122ψe2+γ1ζrΔαrcos⁡ϕNote that, in (36),(37)ψeε≤12ψe2+12εM2,c1ψeζr≤12c12ψe2+12ζr2,γ1ζrΔαrcos⁡ϕ≤12γ12Δαrcos⁡ϕ2+12ζr2.Then, we have (38)V˙3≤-k3-12ψe2-k4-1ζr2-ϑeψeχ+ψerecos⁡ϕ+12εM2≤-ς2V3-ϑeψeχ+ψerecos⁡ϕ+12εM2where ς2=min⁡k3-1/2,k4-1.Step 2 (dynamics controller).
Consider the following time-varying symmetric BLF:(39)V4=12log⁡kre2tkre2t-re2where kret:R+→R+. RecallingRemark10, the constrained yaw rate controller will be proposed to guarantee the constraint re<kret. Define the new variable as ξr=re/kre; (39) can be rewritten as(40)V4=12log⁡11-ξr2It can be seen that V4 is positive definite and continuously differentiable in the set ξr<1. The time derivative of V4 is calculated as (41)V˙4=ϑrr˙e-rek˙rekrewith ϑr=ξr/kre1-ξr2.Substituting the third equation of (7) into (41), we have(42)V˙4=ϑrfr+ϑr,1Tφr,1+ϑr,2Tφr,2+τr+dr-α˙r-rek˙rekreThe yaw rate control law for (42) becomes(43)τr=-fr-ϑ^r,1Tφr,1-ϑ^r,2Tφr,2+α˙r-k5re-k-5re-kre2-re2ψecos⁡ϕ-D^rtanh⁡ϑrσrwhere k-5=k˙re/kre2+ϖ, with ϖ>0 a small constant. k5>0 and σr>0 are design constants. α˙r is directly obtained from the output of the command filter. ϑ^r,1T, ϑ^r,2T, and D^r are adaptive estimations of ϑr,1T, ϑr,2T, and Dr, respectively.The adaptation laws forϑ^r,1, ϑ^r,2, and D^r are designed as follows:(44)ϑ^˙r,1=ϑrφr,1-kϑ,1ϑ^r,1ϑ^˙r,2=ϑrφr,2-kϑ,2ϑ^r,2D^˙r=ϑrtanh⁡ϑrσr-k6D^rwhere kϑ,1, kϑ,2, and k6 are positive design constants.Next, we construct the candidate Lyapunov function as follows:(45)V5=V4+12ϑ~r,1Tϑ~r,1+12ϑ~r,2Tϑ~r,2+12D~r2Taking the time derivative of (45) and using (42) and (44), we obtain(46)V˙5=V˙4-ϑ~r,1Tϑ^˙r,1-ϑ~r,2Tϑ^˙r,2-D~rD^˙r≤-k5ξr21-ξr2-k-5+k˙rekrereϑr+kϑ,1ϑ~r,1Tϑ^r,1+kϑ,2ϑ~r,2Tϑ^r,2+ϑrdr-Drtanh⁡ϑrσr+k6D~rD^r-ψerecos⁡ϕNote that (47)k-5+k˙rekre≥k-5-k˙rekre>0(48)ϑ~r,1Tϑ^r,1≤ϑr,12-ϑ~r,122,ϑ~r,2Tϑ^r,2≤ϑr,22-ϑ~r,222,D~rD^r≤Dr2-D~r22and(49)ϑrdr-Drtanh⁡ϑrσr≤0.2785σrDrSubstituting the inequalities (47)-(49) and (2) into (46), in the set, ξr<1 we can obtain that(50)V˙5≤-2k5V4-k-5+k˙rekreξr21-ξr2-kϑ,12ϑ~r,12-kϑ,22ϑ~r,22-k62D~r2+μr-ψerecos⁡ϕ≤-2k5V4-kϑ,12ϑ~r,12-kϑ,22ϑ~r,22-k62D~r2+μr-ψerecos⁡ϕ≤-ς3V5+μr-ψerecos⁡ϕwhere ς3=min⁡2k5,kϑ,1,kϑ,2,k6, with μr a positive constant defined as(51)μr=kϑ,12ϑr,12+kϑ,22ϑr,22+k62Dr2+0.2785σrDr.


### 3.3. Velocity Tracking Control

In this subsection, the designed control forceτu should enforce the following constraint on the velocity: limt→∞⁡ut→udt. In addition, the constraint ue<kue should also be strictly guaranteed. Using the first equation in (7), the time derivative of ue is calculated as(52)u˙e=vr+fu+ϑu,1Tφu,1+ϑu,2Tφu,2+τu+du-u˙dChoosing the barrier Lyapunov function(53)V6=12log⁡kue2kue2-ue2and differentiating (53) yield(54)V˙6=ϑuu˙ewhere ϑu=ue/kue2-ue2.We design the surge velocity tracking controller forτu as follows:(55)τu=-vr-ϑ^u,1Tφu,1-ϑ^u,2Tφu,2-D^utanh⁡ϑuσu-fu-k7ue+u˙d-ϑekue2-ue2sin⁡ψ-γpwhere k7>0 and σu>0 are design constants.The adaptation laws forϑ^u,1, ϑ^u,2, and D^u are given as(56)ϑ^˙u,1=ϑuφu,1-kϑ,3ϑ^u,1ϑ^˙u,2=ϑuφu,2-kϑ,4ϑ^u,2D^˙u=ϑutanh⁡ϑuσu-k8D^uwhere kϑ,3, kϑ,4, and k8 are positive design constants.Construct the Lyapunov functionV7 as(57)V7=V6+12ϑ~u,1Tϑ~u,1+12ϑ~u,2Tϑ~u,2+12D~u2Taking the time derivative of (57) and invoking (54)-(56), we obtain (58)V˙7=V˙6-ϑ~u,1Tϑ^˙u,1-ϑ~u,2Tϑ^˙u,2-D~uD^˙u≤-k7ue2kue2-ue2+kϑ,3ϑ~u,1Tϑ^u,1+kϑ,4ϑ~u,2Tϑ^u,2+k8D~uD^u+ϑudu-Dutanh⁡ϑuσu-ϑeuesin⁡ψ-γpNote also that ϑ~u,1Tϑ^u,1≤ϑu,12-ϑ~u,12/2, ϑ~u,2Tϑ^u,2≤ϑu,22-ϑ~u,22/2, and D~uD^u≤Du2-D~u2/2.We then have(59)V˙7≤-2k7V6-kϑ,32ϑ~u,12-kϑ,42ϑ~u,22-k82D~u2+μu-ϑeuesin⁡ψ-γp≤-ς4V7+μu-ϑeuesin⁡ψ-γpwhere ς4=min⁡2k7,kϑ,3,kϑ,4,k8 and μu is a positive constant defined as(60)μu=kϑ,32ϑu,12+kϑ,42ϑu,22+k82Du2+0.2578σuDu.Remark 12.
In (55), the desired surge speed udt is required to be continuously differentiable. In previous works [10, 11, 45], the desired surge speed ud was chosen as a constant for all t>0. However, in practice, the desired surge speed of a hovercraft is often directly switched to another value, which will lead to the nonexistence of u˙dt. Thus, to ensure that the desired surge speed is differentiable when ud changes from ud1 to ud2 between any time interval ta and tb, we give a speed smoother as follows: (61)ud=ud1+ht,ta,tb,γtud2-ud1The scalar function ht,ta,tb,γt is defined as(62)ht,ta,tb,γt=fτfτ+γtf1-τwhere τ=t-ta/tb-γt. fτ=0 if τ≤0, and fτ=e-1/τ if τ>0, with ta<tb and γt a positive constant. The speed smoother is shown in Figure 4.Figure 4
The desired surge speed and its derivative.


### 3.4. Stability Analysis

Theorem 13.
Consider the parameterized hovercraft models (7) and (10) in the presence of parametric uncertainties and external disturbances and assume that Assumptions 5–8 are satisfied. If the update law for the path variable ω is chosen as (22), the desired heading angle of the hovercraft is calculated by (23), the auxiliary system is designed as (33), the adaptation laws for unknown parameters and external disturbances are given by (44) and (56), and the controllers are obtained from (43) and (55); then the following properties hold:
(i) All the error signals in the closed-loop system are uniformly ultimately bounded (UUB).
(ii) The static constraints on position and surge velocity tracking errors and the time-varying constraints on the yaw rate are never violated, i.e.,ξs<ks, ξe<ke, ue<kue and rt<krt hold for all t>0.Proof.
(i) Construct the following candidate Lyapunov function:(63)V=V1+V3+V5+V7By virtue of (24), (38), (50), and (59), the time derivative of V is given by(64)V˙≤-ς1V1-ς2V3-ς3V5-ς4V7+ϑeψeχ+μr+ϑeuesin⁡ψ-γp-ϑeψeχ+ψerecos⁡ϕ-ψerecos⁡ϕ-ϑeuesin⁡ψ-γp+12εM2+μu≤-ς1V1-ς2V3-ς3V5-ς4V7+μu+μr+12εM2≤-ςVV+μVwhere ςV=min⁡ς1,ς2,ς3,ς4 and μV=μu+μr+1/2εM2.
If the initial conditions satisfyAssumption6, then the constructed Lyapunov function V satisfies the inequality(65)Vt≤Γt=V0-μVςVexp⁡-ςVt+μVςVFrom (65), we know that limt→∞⁡Vt≤limt→∞⁡Γt=μV/ςV. Therefore, all the error signals in the closed-loop system are uniformly ultimately bounded. Noting the expressions of μV and ςV, the ultimate bounds can be made arbitrarily small by adjusting the design parameters.
(ii)∀t>0, the BLF (14), (40), and (53) satisfy the following inequalities:(66)12log⁡ki2ki2-ξi2≤Vt≤Γt12log⁡kue2kue2-ue2≤Vt≤Γt12log⁡11-ξr2≤Vt≤Γtwhere i=s,e.
Then, from (66), we obtain(67)ξs≤ks1-exp⁡-2Γt<ksξe≤ke1-exp⁡-2Γt<keue≤kue1-exp⁡-2Γt<kueξr≤1-exp⁡-2Γt<1In (67), from ξr<1, ∀t>0, we know that ret<kret,∀t>0. RecallingRemark10, we conclude that rt<krt. Therefore, all the prescribed constraints of the hovercraft system can be guaranteed for ∀t>0. This completes the proof.


## 3.1. Position-Constrained LOS Guidance

In this subsection, the barrier Lyapunov function will be introduced into the LOS guidance to guarantee the position constraints. The Lyapunov function is defined as follows:(14)V1=12log⁡ks2ks2-ξs2+12log⁡ke2ke2-ξe2The time derivative of V1 is calculated as(15)V˙1=ϑsξ˙s+ϑeξ˙ewhere ϑs=ξs/ks2-ξs2 and ϑe=ξe/ke2-ξe2.Substituting (13) into (15), we obtain (16)V˙1=ϑsU-cos⁡ψ-γp+β-ω˙xp′2+yp′2+γ˙pξe+ϑeU-sin⁡ψ-γp+β-γ˙pξs=ϑsU-cos⁡ψ-γp+β-ω˙xp′2+yp′2+ϑsγ˙pξe1-ks2-ξs2ke2-ξe2+ϑeU-sin⁡ψ-γp+βTaking the time derivative of γp yields(17)γ˙p=ω˙yp′′xp′-xp′′yp′xp′2+yp′2Invoking (17) in (16), we have(18)V˙1=ϑsU-cos⁡ψ-γp+β-ω˙σ+ϑeU-sin⁡ψ-γp+βwith(19)σ=xp′2+yp′2-ξeyp′′xp′-xp′′yp′xp′2+yp′21-ks2-ξs2ke2-ξe2Defining the tracking error as (20)ψe=ψ-ψdand substituting (20) into (18) yield(21)V˙1=ϑsU-cos⁡ψ-γp+β-ω˙σ+ϑeudsin⁡ψd-γp+v-cos⁡ψd-γpcos⁡ψe+uesin⁡ψ-γp+udcos⁡ψd-γp-v-sin⁡ψd-γpsin⁡ψe=ϑsU-cos⁡ψ-γp+β-ω˙σ+ϑeU-dsin⁡ψd-γp+β-dcos⁡ψe+uesin⁡ψ-γp+U-dcos⁡ψd-γp+β-dsin⁡ψewith U-d=ud2+v-2 and β-d=atan⁡2v-,ud.From (21), we can obtain the update law for ω:(22)ω˙=U-cos⁡ψ-γp+β+k1ξsσNote that σ is the denominator of (22); to avoid σ=0, we can take σ=σ0sgn⁡σ when σ<σ0, where σ0 is a small positive constant.The desired heading angle of the hovercraft is chosen to be(23)ψd=γp-β-d+atan⁡2-ξe,k2where k1,k2>0 are positive design constants and k2 is the look-ahead distance.By invoking (22) and (23), (21) becomes(24)V˙1=-k1ξs2ks2-ξs2+ϑe-cos⁡ψeU-dξe+sin⁡ψeU-dk2k22+ξe2+uesin⁡ψ-γp=-k1ξs2ks2-ξs2-U-dk22+ξe2ξe2ke2-ξe2+ϑeψeχ+ϑeuesin⁡ψ-γp≤-ς1V1+ϑeψeχ+ϑeuesin⁡ψ-γpwhere (25)χ=-U-dξek22+ξe2cos⁡ψe-1ψe+k2U-dk22+ξe2sin⁡ψeψewith ς1=min⁡k1,U-d/k22+ξe2.Remark 9.
In (25), we have ξe/k22+ξe2≤1 and k2/k22+ξe2≤1. By using L’Hopital’s rule, we obtain limψe→0⁡cos⁡ψe-1/ψe=0 and limψe→0⁡sin⁡ψe/ψe=1. Thus, singularity will not occur in (25).


## 3.2. Attitude Control

In this subsection, the attitude kinematic controller for the hovercraft will be designed to stabilize the yaw angle tracking errorψe. Subsequently, the yaw rate controller will be developed via BLF-based adaptive control, while the time-varying constraint on the yaw rate, r<krt, will be strictly guaranteed.Suppose that the virtual yaw rateαr satisfies(26)αr<kαrt<krtwhere kαrt:R+→R+and krt:R+→R+. The yaw rate tracking error is defined as re=r-αr. If a yaw rate controller exists that can ensure the inequality(27)re<kret≤krt-kαrtwhere kret:R+→R+ and then the time-varying constraint krt on yaw rate rt can be guaranteed.Remark 10.
If the designed yaw rate controller can makere<kret, then we have r-αr<krt-kαrt and r-αr<krt-kαrt. Using (26), we obtain r<krt-kαrt-αr<krt.However, one problem is how to ensure the inequalityαr<kαrt in (26) because there is no effective mechanism to guarantee it. Motivated by [49], the command filter is employed to constrain the magnitude of the virtual yaw rate by the time-varying constraint kαrt. The command filter is shown in Figure 3.Figure 3
Configuration of the command filter.Here, the nominal virtual yaw rateαr,c is filtered to provide the magnitude and bandwidth limited virtual yaw rate αr and its derivative α˙r. In addition, an auxiliary system will be designed to compensate for the estimation error Δαr=αr-αr,c.Step 1 (kinematics controller).
Choose the following candidate Lyapunov functionV2:(28)V2=12ψe2Taking the time derivative of (28), we obtain (29)V˙2=ψerecos⁡ϕ+Δαrcos⁡ϕ+αr,ccos⁡ϕ-ψ˙dTo avoid the complex derivative computations of ψd included in the nominal virtual control law αr,c, a second-order tracking-differentiator [50, 51] will be used for the estimation of ψ˙d, which is described as(30)v˙1=v2v˙2=-R2a1v1-ψd+a2v2Rwhere R, a1, and a2 are positive design constants; ψd is the input signal of (30); v1 and v2 denote the respective estimations of ψd and ψ˙d. Let v2=ψ˙d,c and define the estimation error as(31)ε=ψ˙d,c-ψ˙dAccording to Theorem 2 presented in [50], we have limR→+∞⁡ε=0, which means that ε can be sufficiently small by choosing a large R. Hence, it can be concluded that there exists arbitrarily small positive constant εM such that ε≤εM.As such, the nominal virtual control lawαr,c for (29) is given by(32)αr,c=-k3ψe+ψ˙d,c-ϑeχ+c1ζr-c12/2ψecos⁡ϕwhere k3>1/2 and c1>0 denote the design constants. ϕ≠±π/2. ζr is the state of the auxiliary system and compensates for the constraint effect Δαr. The auxiliary system is designed as(33)ζ˙r=-k4ζr-f1ζr+γ1Δαrcos⁡ϕ,ζr>ρ10,ζr≤ρ1where k4>1 and γ1>0. ρ1 is a small positive design constant that satisfies the performance requirement. f1 is given by(34)f1=ψeΔαrcos⁡ϕ+1/2γ12Δαrcos⁡ϕ2ζr2Remark 11.
Note that, in (33), ζr will exponentially converge onto the small set ζr≤ρ1 if Δαr=0. If the state ζr in (33) satisfies ζr>ρ1, then saturation effects occur again and the initial value of the auxiliary system should be properly reset to ensure that ζr can respond to the case Δαr≠0.Consider the candidate Lyapunov function(35)V3=V2+12ζr2Invoking (32) and (33), the time derivative of V3 is given by(36)V˙3=ψere+Δαrcos⁡ϕ-k3ψe+ψ˙d,c-ψ˙d-ϑeχ+c1ζr-c122ψe+ζr-k4ζr-f1ζr+γ1Δαrcos⁡ϕ=ψerecos⁡ϕ+ψeΔαrcos⁡ϕ+ψeε-ϑeψeχ+c1ψeζr-k3ψe2-k4ζr2-ψeΔαrcos⁡ϕ-12γ12Δαrcos⁡ϕ2-c122ψe2+γ1ζrΔαrcos⁡ϕNote that, in (36),(37)ψeε≤12ψe2+12εM2,c1ψeζr≤12c12ψe2+12ζr2,γ1ζrΔαrcos⁡ϕ≤12γ12Δαrcos⁡ϕ2+12ζr2.Then, we have (38)V˙3≤-k3-12ψe2-k4-1ζr2-ϑeψeχ+ψerecos⁡ϕ+12εM2≤-ς2V3-ϑeψeχ+ψerecos⁡ϕ+12εM2where ς2=min⁡k3-1/2,k4-1.Step 2 (dynamics controller).
Consider the following time-varying symmetric BLF:(39)V4=12log⁡kre2tkre2t-re2where kret:R+→R+. RecallingRemark10, the constrained yaw rate controller will be proposed to guarantee the constraint re<kret. Define the new variable as ξr=re/kre; (39) can be rewritten as(40)V4=12log⁡11-ξr2It can be seen that V4 is positive definite and continuously differentiable in the set ξr<1. The time derivative of V4 is calculated as (41)V˙4=ϑrr˙e-rek˙rekrewith ϑr=ξr/kre1-ξr2.Substituting the third equation of (7) into (41), we have(42)V˙4=ϑrfr+ϑr,1Tφr,1+ϑr,2Tφr,2+τr+dr-α˙r-rek˙rekreThe yaw rate control law for (42) becomes(43)τr=-fr-ϑ^r,1Tφr,1-ϑ^r,2Tφr,2+α˙r-k5re-k-5re-kre2-re2ψecos⁡ϕ-D^rtanh⁡ϑrσrwhere k-5=k˙re/kre2+ϖ, with ϖ>0 a small constant. k5>0 and σr>0 are design constants. α˙r is directly obtained from the output of the command filter. ϑ^r,1T, ϑ^r,2T, and D^r are adaptive estimations of ϑr,1T, ϑr,2T, and Dr, respectively.The adaptation laws forϑ^r,1, ϑ^r,2, and D^r are designed as follows:(44)ϑ^˙r,1=ϑrφr,1-kϑ,1ϑ^r,1ϑ^˙r,2=ϑrφr,2-kϑ,2ϑ^r,2D^˙r=ϑrtanh⁡ϑrσr-k6D^rwhere kϑ,1, kϑ,2, and k6 are positive design constants.Next, we construct the candidate Lyapunov function as follows:(45)V5=V4+12ϑ~r,1Tϑ~r,1+12ϑ~r,2Tϑ~r,2+12D~r2Taking the time derivative of (45) and using (42) and (44), we obtain(46)V˙5=V˙4-ϑ~r,1Tϑ^˙r,1-ϑ~r,2Tϑ^˙r,2-D~rD^˙r≤-k5ξr21-ξr2-k-5+k˙rekrereϑr+kϑ,1ϑ~r,1Tϑ^r,1+kϑ,2ϑ~r,2Tϑ^r,2+ϑrdr-Drtanh⁡ϑrσr+k6D~rD^r-ψerecos⁡ϕNote that (47)k-5+k˙rekre≥k-5-k˙rekre>0(48)ϑ~r,1Tϑ^r,1≤ϑr,12-ϑ~r,122,ϑ~r,2Tϑ^r,2≤ϑr,22-ϑ~r,222,D~rD^r≤Dr2-D~r22and(49)ϑrdr-Drtanh⁡ϑrσr≤0.2785σrDrSubstituting the inequalities (47)-(49) and (2) into (46), in the set, ξr<1 we can obtain that(50)V˙5≤-2k5V4-k-5+k˙rekreξr21-ξr2-kϑ,12ϑ~r,12-kϑ,22ϑ~r,22-k62D~r2+μr-ψerecos⁡ϕ≤-2k5V4-kϑ,12ϑ~r,12-kϑ,22ϑ~r,22-k62D~r2+μr-ψerecos⁡ϕ≤-ς3V5+μr-ψerecos⁡ϕwhere ς3=min⁡2k5,kϑ,1,kϑ,2,k6, with μr a positive constant defined as(51)μr=kϑ,12ϑr,12+kϑ,22ϑr,22+k62Dr2+0.2785σrDr.


## 3.3. Velocity Tracking Control

In this subsection, the designed control forceτu should enforce the following constraint on the velocity: limt→∞⁡ut→udt. In addition, the constraint ue<kue should also be strictly guaranteed. Using the first equation in (7), the time derivative of ue is calculated as(52)u˙e=vr+fu+ϑu,1Tφu,1+ϑu,2Tφu,2+τu+du-u˙dChoosing the barrier Lyapunov function(53)V6=12log⁡kue2kue2-ue2and differentiating (53) yield(54)V˙6=ϑuu˙ewhere ϑu=ue/kue2-ue2.We design the surge velocity tracking controller forτu as follows:(55)τu=-vr-ϑ^u,1Tφu,1-ϑ^u,2Tφu,2-D^utanh⁡ϑuσu-fu-k7ue+u˙d-ϑekue2-ue2sin⁡ψ-γpwhere k7>0 and σu>0 are design constants.The adaptation laws forϑ^u,1, ϑ^u,2, and D^u are given as(56)ϑ^˙u,1=ϑuφu,1-kϑ,3ϑ^u,1ϑ^˙u,2=ϑuφu,2-kϑ,4ϑ^u,2D^˙u=ϑutanh⁡ϑuσu-k8D^uwhere kϑ,3, kϑ,4, and k8 are positive design constants.Construct the Lyapunov functionV7 as(57)V7=V6+12ϑ~u,1Tϑ~u,1+12ϑ~u,2Tϑ~u,2+12D~u2Taking the time derivative of (57) and invoking (54)-(56), we obtain (58)V˙7=V˙6-ϑ~u,1Tϑ^˙u,1-ϑ~u,2Tϑ^˙u,2-D~uD^˙u≤-k7ue2kue2-ue2+kϑ,3ϑ~u,1Tϑ^u,1+kϑ,4ϑ~u,2Tϑ^u,2+k8D~uD^u+ϑudu-Dutanh⁡ϑuσu-ϑeuesin⁡ψ-γpNote also that ϑ~u,1Tϑ^u,1≤ϑu,12-ϑ~u,12/2, ϑ~u,2Tϑ^u,2≤ϑu,22-ϑ~u,22/2, and D~uD^u≤Du2-D~u2/2.We then have(59)V˙7≤-2k7V6-kϑ,32ϑ~u,12-kϑ,42ϑ~u,22-k82D~u2+μu-ϑeuesin⁡ψ-γp≤-ς4V7+μu-ϑeuesin⁡ψ-γpwhere ς4=min⁡2k7,kϑ,3,kϑ,4,k8 and μu is a positive constant defined as(60)μu=kϑ,32ϑu,12+kϑ,42ϑu,22+k82Du2+0.2578σuDu.Remark 12.
In (55), the desired surge speed udt is required to be continuously differentiable. In previous works [10, 11, 45], the desired surge speed ud was chosen as a constant for all t>0. However, in practice, the desired surge speed of a hovercraft is often directly switched to another value, which will lead to the nonexistence of u˙dt. Thus, to ensure that the desired surge speed is differentiable when ud changes from ud1 to ud2 between any time interval ta and tb, we give a speed smoother as follows: (61)ud=ud1+ht,ta,tb,γtud2-ud1The scalar function ht,ta,tb,γt is defined as(62)ht,ta,tb,γt=fτfτ+γtf1-τwhere τ=t-ta/tb-γt. fτ=0 if τ≤0, and fτ=e-1/τ if τ>0, with ta<tb and γt a positive constant. The speed smoother is shown in Figure 4.Figure 4
The desired surge speed and its derivative.


## 3.4. Stability Analysis

Theorem 13.
Consider the parameterized hovercraft models (7) and (10) in the presence of parametric uncertainties and external disturbances and assume that Assumptions 5–8 are satisfied. If the update law for the path variable ω is chosen as (22), the desired heading angle of the hovercraft is calculated by (23), the auxiliary system is designed as (33), the adaptation laws for unknown parameters and external disturbances are given by (44) and (56), and the controllers are obtained from (43) and (55); then the following properties hold:
(i) All the error signals in the closed-loop system are uniformly ultimately bounded (UUB).
(ii) The static constraints on position and surge velocity tracking errors and the time-varying constraints on the yaw rate are never violated, i.e.,ξs<ks, ξe<ke, ue<kue and rt<krt hold for all t>0.Proof.
(i) Construct the following candidate Lyapunov function:(63)V=V1+V3+V5+V7By virtue of (24), (38), (50), and (59), the time derivative of V is given by(64)V˙≤-ς1V1-ς2V3-ς3V5-ς4V7+ϑeψeχ+μr+ϑeuesin⁡ψ-γp-ϑeψeχ+ψerecos⁡ϕ-ψerecos⁡ϕ-ϑeuesin⁡ψ-γp+12εM2+μu≤-ς1V1-ς2V3-ς3V5-ς4V7+μu+μr+12εM2≤-ςVV+μVwhere ςV=min⁡ς1,ς2,ς3,ς4 and μV=μu+μr+1/2εM2.
If the initial conditions satisfyAssumption6, then the constructed Lyapunov function V satisfies the inequality(65)Vt≤Γt=V0-μVςVexp⁡-ςVt+μVςVFrom (65), we know that limt→∞⁡Vt≤limt→∞⁡Γt=μV/ςV. Therefore, all the error signals in the closed-loop system are uniformly ultimately bounded. Noting the expressions of μV and ςV, the ultimate bounds can be made arbitrarily small by adjusting the design parameters.
(ii)∀t>0, the BLF (14), (40), and (53) satisfy the following inequalities:(66)12log⁡ki2ki2-ξi2≤Vt≤Γt12log⁡kue2kue2-ue2≤Vt≤Γt12log⁡11-ξr2≤Vt≤Γtwhere i=s,e.
Then, from (66), we obtain(67)ξs≤ks1-exp⁡-2Γt<ksξe≤ke1-exp⁡-2Γt<keue≤kue1-exp⁡-2Γt<kueξr≤1-exp⁡-2Γt<1In (67), from ξr<1, ∀t>0, we know that ret<kret,∀t>0. RecallingRemark10, we conclude that rt<krt. Therefore, all the prescribed constraints of the hovercraft system can be guaranteed for ∀t>0. This completes the proof.


## 4. Simulation

In this section, numerical simulations will be implemented to demonstrate the effectiveness and safety of the proposed control scheme for hovercraft navigation and to compare our scheme with the results presented in [11, 45]. A general curvilinear path simulation scenario is considered. In simulations, the uncertainty parameters in (7), including Sxa, Sya, Sza, Sh, and C∗#, have 20% maximum relative uncertainties according to their nominal values; i.e., their real values are chosen according to(68)S∗=S∗nominal+λS∗nominal,C∗#=C∗nominal#+λC∗nominal#,where λ is randomly selected from -0.2,0.2. The nominal parameters of the hovercraft are shown in Appendix A, Table 1. For numerical simulations, the time-varying bounded disturbances in (7) are generated by using the first-order Gauss-Markov process [52]:(69)d˙u+a1du=ω1d˙v+a2dv=ω2d˙r+a3dr=ω3d˙p+a4dp=ω4where ωi(i=1,2,3,4) are zero-mean Gaussian white noise processes and ai≥0(i=1,2,3,4) are constants.Table 1
Nominal values of the hovercraft in the simulation.
Parameter Nominal Value SI-Unit Parameter Nominal Value SI-Unit C x a 0 -0.345 - C y h 0 -0.00011 - C x a δ r -0.00516 1/rad C y h ϕ -0.0012 1/rad C x a β -0.2504 1/rad C y h β 0.04582 1/rad C x a β ˙ -0.0001 s/rad C m z a 0 -0.000356 - C x h 0 -0.1049 - C m z a δ r -0.001534 1/rad C x h ϕ 0.00014 1/rad C m z a β -0.05412 1/rad C x h β 0.191 1/rad C m z a β ˙ -0.00021 s/rad C y a 0 -0.001 - C m z h ϕ 0 1/rad C y a δ r 0.002614 1/rad C m z h β 0 1/rad C y a β 0.1645 1/rad m 40000 kg C y a β ˙ 0.00042 s/rad I x 247810 k g m 2 S x a 45.27 m 2 I z 1746347 k g m 2 S y a 93 m 2 S z a 268 m 2 S h 21 m 2To demonstrate that the control objective can be achieved by the proposed controller, we first set the constant constraints on position and surge velocity to beks=30m, ke=30m, and kue=1m/s. The time-varying yaw rate constraint in our simulations is shown in Figure 5, which is similar to the yaw rate constraint in [33]. We can see that the yaw rate constraint of the hovercraft decreases with increasing surge speed. To adapt to practical applications, it is determined that the yaw rate should be further restricted to guarantee the navigation safety of a hovercraft with a higher speed. The yaw rate constraint krt can be written as krt=-0.25ut+7, with ut≥8m/s. The maximum yaw rate tracking error is limited by re≤1deg/s. Thus, the time-varying magnitude of the limiter in Figure 3 is set as kαrt=-0.25ut+6, with ut≥8m/s, and the yaw rate tracking error constraints should satisfy the conditions: kret≤krt-kαrt=1deg/s. The control parameters are listed in Table 2.Table 2
Control parameters in hovercraft simulations.
Parameter Value Parameter Value k 1 2 σu 0.1 k2 80 σr 0.1 k3 1 kϑ,1 5 c 1 0.1 k ϑ , 2 3 k 4 1.5 k ϑ , 3 1.5 k5 0.01 k ϑ , 4 4 k - 5 3 k 7 1 k 6 20 k 8 0.2Figure 5
The yaw rate constraints on the hovercraft.In addition to our proposed controller, we simulate two different controllers for comparison, which are assigned as follows.Case 1.
This comparative controller is derived from [11] by using traditional adaptive LOS guidance. The yaw rate constraint is not considered.Case 2.
This comparative controller is designed by using the strategy in [45], which can only guarantee the position constraints. However, the yaw rate constraint is not considered.Case 3.
This controller in this case is our proposed control scheme.The desired path for a curvilinear path following simulation is given asXp=ω,2000sin⁡ω/550T m. The initial positions of the hovercraft are chosen as x0,y0,ψ0=29m,29m,0rad, the initial velocities are u0,v0=7.5m/s,0m/s, and r0=0.0087rad/s. The initial states of the hovercraft satisfy Assumption 6, and according to (61), the desired surge speed udt varies from 8 m/s to 10 m/s between ta=406s and tb=410s. For comparison purposes, the desired surge speed in Cases 1 and 2 is directly switched from 8 m/s to 10 m/s at ts=408s. To analyse the control performances under these three controllers, simulation results are given as follows.The path following results in Figure6 demonstrate that the proposed controller has improved performance in terms of the tracking accuracy and convergence rate. Figure 7 shows that the path following error constraints cannot be violated under the proposed controller. Neither the along-track error nor the cross-track error exceeds the position constraints ks and ke. However, for the Case 1 method, the position tracking errors transgress the predefined constraints. The position constraint problem is also handled by the Case 2 method, in which the path following errors are also regulated within the prescribed bounds. Figure 8 shows the surge speed tracking results under the different control schemes. We can see that the surge speed smoothly and continuously tracks the desired surge speed ud under the proposed speed controller. However, the surge speed under Cases 1 and 2 is discontinuous and has a large overshoot.Figure 6
Curvilinear path following.Figure 7
Position errors of curvilinear path following.Figure 8
The surge speed of the curvilinear path following.The yaw rate tracking results under the three controllers are presented in Figure9. When t<ts=408s, the desired surge speed is ud=8m/s; thus, the yaw rate constraint is calculated as kr=5deg/s and the virtual yaw rate constraint is kαr=4deg/s. After ts, ud increases to 10 m/s; therefore the yaw rate constraint is calculated as kr=4.5deg/s and kαr=3.5deg/s. The yaw rate tracking error constraint is set as kre=1deg/s≤krt-kαrt.Figure 9
The yaw rate of the curvilinear path following.In the first subfigure of Figure9, we observe that the magnitude of the virtual yaw rate αrt is limited by the designed command filter. The yaw rate of the hovercraft has smaller tracking errors in both transient and steady-state performances, as expected. However, in the second and third figures of Figure 10, it can be observed that the yaw rate tracking errors have large transient and steady-state error. The yaw rate tracking errors under the three controllers are illustrated in Figure 10. The dotted line denotes the yaw rate constraint kret. It is shown that the yaw rate constraints are guaranteed under the proposed controller, which indicates that the yaw rate of the hovercraft can track the desired yaw rate with the prescribed precision. Therefore, the yaw rate of the hovercraft will never violate the corresponding safety boundary. In contrast, the yaw rate tracking errors under the methods of Cases 1 and 2 both violate the prescribed constraints, because the yaw rate tracking error constraint cannot be guaranteed by using the conventional quadratic Lyapunov function in theory. Due to the page limitation, in Figure 11, the estimate values of the uncertainty parameters in control laws (43) and (55) are shown in the form ξ1=mϑ^u,2Tφu,2, ξ2=Izϑ^r,2Tφr,2, ξ3=mϑ^u,1Tφu,1, and ξ4=Izϑ^r,1Tφr,1. Figure 12 shows the estimate values D^u and D^r. The control inputs τu and τr are shown in Figure 13; we see that the control signals under our proposed method perform better than another two controllers in terms of oscillations.Figure 10
The yaw rate tracking errors of the curvilinear path following.Figure 11
Estimated values of the uncertain parameters.Figure 12
Estimated values of the external disturbances.Figure 13
Control inputs of the curvilinear path following.


## 5. Conclusion

A constrained adaptive path following control approach has been proposed for a hovercraft under parametric uncertainties and external disturbances. It is shown that the proposed controller can strictly ensure the satisfaction of the prescribed position and velocity constraints by resorting to the barrier Lyapunov function design method. The command filter with a time-varying magnitude limit is employed to generate the desired virtual yaw rate and to reduce the computational complexity, while an auxiliary system is designed to compensate for the estimation error. The uncertain parameters and the upper bounds of the external disturbances are well estimated by designing parameter adaptation laws. Simulations have indicated the superiority of the proposed control scheme. Because the surge speed of the hovercraft is usually high, the time-varying delay of the measurements may happen. In our future work, we will take into account the time delays in the feedback loop.



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*Source: 1940784-2019-02-12.xml*