research article longitudinal motion control of auv based

Research ArticleLongitudinal Motion Control of AUV Based onFuzzy Sliding Mode Method

Duo Qi, Jinfu Feng, and Jian Yang

Aeronautics and Astronautics Engineering College, Air Force Engineering University, Xi’an 710038, China

Correspondence should be addressed to Duo Qi; [email protected]

Received 22 October 2015; Accepted 21 January 2016

Academic Editor: Lifeng Ma

Copyright © 2016 Duo Qi et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

According to the characteristics of AUVmovement, a fuzzy slidingmode controller was designed, inwhich fuzzy ruleswere adoptedto estimate the switching gain to eliminate disturbance terms and reduce chattering. The six-degree-of-freedom model of AUVwas simplified and longitudinal motion equations were established on the basis of previous research. The influences of first-orderwave force and torque were taken into consideration. The REMUS was selected to simulate the control effects of conventionalsliding mode controller and fuzzy sliding mode controller. Simulation results show that the fuzzy sliding mode controller can meetthe requirements and has higher precision and stronger antijamming performances compared with conventional sliding modecontroller.

1. Introduction

Nowadays the ocean space is an important competition fieldof military and economic powers in the world, and manycountries are vigorously developing deep sea explorationtechnology. As an intercrossed subject of ocean engineer-ing and robot technology, autonomous underwater vehicles(AUV) are playing increasingly significant roles in underwa-ter activities, such as offshore oil exploitation, underwater tar-get search, marine science research, and military application[1–5].

The stable and efficient control of AUV is very difficultfor its inherent highly nonlinear, uncertain hydrodynamicparameters and external disturbances.Wang et al. [6] adoptedS-surface control method to simulate the heading con-trol and depth control of a mini AUV, and, furthermore,they simulated long distance traveling following a plannedpath. The results showed that the AUV has good spatialmaneuverability and verified the feasibility and reliability ofcontrol method. Ma and Cui [7] proposed a robust path-following control method for nonlinear and underactuatedAUV based on a fuzzy hybrid control strategy. Jia et al.[8] presented a nonlinear iterative sliding mode controllerbased on the virtual guide method. It can decrease the static

error and overshoot and achieves high tracking precision.Sahu and Subudhi [9] developed an adaptive control lawfor the AUV to track the desired trajectory and verified thestability of the controller using Lyapunov’s directmethod.Thesimulation results demonstrate that the controller is feasiblefor the tracking of uncertain parameters model. Lapierre andSoetanto [10] designed a new backstepping controller, whichcan get rid of the limits of initial conditions and make thetracking error converge to zero.

The sliding mode control has been successfully appliedto dynamic positioning and motion control of underwatervehicle [11, 12], due to its simple algorithm, robustness againstmodeling imprecision, and external disturbances. However,the discontinuous switching characteristics of sliding modecontrol will cause chattering, which not only affects the con-trol accuracy but also degrades the system performance andeven severely damages the control units. Many researchershave put forward solutions to eliminate the chattering phe-nomenon fromdifferent angles, such as adaptivemethod [13],neural network method [14], feedback linearization method[15], and fuzzy method [16]. According to the experience,a proper switching gain can reduce chattering [17]. Thefuzzy control has many advantages; for example, it needsno accurate mathematical model and has good robustness.

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016, Article ID 7428361, 7 pageshttp://dx.doi.org/10.1155/2016/7428361

2 Journal of Control Science and Engineering

Table 1: Motion mode and attitude parameters of AUV.

Degree offreedom Motion modes

Force/torque(in the body-fixed

coordinate)

Linear velocity/angularvelocity

(in the body-fixed coordinate)

Location/Euler angles(in the earth-fixed

coordinate)1 Back/forward (movement along the 𝑥-axis) 𝐹

𝑥𝑢 𝑥

2 Sway (movement along the 𝑦-axis) 𝐹𝑦

V 𝑦

3 Lift/dive (movement along the 𝑧-axis) 𝐹𝑧

𝑤 𝑧

4 Roll (rotation along the 𝑥-axis) 𝑀𝑥

𝑝 𝜙

5 Pitch (rotation along the 𝑦-axis) 𝑀𝑦

𝑞 𝜃

6 Yaw (rotation along the 𝑧-axis) 𝑀𝑧

𝑟 𝜓

The fuzzy sliding mode control combines the advantages ofslidingmode control and fuzzy control and canmake discretecontrol signals continuous to reduce chattering effectively.

The main contribution of this paper is to design a fuzzysliding mode controller for the longitudinal motion controlof AUV with the consideration of first-order wave force andtorque. Based on the conventional slidingmode control, fuzzyrules are adopted to estimate the switching gain to eliminatedisturbance terms and reduce chattering. The simulationresults show that the fuzzy sliding mode controller can meetthe requirements. Compared with conventional slidingmodecontroller, it has higher precision and stronger antijammingperformances, which has good practical values.

2. Longitudinal Motion Model ofAUV and Wave Disturbance

Earth-fixed coordinate and AUV body-fixed coordinate areshown in Figure 1. Six-DOF kinematic modes and attitudeparameters are defined in the coordinate system as shown inTable 1.

According to the parameters in Table 1, we define vectorsas follows: 𝜂

1= (𝑥, 𝑦, 𝑧)

𝑇, 𝜂2= (𝜙, 𝜃, 𝜓)

𝑇, 𝜂 = (𝜂1, 𝜂2)𝑇,

𝜏1= (𝐹𝑥, 𝐹𝑦, 𝐹𝑧)𝑇, 𝜏2= (𝑀

𝑥,𝑀𝑦,𝑀𝑧)𝑇, 𝜏 = (𝜏

1, 𝜏2)𝑇, 𝑉 =

(𝑢, V, 𝑤)𝑇, 𝜔 = (𝑝, 𝑞, 𝑟)𝑇, V = (𝑉, 𝜔)

𝑇, the center of gravityposition is 𝑟

𝐺= (𝑥𝑔, 𝑦𝑔, 𝑧𝑔)𝑇, and the center of buoyancy

position is 𝑟𝐵= (𝑥𝐵, 𝑦𝐵, 𝑧𝐵)𝑇.

The longitudinal motion equations of AUV with respectto the body-fixed moving frame are described by a set ofnonlinear differential equations as follows:

𝑚(�� + 𝑤𝑞 − 𝑥𝑔𝑞2

+ 𝑧𝑔��) = 𝐹

𝑥,

𝑚 (�� − 𝑢𝑞 − 𝑧𝑔𝑞2

− 𝑥𝑔��) = 𝐹

𝑧,

𝐼𝑦𝑦�� + 𝑚 [𝑧

𝑔(�� + 𝑤𝑞) − 𝑥

𝑔(�� − 𝑢𝑞)] = 𝑀

𝑦.

(1)

Ignore all the high-order terms under the condition of lowspeed; the mathematical model above can be simplified as

𝑚(�� + 𝑧𝑔��) = 𝐹

𝑥,

𝑚 (�� − 𝑈𝑞 − 𝑥𝑔��) = 𝐹

𝑧,

𝐼𝑦𝑦�� + 𝑚 [𝑧

𝑔�� − 𝑥𝑔(�� − 𝑢𝑞)] = 𝑀

𝑦,

(2)

Body-fixed coordinateEarth-fixed coordinate

Pitch: q,My

Sway: �, Fy

y, 𝜃x, 𝜑

Roll: p,Mx

Surge: u, Fx

z, 𝜓

Heave: w, FzYaw: r,Mz

Figure 1: Reference coordinates and 6-DOF coordinates of AUV.

where

𝐹𝑥= 𝑋�� + 𝑋

𝑢𝑢 + 𝑋

𝑞𝑞 + 𝑋

𝜃𝜃,

𝐹𝑧= 𝑍�� + 𝑍

�� + 𝑍𝑤𝑤 + 𝑍

𝑞𝑞 + 𝑍𝛿𝑠

𝛿𝑠,

𝑀𝑦= 𝑀�� + 𝑀

�� + 𝑀

𝑤𝑤 +𝑀

𝑞𝑞 +𝑀

𝛿𝑠

𝛿𝑠,

(3)

and𝑋��, 𝑍��,𝑀��. . . are hydrodynamic parameters.

According to the practical situation, we suppose thatthe AUV moves with a constant velocity 𝑢 = 𝑢

0, and the

longitudinal motion equations of AUV can be describedfurther [18]:

[

��

𝜃

] = [

cos 𝜃 0

0 1

] [

𝑤

𝑞

] − [

𝑢0sin 𝜃0

] = 𝐴1[

𝑤

𝑞

] − 𝐴𝜃1, (4)

[

��

��

] = 𝑀−1

𝑢0[

𝑍𝑤

𝑍𝑞+ 𝑚

𝑀𝑤

𝑀𝑞

][

𝑤

𝑞

]

+𝑀−1

𝑢0

2

[

𝑍𝛿

𝑀𝛿

] 𝛿𝑠+𝑀−1

[(𝑊 − 𝐵0) cos 𝜃

− (𝑥𝑔𝑊− 𝑥

𝐵𝐵0) cos 𝜃 − (𝑧

𝑔𝑊− 𝑧

𝐵𝐵0) sin 𝜃]

+𝑀−1

[

𝑍𝑑

𝑀𝑑

] = 𝐴2[

𝑤

𝑞

] + 𝐴𝛿𝛿𝑠+ 𝐴𝑚𝑥𝜃+ 𝐴𝑑,

(5)

where𝑀−1 = [𝑚−𝑍�� −𝑍��−𝑀��𝐼𝑦−𝑀��

] .

The first-order wave force mainly affects the AUV duringits moving near water surface. The first-order wave force,which is of high frequency and periodical, has amplitude

Journal of Control Science and Engineering 3

0 10 20 30 40 50

Forc

e (N

)

Time (s)

2.5

2

1.5

1

0.5

0

−0.5

−1

−1.5

−20 10 20 30 40 50

Time (s)

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2.5

−2

Mom

ent (

N∗

m)

Figure 2: Wave force and wave torque.

proportional to wave height. In this paper, theHirom approx-imation formula is adopted to calculate the first-order waveforce and torque, and the concrete form is

𝑍 = (780 − 145∑𝐹 sin𝜔𝑎𝑡)∑𝐹 sin𝜔,

𝑀 = 1070𝐹 cos𝜔𝑎𝑡,

(6)

where 𝐹 = 𝑎𝜔2

𝑒−𝜔2

(𝐻+ℎ(𝑡)/𝑔); 𝜔𝑎= −𝜔(1 − 𝜔𝑈 cos𝛽/𝑔);

𝑎𝑖= √2𝑆(𝜔)𝛿𝜔; ℎ(𝑡) is the wave height; 𝑔 is gravitational

acceleration; 𝜔 is wave frequency; 𝛽 is wave to course angle;ℎ(𝑡) is the distance between the AUV and sea surface. Thefirst-order wave force and torque under the conditions that𝐻 = 10m, 𝑢 = 2m/s, and 𝛽 = 0

∘ are shown in Figure 2,respectively.

From (5) and (6), the model can be expressed as

�� = 𝐴𝑥 + 𝐵𝑢 + 𝐷, (7)

where state vector is 𝑥 = [𝑧 𝜃 𝑤 𝑞]

𝑇, input is 𝑢 = 𝛿𝑠, 𝐴 =

[𝑂 𝐴1

𝑂 𝐴2

], 𝐵 = [𝑂

𝐴𝛿

], 𝛿𝑠is rudder angle, and 𝐷 is the sum of

disturbance terms and uncertain terms with wave force andtorque.

3. Design of Fuzzy Sliding Mode Controller

The control of AUV, a typical underactuated system, isdifficult in complex and variable underwater environment.Sliding mode control has the characteristic of discontinuitywhich forces the system to make a small range and highfrequency sliding motion along a certain state. When thesystem is in the sliding mode, the control plant is invari-ant to uncertain parameters and disturbance. However, theinvariance comes at the cost of high chattering, and it severelyimpacts the practical application of slidingmode control. It isone of themost effectiveways to determine the switching gainin order to reduce the chattering by fuzzy method.

Let us suppose that the desired target state is 𝑥𝑑

=

[𝑧𝑑𝜃𝑑𝑤𝑑𝑞𝑑]

𝑇, and the control error is defined as follows:

𝑒 = 𝑥 − 𝑥𝑑. (8)

The control target is to find a design of 𝑢 to minimize thecontrol error. We choose the switching function as

𝑠 = 𝐶𝑒, (9)

where 𝐶 = [𝑐1, 𝑐2, 𝑐3, 1], which satisfies the Hurwitz stability

condition.With uncertain disturbance, we define the sliding mode

control law as

𝑢 = 𝑢eq + 𝑢𝑠, (10)

where 𝑢eq is the equivalent control and 𝑢𝑠is the switching

control.Let us suppose that, after a period of time, the system

reaches sliding mode surface, and, in an ideal situation, thecontrol system will meet

𝑠 = 𝐶𝑒 = 0,

𝑠 = 𝐶 𝑒 = 0.

(11)

The equivalent control is

𝑢eq = − (𝐶𝐵)−1

(𝐶𝐴𝑥) + (𝐶𝐵)−1

𝐶��𝑑. (12)

We set the switching controller as

𝑢𝑠= 𝑘 sgn (𝑠) , (13)


Table 2: Fuzzy table of output.

𝑠 𝑠 NB NM NS ZO PS PM PBΔ𝑘 NB NM NS ZO PS PM PB

where 𝑘 is switching gain; sgn(𝑠) is sign function. Consider

sgn (𝑠) ={{{{

{{{{

{

+1, 𝑠 > 0,

0, 𝑠 = 0,

−1, 𝑠 < 0,

𝑠 𝑠 = 𝑠 [𝐶 𝑒] = 𝑠 [𝐶 (𝐴𝑥 + 𝐵𝑢 + 𝐷 − ��𝑑)] = 𝑠 (𝐶𝐴𝑥

+ 𝐶𝐵𝑢 + 𝐶𝐷 − 𝐶��𝑑) = 𝑠 (𝐶𝐴𝑥

+ 𝐶𝐵 [− (𝐶𝐵)−1

(𝐶𝐴𝑥) + (𝐶𝐵)−1

𝐶��𝑑+ 𝑘 sgn (𝑠)]

+ 𝐶𝐷 − 𝐶��𝑑) = 𝑠 (𝐶𝐴𝑥 − 𝐶𝐴𝑥 + 𝐶��

𝑑

+ 𝐶𝐵𝑘 sgn (𝑠) + 𝐶𝐷 − 𝐶��𝑑) = 𝑠 (𝐶𝐵𝑘 sgn (𝑠)

− 𝐶𝐷) .

(14)

Let 𝑘 = −𝜀(𝐶𝐵)−1, where 𝜀 > max (|𝐶𝐷|); thus 𝑠 𝑠 < 0.The reachability is verified. When the system enters into

the sliding mode, it is effective to deal with the uncertaindisturbance.

Appropriate switching gain can reduce the chatteringefficiently. 𝑠 𝑠 is the input, andΔ𝑘 is the output.Define both thefuzzy sets of 𝑠 𝑠 and Δ𝑘 as [NB,NM,NS,ZO,PS,PM,PB], theuniverse is [−3, −2, −1, 0, 1, 2, 3], and membership functionadopts the triangle function. Fuzzy control rules are asfollows:

(1) If 𝑠 𝑠 > 0, 𝑘 should be increased.

(2) If 𝑠 𝑠 < 0, 𝑘 should be decreased.

It is known that when the system goes out of the slidingmode surface, the switching gain should be increased tomakethe system reach the surface as quickly as possible; otherwise,if the system is on the sliding mode surface, the switchinggain should be increased to reduce chattering. Fuzzy table ofoutput is shown in Table 2.

Centroidmethod is adopted to achieve defuzzification, inwhich the enclosed area centroid by membership functionand 𝑥-axis is calculated. We define 𝜇 as the fuzzy set ofvariable 𝑥; then the defuzzification result is

𝜀 =

∫ 𝑥𝜇 𝑑𝑥

∫𝜇𝑑𝑥

. (15)

AUV model

d/dt FSMCDisturbance

Output

Input+

−⨂ u = ueq + uss = Ce

Figure 3: Diagram of fuzzy sliding mode control system.

We define the Lyapunov function as

𝑉 =

1

2

𝑠2

,

𝑠 𝑠 = 𝑠 [𝐶 𝑒] = 𝑠 [𝐶 (𝐴𝑥 + 𝐵𝑢 + 𝐷 − ��𝑑)] = 𝑠 (𝐶𝐴𝑥

+ 𝐶𝐵𝑢 + 𝐶𝐷 − 𝐶��𝑑) = 𝑠 (𝐶𝐴𝑥

+ 𝐶𝐵 [− (𝐶𝐵)−1

(𝐶𝐴𝑥) + (𝐶𝐵)−1

𝐶��𝑑+ 𝑘 sgn (𝑠)]

+ 𝐶𝐷 − 𝐶��𝑑) = 𝑠 (𝐶𝐴𝑥 − 𝐶𝐴𝑥 + 𝐶��

𝑑

+ 𝐶𝐵𝑘 sgn (𝑠) + 𝐶𝐷 − 𝐶��𝑑) = 𝑠 (𝐶𝐵𝑘 sgn (𝑠)

− 𝐶𝐷) .

(16)

Let 𝑘 = −𝜀(𝐶𝐵)−1, where 𝜀 > max (|𝐶𝐷|); thus 𝑠 𝑠 < 0.Asymptotically stable condition is met.

4. Analysis of Simulation Result

The process of AUV longitudinal motion control is simulatedto validate the effectiveness of this sliding mode control. Thesimulation environment is Matlab (R2011a)/Simulink, andthe external disturbance is normal sea condition. ChooseREMUS as the control plant. Hydrodynamic and physicalparameters of REMUS when the AUVmoves underwater areas follows:

𝑀��= −5.30 kg⋅m2/rad; 𝑍

��= −2.24 kg⋅m/rad;

𝑀��= −2.35 kg⋅m; 𝑍

��= −47.9 kg;

𝑀𝑞= −23.2 kg⋅m/rad; 𝑍

𝑞= −26.6 kg/rad;

𝑀𝑤= 15.9 kg; 𝑍

𝑤= −45.6 kg/m;

𝑀𝛿= −6.51 kg/rad; 𝑍

𝛿= −6.51 kg/(m⋅rad);

𝑥𝑔= 0; 𝑦

𝑔= 0; 𝑧

𝑔= 0;

𝑥𝐵= 0; 𝑦

𝐵= 0; 𝑧

𝐵= −0.02m;

𝑊 = 363N; 𝐵0= 371N.

The diagram of fuzzy sliding mode control system isshown in Figure 3.

Initial depth of AUV is 𝑍0= 0m, and target depth is 𝑍 =

10m; initial pitch angle is 𝜃0= 0.18 rad, and target pitch angle

is 𝜃0= 0.01 rad; initial pitch angle rate is 𝑞

0= −0.01 rad/s,

and initial velocity along the 𝑧 direction 𝑤0= 0.15m/s. The

switching function matrix is 𝐶 = [25 10 8 1].First, the sine wave response is used to test the perfor-

mance of the controller, and the simulation result is shown in


0 10 20 30 40 50Time (s)

Out

put

SineFSMC

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

Figure 4: Sine wave response.

0 10 20 30 40 500

2

4

6

8

10

12

Time (s)

SMCFSMC

z(m

)

0 10 20 30 40 50Time (s)

SMCFSMC

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4

𝜃(r

ad)

Figure 5: Changing curves of depth and pitch angle.

Figure 4. It can be seen that the fuzzy sliding mode controllerhas a good performance to track a varying state trajectory.

The conventional sliding mode control (SMC) and fuzzysliding mode control (FSMC) are adopted, respectively, tocontrol the longitudinal motion of AUV.The depth changingcurves and pitch angle changing curve are shown in Figure 5,and the velocity changing along 𝑧 direction curve and pitchangle changing rate are shown in Figure 6.

As shown in the figures, under the normal sea conditions,both control methods are robust and can meet the require-ments, whichmeans that they can reach the control target andkeep the system stable. Compared with conventional slidingmode control, fuzzy sliding mode control has a smallerovershoot and shorter adjusting time. Overshoot of SMC isabout 18% while that of FSMC is less than 2%; adjusting time

of SMC is 25.53 s and that of FSMC is 19 s. In addition, FSMCreduces the chattering phenomenon and control error, andthe overall performance of FSMC is superior to that of SMC.

Figure 7 shows the changing curve of rudder angel. SMChas a longer adjusting time and bigger overshoot. A slightchattering will appear when the system reaches stable state.FSMC can adjust the horizontal rudder angles in a short timeto reach steadiness without chattering.

5. Conclusions

A fuzzy sliding mode controller is designed to improve thecontrol precision and antijamming capability of AUV in thispaper, which combines the sliding mode control and fuzzy


0 10 20 30 40 50Time (s)

SMCFSMC

q(r

ad/s

)

0.15

0.1

0.05

0

−0.05

−0.10 10 20 30 40 50

Time (s)

w(m

/s)

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

SMCFSMC

Figure 6: Changing curves of pitch rate and heave velocity.

0 5 10 15 20 25 30 35 40 45 50

0

0.1

0.2

0.3

0.4

0.5

Time (s)

SMCFSMC

8 9 10 11 12 13 14 15

0−0.01−0.02−0.03−0.04−0.05−0.06−0.07−0.08−0.09−0.1

𝛿s

(rad

)

−0.1

−0.2

−0.3

−0.4

Figure 7: Changing curves of rudder angle.

control. Fuzzy rules are adopted to estimate the switchinggain to eliminate disturbance terms and reduce chattering.The simulation results show that the fuzzy sliding modecontroller can meet our requirements and has a higher preci-sion and stronger antijamming performances compared withconventional slidingmode controller. In the further research,membership functions will be taken into consideration toimprove the performance of fuzzy sliding mode controller.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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