# Safety-Guaranteed Course Control of Air Cushion Vehicle with Dynamic Safe Space Constraint

Authors: Mingyu Fu; Shuang Gao; Chenglong Wang
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper develops a safety-guaranteed course controller for air cushion vehicle (ACV). As the safety criterion, the unique safety limit of ACV including turn rate (TR) and sideslip angle (SA) changes with the speed when ACV is turning. To be more intuitive to show the change of safety limit and more convenient for safety monitoring and control, dynamic safe space of ACV is proposed. If the work point is within the safe space during the manual operation or automatic control, the sailing of ACV is safe. Then, the safety-guaranteed controller is designed to keep TR and SA within the safe space during the course control process based on the dynamic safe space constraint, terminal sliding mode control, and adaptive mechanism. The adaptive mechanism can effectively estimate the system uncertainty and external disturbances online without the requirement of their upper bounds. The proposed controller guarantees the convergence of tracking error. Simulations are implemented to demonstrate the efficacy of the designed controller.


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## Body

## 1. Introduction

An air cushion vehicle (ACV, as shown in Figure1) is supported totally by its air cushion, with a flexible skirt system around its periphery to seal the cushion air [1]. The ACV 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 motion control of ACV moving with high velocities is meaningful to reduce the burden of drivers.Figure 1
A 3D model of hovercraft. Photo is from the international cooperation project described in the Acknowledgment.From a detailed review of the available literatures about the motion control of ACV such as Antonio Pedro Aguiar [2], Hebertt Sira-Ramirez [3], Tanaka K [4], Han B [5], Aranda [6], Zhao Jingbo [7], Morales R [8], Gerasimos G. Rigatos[9], and Shojaei Khoshnam [10, 11], the designs only concern the success of control task and the improvement of control performance. And the safety of ACV which is the necessary precondition to complete the control task is neglected by them.The turn rate (TR) and sideslip angle (SA) play key roles for the safety of ACV in the high-speed moving process [12, 13]. If the SA exceeds the angle of drift which corresponds to the maximum of hydrodynamic forces, the behavior of hovercraft will be nonstable [14]. The dangers caused by the TR and SA include stern-kickoff, plough-in, and great heeling [12]. Hence, safety limit of TR and SA must be strictly obeyed in the high-speed moving process to ensure safe maneuvers of ACV [13]. For instance, safety limit of an ACV in [13] shows that if speed is 35 knots, the limits of TR and SA are 1.2°/s and 2°. If speed is 14 knots, the limits of TR and SA are 3.5°/s and 20°. The safety limit of TR and SA has been considered in the course controller and trajectory controller by Mingyu Fu [1, 15]. However, the constant safety limit is used in them. In practice, the safety limit changes with the speed of ACV. Hence in this paper, the changed safety limit is considered. And a dynamic safe space is established based on the relationship of the speed and the safety limit.Moreover, sliding mode control (SMC) has attracted significant amount of interest due to its fast global convergence, simplicity of implementation, and high robustness to external disturbances [16, 17]. Compared to the conventional linear sliding mode (LSM) control, terminal sliding mode (TSM) control has superior properties such as fast and finite-time convergence and smaller steady-state tracking errors [18–22]. So a safety-guaranteed course controller is designed based on the dynamic safe space constraint and TSM method. The system uncertainty and external disturbances can be estimated online by the designed adaptive mechanism in this paper without the requirement of their upper bounds.Remark 1.
The control task in this paper is chosen as course control. It is an important and common control task for ACV due to the difference with heading control as shown in Figure2. But it needs to be noted that the dynamic safe space constraint method is suitable to other control task of ACV for safety.Figure 2
Diagram of ACV.


## 2. Problem Formulation

### 2.1. ACV Model Description

The three degrees of freedom (DOF) model is(1)η˙=RψνMν˙+fν=τc+τw+dtwhere(2)Rψ=cos⁡ψ-sin⁡ψ0sin⁡ψcos⁡ψ0001M is the inertia matrix of ACV, η=ξ,η,ψT and ν=u,v,rT are position/Euler angles and velocities, d(t)=dut,dvt,drtT is the bounded input disturbance, dt<d0, where d0>0, τc=τcu,0,τcrT is the forces and moments provide by the actuators, and τw=τwu,τwv,τwrT is the external environment forces written as(3)Fua=-0.5ρaVa2CuaSPPFva=-0.5ρaVa2CvaSLPMra=-0.5ρaVa2CraSHPHhov+Fvaxa+FuayaFm=ρaVaQFwm=Cwmpc2BcρwgFsk=0.5ρwVa2Cskhlsk-0.34lskSc0.5+2.8167pclc-0.259-1Fwmτwu=Fua-Fmcos⁡β-Fwmcos⁡β-Fskcos⁡βτwv=Fva-Fmsin⁡β-Fwmsin⁡β-Fsksin⁡βτwr=Mra-Fmxmsin⁡β+Fmymcos⁡β-Fwmxwmsin⁡β+Fwmywmcos⁡β-Fskxsksin⁡β+Fskyskcos⁡βwhere the suffixes ua, va, and ra mean aerodynamic profile drags in three DOF, respectively. The suffixes m, wm, and sk mean wave-making drag, air momentum drag, and skirt drag, respectively. lc and Bc denote cushion length and beam, SPP, SLP, and SHP are positive, lateral, and horizontal projection areas, Sc means cushion area, β means drift angle, h is the average clearance for air leakage in static hovering mode, lsk is the total peripheral length of the skirts, Hhov is the height of the hull, ρa and ρw are air and water density, and (xa,ya), (xm,ym),(xwm,ywm), and (xsk,ysk) are the coordinates of these acting points. Va can be obtained by(4)Va=u+Vwcos⁡βw-ψ2+v+Vwsin⁡βw-ψ2βa=tan-1⁡v+Vwsin⁡βw-ψu+Vwcos⁡βw-ψin which Vw and βw are absolute wind speed and direction. For more details can see [1, 17, 23].Considering the modeling errors and the variations of parameters,M and fν can be rewritten as follows:(5)M=M0+ΔMfν=f0ν+Δfνwhere(6)M0=m000m000Iz(7)f0ν=-mvrmur0Substituting (5) into (3) yields (8)M0ν˙+f0ν=τc+τw+δtwhere δt=δ1t,δ2t,δ3tT is defined as follows:(9)δt=-ΔMν˙-Δfν+dtThen the following assumptions are made.Assumption 2.
The norm of inertia matrixM is upper bounded by a positive number a0.(10)M<a0Assumption 3.
The vectorfν is upper bounded by a positive function(11)fν<ρ0+ρ1ν+ρ2ν˙2where ρ0, ρ1, and ρ2 are positive numbers.
Then the system uncertainty will be bounded in the following form:(12)δt<b0+b1ν+b2ν˙2This bounded property has been used by some researchers of [24–26].


### 2.2. Dynamic Safe Space of ACV

Lemma 4.
Assuming thatε is any positive value, then the open interval a-ε,a+ε is ε neighborhood of a, write for Na,ε, that is,(13)Na,ε=x∣a-ε<x<a+εDefinition 5.
From Lemma4, we define the following expression:(14)Sa,ε1,ε2=x,y∣x∈a-ε1,a+ε1,y∈a-ε2,a+ε2Defining the interval of speed of ACV as ϕu, for any ui∈ϕu, the safety limit values of TR and SA are written as εuirsafe and εuiβsafe, and then the safety area of ACV at the speed ui is(15)S0,εuirsafe,εuiβsafeThen the sailing is safe if the following condition is satisfied.(16)rui,βui∈S0,εuirsafe,εuiβsafeThe green limit values of TR and SA at the speed of ui∈ϕu are defined as εuirgs and εuiβgs which satisfy 0≤εuirgs<εuirsafe and 0≤εuiβgs<εuiβsafe.
From Definition5, we know that(17)S0,εuirgs,εuiβgs⊂S0,εuirsafe,εuiβsafeThen the dynamic safe space χϕusafe is obtained as follows:(18)χϕusafe=S0,εrsafe,εβsafe=⋃ui∈ϕuS0,εuirsafe,εuiβsafeThe dynamic green space χϕugs is defined as follows:(19)χϕugs=S0,εrgs,εβgs=⋃ui∈ϕuS0,εuirgs,εuiβgsThen the safe buffer space χϕubs is gotten by(20)χϕubs=χϕusafe-χϕugsThe relationships of χϕugs, χϕusafe, and χϕubs are shown in Figure 3. If the speed of ACV exceeds Vmax, the motion of turning is not allowed. If the speed is lower than 0.2Vmax, the TR and SA are not required to be limited. The inside space of green boundary is green space. The inside space of yellow boundary is safe space. The space between green boundary and yellow boundary is buffer space.Figure 3
The relationship of safe space, green space, and buffer space.Theorem 6.
For anyuk∈ϕu, if ruk,βuk∈χϕugs or ruk,βuk∈χϕubs, the sailing of ACV is safe.Proof.
r u k , β u k ∈ χ ϕ u g s or ruk,βuk∈χϕubs means ruk,βuk∈χϕusafe.
It is obvious from (18) that ruk,βuk∈S0,εukrsafe,εukβsafe, where εukrsafe and εukβsafe are the safety limit of TR and SA at the speed uk. Hence, the sailing of ACV is safe and stability.


## 2.1. ACV Model Description

The three degrees of freedom (DOF) model is(1)η˙=RψνMν˙+fν=τc+τw+dtwhere(2)Rψ=cos⁡ψ-sin⁡ψ0sin⁡ψcos⁡ψ0001M is the inertia matrix of ACV, η=ξ,η,ψT and ν=u,v,rT are position/Euler angles and velocities, d(t)=dut,dvt,drtT is the bounded input disturbance, dt<d0, where d0>0, τc=τcu,0,τcrT is the forces and moments provide by the actuators, and τw=τwu,τwv,τwrT is the external environment forces written as(3)Fua=-0.5ρaVa2CuaSPPFva=-0.5ρaVa2CvaSLPMra=-0.5ρaVa2CraSHPHhov+Fvaxa+FuayaFm=ρaVaQFwm=Cwmpc2BcρwgFsk=0.5ρwVa2Cskhlsk-0.34lskSc0.5+2.8167pclc-0.259-1Fwmτwu=Fua-Fmcos⁡β-Fwmcos⁡β-Fskcos⁡βτwv=Fva-Fmsin⁡β-Fwmsin⁡β-Fsksin⁡βτwr=Mra-Fmxmsin⁡β+Fmymcos⁡β-Fwmxwmsin⁡β+Fwmywmcos⁡β-Fskxsksin⁡β+Fskyskcos⁡βwhere the suffixes ua, va, and ra mean aerodynamic profile drags in three DOF, respectively. The suffixes m, wm, and sk mean wave-making drag, air momentum drag, and skirt drag, respectively. lc and Bc denote cushion length and beam, SPP, SLP, and SHP are positive, lateral, and horizontal projection areas, Sc means cushion area, β means drift angle, h is the average clearance for air leakage in static hovering mode, lsk is the total peripheral length of the skirts, Hhov is the height of the hull, ρa and ρw are air and water density, and (xa,ya), (xm,ym),(xwm,ywm), and (xsk,ysk) are the coordinates of these acting points. Va can be obtained by(4)Va=u+Vwcos⁡βw-ψ2+v+Vwsin⁡βw-ψ2βa=tan-1⁡v+Vwsin⁡βw-ψu+Vwcos⁡βw-ψin which Vw and βw are absolute wind speed and direction. For more details can see [1, 17, 23].Considering the modeling errors and the variations of parameters,M and fν can be rewritten as follows:(5)M=M0+ΔMfν=f0ν+Δfνwhere(6)M0=m000m000Iz(7)f0ν=-mvrmur0Substituting (5) into (3) yields (8)M0ν˙+f0ν=τc+τw+δtwhere δt=δ1t,δ2t,δ3tT is defined as follows:(9)δt=-ΔMν˙-Δfν+dtThen the following assumptions are made.Assumption 2.
The norm of inertia matrixM is upper bounded by a positive number a0.(10)M<a0Assumption 3.
The vectorfν is upper bounded by a positive function(11)fν<ρ0+ρ1ν+ρ2ν˙2where ρ0, ρ1, and ρ2 are positive numbers.
Then the system uncertainty will be bounded in the following form:(12)δt<b0+b1ν+b2ν˙2This bounded property has been used by some researchers of [24–26].


## 2.2. Dynamic Safe Space of ACV

Lemma 4.
Assuming thatε is any positive value, then the open interval a-ε,a+ε is ε neighborhood of a, write for Na,ε, that is,(13)Na,ε=x∣a-ε<x<a+εDefinition 5.
From Lemma4, we define the following expression:(14)Sa,ε1,ε2=x,y∣x∈a-ε1,a+ε1,y∈a-ε2,a+ε2Defining the interval of speed of ACV as ϕu, for any ui∈ϕu, the safety limit values of TR and SA are written as εuirsafe and εuiβsafe, and then the safety area of ACV at the speed ui is(15)S0,εuirsafe,εuiβsafeThen the sailing is safe if the following condition is satisfied.(16)rui,βui∈S0,εuirsafe,εuiβsafeThe green limit values of TR and SA at the speed of ui∈ϕu are defined as εuirgs and εuiβgs which satisfy 0≤εuirgs<εuirsafe and 0≤εuiβgs<εuiβsafe.
From Definition5, we know that(17)S0,εuirgs,εuiβgs⊂S0,εuirsafe,εuiβsafeThen the dynamic safe space χϕusafe is obtained as follows:(18)χϕusafe=S0,εrsafe,εβsafe=⋃ui∈ϕuS0,εuirsafe,εuiβsafeThe dynamic green space χϕugs is defined as follows:(19)χϕugs=S0,εrgs,εβgs=⋃ui∈ϕuS0,εuirgs,εuiβgsThen the safe buffer space χϕubs is gotten by(20)χϕubs=χϕusafe-χϕugsThe relationships of χϕugs, χϕusafe, and χϕubs are shown in Figure 3. If the speed of ACV exceeds Vmax, the motion of turning is not allowed. If the speed is lower than 0.2Vmax, the TR and SA are not required to be limited. The inside space of green boundary is green space. The inside space of yellow boundary is safe space. The space between green boundary and yellow boundary is buffer space.Figure 3
The relationship of safe space, green space, and buffer space.Theorem 6.
For anyuk∈ϕu, if ruk,βuk∈χϕugs or ruk,βuk∈χϕubs, the sailing of ACV is safe.Proof.
r u k , β u k ∈ χ ϕ u g s or ruk,βuk∈χϕubs means ruk,βuk∈χϕusafe.
It is obvious from (18) that ruk,βuk∈S0,εukrsafe,εukβsafe, where εukrsafe and εukβsafe are the safety limit of TR and SA at the speed uk. Hence, the sailing of ACV is safe and stability.


## 3. Dynamic Safe Space Constraint Controller

### 3.1. Controller Design

For course safe space constraint controller design, definee1=σ-σd and then(21)e2=e˙1=σ˙-σ˙d=r+β˙-σ˙d(22)e˙2=r˙+β¨-σ¨d=1Izτcr+τwr+δ3t+β¨-σ¨dChoose the sliding mode surface and approaching law as follows based on the fast terminal sliding mode approach:(23)s=e1+1α1e1g/h+1α2e2p/q(24)s˙=-ϕs-γsm/ne2p/q-1where α1,α2∈R+,p,q,g,h∈N are odd numbers and 1<p/q<2,g/h>p/q. ϕ∈R+,γ∈R+,m,n∈N are odd numbers, and 0<m/n<1.After considering the safe space constraint, the control law is given by(25)τcr=-IzY-ψ¨d-τwr-k1sξgs-b^02+b^12r+b^22r˙2where (26)Y=α2qpϕs+γsm/n+α2qpe22-p/q1+gα1he1g/h-1ξgs=krflockr-δuirgs+kβflockβ-δuiβgsflockx=x,x>00,x≤0and k1>0,kr>0,kβ>0.The adaptive laws are(27)b^˙02=x0spIzβqe2p/q-1b^˙12=x1spIzβqe2p/q-1rb^˙22=x2spIzβqe2p/q-1r˙2where x0, x1, and x2 are arbitrary positive constants.


### 3.2. Stability Analysis

Theorem 1.
If the control law (25), the sliding mode surface (23), approaching law (24), and adaptive laws (27) are applied to ACV, the controller is stable and the course error will converge to zero in finite time.Proof.
The following Lyapunov function is defined:(28)V=12s2+12∑i=02xi-1b~i22where b~i2=bi2-b^i2,i∈0,1,2.
Then(29)V˙=ss˙-∑i=02xi-1b~i2b^˙i2=se˙1+gα1hse1g/h-1e˙1-∑i=02xi-1b~i2b^˙i2+spe2p/q-1α2q1Izτcr+τwr+δ3t+β¨-σ¨dSubstituting (25) into (29) yields (30)V˙=spe2p/q-1Izα2qδ3t-b^02+b^12r+b^22r˙2-V˙1-V˙2-∑i=02xi-1b~i2b^˙i2≤spIzα2qe2p/q-1δ3t-spIzα2qe2p/q-1b^02+b^12r+b^22r˙2-V˙1-V˙2-∑i=02xi-1b~i2b^˙i2where(31)V˙1=e2p/q-1ϕs2+γsm/n+1(32)V˙2=k1s2pIzα2qe2p/q-1ξgsUsing (27), we rewrite (30) as(33)V˙≤-V˙1-V˙2+spIzα2qe2p/q-1δ3t-spIzα2qe2p/q-1b^02+b^12r+b^22r˙2-spIzα2qe2p/q-1b~02+b~12r+b~22r˙2From b~i=bi-b^i,i∈0,1,2, we have(34)V˙≤-V˙1-V˙2+spIzα2qe2p/q-1δ3t-spIzα2qe2p/q-1b02+b12r+b22r˙2≤-V˙1-V˙2-V˙3From [27], we know that V˙1≥0. It is obvious from (26), (31)-(32), and (34) that V˙≤0. Hence, the closed-loop system is stable.
Integrating both sides of (34), we have(35)Vt-V0=-∫0tV˙1+V˙2+V˙3dτThen(36)Vt=V0-∫0tV˙1+V˙2+V˙3dτ≤V0<∞We can see that(37)V0=Vt+∫0tV˙1+V˙2+V˙3dτ≥∫0tV˙1+V˙2+V˙3dτTherefore(38)limt→∞⁡∫0tV˙1+V˙2+V˙3dτ≤V0<∞According to Barbalat’s Lemma, we have(39)limt→∞⁡s=0,limt→∞⁡e2=0From (23), we have(40)limt→∞⁡e1=0This concludes the proof.


## 3.1. Controller Design

For course safe space constraint controller design, definee1=σ-σd and then(21)e2=e˙1=σ˙-σ˙d=r+β˙-σ˙d(22)e˙2=r˙+β¨-σ¨d=1Izτcr+τwr+δ3t+β¨-σ¨dChoose the sliding mode surface and approaching law as follows based on the fast terminal sliding mode approach:(23)s=e1+1α1e1g/h+1α2e2p/q(24)s˙=-ϕs-γsm/ne2p/q-1where α1,α2∈R+,p,q,g,h∈N are odd numbers and 1<p/q<2,g/h>p/q. ϕ∈R+,γ∈R+,m,n∈N are odd numbers, and 0<m/n<1.After considering the safe space constraint, the control law is given by(25)τcr=-IzY-ψ¨d-τwr-k1sξgs-b^02+b^12r+b^22r˙2where (26)Y=α2qpϕs+γsm/n+α2qpe22-p/q1+gα1he1g/h-1ξgs=krflockr-δuirgs+kβflockβ-δuiβgsflockx=x,x>00,x≤0and k1>0,kr>0,kβ>0.The adaptive laws are(27)b^˙02=x0spIzβqe2p/q-1b^˙12=x1spIzβqe2p/q-1rb^˙22=x2spIzβqe2p/q-1r˙2where x0, x1, and x2 are arbitrary positive constants.


## 3.2. Stability Analysis

Theorem 1.
If the control law (25), the sliding mode surface (23), approaching law (24), and adaptive laws (27) are applied to ACV, the controller is stable and the course error will converge to zero in finite time.Proof.
The following Lyapunov function is defined:(28)V=12s2+12∑i=02xi-1b~i22where b~i2=bi2-b^i2,i∈0,1,2.
Then(29)V˙=ss˙-∑i=02xi-1b~i2b^˙i2=se˙1+gα1hse1g/h-1e˙1-∑i=02xi-1b~i2b^˙i2+spe2p/q-1α2q1Izτcr+τwr+δ3t+β¨-σ¨dSubstituting (25) into (29) yields (30)V˙=spe2p/q-1Izα2qδ3t-b^02+b^12r+b^22r˙2-V˙1-V˙2-∑i=02xi-1b~i2b^˙i2≤spIzα2qe2p/q-1δ3t-spIzα2qe2p/q-1b^02+b^12r+b^22r˙2-V˙1-V˙2-∑i=02xi-1b~i2b^˙i2where(31)V˙1=e2p/q-1ϕs2+γsm/n+1(32)V˙2=k1s2pIzα2qe2p/q-1ξgsUsing (27), we rewrite (30) as(33)V˙≤-V˙1-V˙2+spIzα2qe2p/q-1δ3t-spIzα2qe2p/q-1b^02+b^12r+b^22r˙2-spIzα2qe2p/q-1b~02+b~12r+b~22r˙2From b~i=bi-b^i,i∈0,1,2, we have(34)V˙≤-V˙1-V˙2+spIzα2qe2p/q-1δ3t-spIzα2qe2p/q-1b02+b12r+b22r˙2≤-V˙1-V˙2-V˙3From [27], we know that V˙1≥0. It is obvious from (26), (31)-(32), and (34) that V˙≤0. Hence, the closed-loop system is stable.
Integrating both sides of (34), we have(35)Vt-V0=-∫0tV˙1+V˙2+V˙3dτThen(36)Vt=V0-∫0tV˙1+V˙2+V˙3dτ≤V0<∞We can see that(37)V0=Vt+∫0tV˙1+V˙2+V˙3dτ≥∫0tV˙1+V˙2+V˙3dτTherefore(38)limt→∞⁡∫0tV˙1+V˙2+V˙3dτ≤V0<∞According to Barbalat’s Lemma, we have(39)limt→∞⁡s=0,limt→∞⁡e2=0From (23), we have(40)limt→∞⁡e1=0This concludes the proof.


## 4. Simulations

Simulations are implemented to verify the effectiveness and superiority of the proposed controller. In simulations, the main particulars of ACV are shown in Table1.Table 1
Main particulars of hovercraft.
m ( k g ) 40000 l s k ( m ) 65 I z ( k g m 2 ) 1.8 × 1 0 6 B c ( m ) 8.9 S P P ( m 2 ) 45 l c ( m ) 23.6 S L P ( m 2 ) 93 h ( m ) 1 S H P ( m 2 ) 260 H h o v ( m ) 5.9 Q ( m 3 / s ) 140.8 S c ( m 2 ) 212 V w ( k n o t s ) 10 β w ( d e g ) 45The initial course angle is0°, and the desired course angle is -55°. The bounded input disturbance is chosen as dr(t)=1100sin⁡(0.02t).You can see from Figures4 and 6–8 that the proposed dynamic safe space can effectively reflect the variations of safety limit following the speed. The course controller is verified to make the error converge to zero from Figure 5. From the comparison with general unsafe controller without considering the safety limit of ACV in Figures 6–8, the proposed safety-guaranteed course controller can effectively keep TR and SA in the safe space during the course control process.Figure 4
The speed of ACV during the control process.Figure 5
The course angle of ACV.Figure 6
The variations ofTR and SA.Figure 7
The variation ofSA.Figure 8
The variation ofTR.


## 5. Conclusion

In this paper, dynamic safe space of ACV is proposed to be more intuitive to show the change of safety limit and more convenient for safety monitoring and control. For the safety of ACV, a safety-guaranteed course controller is designed based on the dynamic safe space constraint. In the controller, the adaptive mechanism is designed to estimate the system uncertainty and external disturbances online without the requirement of their upper bounds. The proposed controller guarantees the safe space constraint and the convergence of the tracking error.



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