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

1. Introduction

An air cushion vehicle (ACV, as shown in Figure 1) 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.

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[degrees]/s and 2[degrees]. If speed is 14 knots, the limits of TR and SA are 3.5[degrees]/s and 20[degrees]. 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 Figure 2. But it needs to be noted that the dynamic safe space constraint method is suitable to other control task of ACV for safety.

2. Problem Formulation

2.1. ACV Model Description. The three degrees of freedom (DOF) model is

[mathematical expression not reproducible] (1)

where

[mathematical expression not reproducible] (2)

M is the inertia matrix of ACV, [eta] = [([xi], [eta], [psi]).sup.T] and v = [(u, v, r).sup.T] are position/Euler angles and velocities, d(t) = [([d.sub.u](t), [d.sub.v](t), [d.sub.r](t)).sup.T] is the bounded input disturbance, [parallel]d(t)[parallel] < [d.sub.0], where [d.sub.0] > 0, [[tau].sub.c] = [([[tau].sub.cu], 0, [[tau].sub.cr]).sup.T] is the forces and moments provide by the actuators, and [[tau].sub.w] = [([[tau].sub.wu], [[tau].sub.wv], [[tau].sub.wr]).sup.T] is the external environment forces written as

[mathematical expression not reproducible] (3)

where 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. [l.sub.c] and [B.sub.c] denote cushion length and beam, [S.sub.PP], [S.sub.LP], and [S.sub.HP] are positive, lateral, and horizontal projection areas, [S.sub.c] means cushion area, [beta] means drift angle, h is the average clearance for air leakage in static hovering mode, [l.sub.sk] is the total peripheral length of the skirts, [H.sub.how] is the height of the hull, [[rho].sub.a] and [[rho].sub.w] are air and water density and ([x.sub.a], [y.sub.a]) ([x.sub.m], [y.sub.m]) ([x.sub.wm], [y.sub.wm]) and ([x.sub.sk], [y.sub.sk]) are the coordinates of these acting points. Va can be obtained by

[mathematical expression not reproducible] (4)

in which [V.sub.w] and [[beta].sub.w] 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 f(v) can be rewritten as follows:

M = [M.sub.0] + [DELTA]M

(5)

f(v) = [f.sub.0] (v) + [DELTA]f(v)

where

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

Substituting (5) into (3) yields

[M.sub.0] [??] + [f.sub.0] (v) = [[tau].sub.C] + [[tau].sub.w] + [delta] (t) (8)

where [delta](t) = [([[delta].sub.1] (t), [[delta].sub.2] (t), [[delta].sub.3] (t)).sup.T] is defined as follows:

[delta] (t) = -[DELTA]M[??] - [DELTA]f (v) + d(t) (9)

Then the following assumptions are made.

Assumption 2. The norm of inertia matrix M is upper bounded by a positive number [a.sub.0].

[parallel]M[parallel] < [a.sub.0] (10)

Assumption 3. The vector f(v) is upper bounded by a positive function

[parallel]f(v)[parallel] < [[rho].sub.0] + [[rho].sub.1] [parallel]v[parallel] + [[rho].sub.2] [[parallel][??][parallel].sup.2] (11)

where [[rho].sub.0], [[rho].sub.1], and [[rho].sub.2] are positive numbers.

Then the system uncertainty will be bounded in the following form:

[parallel][delta](t)[parallel] < [b.sub.0] + [b.sub.1] [parallel]v[parallel] + [b.sub.2] [[parallel][??][parallel].sup.2] (12)

This bounded property has been used by some researchers of [24-26].

2.2. Dynamic Safe Space of ACV

Lemma 4. Assuming that e is any positive value, then the open interval (a - [epsilon], a + [epsilon]) is e neighborhood of a, write for N (a, [epsilon]), that is,

N (a, [epsilon]) = {x | a - [epsilon] < x < a + [epsilon]} (13)

Definition 5. From Lemma 4, we define the following expression:

S(a, [[epsilon].sub.1], [[epsilon].sub.2])

(14)

= {(x, y) | x [member of](a - [[epsilon].sub.1],a + [[epsilon].sub.1]), y [member of] (a - [[epsilon].sub.2], a + [[epsilon].sub.2])}

Defining the interval of speed of ACV as [[phi].sub.u], for any [u.sub.i] [member of] [[phi].sub.u], the safety limit values of TR and SA are written as [mathematical expression not reproducible], and then the safety area of ACV at the speed u{ is

[mathematical expression not reproducible] (15)

Then the sailing is safe if the following condition is satisfied.

[mathematical expression not reproducible] (16)

The green limit values of TR and SA at the speed of [u.sub.i] [member of] [[phi].sub.u] are defined as [mathematical expression not reproducible] which satisfy [mathematical expression not reproducible].

From Definition 5, we know that

[mathematical expression not reproducible] (17)

Then the dynamic safe space X$usafe is obtained as follows:

[mathematical expression not reproducible] (18)

The dynamic green space [mathematical expression not reproducible] is defined as follows:

[mathematical expression not reproducible] (19)

Then the safe buffer space [mathematical expression not reproducible] is gotten by

[mathematical expression not reproducible] (20)

The relationships of [mathematical expression not reproducible] are shown in Figure 3. If the speed of ACV exceeds [V.sub.max], the motion of turning is not allowed. If the speed is lower than 0.2[V.sub.max], 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.

Theorem 6. For any [mathematical expression not reproducible], the sailing of ACV is safe.

Proof [mathematical expression not reproducible].

It is obvious from (18) that [mathematical expression not reproducible], where [mathematical expression not reproducible] are the safety limit of TR and SA at the speed [u.sub.k]. 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, define [e.sub.1] = [sigma] - [[sigma].sub.d] and then

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

Choose the sliding mode surface and approaching law as follows based on the fast terminal sliding mode approach:

s = [e.sub.1] + 1/[[alpha].sub.1] [e.sup.g/h.sub.1] + 1/[[alpha].sub.2] [e.sup.p/q.sub.2] (23)

[??] = (-[phi]s - [gamma][s.sup.m/n]) [e.sup.p/q-1.sub.2] (24)

where [[alpha].sub.1], [[alpha].sub.2] [member of] [R.sup.+],p,q,g,h [member of] N are odd numbers and 1 < p/q < 2,g/h > p/q. [phi] [member of] [R.sup.+], [gamma] [member of] [R.sup.+], m,n [member of] N are odd numbers, and 0 < m/n < 1.

After considering the safe space constraint, the control law is given by

[mathematical expression not reproducible] (25)

where

[mathematical expression not reproducible] (26)

and [k.sub.1] >0, [k.sub.r] > 0, [k.sub.[beta]] > 0.

The adaptive laws are

[mathematical expression not reproducible] (27)

where [x.sub.0], [x.sub.1], and [x.sub.2] 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 infinite time.

Proof. The following Lyapunov function is defined:

[mathematical expression not reproducible] (28)

where [mathematical expression not reproducible].

Then

[mathematical expression not reproducible] (29)

Substituting (25) into (29) yields

[mathematical expression not reproducible] (30)

where

[[??].sub.1] = [e.sup.p/q-1.sub.2] ([phi][s.sup.2] + [gamma][s.sup.m/n+1]) (31)

[[??].sub.2] = [k.sub.1] [s.sup.2]p/[I.sub.z][[alpha].sub.2]q [e.sup.p/q-1.sub.2] [[xi].sub.gs] (32)

Using (27), we rewrite (30) as

[mathematical expression not reproducible] (33)

From [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (34)

From [27], we know that [[??].sub.1] [greater than or equal to] 0. It is obvious from (26), (31)-(32), and (34) that [??] [less than or equal to] 0. Hence, the closed-loop system is stable.

Integrating both sides of (34), we have

[mathematical expression not reproducible] (35)

Then

[mathematical expression not reproducible] (36)

We can see that

[mathematical expression not reproducible] (37)

Therefore

[mathematical expression not reproducible] (38)

According to Barbalats Lemma, we have

[mathematical expression not reproducible] (39)

From (23), we have

[mathematical expression not reproducible] (40)

This 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 Table 1.

The initial course angle is 0[degrees], and the desired course angle is -55[degrees]. The bounded input disturbance is chosen as [d.sub.r] (t) = 1100 sin(0.02f).

You can see from Figures 4 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.

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.

Data Availability

The model used in the simulations is one of the research achievements of our team. If the simulation data is provided, it may reveal information about the characteristics of our model and affect the protection of our research achievements. So we are sorry that we cannot support the data.

https://doi.org/10.1155/2018/9687282

Conflicts of Interest

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

Acknowledgments

The supports from the National Natural Science Foundation of China (Grant no. 51309062) and the project "Research on Maneuverability of High Speed Hovercraft" (Project no. 2007DFR80320) are gratefully acknowledged.

References

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Mingyu Fu, Shuang Gao, and Chenglong Wang

College of Automation, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Shuang Gao; gaussian@hrbeu.edu.cn

Received 13 May 2018; Accepted 11 August 2018; Published 2 September 2018

Academic Editor: Sing Kiong Nguang

Caption: FIGURE 1: A 3D model of hovercraft. Photo is from the international cooperation project described in the Acknowledgment.

Caption: FIGURE 2: Diagram of ACV.

Caption: FIGURE 3: The relationship of safe space, green space, and buffer space.

Caption: FIGURE 4: The speed of ACV during the control process.

Caption: FIGURE 5: The course angle of ACV.

Caption: FIGURE 6: The variations of TR and SA.

Caption: FIGURE 7: The variation of SA.

Caption: FIGURE 8: The variation of TR.
TABLE 1: Main particulars of hovercraft.

m(kg)                         40000            [l.sub.sk](m)        65
[I.sub.z](kg[m.sup.2])   1.8 x [10.sup.6]      [B.sub.c] (m)       8.9
[S.sub.PP] ([m.sup.2])          45              [l.sub.c](m)       23.6
[S.sub.LP]([m.sup.2])           93                  h(m)            1

[S.sub.HP]([m.sup.2])          260             [H.sub.hov](m)      5.9
Q([m.sup.3]/s)                140.8         [S.sub.c]([m.sup.2])   212
[V.sub.w](knots)                10          [[beta].sub.w](deg)     45
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Title Annotation:Research Article
Author:Fu, Mingyu; Gao, Shuang; Wang, Chenglong
Publication:Journal of Control Science and Engineering
Date:Jan 1, 2018
Words:3205
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