# Safety-Guaranteed Trajectory Tracking Control for the Underactuated Hovercraft with State and Input Constraints

Authors: Mingyu Fu; Shuang Gao; Chenglong Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper develops a safety-guaranteed trajectory tracking controller for hovercraft by using a safety-guaranteed auxiliary dynamic system, an integral sliding mode control, and an adaptive neural network method. The safety-guaranteed auxiliary dynamic system is designed to implement system state and input constraints. By considering the relationship of velocity and resistance hump, the velocity of hovercraft is constrained to eliminate the effect of resistance hump and obtain better stability. And the safety limit of drift angle is well performed to guarantee the light safe maneuvers of hovercraft tracking with high velocities. In view of the natural capabilities of actuators, the control input is constrained. High nonlinearity and model uncertainties of hovercraft are approximated by employing adaptive radical basis function neural networks. The proposed controller guarantees the boundedness of all the closed-loop signals. Specifically, the tracking errors are uniformly ultimately bounded. Numerical simulations are implemented to demonstrate the efficacy of the designed controller.


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## Body

## 1. Introduction

A hovercraft (Figure1) is supported totally by its air cushion, with a flexible skirt system around its periphery to seal the cushion air [1]. The hovercraft is able to run at high speed over shallow water, rapids, ice, and swamp where no other craft can go. These “special abilities” have attracted many military and civil users with particular mission requirements. The study about the safety-guaranteed trajectory tracking control of underactuated hovercraft moving with high velocities is meaningful and challenging to reduce the burden of pilot.Figure 1
A 3D model of hovercraft. Photo is from the international cooperation project described in Acknowledgments.From a detailed review of the available literatures about the trajectory tracking control of hovercraft [2–8], only position errors were considered and the velocities of hovercraft were not controlled. However, the velocity is related to the resistance hump of hovercraft. From [9], the resistance hump occurs in the vicinity of Froude number Fr=1 which can be calculated by Fr=V/glc, where V is the velocity of hovercraft and lc is the cushion length. From [10], two resistance humps (mainly caused by wave-making drag) are encountered for hovercraft during the acceleration process. It is shown in [1] that the resistance hump will be crossed as Fr increases and the craft will travel with better course stability and transverse stability. Hence, the velocity of hovercraft needs to be large enough, corresponding to large Fr, to avoid the resistance hump and hold the better stability.Moreover, drift angle plays a key role in the high-speed moving process of hovercraft [11, 12]. If the drift angle exceeds the angle of drift which corresponds to the maximum of hydrodynamic forces, the behavior of hovercraft will be nonstable [13]. The dangers caused by the drift angle include stern kickoff, plough-in, and great heeling [11]. Hence, safety limit of drift angle must be strictly obeyed in the high-speed tracking process to ensure safe maneuvers of hovercraft [12]. For instance, safety limit of a hovercraft shows if speed exceeds 40 knots, turning is not allowed; if speed is in the range of 25 knots~35 knots, drift angle needs to be within the limits of 7.5°~2°.Besides, from a practical viewpoint, the control input is restrained to prevent the actuators from going beyond their natural capabilities [14, 15]. And radical basis function neural networks (RBFNNs) are used to stabilize complex nonlinear dynamic systems and deal with model uncertainties [16–18].The contributions of this paper are as follows:(i)
The velocity of hovercraft is controlled within a reasonable range to avoid the effect of resistance hump and keep better stability.(ii)
The safety limit of drift angle is obeyed to get light safe maneuvers of hovercraft moving with high velocities.(iii)
The control input is constrained to handle input saturation.This paper is organized as follows. Section2 establishes a nonlinear model of underactuated hovercraft and the transformation of it. Section 3 proposes a safety-guaranteed auxiliary dynamic system for state and input constraints. Moreover, the controller is designed and analyzed in this section. Numerical simulation results are shown in Section 4, and the conclusion is discussed in Section 5.


## 2. Problem Formulation

### 2.1. Hovercraft Model Description

In general, only air propellers and rudders are available for hovercraft as shown in Figure2. It means only the surge and yaw can be regulated directly, but without any actuators for their sway motion [19, 20].Figure 2
Diagram of underactuated hovercraft.The nonlinear model of hovercraft is obtained by neglecting the roll and pitch motions.(1)x˙y˙z˙ψ˙T=SψuvwrT,u˙v˙w˙r˙=vr-ur00+MRu+τuRvRw-pcScRr+τr,where(2)Sψ=cos⁡ψ-sin⁡ψ00sin⁡ψcos⁡ψ0000100001,M=1m00001m00001m00001Jz;the signals u, v, and w represent the surge, sway, and heave velocities, r is the turn rate, x, y, and z denote the position of hovercraft’s mass center in the earth fixed frame, ψ describes yaw angle, m and Jz are hovercraft’s mass and moments of inertia, τu and τr are the control inputs which are provided by the actuators, pc is the average cushion pressure, Sc is the cushion area, and RuRvRwRrT are the total drags written as(3)Rm=ρaV-aQ,Rwm=Cwmpc2Bcρwg,Rsk=0.5ρwVa2Cskhlsk-0.34lskSc0.5+2.8167pclc-0.259-1Rwm,Ru=Rua+Rum+Ruwm+Rusk=-0.5ρaVa2CuaSPP-Rmcos⁡β-Rwmcos⁡β-Rskcos⁡β,Rv=Rva+Rvm+Rvwm+Rvsk=-0.5ρaVa2CvaSLP-Rmsin⁡β-Rwmsin⁡β-Rsksin⁡β,Rw=Rwa+Rwwm+G=-0.5ρaVa2CwaSHP-ρawQ+G,Rr=Rra+Rrm+Rrwm+Rrsk=-0.5ρaVa2CraSHPHhov-Rva×xa+Rua×ya-Rvmxm+Rumym-Rvwmxwm+Ruwmywm-Rvskxsk+Ruskysk,where the drag’s suffix a is the aerodynamic profile drag, wm is the wave-making drag, m is the air momentum drag, sk is the skirt drag, Cua, Cva, Cpa, Cra, Cwm, and Csk are the drag coefficients, Bc is the cushion beam, lc is the cushion length, G is the weight, SPP, SLP, and SHP are the positive, lateral, and horizontal projection areas, β is the drift angle, Q is the fan flow rate of cushion fan, h is the distance between baffle and the bottom of skirt’s finger, lsk is the total peripheral length of the skirts, Hhov is the height of hovercraft, ρa and ρw are air and water density, and (xa,ya), (xm,ym), (xwm,ywm), and (xsk,ysk) are the coordinates of force’s acting points.V
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a and Va in (3) can be obtained by(4)V-a=u+Vwcos⁡βw-ψ2+v+Vwsin⁡βw-ψ2,Va=V-a2+w2,in which Vw and βw are absolute wind speed and direction. More details can be found in [1, 9, 21].Remark 1.
When a hovercraft is moving on a calm water surface, cushion pressurepc varies within a narrow range and the heave motion is stable. This paper is the research about the horizontal motion of the hovercraft. Hence, the heave motion is not discussed and the cushion pressure pc is assumed to be a constant.
From Figure2, we have(5)v=utan⁡β.In order to make β be the system state and more convenient for the constraint and control of β, an improved model is derived from (1) and (5); that is,(6)x˙y˙z˙ψ˙T=Sψuutan⁡βwrT,u˙β˙w˙r˙=urtan⁡β-u˙sin⁡2β2u-rcos2β00+MRu+τuRvcos2βuRw-pcScRr+τr.


### 2.2. State and Input Constraints

Saturation nonlinearities of actuators can be described by(7)τϖ=satτϖc,τϖcmax,τϖcmin,ϖ=u,r,where τϖcmax and τϖcmin are the maximum and minimum limitations of actuators, τϖc are the designed control laws, and sat(·) is a generalized saturation function with the following form:(8)satα,αM,αm=αM,ifα>αM,α,ifαm≤α≤αM,αm,ifα<αm.Assumption 2.
All position, orientation, velocity, and acceleration values of hovercraft are available for feedback.
Safety limit ofβ and hump speed of hovercraft need to be obtained from model and real ship tests [11, 12]. In this paper, they are assumed to be known and available for the state constraint. Then the safe constraints of system state are defined as(9)umin≤u≤umax,βmin≤β≤βmax.


## 2.1. Hovercraft Model Description

In general, only air propellers and rudders are available for hovercraft as shown in Figure2. It means only the surge and yaw can be regulated directly, but without any actuators for their sway motion [19, 20].Figure 2
Diagram of underactuated hovercraft.The nonlinear model of hovercraft is obtained by neglecting the roll and pitch motions.(1)x˙y˙z˙ψ˙T=SψuvwrT,u˙v˙w˙r˙=vr-ur00+MRu+τuRvRw-pcScRr+τr,where(2)Sψ=cos⁡ψ-sin⁡ψ00sin⁡ψcos⁡ψ0000100001,M=1m00001m00001m00001Jz;the signals u, v, and w represent the surge, sway, and heave velocities, r is the turn rate, x, y, and z denote the position of hovercraft’s mass center in the earth fixed frame, ψ describes yaw angle, m and Jz are hovercraft’s mass and moments of inertia, τu and τr are the control inputs which are provided by the actuators, pc is the average cushion pressure, Sc is the cushion area, and RuRvRwRrT are the total drags written as(3)Rm=ρaV-aQ,Rwm=Cwmpc2Bcρwg,Rsk=0.5ρwVa2Cskhlsk-0.34lskSc0.5+2.8167pclc-0.259-1Rwm,Ru=Rua+Rum+Ruwm+Rusk=-0.5ρaVa2CuaSPP-Rmcos⁡β-Rwmcos⁡β-Rskcos⁡β,Rv=Rva+Rvm+Rvwm+Rvsk=-0.5ρaVa2CvaSLP-Rmsin⁡β-Rwmsin⁡β-Rsksin⁡β,Rw=Rwa+Rwwm+G=-0.5ρaVa2CwaSHP-ρawQ+G,Rr=Rra+Rrm+Rrwm+Rrsk=-0.5ρaVa2CraSHPHhov-Rva×xa+Rua×ya-Rvmxm+Rumym-Rvwmxwm+Ruwmywm-Rvskxsk+Ruskysk,where the drag’s suffix a is the aerodynamic profile drag, wm is the wave-making drag, m is the air momentum drag, sk is the skirt drag, Cua, Cva, Cpa, Cra, Cwm, and Csk are the drag coefficients, Bc is the cushion beam, lc is the cushion length, G is the weight, SPP, SLP, and SHP are the positive, lateral, and horizontal projection areas, β is the drift angle, Q is the fan flow rate of cushion fan, h is the distance between baffle and the bottom of skirt’s finger, lsk is the total peripheral length of the skirts, Hhov is the height of hovercraft, ρa and ρw are air and water density, and (xa,ya), (xm,ym), (xwm,ywm), and (xsk,ysk) are the coordinates of force’s acting points.V
-
a and Va in (3) can be obtained by(4)V-a=u+Vwcos⁡βw-ψ2+v+Vwsin⁡βw-ψ2,Va=V-a2+w2,in which Vw and βw are absolute wind speed and direction. More details can be found in [1, 9, 21].Remark 1.
When a hovercraft is moving on a calm water surface, cushion pressurepc varies within a narrow range and the heave motion is stable. This paper is the research about the horizontal motion of the hovercraft. Hence, the heave motion is not discussed and the cushion pressure pc is assumed to be a constant.
From Figure2, we have(5)v=utan⁡β.In order to make β be the system state and more convenient for the constraint and control of β, an improved model is derived from (1) and (5); that is,(6)x˙y˙z˙ψ˙T=Sψuutan⁡βwrT,u˙β˙w˙r˙=urtan⁡β-u˙sin⁡2β2u-rcos2β00+MRu+τuRvcos2βuRw-pcScRr+τr.


## 2.2. State and Input Constraints

Saturation nonlinearities of actuators can be described by(7)τϖ=satτϖc,τϖcmax,τϖcmin,ϖ=u,r,where τϖcmax and τϖcmin are the maximum and minimum limitations of actuators, τϖc are the designed control laws, and sat(·) is a generalized saturation function with the following form:(8)satα,αM,αm=αM,ifα>αM,α,ifαm≤α≤αM,αm,ifα<αm.Assumption 2.
All position, orientation, velocity, and acceleration values of hovercraft are available for feedback.
Safety limit ofβ and hump speed of hovercraft need to be obtained from model and real ship tests [11, 12]. In this paper, they are assumed to be known and available for the state constraint. Then the safe constraints of system state are defined as(9)umin≤u≤umax,βmin≤β≤βmax.


## 3. Controller Design

### 3.1. Safety-Guaranteed Auxiliary Dynamic System

Proposition 3.
A constraint error function is designed as follows:(10)Δκ=kw1satx,xmax,xmin-x+kw2satuc,ucmax,ucmin-uc,where kw1>0, kw2>0, x is the system state, and uc is the designed control input.
Then an auxiliary dynamic system is designed by(11)ξ˙=-kξ1ξ-ϑ·+kξ2Δκ2ξ2ξ+Δκ,ξ≥σ,deadzoneΔκ,σ-,ξ<σ,where ξ is the state of the auxiliary dynamic system, kξ1, kξ2 are positive constants, σ, σ- are positive small design constants, ϑ(·) can be derived from the stability analysis, and deadzone(Δκ,σ-) is a dead zone function given by(12)deadzoneΔκ,σ-=Δκ,ifΔκ>σ-,0,if-σ-<Δκ≤σ-,Δκ,ifΔκ≤σ-.


### 3.2. Design of the Desired States

Desired reference trajectory is generated by a virtual ship described in the following form:(13)x˙dy˙dψ˙d=cos⁡ψd-sin⁡ψd0sin⁡ψdcos⁡ψd0001udsetvdsetrdset.Then the trajectory tracking errors are defined as(14)xe,ye=x-xd,y-yd.For the position tracking, the desired states are designed by(15)ud=x˙d-kxxesignxecos⁡ψ+y˙d-kyyesignyesin⁡ψ,βd=arctan⁡-x˙d-kxxesignxeudsin⁡ψ+y˙d-kyyesignyeudcos⁡ψ,where kx>0 and ky>0 are control gains.Remark 4.
To guarantee the traceability of the reference trajectory under state constraints, the desired reference trajectory needs to satisfy the following conditions:(C1)
ηd, η˙d, and η¨d are all bounded, in which ηd={xd,yd,ψd}.(C2)
There existsTr>0 such that, for all t>Tr,(16)umin<ud<umax,βmin<βd<βmax.


### 3.3. Controller Design

State tracking errors are defined as(17)eu=u-ud,eβ=β-βd.Then two integral sliding mode manifolds are given by(18)su=eu+λ1∫0teudτ,sr=e˙β+λ2eβ+λ3∫0teβdτ,where λ1, λ2, and λ3 are positive constants.Using (6), (17), and (18), the time derivatives of su and sr are expressed as(19)s˙u=e˙u+λ1eu=fuxu+Muτu+MuRu-u˙d+λ1eu,(20)s˙r=e¨β+λ2e˙β+λ3eβ=frxr+Mrτr+Drw-β¨d+λ2e˙β+λ3eβ,where(21)fuxu=urtan⁡β,xu=u,β,rT,frxr=-u¨u-u˙22u2sin⁡2β-u˙β˙ucos⁡2β+β˙rsin⁡2β,xr=u,u˙,u¨,r,β,β˙T,Mu=1m,Mr=-cos2βJz,Drw=-β˙Rvsin⁡2βum+uR˙v-u˙Rvu2mcos2β-Rrcos2βJz.To deal with high nonlinearity and model uncertainties, fu(xu) and fr(xr) are approximated by RBFNNs.(22)fϖxϖ=Wϖ∗THϖxϖ+εϖ,ϖ=u,r,where xϖ∈RI is the input vector and Wϖ∗∈Rn is the ideal weight vector. Hϖ(xϖ):RI→Rn is the basis function vector with element hiϖ(xϖ) shown as follows:(23)hiϖxϖ=exp⁡-xϖ-μiϖ2σi2,where μiϖ is the center of the receptive field and σi is the width of the Gaussian function. The approximation error εϖ satisfies εϖ≤εN.Using (10) and (11), constraint error functions are designed as follows:(24)Δκu=kwu1satu,umax,umin-u+kwu2satτuc,τucmax,τucmin-τuc,Δκr=kwr1satβ,βmax,βmin-β+kwr2satτrc,τrcmax,τrcmin-τrc,where kwu1>0, kwu2>0, kwr1>0, and kwr2>0.Then the auxiliary dynamic system is designed as(25)ξ˙ϖ=-kξϖ1ξϖ-ϑϖ·+kξϖ2Δκϖ2ξϖ2ξϖ+Δκϖ,ξϖ≥σ,deadzoneΔκϖ,σ-,ξϖ<σ,where ϑϖ·=Mϖsϖτϖ-τϖc and ϖ=u,r.Finally, the control laws are given by(26)τuc=1Mu-k1su-η1signsu+u˙d-λ1eu-MuRu-W^uTHuxu-εNsignsu+ksuξu,(27)τrc=1Mrβ¨d-k2sr-η2signsr-λ2e˙β-Drw-λ3eβ-W^rTHrxr-εNsignsr+ksrξr,where k1, k2, η1, η2, ksu, and ksr are positive constants, W^u=W~u+Wu∗, W^r=W~r+Wr∗, and Drw is defined in (20).And the adaptive laws are(28)W^˙ϖ=γϖsϖHϖxϖϖ=u,r,where γϖ is the adaptive coefficient.


### 3.4. Stability Analysis

Theorem 5.
If the state tracking errors (17), the desired states (15), the auxiliary dynamic system (25), the surge control law τuc (26), the yaw control law τrc (27), and the adaptive laws (28) are applied to the hovercraft system represented by (6) and, for any bounded initial condition, the closed-loop control system signals su, sr, ξu, ξr, eu, eβ, xe, ye and w~iu and w~ir, i=1,2,…,n, are uniformly ultimately bounded (UUB). The position and desired state tracking errors can be made arbitrarily small by appropriately selecting design parameters. And the yaw motion will remain bounded.Proof.
The following Lyapunov function is defined:(29)V=12su2+12sr2+12γuW~uTW~u+12γrW~rTW~r+12ξu2+12ξr2.From (19), (20), and (29), V˙ can be expressed as(30)V˙=sufuxu+Muτu+MuRu-u˙d+λ1eu+srfrxr+Mrτr+Drw-β¨d+λ2e˙β+λ3eβ+W~uTW^˙uγu+W~rTW^˙rγr+ξuξ˙u+ξrξ˙r.By defining τϖ=Δτϖ+τϖc and ϖ=u,r and using (22), (26), and (27), we have(31)V˙=-k1su2-η1su+su-W~uTHuxu+MusuΔτu+ksusuξu+-εNsu+εusu-k2sr2-η2sr+sr-W~rTHrxr+MrsrΔτr+ksrsrξr+-εNsr+εrsr+1γuW~uTW^˙u+1γrW~rTW^˙r+ξuξ˙u+ξrξ˙r.By using εϖ≤εN and ϖ=u,r, (31) can be rewritten as(32)V˙≤-k1su2-η1su+su-W~uTHuxu-k2sr2-η2sr+sr-W~rTHrxr+MusuΔτu+ksusuξu+MrsrΔτr+ksrsrξr+1γuW~uTW^˙u+1γrW~rTW^˙r+ξuξ˙u+ξrξ˙r.Substituting adaptive laws (28) into it yields(33)V˙≤-k1su2+MusuΔτu+ksusuξu+ξuξ˙u-k2sr2+MrsrΔτr+ksrsrξr+ξrξ˙r.The process of stability analysis is, respectively, discussed in the following two cases.
Case  1. When ξϖ≥σ, in light of (25) and Young’s inequality, we have(34)ξϖξ˙ϖ=ξϖΔκϖ-kξϖ1ξϖ2-kξϖ2Δκϖ2-Mϖsϖτϖ-τϖc≤kξϖ2Δκϖ2+14kξϖ2ξϖ2-kξϖ1ξϖ2-kξϖ2Δκϖ2-Mϖsϖτϖ-τϖc=-kξϖ1-14kξϖ2ξϖ2-Mϖsϖτϖ-τϖc.Substituting τϖ=Δτϖ+τϖc and (34) into (33) and using Young’s inequality yield(35)V˙≤-k1su2+ksusuξu-kξu1-14kξu2ξu2-k2sr2+ksrsrξr-kξr1-14kξr2ξr2≤-k1-0.5su2-kξu1-14kξu2-0.5ksu2ξu2-k2-0.5sr2-kξr1-14kξr2-0.5ksr2ξr2-k1W~uTW~uγu-k2W~rTW~rγr+k1W~uTW~uγu+k2W~rTW~rγr≤-2μ1V+ρ1,where(36)μ1=min⁡k1-0.5,kξu1-14kξu2-0.5ksu2,k2-0.5,kξr1-14kξr2-0.5ksr2,ρ1=k1γuW~uTW~u+k2γrW~rTW~r.
Case  2. When ξϖ<σ, in light of (25) and Young’s inequality, we have(37)ξϖξ˙ϖ=ξϖdeadzoneΔκϖ,σ-≤0.5ξϖ2+0.5Δκϖ2,0.5ksϖ2ξϖ2=ksϖ2ξϖ2-0.5ksϖ2ξϖ2≤-0.5ksϖ2ξϖ2+ksϖ2σ2.Substituting (37) into (33) and using Young’s inequality yield(38)V˙≤-k1su2+MusuΔτu+ksusuξu+0.5ξu2+0.5Δκu2-k2sr2+MrsrΔτr+ksrsrξr+0.5ξr2+0.5Δκr2≤-k1-1su2-k2-1sr2+0.5Mu2Δτu2+0.5Mr2Δτr2+0.5ksu2ξu2+0.5ksr2ξr2+0.5ξu2+0.5Δκu2+0.5ξr2+0.5Δκr2≤-k1-1su2-k2-1sr2-0.5ξu2ksu2-1-0.5ξr2ksr2-1+0.5Mu2Δτu2+0.5Mr2Δτr2+ksu2σ2+ksr2σ2+0.5Δκu2+0.5Δκr2≤-2μ2V+ρ2,where(39)μ2=min⁡k1-1,0.5ksu2-0.5,k2-1,0.5ksr2-0.5,ρ2=k1γuW~uTW~u+k2γrW~rTW~r+0.5Mu2Δτu2+0.5Mr2Δτr2+ksu2+ksr2σ2+Δκu2+Δκr22.Synthesizing (35) and (38), we have(40)V˙≤-2μV+ρ,where μ=min⁡μ1,μ2 and ρ=max⁡ρ1,ρ2 with the design parameters k1, k2, ksu, ksr, kξu1, kξu2, kξr1, and kξr2 satisfying(41)k1>1,k2>1,ksu>1,ksr>1,kξu1-14kξu2-0.5ksu2>0,kξr1-14kξr2-0.5ksr2>0.Solving (40), we have(42)0≤Vt≤ρ2μ+V0-ρ2μe-2μt.It is obviously seen that V(t) is UUB for all V(0)≤B0 with B0 being any positive constant. Therefore, in the light of (29), we know that su, sr, ξu, ξr and w~iu and w~ir, i=1,2,…,n, are UUB for all V(0)≤B0. It can be expressed as (43)χ≤ρμ+2V0-ρ2μe-2μt,where χ=su,sr,ξu,ξr,w~iu,w~ir.
It implies that there existsT>0 such that, for all t>T,(44)χ≤ρμ,where ρ/μ can be made arbitrarily small by appropriately selecting the design parameters.
Further, the following dynamics are obtained from (18) and (44):(45)eu+λ1∫0teudτ=εu,e˙β+λ2eβ+λ3∫0teβdτ=εβ,where εu and εβ are arbitrarily small errors.
To prove thateu and eβ are UUB, the Lyapunov function is given by(46)Vs=12∫0teudτ2+12eβ2.The time derivative of this Lyapunov function along the dynamics in (45) is such that(47)V˙s=eu∫0teudτ+eβe˙β=-λ1∫0teudτ2-λ2eβ2+εu∫0teudτ-λ3eβ∫0teβdτ+εβeβ.From Young’s inequality, the following fact is obtained:(48)εu∫0teudτ≤0.5λ1∫0teudτ2+0.5εu2λ1,-λ3eβ∫0teβdτ≤λ2eβ22+λ32λ2∫0teβdτ2,εβeβ≤0.25λ2eβ2+εβ2λ2.Then (49)V˙s≤-0.5λ1∫0teudτ2-0.25λ2eβ2+0.5εu2λ1+0.5λ3λ2∫0teβdτ2+εβ2λ2=-2μsVs+ρs,where(50)μs=min⁡0.5λ1,0.25λ2,ρs=0.5εu2λ1+0.5λ3λ2∫0teβdτ2+εβ2λ2.Similar to the analysis in (42)~(44), there exists Ts>0 such that, for all t>Ts,(51)χs≤ρsμs,where χs=∫0teudτ,eβ.
It is obvious that∫0teudτ and eβ are UUB and will be arbitrarily small by choosing suitable parameters.
Fromsu=eu+λ1∫0teudτ, we know eu will be arbitrarily small. From (6) and (15), we have(52)u-udutan⁡β-udtan⁡βd=R-x˙e+kxxesignxey˙e+kyyesignye,where(53)R-=cos⁡ψsin⁡ψ-sin⁡ψcos⁡ψ.It is clear that R-=1 which indicates that it is nonsingular. Then we have(54)x˙e=-kxxesignxe+εxb,y˙e=-kyyesignye+εyb,where εxb, εyb are arbitrarily small errors.
Furthermore, consider the following Lyapunov function candidate:(55)Vp=12xe2+12ye2.The time derivative of Vp along the dynamics in (54) is such that (56)V˙p=xex˙e+yey˙e=xe-kxxesignxe+εxb+ye-kyyesignye+εyb=-kxxe2-kyye2+xeεxb+yeεyb≤-0.5kxxe2-0.5kyye2+εxb22kx+εyb22ky=-2μpVp+ρp,where(57)μp=min⁡0.5kx,0.5ky,ρp=εxb22kx+εyb22ky.Also similar to the analysis in (42)~(44), there exists Tp>0 such that, for all t>Tp,(58)χp≤ρpμp,where χp=xe,ye.
From the reason thatεxb and εyb are arbitrarily small errors, we know that xe and ye are UUB and will be arbitrarily small.
Also,u is continuously differentiable in the moving process of hovercraft. Hence, u˙ is bounded. From (6), we have(59)u˙=urtan⁡β+Ru+τum.From (15), Remark 4, and the boundedness of xe and ye, we have that ud and βd are bounded. Then u, v, and β are bounded from (5) and (17). Furthermore, Ru is bounded from (3). Therefore, it can be concluded that r will remain bounded from (7) and (59). This concludes the proof.Remark 6.
In order to avoid the well-known chattering problem, the sign function used in the control laws (26) and (27) can be replaced by hyperbolic tangent function which is continuous such that sign(s)=tanh⁡(kHs), where kH is a positive scalar which can be chosen to get a very good approximation.


## 3.1. Safety-Guaranteed Auxiliary Dynamic System

Proposition 3.
A constraint error function is designed as follows:(10)Δκ=kw1satx,xmax,xmin-x+kw2satuc,ucmax,ucmin-uc,where kw1>0, kw2>0, x is the system state, and uc is the designed control input.
Then an auxiliary dynamic system is designed by(11)ξ˙=-kξ1ξ-ϑ·+kξ2Δκ2ξ2ξ+Δκ,ξ≥σ,deadzoneΔκ,σ-,ξ<σ,where ξ is the state of the auxiliary dynamic system, kξ1, kξ2 are positive constants, σ, σ- are positive small design constants, ϑ(·) can be derived from the stability analysis, and deadzone(Δκ,σ-) is a dead zone function given by(12)deadzoneΔκ,σ-=Δκ,ifΔκ>σ-,0,if-σ-<Δκ≤σ-,Δκ,ifΔκ≤σ-.


## 3.2. Design of the Desired States

Desired reference trajectory is generated by a virtual ship described in the following form:(13)x˙dy˙dψ˙d=cos⁡ψd-sin⁡ψd0sin⁡ψdcos⁡ψd0001udsetvdsetrdset.Then the trajectory tracking errors are defined as(14)xe,ye=x-xd,y-yd.For the position tracking, the desired states are designed by(15)ud=x˙d-kxxesignxecos⁡ψ+y˙d-kyyesignyesin⁡ψ,βd=arctan⁡-x˙d-kxxesignxeudsin⁡ψ+y˙d-kyyesignyeudcos⁡ψ,where kx>0 and ky>0 are control gains.Remark 4.
To guarantee the traceability of the reference trajectory under state constraints, the desired reference trajectory needs to satisfy the following conditions:(C1)
ηd, η˙d, and η¨d are all bounded, in which ηd={xd,yd,ψd}.(C2)
There existsTr>0 such that, for all t>Tr,(16)umin<ud<umax,βmin<βd<βmax.


## 3.3. Controller Design

State tracking errors are defined as(17)eu=u-ud,eβ=β-βd.Then two integral sliding mode manifolds are given by(18)su=eu+λ1∫0teudτ,sr=e˙β+λ2eβ+λ3∫0teβdτ,where λ1, λ2, and λ3 are positive constants.Using (6), (17), and (18), the time derivatives of su and sr are expressed as(19)s˙u=e˙u+λ1eu=fuxu+Muτu+MuRu-u˙d+λ1eu,(20)s˙r=e¨β+λ2e˙β+λ3eβ=frxr+Mrτr+Drw-β¨d+λ2e˙β+λ3eβ,where(21)fuxu=urtan⁡β,xu=u,β,rT,frxr=-u¨u-u˙22u2sin⁡2β-u˙β˙ucos⁡2β+β˙rsin⁡2β,xr=u,u˙,u¨,r,β,β˙T,Mu=1m,Mr=-cos2βJz,Drw=-β˙Rvsin⁡2βum+uR˙v-u˙Rvu2mcos2β-Rrcos2βJz.To deal with high nonlinearity and model uncertainties, fu(xu) and fr(xr) are approximated by RBFNNs.(22)fϖxϖ=Wϖ∗THϖxϖ+εϖ,ϖ=u,r,where xϖ∈RI is the input vector and Wϖ∗∈Rn is the ideal weight vector. Hϖ(xϖ):RI→Rn is the basis function vector with element hiϖ(xϖ) shown as follows:(23)hiϖxϖ=exp⁡-xϖ-μiϖ2σi2,where μiϖ is the center of the receptive field and σi is the width of the Gaussian function. The approximation error εϖ satisfies εϖ≤εN.Using (10) and (11), constraint error functions are designed as follows:(24)Δκu=kwu1satu,umax,umin-u+kwu2satτuc,τucmax,τucmin-τuc,Δκr=kwr1satβ,βmax,βmin-β+kwr2satτrc,τrcmax,τrcmin-τrc,where kwu1>0, kwu2>0, kwr1>0, and kwr2>0.Then the auxiliary dynamic system is designed as(25)ξ˙ϖ=-kξϖ1ξϖ-ϑϖ·+kξϖ2Δκϖ2ξϖ2ξϖ+Δκϖ,ξϖ≥σ,deadzoneΔκϖ,σ-,ξϖ<σ,where ϑϖ·=Mϖsϖτϖ-τϖc and ϖ=u,r.Finally, the control laws are given by(26)τuc=1Mu-k1su-η1signsu+u˙d-λ1eu-MuRu-W^uTHuxu-εNsignsu+ksuξu,(27)τrc=1Mrβ¨d-k2sr-η2signsr-λ2e˙β-Drw-λ3eβ-W^rTHrxr-εNsignsr+ksrξr,where k1, k2, η1, η2, ksu, and ksr are positive constants, W^u=W~u+Wu∗, W^r=W~r+Wr∗, and Drw is defined in (20).And the adaptive laws are(28)W^˙ϖ=γϖsϖHϖxϖϖ=u,r,where γϖ is the adaptive coefficient.


## 3.4. Stability Analysis

Theorem 5.
If the state tracking errors (17), the desired states (15), the auxiliary dynamic system (25), the surge control law τuc (26), the yaw control law τrc (27), and the adaptive laws (28) are applied to the hovercraft system represented by (6) and, for any bounded initial condition, the closed-loop control system signals su, sr, ξu, ξr, eu, eβ, xe, ye and w~iu and w~ir, i=1,2,…,n, are uniformly ultimately bounded (UUB). The position and desired state tracking errors can be made arbitrarily small by appropriately selecting design parameters. And the yaw motion will remain bounded.Proof.
The following Lyapunov function is defined:(29)V=12su2+12sr2+12γuW~uTW~u+12γrW~rTW~r+12ξu2+12ξr2.From (19), (20), and (29), V˙ can be expressed as(30)V˙=sufuxu+Muτu+MuRu-u˙d+λ1eu+srfrxr+Mrτr+Drw-β¨d+λ2e˙β+λ3eβ+W~uTW^˙uγu+W~rTW^˙rγr+ξuξ˙u+ξrξ˙r.By defining τϖ=Δτϖ+τϖc and ϖ=u,r and using (22), (26), and (27), we have(31)V˙=-k1su2-η1su+su-W~uTHuxu+MusuΔτu+ksusuξu+-εNsu+εusu-k2sr2-η2sr+sr-W~rTHrxr+MrsrΔτr+ksrsrξr+-εNsr+εrsr+1γuW~uTW^˙u+1γrW~rTW^˙r+ξuξ˙u+ξrξ˙r.By using εϖ≤εN and ϖ=u,r, (31) can be rewritten as(32)V˙≤-k1su2-η1su+su-W~uTHuxu-k2sr2-η2sr+sr-W~rTHrxr+MusuΔτu+ksusuξu+MrsrΔτr+ksrsrξr+1γuW~uTW^˙u+1γrW~rTW^˙r+ξuξ˙u+ξrξ˙r.Substituting adaptive laws (28) into it yields(33)V˙≤-k1su2+MusuΔτu+ksusuξu+ξuξ˙u-k2sr2+MrsrΔτr+ksrsrξr+ξrξ˙r.The process of stability analysis is, respectively, discussed in the following two cases.
Case  1. When ξϖ≥σ, in light of (25) and Young’s inequality, we have(34)ξϖξ˙ϖ=ξϖΔκϖ-kξϖ1ξϖ2-kξϖ2Δκϖ2-Mϖsϖτϖ-τϖc≤kξϖ2Δκϖ2+14kξϖ2ξϖ2-kξϖ1ξϖ2-kξϖ2Δκϖ2-Mϖsϖτϖ-τϖc=-kξϖ1-14kξϖ2ξϖ2-Mϖsϖτϖ-τϖc.Substituting τϖ=Δτϖ+τϖc and (34) into (33) and using Young’s inequality yield(35)V˙≤-k1su2+ksusuξu-kξu1-14kξu2ξu2-k2sr2+ksrsrξr-kξr1-14kξr2ξr2≤-k1-0.5su2-kξu1-14kξu2-0.5ksu2ξu2-k2-0.5sr2-kξr1-14kξr2-0.5ksr2ξr2-k1W~uTW~uγu-k2W~rTW~rγr+k1W~uTW~uγu+k2W~rTW~rγr≤-2μ1V+ρ1,where(36)μ1=min⁡k1-0.5,kξu1-14kξu2-0.5ksu2,k2-0.5,kξr1-14kξr2-0.5ksr2,ρ1=k1γuW~uTW~u+k2γrW~rTW~r.
Case  2. When ξϖ<σ, in light of (25) and Young’s inequality, we have(37)ξϖξ˙ϖ=ξϖdeadzoneΔκϖ,σ-≤0.5ξϖ2+0.5Δκϖ2,0.5ksϖ2ξϖ2=ksϖ2ξϖ2-0.5ksϖ2ξϖ2≤-0.5ksϖ2ξϖ2+ksϖ2σ2.Substituting (37) into (33) and using Young’s inequality yield(38)V˙≤-k1su2+MusuΔτu+ksusuξu+0.5ξu2+0.5Δκu2-k2sr2+MrsrΔτr+ksrsrξr+0.5ξr2+0.5Δκr2≤-k1-1su2-k2-1sr2+0.5Mu2Δτu2+0.5Mr2Δτr2+0.5ksu2ξu2+0.5ksr2ξr2+0.5ξu2+0.5Δκu2+0.5ξr2+0.5Δκr2≤-k1-1su2-k2-1sr2-0.5ξu2ksu2-1-0.5ξr2ksr2-1+0.5Mu2Δτu2+0.5Mr2Δτr2+ksu2σ2+ksr2σ2+0.5Δκu2+0.5Δκr2≤-2μ2V+ρ2,where(39)μ2=min⁡k1-1,0.5ksu2-0.5,k2-1,0.5ksr2-0.5,ρ2=k1γuW~uTW~u+k2γrW~rTW~r+0.5Mu2Δτu2+0.5Mr2Δτr2+ksu2+ksr2σ2+Δκu2+Δκr22.Synthesizing (35) and (38), we have(40)V˙≤-2μV+ρ,where μ=min⁡μ1,μ2 and ρ=max⁡ρ1,ρ2 with the design parameters k1, k2, ksu, ksr, kξu1, kξu2, kξr1, and kξr2 satisfying(41)k1>1,k2>1,ksu>1,ksr>1,kξu1-14kξu2-0.5ksu2>0,kξr1-14kξr2-0.5ksr2>0.Solving (40), we have(42)0≤Vt≤ρ2μ+V0-ρ2μe-2μt.It is obviously seen that V(t) is UUB for all V(0)≤B0 with B0 being any positive constant. Therefore, in the light of (29), we know that su, sr, ξu, ξr and w~iu and w~ir, i=1,2,…,n, are UUB for all V(0)≤B0. It can be expressed as (43)χ≤ρμ+2V0-ρ2μe-2μt,where χ=su,sr,ξu,ξr,w~iu,w~ir.
It implies that there existsT>0 such that, for all t>T,(44)χ≤ρμ,where ρ/μ can be made arbitrarily small by appropriately selecting the design parameters.
Further, the following dynamics are obtained from (18) and (44):(45)eu+λ1∫0teudτ=εu,e˙β+λ2eβ+λ3∫0teβdτ=εβ,where εu and εβ are arbitrarily small errors.
To prove thateu and eβ are UUB, the Lyapunov function is given by(46)Vs=12∫0teudτ2+12eβ2.The time derivative of this Lyapunov function along the dynamics in (45) is such that(47)V˙s=eu∫0teudτ+eβe˙β=-λ1∫0teudτ2-λ2eβ2+εu∫0teudτ-λ3eβ∫0teβdτ+εβeβ.From Young’s inequality, the following fact is obtained:(48)εu∫0teudτ≤0.5λ1∫0teudτ2+0.5εu2λ1,-λ3eβ∫0teβdτ≤λ2eβ22+λ32λ2∫0teβdτ2,εβeβ≤0.25λ2eβ2+εβ2λ2.Then (49)V˙s≤-0.5λ1∫0teudτ2-0.25λ2eβ2+0.5εu2λ1+0.5λ3λ2∫0teβdτ2+εβ2λ2=-2μsVs+ρs,where(50)μs=min⁡0.5λ1,0.25λ2,ρs=0.5εu2λ1+0.5λ3λ2∫0teβdτ2+εβ2λ2.Similar to the analysis in (42)~(44), there exists Ts>0 such that, for all t>Ts,(51)χs≤ρsμs,where χs=∫0teudτ,eβ.
It is obvious that∫0teudτ and eβ are UUB and will be arbitrarily small by choosing suitable parameters.
Fromsu=eu+λ1∫0teudτ, we know eu will be arbitrarily small. From (6) and (15), we have(52)u-udutan⁡β-udtan⁡βd=R-x˙e+kxxesignxey˙e+kyyesignye,where(53)R-=cos⁡ψsin⁡ψ-sin⁡ψcos⁡ψ.It is clear that R-=1 which indicates that it is nonsingular. Then we have(54)x˙e=-kxxesignxe+εxb,y˙e=-kyyesignye+εyb,where εxb, εyb are arbitrarily small errors.
Furthermore, consider the following Lyapunov function candidate:(55)Vp=12xe2+12ye2.The time derivative of Vp along the dynamics in (54) is such that (56)V˙p=xex˙e+yey˙e=xe-kxxesignxe+εxb+ye-kyyesignye+εyb=-kxxe2-kyye2+xeεxb+yeεyb≤-0.5kxxe2-0.5kyye2+εxb22kx+εyb22ky=-2μpVp+ρp,where(57)μp=min⁡0.5kx,0.5ky,ρp=εxb22kx+εyb22ky.Also similar to the analysis in (42)~(44), there exists Tp>0 such that, for all t>Tp,(58)χp≤ρpμp,where χp=xe,ye.
From the reason thatεxb and εyb are arbitrarily small errors, we know that xe and ye are UUB and will be arbitrarily small.
Also,u is continuously differentiable in the moving process of hovercraft. Hence, u˙ is bounded. From (6), we have(59)u˙=urtan⁡β+Ru+τum.From (15), Remark 4, and the boundedness of xe and ye, we have that ud and βd are bounded. Then u, v, and β are bounded from (5) and (17). Furthermore, Ru is bounded from (3). Therefore, it can be concluded that r will remain bounded from (7) and (59). This concludes the proof.Remark 6.
In order to avoid the well-known chattering problem, the sign function used in the control laws (26) and (27) can be replaced by hyperbolic tangent function which is continuous such that sign(s)=tanh⁡(kHs), where kH is a positive scalar which can be chosen to get a very good approximation.


## 4. Simulations

Two different cases are implemented to verify the effectiveness and superiority of the proposed controller. In simulations, the main particulars and constraints of hovercraft are shown in Tables1 and 2. The water surface is calm without waves.Table 1
Main particulars of hovercraft.
m (kg)
40000
J
z (kgm2)
1.8
×
1
0
6
S
P
P (m2)
45
S
L
P (m2)
93
S
H
P (m2)
260
Q (m3/s)
140.8
V
w (knots)
10
l
s
k (m)
65
B
c (m)
8.9
l
c (m)
23.6
h (m)
1
H
h
o
v (m)
5.9
S
c (m2)
212
β
w (deg)
45Table 2
State and input constraints.
Variable
Maximum
Minimum
u (knots)
40
25
β (deg)
8
-
8
τ
u (N)
80000
-
80000
τ
r (Nm)
1
×
1
0
5
-
1
×
1
0
5The comparisons of three different methods are carried out in each case. The legend “Method A” means the method in [14]; the legend “Method B” means the method without state and input saturation constraints.Saturation coefficients are(60)ksu=1.2,kwu1=1,kwu2=903000,kξu1=31,ksr=1.2,kwr1=1,kwr2=650300,kξr1=38,kξu2=0.08,kξr2=0.08,σ-=3,σ=1.Case 1.
The reference trajectory parameters are(61)xd0=300m,yd0=100m,ψd0=45°,udsett=35knots,vdsett=0,rdsett=0.The initial values of hovercraft model are (62)x0=0,y0=0,ψ0=70°,u0=35knots,β0=0,r0=0.The controller parameters are(63)kx=0.6,ky=0.6,k1=1.1,η1=0.25,λ1=2,k2=1.2,η2=0.5,λ2=30,λ3=6,εN=0.5.It is observed from Figures 3–8 that the proposed trajectory tracking controller is effective. Tracking errors of all three methods converge to arbitrarily small values and heading angle is bounded. From the comparisons with Methods A and B in Figures 9 and 10, the proposed controller can limit the surge speed and the drift angle into the safe range effectively. Figures 11 and 12 show that the input saturation is also handled by the proposed controller.Figure 3
The actual and desired trajectory of hovercraft.Figure 4
The heading angle of hovercraft.Figure 5
The surge position tracking error.Figure 6
The sway position tracking error.Figure 7
The surge velocity tracking error.Figure 8
The drift angle tracking error.Figure 9
The surge velocity of hovercraft.Figure 10
The drift angle of hovercraft.Figure 11
The surge control law of hovercraft.Figure 12
The yaw control law of hovercraft.Case 2.
The reference trajectory parameters are(64)xd0=-200m,yd0=50m,ψd0=45°,udsett=35knots,vdsett=0,rdsett=0.1°/s.The initial values of hovercraft model are(65)x0=0,y0=0,ψ0=30°,u0=35knots,β0=0,r0=0.The controller parameters are(66)kx=0.6,ky=0.6,k1=1.53,η1=0.1,λ1=4,k2=1.2,η2=0.5,λ2=30,λ3=2,εN=0.5.It is obvious from Figures 14–19 that only the proposed controller can track the trajectory successfully. Figures 20 and 21 show that the surge speed changes to negative values and the drift angle exceeds the safety limit in the tracking control process of Methods A and B. These are absolutely not allowed for hovercraft. In contrast, you can see from Figure 20 that surge speed can still be positive, even large enough to avoid the influence of the resistance humps under the control of the proposed controller. And drift angle is also within the safety limit from Figure 21. From Figures 22 and 23, we know that the input constraint ability of the proposed controller is effective. Figures 13 and 24 show the heave position of hovercraft.Figure 13
The heave position of hovercraft.Figure 14
The actual and desired trajectory of hovercraft.Figure 15
The heading angle of hovercraft.Figure 16
The surge position tracking error.Figure 17
The sway position tracking error.Figure 18
The surge velocity tracking error of hovercraft.Figure 19
The drift angle tracking error of hovercraft.Figure 20
The surge velocity of hovercraft.Figure 21
The drift angle of hovercraft.Figure 22
The surge control law of hovercraft.Figure 23
The yaw control law of hovercraft.Figure 24
The heave position of hovercraft.


## 5. Conclusion

A safety-guaranteed trajectory tracking controller has been proposed for underactuated hovercraft in this paper. The safety-guaranteed auxiliary dynamic system is designed to deal with state and input constraints. The velocity of hovercraft is constrained to eliminate the effect of resistance hump and obtain better stability. The safety limit of drift angle is carried out effectively for safety in the high-speed trajectory tracking process of hovercraft. The input saturation is handled. High nonlinearity and model uncertainties are approximated by RBFNNs. Simulation results have been presented to illustrate the effectiveness of the proposed controller.



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