design of a sliding mode fuzzy controller for auv

Ocean Engineering 30 (2003) 2137–2155www.elsevier.com/locate/oceaneng

Design of a sliding mode fuzzy controller forthe guidance and control of an autonomous

underwater vehicle

J. Guo∗, F.-C. Chiu, C.-C. HuangDepartment of Naval Architecture and Ocean Engineering, National Taiwan University,

73 Chou-Shan Road, Taipei, Taiwan, ROC

Received 21 August 2002; accepted 27 November 2002

Abstract

This work demonstrates the feasibility of applying a sliding mode fuzzy controller to motioncontrol and line of sight guidance of an autonomous underwater vehicle. The design methodof the sliding mode fuzzy controller offers a systematical means of constructing a set of shrink-ing-span and dilating-span membership functions for the controller. Stability and robustnessof the control system are guaranteed by properly selecting the shrinking and dilating factorsof the fuzzy membership functions. Control parameters selected for a testbed vehicle, AUV-HM1, are evaluated through tank and field experiments. Experimental results indicate the effec-tiveness of the proposed controller in dealing with model uncertainties, non-linearities of thevehicle dynamics, and environmental disturbances caused by ocean currents and waves. 2003 Elsevier Ltd. All rights reserved.

Keywords: AUV; Sliding mode control; Fuzzy control; Guidance

1. Introduction

Control problems involving autonomous underwater vehicles (AUVs) present sev-eral difficulties, owing to their non-linear dynamics, the presence of disturbance, andobservation noises. Ocean exploration and the utilization of oceanic resources inshallow, confined water areas have received increasing interest in recent years. In

∗ Corresponding author. Fax:+886-2-23929885.E-mail address: [email protected] (J. Guo).

0029-8018/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0029-8018(03)00048-9

2138 J. Guo et al. / Ocean Engineering 30 (2003) 2137–2155

such regions, shallow water phenomena from the interaction among wave dynamics,tidal currents, coastal currents, and artificial objects create a complex environmentfor operating unmanned underwater vehicles. Therefore, controlling AUVs to satis-factorily track trajectories in shallow waters remains a challenge.

Several control strategies have been developed for controlling the motion of under-water vehicles, among them are, supervisory control (Yoerger et al., 1986), neuralnetwork control (Yuh, 1990), self-turning control (Goheen and Jefferys, 1990),LQG/LTR (Triantafyllou and Grosenbaugh, 1991), sliding mode control (Yoergerand Slotine, 1985; Fossen and Sagatun, 1991; da Cunha et al., 1995; Christi et al.,1990; Healey and Lienard, 1993; Lam and Ura, 1996; Lee et al., 1999), fuzzy logiccontrol (Kato et al., 1993; Smith et al., 1994), and recently, the sliding mode fuzzylogic control (Song and Smith, 2000). More references could be found from theunderwater robotics community, for example (Yuh, 1994; Yuh et al., 1996). Thispaper presents a design method based on sliding mode fuzzy logic control. As wellrecognized, fuzzy logic controllers are effective robust controllers for various appli-cations. A merit of using fuzzy logic to design a controller is that the dynamics ofthe controlled system need not be fully known. However, the linguistic expressionof the fuzzy controller makes it difficult to guarantee the stability and robustness ofthe control system. Sliding mode control can also be applied effectively in the pres-ence of model uncertainties, parameter variations, and disturbances. A boundary layeris generally used to avoid chattering on the sliding surface (Slotine and Li, 1991).Designing a fuzzy logic controller based on the sliding mode theory assures perform-ance and stability, while simultaneously reducing the number of fuzzy rules. Further-more, fuzzy partition of the manipulated variables avoids the chattering problem ofthe sliding mode control method (Palm, 1994; Palm et al., 1996). The ‘sliding modefuzzy logic controller (SMFLC)’ is thus adopted herein as the basic controller struc-ture.

2. Background

In Yoerger and Slotine (1985), sliding mode control methodology is applied tocontrol trajectories of remotely operated vehicles (ROVs). It is demonstrated that asliding mode controller can be designed using a simple nonlinear model of thevehicle. The controller effectively deals with nonlinearities, and is robust even withimprecise models. Furthermore, the trade-off between performance and model uncer-tainty is predictable. In Fossen and Sagatun (1991), a hybrid controller combiningan adaptive scheme and a sliding-mode term was designed to control the motion ofan ROV. Vehicle dynamics, such as inertia, hydrodynamic forces, and restoringforces, were proven able to be estimated on-line, while the thruster nonlinearitiesand unknown thruster dynamics can be compensated by the switching controller.Meanwhile, in da Cunha et al. (1995), an adaptive control scheme for dynamic pos-itioning of an ROV was developed based on a sliding mode control algorithm thatonly used position measurements. The applications of the sliding mode can beextended to the motion control of AUVs. Furthermore, the problem of controlling

2139J. Guo et al. / Ocean Engineering 30 (2003) 2137–2155

an AUV in the dive plane through the sliding mode based on observed states wasaddressed in Christi et al. (1990). Additionally, parameter variations of the motionequation resulting from speed changes are estimated on-line, while unmodeleddynamics are compensated by the sliding mode. In Healey and Lienard (1993), multi-variable sliding mode design of an AUV autopilot system for the maneuvering ofcombined steering, diving, and speed was presented. Meanwhile, the simulation ofa line-of-sight scheme for guiding the AUV between waypoints in ocean currentswas also considered. Moreover, a switched control law that allowed a non-cruisingtype AUV to hover precisely was presented in Lam and Ura (1996). The controlleruses a dead zone as well as a switching law to avoid the effects of measurementnoises and the non-linearity of the thruster. In Lee et al. (1999), a discrete-timequasi-sliding mode controller is used for controlling the depth of an AUV, and thecontrol algorithm is confirmed to be effective with model uncertainties and largesampling periods. In Kato et al. (1993), the fuzzy algorithm was applied to managethe guidance and control of a cable tracking AUV in both attitude control and cabletracking. The optimization of fuzzy rules based on the dynamic model of the AUVwas also investigated. In Smith et al. (1994), fuzzy logic controllers were proposedfor controlling and guiding a low-speed torpedo shaped vehicle. Heading, pitch, anddepth are controlled simultaneously via three single-axis fuzzy logic controllers. Adocking law based fuzzy logic was also assessed using a nonlinear simulation model.Recently, a time optimal design method base on SMFLC was presented in Song andSmith (2000). The shape of the AUV open loop step response was used as theswitching surface for the SMFLC controller. It is argued that the design methodprovides time optimality and robustness, and it does not require system model ofthe AUV.

This work presents an efficient controller for controlling the direction of a flat,streamlined AUV, AUV-Hai-Min. The designated mission of the AUV-HM1 is areasearch and survey in shallow water. Typical search/survey scenarios covering largeareas require route stability and good turning performance in the horizontal planeof motion. Directional control is thus fundamental to the system presented herein.The proposed design method requires designation of a shrinking or dilating factorfor fuzzy input/output membership functions. We select shrinking and dilating factorsbased on specifications on control precision and the robustness to external disturb-ance and system modeling error. This procedure is derived using the formulation ofthe sliding mode control described in Slotine and Li (1991). Compared to conven-tional fuzzy logic controllers and SMFLCs, this method has less number of controlparameters, and is therefore much easier to implement. The paper first describes theguidance law for the AUV-HM1. Then the design method based on fuzzy slidingmode is shown, and guidelines for selecting control parameters are illustrated. Guid-ance and control experiments in a water tank and in shallow sea were conducted todemonstrate the effectiveness of the method.


3. Guidance law

A one-degree-of-freedom vehicle model is used herein to describe the horizontalturning behavior of the AUV. The model includes drag, added mass, and thrustmoment for yaw motion,

I r � br�r� � u � d (1)

where I denotes the vehicle’s mass moment of inertia plus the added inertia of thebody about the body-fixed z-axis, r represents the body-fixed rate for heading direc-tion, b denotes the square-law damping coefficient, u is the moment generated bycommanding differential thrust force on the left and right thrusters, and d representsthe disturbance caused by ocean currents, modeling errors, and unmodeled dynamics.

The line-of-sight guidance procedure is illustrated in Figs. 1 and 2. The guidancelaw comprises of the following components.

3.1. Inputs

The heading error, ye(t), and rate of heading error, re(t), where

ye(t) � yi(t)�yd(t) (2)

and yi(t) is the vehicle heading, while yd(t) denotes the desired vehicle heading.

yd(t) � tan�1�yi(t)�yd(t)xi(t)�xd(t)

�,0�yd(t) � 2p (3)

where (xi(t),yi(t)), (xd(t),yd(t)) represent the vehicle’s position and desired position,respectively.

Fig. 1. Signal flow diagram of the guidance law.


Fig. 2. Definition of parameters in the guidance law.

3.2. Error and control signals

The sliding error, SE∗, control command, u∗

SE∗(t) � (1�l)y∗e (t) � ly∗

e (t),0 � l�1 (4)

where the superscript ∗ represents the normalized variables. The normalization fac-tors,Gy, and Guare used to map the signals on to a defined domain.

SE∗ �SEGy

;y∗e (t) �

ye(t)Gy

;r∗e (t) � y∗

e �re(t)Gy

;u∗(t) �u(t)Gu

(5)

3.3. The fuzzy rule-base

Rule i in the rule base is written as

If SE∗�Fsithen u∗�Fu�i

, and msi(SE∗) � mu�i

(u∗) (6)

Where i = �m,%,�1, 0, 1%,m, and Fsiand Fui

are the ith linguistic members ofthe input and output variables, respectively. Here, we use 2m + 1 linguistic membersto partition input and output variables. Both positive/negative sides have the same


number of linguistic members. Triangular membership functions are used to rep-resent linguistic members. The degree-of-membership function, mXi

(X∗) is calcu-lated by

mXi(X∗) � �

X∗�Xi�1

Xi�Xi�1for Xi�1�X∗�Xi

Xi+1�X∗

Xi+1�Xi

for Xi�X∗�Xi+1

0 others

(7)

where X∗ represents the sliding error, SE∗, or the control command, u∗.For example, if five sections are used to partition the domain of input and output

variables, the following rule-base can be obtained,

If SE∗ is PB, then u∗ is NB.

If SE∗ is PM, then u∗ is NM.

If SE∗ is ZO, then u∗ is ZO.

If SE∗ is NM, then u∗ is PM.

If SE∗ is NB, then u∗ is PB.

where PB represents positive big, PM denotes positive medium, ZO represents zero,NM is negative medium, and NB denotes negative big, respectively.

3.4. Arrangement of membership functions

The ‘core’ (or the highest) values Si and Ui for the ith membership functions aredesigned such that

Xi �im

sm�| i |f (8)

where Xi = Si or Ui, �1 = X�ms� % � X�1 � X0 � X1 � % � Xms

= 1, andmXi

(X∗) = 1. Here, sf�(0,1] is the ‘shrinking factor’ (Chen and Hsieh, 1996). Thisset up results in a series of shrinking-span membership functions. The same ideacan be used to form a series of dilating-span membership functions by assigningcore values as

Xi �i�msgn(i)

mdm�| i�msgn(i) |

f � sgn(i) (9)

where df� ( 0,1] is termed the ‘dilating factor’ . Fig. 3 illustrates examples of theshrinking and dilating-span membership functions. Note that when the shrinking fac-tor sf = 1 , or the dilating factor df = 1, the membership functions are equal-spanisosceles triangular membership functions.


Fig. 3. Examples of shriking-span and dilating-span membership functions.

This arrangement brings out the fact that there exist at most two rules being firedat the same time. Furthermore, the two membership values have a sum of 1,

mXi(X∗) � mXi+1

(X∗) � 1 ,X∗�[Xi,Xi+1] (10)

3.5. Defuzzification

For a fired rule, and the corresponding linguistic member Fui, define

a � Ui�1

b � mui(u∗)(Ui�Ui�1) � Ui�1

c � Ui+1�mui(u∗)(Ui+1�Ui)

d � Ui+1

h � mui(u∗)

(11)

Since there are two linguistic members being fired at the same time, Fig. 4 illus-

Fig. 4. Shaded regions represent areas covered by membership functions for defuzzification.


trates the area obtained after the fuzzy inference. The commonly used center of areamethod (Palm et al., 1996) is employed herein to obtain the crisp value of the controlsignal u∗.

Aui�

c � d�a�b2

h (12)

Cui� a (13)

�

23

(b�a)2 � 2(b�a)(c�b) � (c�b)2 � (c�a)(d�c) �13

(d�c)2

c � d�a�b

u∗ ��Cui

·Aui

�Aui

(14)

where Aui, Cui

are the centroid and area covered by the ith membership function ofthe output linguistic variable.

4. Controller design

A sliding mode formulation is used to design controller parameters. The controllaw u(t) is designed so that the system trajectories are ultimately bounded under theregion B = {SE,�SE� �} ; � � 0. This goal can be achieved by satisfying thefollowing sliding condition outside region B (Slotine and Li, 1991),

SE·SE��h(|SE|��) ; h � 0 (15)

where the sliding error SE is defined in Eq. (5). The constant h is a design parameterthat is inversely proportional to the time required for the SE to reach the boundaryof region B from outside of B. The time derivative of the sliding error is determinedfrom Eq. (4), in which the acceleration of yawing angle can be obtained from Eq. (1),

SE �1�l

I �u�I�yd�l

1�lye��br�r� � d� (16)

Define r = yd�l

1�lye, Eq. (16) can be expressed as

SE �1�l

I[ u � d�(I r � br|r| )] �

1�lI

[ u � f(r,r, d)] (17)

where f(r,r, d) = d�(I r + br�r�). If no sufficient model of r and d exists, the estimateof the following upper bound must be determined


| f(r,r,d)|max � F (18)

We re-express the sliding condition as

SE·1�l

I·[u � f(r,r,d)]�SE·�1�l

I �·u � |SE|·�1�lI �·F��h (|SE|�� )

or

SE·�1�lI �·u��h(�SE��)��SE�·�1�l

I �·F � ��h � �1�lI �F��SE� � h�

(19)

The fuzzy rule base is designed so that the fuzzy controller has a similar form tothe sliding mode controller with a boundary layer,

u � �K(SE,l)·sat(SE�

) (20)

where

sat(SE�

) � �SE /� if �SE� � �

sgn(SE /�) if �SE� � �

and K(SE,l) � 0. When �SE� ��, the sliding condition holds, replacing u in Eq.(19) with the u in Eq. (20) obtains a stability and robustness bound for the magnitudeof K∗(SE∗,l),

K∗(SE∗,l)� �I /Gu

1�l��h � �1�lI �F��

h� /Gy�SE∗� (21)

where K∗(SE,l) = K(SE,l) /Gu. The method of determining the stability and robust-ness bound offers simplicity in selecting the input/output membership functions, andis similar to the technique presented in (Emami et al., 2000). When �SE� ��, thetransfer function from SE∗ to u∗ can be tuned by designing the shape of the inputand output membership functions. According to the design principle of sliding modecontrollers, the ‘balance condition’ (Slotine and Li, 1991) should be fulfilled insidethe boundary layer to avoid the influence of high-frequency unmodeled dynamics.Substituting Eq. (20) into Eq. (17) obtains the dynamics of SE inside the bound-ary layer

(1�l

I)·(

K(SE,l)�

)·SE � SE � (1�l

I)·f(r,r,d) (22)

Letting the corner frequency of Eq. (22) be less than or equal to l /1�l, thedesign criterion for the slope in the vicinity of the origin of the SE∗�u∗ phase spaceis obtained. Define the K∗(0,l) to be the magnitude of K∗(SE,l) near the origin.K∗(0,l) shall satisfy the following inequality,


K∗(0,l)�∗ � � I

1 � l�� l1 � l�·�Gy

Gu� (23)

The slope of the transfer function at the origin can be adjusted by careful selectionof shrinking and dilating factors. For example, Fig. 5 illustrates the transfer functionof a normalized fuzzy logic controller with its input–output membership functions.Approximating the shape of the transfer function with piece-wise linear functions,discontinuous points of the piece-wise linear functions appear at core values of theinput triangular membership functions (Palm, 1994). A reasonable selection of theparameter � will be at the first core values (i = 1 and �1, in Eq. (8) and Eq. (9))of the input membership function. For example, if an input shrinking factor 0.5 isused, then � = U1 = 0.25Gy, or �∗ = 0.25. Fig. 6 illustrates constant-slope contoursat the origin of the SE∗�u∗ phase plane in terms of input/output shrinking and dilat-ing factors. A plot such as Fig. 6 is useful for the selection of input/output shrinkingand dilating factors. Five sections are used in Fig. 6 to partition the domain of inputand output variables.

From Eq. (21), the parameter Gu can be determined by letting SE∗→, u∗→�1,

Gu � � I1�l��h � �1�l

I �F� (24)

To design the fuzzy controller, parameter values of h,Gu,Gy, and l are needed.Parameter h is inversely proportional to the time required to reach region B froman outside initial condition. Parameters Gu and Gy influence the slope of the control-ler’s transfer function, thus determining the behavior of the system around the origin.The value of l / 1�l specifies the rate of convergence on the sliding surface. InSlotine and Li (1991), a suggested choice for l /1�l is one fifth of the sampling fre-quency.

Fig. 5. The normalized input-output relationship of the fuzzy logic controller with its input/output mem-bership functions.


Fig. 6. Constant-slope contours at the origin of the SE∗�u∗ phase space in terms of input-output shrink-ing and dilating factors. The numerical numbers on the curves indicates the magnitude of the slope.

5. Experiments

Tank tests and open sea trials were conducted to determine the effects of thedesign parameters. The testing tank was 120 × 8×4 m (L × W×D). Meanwhile, thetestbed vehicle AUV-HM1 had dimensions of 2 × 1×0.6 m (L × W×D), as displayedin Fig. 7. The vehicle employed a 300 kHz Doppler velocity log and a fiber-opticsgyro as navigation sensors. The sea trial area had a water depth of 60 to 80 m. Totest the effect of surface currents and waves on the effectiveness of the controlalgorithm, the vehicle was submerged at a depth of 2 m below the surface.

To design the controller, the balance condition shall be satisfied at the origin ofthe SE∗�u∗ phase plane. One can select proper shrinking/dilating factors to fulfillthe balance condition. Gy is designed according to the tracking precision required.The design parameter h is determined to set the time required to reach the boundary

Fig. 7. The testbed vehicle AUV-HM1.


layer from an outside of the boundary layer initial condition. Gu sets the maximumoutput range of thrust forces. Gu is determined from Eq. (24), in which the uncer-tainty bound F represents the maximum uncertainty the system is able to tolerate.Finally, the stability and robustness property of the control system shall be checkedusing Eq. (21).

5.1. Effects of Gy, Gu, and l

A directional command is given while the vehicle is moving with a forward velo-city of 0.8 m/s in the water tank. Figs. 8–10 present the effects of Gy, Gu, l ontransient responses and tracking precision. Gu is a scale factor to determine the mag-nitude of the thrust force. While advancing at higher speeds, the vehicle encounterslarger disturbances and model uncertainties, higher Gu is needed to maintain satisfac-tory control performance. Higher Gu lead to faster transient responses and moreprecise tracking. Gy is proportional to the thickness of the boundary layer. HigherGy leads to less tracking precision. Higher l lead to faster transient responses, andless steady state tracking errors. The parameter h is kept constant in these tests. Figs.8–10 show different initial conditions. The time required to reach region B from anoutside initial condition is inversely proportional to the value of h and the initialdistance from region B.

5.2. The effect of xo

During sea trials, the vehicle was ordered to follow a series of waypoint circlesof origin [xkd,ykd], and radius xo. If the present vehicle position [xc,yc] satisfied thefollowing condition

Fig. 8. Transient responses of vehicle heading with different Gys.


Fig. 9. Transient responses of vehicle heading with different Gus.

Fig. 10. Transient responses of vehicle heading with different ls.

[xc�xkd]2 � [yc�ykd]2 � x2o (25)

the guidance algorithm triggered the selection of the next waypoint. Fig. 11 presentsthe results of tracking experiments in ocean currents. Waypoints are arranged on astraight line. The controller is selected to be the same for all three experiments. Weexamine the tracking precision using three different waypoint circles radius xo under


Fig. 11. Waypoint tracking in ocean currents.


the influence of ocean currents and waves. Since precise tracking is difficult to achi-eve given model uncertainties and external disturbances, the radius of the waypointcircle must not be less than the achievable tracking precision. Achieving precisetracking obviously comes at the cost of poor transient performance. Consequently,a tradeoff between stability and tracking precision must be found.

5.3. The effect of the control gain

Fig. 12 illustrates the state dependent control gains used in the sea trials. Theupper plot of Fig. 12 represents a controller that satisfies the balance condition, while

Fig. 12. Controller gain of the sliding mode fuzzy controllers and the stability and robustness boundsfor the sea experiments presented in Fig. 13.


Fig. 13. Square path tracking in ocean currents.

the lower plot does not. Both controllers satisfy the stability and robustness condition.Fig. 13 presents waypoint-tracking results using the above controllers. Waypointsare arranged at the corner of a square route. The sampling frequency is 12.5 rad/s,and l is set at 0.7. The upper bound of the slope for the control gain at the originis calculated using Eq. (23), yielding a value of 5.6. Two experimental results werechosen to demonstrate the effect of the control gain. The control gain at the origincorresponding to the upper plot of Fig. 12 is designed to be 5.5, while that corre-sponding to the lower plot of Fig. 12 is 1. Notably, while the AUV moved fromone side of the square path to the other, the uncertainty bound F varied significantly.At the moment of transition, the state and the sliding surface are separated by asignificant distance, and the unmodeled dynamics cannot change the sign of thecontrol input. Consequently, we evaluated this bound only in the vicinity of the


Fig. 14. The uncertainty f, and its bound F, calculated from the experimental data shown in Fig. 13.

origin of the sliding error. The magnitude of uncertainty bound F is influenced bythe waypoint selection, magnitude and direction of ocean currents, Vc, Dc, and thevehicle’s unmodeled dynamics. To check the stability and robustness of the controlsystem, we need to measure F. The Fs corresponding to these control gains can becalculated using experimental data. The following parameter values are used: I =24.13 Kgf·m·sec2, b = 32.50 Kgf·m·sec2, �d�

max=1.02Kgf·m. The moment of inertia, I,

including the added moment of inertia, was measured using a Planar Motion Mech-anism (Chiu et al., 1997), the damping coefficient b was derived from yaw step


responses’ data, and the cross flow moment d at current velocity 0.5 m/s was meas-ured by towing tank tests. Fig. 14 shows that F = 4.5Kgf·m, which was calculatedfrom all path segments, ignoring large variations in magnitude at transition periods.Based on Eq. (24), the value of Gu is set at 9.5 Kgf·m, which, after a unit conversionfrom torque to command voltage, is 5 V.

6. Conclusions

This study has demonstrated the feasibility of applying a sliding mode fuzzy con-troller to an AUV in shallow water in order to perform line-of-sight guidance in thehorizontal plane. The state dependent control gain is specified by a set of shrinking-span and dilating-span factors. Design parameters of this controller, such as the slopeat the origin, and the stability and robustness bound, are specified using definedcriteria. The effect of selecting different control parameters is evaluated throughwater tank and open sea experiments using an AUV testbed. The test results alsoconfirm the effectiveness of the method.

Acknowledgements

The authors would like to thank the National Science Council of ROC for financi-ally supporting this research under Contract No. NSC88-2611-E002-020.

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design of a sliding mode fuzzy controller for auv

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