Proceedings ofthe 2009 IEEEInternational Conference onMechatronics and Automation

August 9 - 12, Changchun, China

Research of GA-based PID for AUV Motion ControlQiang Chen

Institute ofNaval VesselsNavy Academy ofArmament

Beijing, China

cq20071 OI8@sina.com

Abstract -This paper describes the research of geneticalgorithm (GA) based PID for autonomous underwater vehicle(AUV) motion control. A simulation of heading controlapplication is considered using GA and improved GA based PID.The dynamic model of AUV is established. The design method ofGA and improved GA based PID is introduced. The headingcontrol design model is obtained by the AUV linearization modelat given operating points which considering the ocean currentdisturb. The heading controller is designed according to themethod of GA and improved GA based PID. The simulationresults show that the controller is effective with good dynamicperformances for AUV motion control.

Index Terms -autonomous underwater vehicle; geneticalgorithm;PID control;motion control

I. INTRODUCTION

The motion control of AUV (such as heading control,depth control, pitch control and so on) is a key techniqueneeds to be researched and resolved exigent. Good motioncontrol is an important technical guarantee for AUV tocomplete missions successfully. However, the kinetics ofAUV has serious nonlinear characteristic, the motion modelhas excessive uncertainty, and there are external interferences(such as ocean current). So, the motion control of AUV iscomplex and difficult. There ate many methods are proposedto carry out motion control of AUV. Ref. [1] adopts neuralnetwork method, Ref. [2] adopts sliding mode method tocontrol without AUV model. A heading controller usingIntegral Variable Structure is design ed to control the AUVheading in [3]. Ref. [4] designed a Nonlinear H-infinityController. Ref. [5] designed a optimization of S-surfacecontroller which using immune-genetic algorithm to optimizeparameters.

These methods mentioned above are effect in a certaincontrol scope. However, in actual situation, the primarycontroller is still PID control. There are two key problems notgood resolved for motion control using PID. The one is thatparameter adjusting is difficult. The other is that selectedparameters can 't suit for nonlinear and diverse system state.The evolutionary mechanism of GA can rapid and reliableresolve complicated problems-". Using GA to adjust PIDparameter has advantages of practical, efficient, and robust.This paper using GA based PID to realize motion control ofAUV and application of heading control is researched andsimulated. The simulation results show that it has a gooddynamic performance and definite effect to restrain theexoteric disturbs.

Tao Chen, Yang ZhangCollege ofAutomation

Harbin Engineering UniversityHarbin, Heilongjiang Province, China

chen tao_7777@ 163.com

Rudder E ave to r Main-T hruster

Aux-Thru ster Planfonn

~ ~ ~S ide elevation

Fig. I Exterior and physical config of vehicle

II. AUV AND DYNAMIC MODEL

A. Researched AUV

Researched AUV is a clipper-built gyroidal style. Itsexterior is symmetrical on left-right and anisomerous on up­down and front-back. An elevator, a pair of rudders and twomain-thrusters configured with left-right are fixed on thestem. two slot aux-thrusters fixed on the midship. The exteriorand physical config are showed in Fig. I. The elevator iscontrolled for depth and pitch. The rudders are controlled forheading. The main-thrusters are controlled for surge velocity.In addition, the left and right main-thrusters can also becontrolled for heading using difference method. The aux­thrusters are controlled for rotation and translational motionswhen surge velocity is low.

B. Dynamic Model ofA UV

The 6 DOF components of the submerged vehicle rigidbody dynamic equations of motion can be writtenincomponents form as in [7]. The form of equation of motionis obtained with body axes coincidentwith the principles axesof inertia, and the origin not at the center of mass center ofgravity (CG). For this case the equation in the dimensionlessform are:

X = m[u-vr+ wq- xG(q2 +r2)+ YG(pq-r)+ zG(pr+q)]

Y = m[v- wp+ur - YG(r2+ p2)+ zG(qr - p)+xG(pq+r)]

Z = m[w-uq+vp - ZG(p2 +q2)+ x(,(pr -q)+ YG(rq + .0)]

K = Ix.o+ IxA+ Ixi + (lzxP+ Izyq + Izr)q-(lyXP+l yq+ Iy.r )r

+ m[YG(w+ vp - uq) - z(,(v+ ur - wp)]

978-1-4244-2693-5/09/$25.00 ©2009 IEEE 4446

III. GA BASED PID

Let (VRx,VRy,VRz) replaces (u,v,w) in (1), then obtain the ocean

current interferential model of AUV.

111111·· ·1111 =2k -1 ~ V z

V -VWhere t5 = z 1

, 2k -1

When decoding, assume that the coding of a individualis bkbk_Ibk_Z ... bzbI , then its decoding formula is:

X = V +(i> .2;-IJ. V z -VI (7)1 i=I 1 2k -1

(6)

VRx=U-Ue

VRy =v-ve

VRz = w-we

The basic idea of GA based PID is using GA to optimizethe proportional, integral and derivative parameters (i.e.kp ' k.,kd ) . The optimization results of the three parameters

will be used for control. Then, the fitness will be calculated toguide farther optimization in next generation. Finally, theresults of last generation are the optimization parameters usedfor PID control. Here, some pivotal operations are introduced.

A. Parameters Initialization and Coding

First, N groups of ( kp ' k.,kd ) are selected based on

symmetrical principle as the initial population. The value rangeof the parameters is based on experience. Then, kp ' kp kd are

coded respectively using M-bit binary codes. Last, a longbinary code is composed of the three codes as the operationalobject in GA.

Supposed that the value range of a parameter is [VI' V z ] . A

binary code with length k is used to code this parameter

which can bring 2k varieties codes. Let the correspondingrelation between this parameter and its code is:

000000· .. 0000 =a~ VI

000000· .. 000 1= 1~ VI + t5

000000· .. 00 10 = 2 ~ VI + 2t5

[

COSIf/cos B sin If/sin Bsin qJ - sin If/cos tp

S = sin If/cos B sin If/sin Bsin qJ+ cos If/cos qJ

-sin B cos Bsin qJ

. B ..] (5)cos If/SIn cos tp+ SInIf/SInqJ

sin If/sin Bcos qJ - sin If/sin qJ

cosBcosqJ

Let VR(VRx' VRy' VRz) is the velocity of AUV relative to ocean

current, then:

(4)

M =IyxP+ IyiJ + Iyzr+ (Ixp + Ixyq + t.r» - u:» + IZyq + Izr)p

+ m[zG (u + wq -vr) - xG(w+ vp - uq)]

N = IzxP+ IZyiJ + Izr + (IyXp + Iyq + Iyzr)p - (Ixp + Ixyq + Ixzr)q

+ m[xG(v+ ur - wp)- YG(u + wp-vr)]

(1)Where, x is surge force, Y is sway force, Z is heave force.K is roll moment, M is pitch moment, N is yaw moment, p

is roll rate, q is pitch rate, r is yaw rate, u is surge velocity,

v is sway velocity, w is heave velocity, x is body fixed axesin positive forward, y is body fixed axes in positive starboard,

z is body fixed axes in positive down, i, I y , I z are vehicle

mass moment of inertia around the x -axis, the y -axis, the z­

axis respectively, XG is longitudinal position of center of

gravity, YG is athwart location of center gravity, ZG is vertical

position of center of gravity, m is the mass of the AUV.The attitude equations are:

ip= p + qsin qJtan B+ r cos qJtan B

iJ = qcos qJ- r sin qJ (2)

iff =q sin qJ / cos B+ r cos qJ / cos BWhere, ffJ is roll angle, () is pitch angle, and 'If is yaw angle.

The motion relation formulas are:

~ =u cos If/cos B+ v(cos If/sin Bsin qJ - sin If/cos qJ)

+ w(cos If/sin Bcos qJ+ sin If/sin qJ)

1/ =u sin If/cos B+ v(sin If/sin Bsin qJ + cos If/cos qJ) (3)

+ w(sin If/sin Bcos qJ - cos If/sin qJ)

¢ =-u sin B+vcos Bsin qJ+ wcos Bcos qJ

Where, ~ , 17 , ; are displacements in inertial coordinate

system.

C. Ocean Current Interferential Model

There is always ocean current in the sea area where AUVnavigates. So, it is necessary to research ocean currentinterference model. The velocity of ocean current is usuallylow, and much lower in deep ocean. But the velocity of oceancurrent is variational in different area, depth and seasons. So,the velocity of ocean current is a complex function anentspace and time. Usually, the sea area where AUV navigatesand navigational time are restricted. So, it is supposed that theocean current is horizontal and its velocity is fixed.

Let V(u~, v17

' we) is the velocity of ocean current in inertial

frame, Uiu, ve' we) is the velocity in frame, then:

Where, S is coordinate transform matrix from kinetic

coordinate system to inertial coordinate system. S can becalculated as belows:

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B. Fitness Function

In this paper, N is 30, M is lO.that is to say, the num ofindividual in population is 30, the precision of kp,ki,kd is 10

bit , then the total length of each individual is 3*I0=30 bit.

Usually, the fitness function of GA is reciprocal ofobjective function. In order to obtain better control effect, theselected objective function contains control variable, error andrise time as the constraint. The objective function can beselected using method below. Considering a control systemcontains multiplication and uncertainty as described in Fig. 2.

Fig. 2 Control system contains multiplication and uncertainty

Where, G(s) is system nominal model, K(s) is typical PID

controller, K(s)=Kp+K;/s+Kds , ~G(s) is uncertain

error, and existing true rational function S(jm) E H cc ,meeting:

I~G (jm)1 ::; Is(jm)l , 'Ifaie [0,00] (8)

J = [(WI Ie(t) I+WZUZ(t) + w4 Iey(t) Ddt + W3 . til if ey(t) > 0

(14)

Where, w4 is weight, w4 > WI ' ey(t) = yet) - yet -1) , yet) is

the output of control object.So, the PID parameter tuning problem can be described as:

searching a group of parameter, let (13) and (14) minimum,meanwhile, meeting the constraint in (12). It is a inequalityconstraints nonlinear planning problem:

{Min J (15)h(m) ::; 0, 'Ifta

Equation (15) is selected as the objective function J , the

fitness function of GA is f = 1/ J 0

C. Improved GA

Now , there is much method to improve GA. This paperadopts fitness zooming and self-adaptive GA operation toimprove GA.(1) Fitness Zooming

The effect of fitness zooming is to prevent prematureconvergence in evolution early stage and accelerateconvergence rate in evolution later stage. Supposed that ~ (k )

is the fitness of individual j in k th population. f(k) is the

average fitness of k th population. J; is the zoomed fitness of

individual j , then:

/ ={~(k)+f.L[!!(k)-7(k)/f.N, ~(k)"C.f(k) (16)

} f(k)-f.L[f(k)-~(k)/f.L]q, ~(k)<f(k)

y (t)G(s)[l +.6G(s)] iv-u (t)

K(s)I

e (t) I+r (t)

The robust stability of the system showed in Fig. 2 needsto meet the inequality below:

IIG(S)K(S)(S)II ::; 1 (9)1+G(s)K(s) ~

The (9) can also be expressed as:

IG(jm)K(jm)((jm)I::; 1 'lfm (10)

I+G(jm)K(jm) ,

Let:

hem) = IG(jm)K(jm)((jm)I_I1+G(jm)K(jm)

(11)

Where, ; is zooming value, it will be increased a zooming

step CJ' after each population, i.e.;(k+I)=;(k)+CJ' (17)

Where, e(t) is system error, u(t) is the control output, til is

rise time, WI ' Wz ' w3 are weights.

In order to prevent overshoot, punishment function is usedto add overshoot value to the performance objective:

If only pursuing the dynamic characteristics of the controlsystem, the obtained parameters are very likely too large. Thenthe system may be unstable for its inherent saturationcharacteristic. So, in order to obtain satisfactory transitiondynamic characteristics, the IAE performance objective isselected as the mini objective function. And, in order toprevent too large control variable, the square of control inputis added to the objective function. So, the performanceobjective is selected as:

J= [(wl le(t) l+wzuZ(t))dt+W3 ·tll (13)

(18)

Then , the (10) is equivalent to:h(m) ::;0, 'Ifio (12)

Where, f.L can be determined by equations below:

f1J =max ~(k)- f(k)}

f.Lz =f(k)-min ~(k)}

f.L = ~min(f1Jf.Lz)2

(2) Self-Adaptive GA OperationThe genetic operations are to operate the initial population

composed ofkp ,ki,kd .They are selection crossing and variation.

Ratio selection is selected in this paper. The thought of ratioselection is the selection probability of each individual isproportional to its fitness. Let the population size is M , the

fitness of individual i IS F , then the selection probability

P;s of individual i is:M

P;s=FJ"I F, i =(l,2,· ··,M) (19)i= l

4448

(27)

(23)

Alone point crossing is adopted in this paper. The crossingprobability is P, . In improved GA, the crossing probability P,is self-adaptive according to (17). The variation probability ofvariation operation is Pm . In improved GA, the variation

probability Pm is self-adaptive according to (18).

{~l - (~l - ~2 ) (f' - r: )I(fmax - faVg) r ~ r:

~ = I (20)~l f <r:

Pm ={P"'l - (Pm l - Pm 2 ) (f - r: )«: - faVg) ', ~ i; (21)

Pm l f < L;Where, ~l' ~2 'Pml and Pm 2 is adjusting parameter. t.; is the

max fitness in the population, faVg is the average fitness in

each population, f' is the larger fitness in the two individuals

will be crossed, f is the fitness of variation individual.

IV. ApPLICATION OF HEADING CONTROL

A. Model ofHeading Control

Considering the two equations of AUV model in (1):

Y =m[v-wp+ur - YG(r2+ p2)+zc(qr - P)+xG(pq+r)]

N = IzxP+ Izyq+ Izr + (Iyxp + Iyq + Iyzr)p - (Ixp + Ixyq + Ixzr)q

+m[xG(v+ur -wp) - YG(it + wp -vr)]

Analyzing the forces and moments AUV endured, can obtainthe detailed Y and N . Then, the two equations above are:

m[v-wp+ur- YG(r2+ p2)+zG(qr- P)+xG(pq+r)]

= t pL4[Y;r +Y;p +Y;jplPlpl +Y;qpq+Y;rqr + ~~r(lrl]

+t p J}[}~:v + Y,:,ur + y,~vq +r:p wp + r:,wr + Y~d ,:,.Jv2 + w2

1r I]

+t pL2[~:ZIU2 + ~:vuv + Y:wvw + Y~vlvv'V2 + w2]+ (W - B)cos Bsin qJ

+tPL3~~l5rUlrI8r +tpL2Y;ru28r +Yprop (22)

Izxp + Izyq + Izi' + (Iyxp + Iyq + Iyzr)p - (Ixp + I.'(}/q + I.-ar)q

+m[xG(v+ur-wp)- YG(u+wp-vr)]

=tpLs[N;i' + N;P +«: Iplp + N:1r( Irl + N;qpq + N;rqr]

+t pL4[N:v+ N:pup+ N:rur + N~rwr + N~pwp + N:qvq + N':lr .Jv2 + w

2r]

+t pL3[N:u u2 + N:vuv+N~vw + N:,vIV.Jv2 + w2

]

+ (xGW -xBB)cosBsin (jJ+ (YG W - yBB)sinB

+t pL4N';'l5rUlrI8r +tpL3N;ru2s. + N prop

Where, Y'(.) is dimensionless hydrodynamic derivatives,

N '(.) is dimensionless hydrodynamic moment derivatives,

xB 'YB are the barycentric coordinates of AUV, W is weight

and B is buoyancy, 8r is the rudder angle, Yprop is thruster

force, N prop is thruster moment.

4449

Now, supposing that the surge velocity of AUV is aconstant u =Uo , ignoring the effects of vertical motion and roll

motion, i.e. let w ~ 0, p ~ 0, q ~ 0 , ignoring the nonlinear

quadratic term on v and r. The obtained linear model ofheading control after simplification is:

( 1 3 ') (1 4 ')m-2 pL J: v+ -2pL Yy r (24)

=(tpL2Y:vUO)v+(-tpL3~:ruo -muo)r+(tpL2Y;ru~)gr +dv

(-tpL4N;)v+(Iz -tpL5N~)r (25)

=(t pL3N~vuo)v +(tpL

4N,:ruO)r +(t pL

3N;ru~)8r + d,

'fr = r + dlff (26)

Where, dlff, d, d, contain the errors brought by linearization,

uncertainty and exoteric disturbs.Let the heading instruction is a constant If/r meeting 'frr =0 .

Let If/e =If/r -If/ , due to (21):

. d(lf/r-If/). dIf/e = dt =-If/ =-r - Iff

Combining (19) and (20), the model for heading control is:

[;]=[~ :: ~][: ]+[{]or (28)

Where,

[ ~ll ~l2] =H-Ip,[~ll] =H-IQa21 a22 b21

[

m _1.pL3y' _1.PL4yl]H= 2 v 2 r

-tpL4N:Iz-tpL5N~

== [t pL2y:Vu

o tpL3y;u

o- muo] Q =[tpL2Y;ru~ ]p I L3N' I L4N' , I L3N' 2"2P uvUO "2P urUO -zp l5r uO

The exoteric steady disturbs (for example, ocean current)may result in static error of heading control. So, Hereintroduces a integral to eliminate the steady disturbs'I':

v, =v. (29)

So, the final model of heading control is:

v, 0 1 0 0 If/I 0

v. 0 0 0 -1 v. 0+ 4 (30)

v 0 0 ~l ~2 V bll

i 0 0 a21 a22 r b21

B. GA Based PID for Heading Control

Here, the GA based PID will be designed according tosection 0 and (30). The selected population size is 30,evolutional time is 300. According to (13) and (14), Theobjective function is set as:

If ey(t) < 0 J =r (0.9991 e(t) I+0.00 1u2 (t»dt + 2.000· tIl

If ey(t) >°J = r (0.999 1e(t) I+0.00 lu 2 (t) + 100 1ey(t) I)dt + 2.000 · til

In improved GA, the fitness function is zoomed accordingto (l6)~(l8) , the zooming value q= 0.3 , the zooming

step (7 = 0.007 .In this paper, the crossing probability p"=0.9, the crossing

probability P",=0.1. In improved GA, the crossing probability

and variation probability are self-adaptive according to (20)and (21). Where, P"I = 0.9 , P"2 = 0.6 , Pml = 0. I.

velocity increase, the control speed becomes rapid with theovershoot a little increase.

TABL E IHEADI NG C ONTROL PERFO RMANCE INDEX WITHOUT CU RRENT

operating point (u) u= 2kn u=4kn u=8kn

rise time tr /s 7 5.5 4

adjusting time ts /s 12 II II

overshoot (7 % 3.4 3.1 3.5

40

Fig. 3 Step response curves of heading control without current

160

160

60 80 100 120 140T ime(s)

60 80 10 0 12 0 140Time(s)

40

40

III~I!\!I::1

l\~" -f-~------------

20

20

Fig. 6 Rudd er angle control curves with current

-5

35

30

40

25

Fig. 5 Step response curves of heading control with current

70

60

!l50 j

-oa i.{j 40

!J~ 30~

20

10

0

Now, the ocean current disturbs are added with velocity1.0kn and direction north. Fig (5) is the step response curvesof heading control on surge velocity operating point (2kn, 4kn,8kn). The heading instruction is step changed from °degree to60 degree at simulation time 20s. Fig (6) is the rudder anglecontrol curves and the rudder angle rang is restrictedin[-35" 35"] .The obtained PID parameters and cost function

value are:u = 2kn [kp,kj,kd ]=[1.4179,0.0166,2.8832],J = 7546.4

u = 4kn [k p' kj, kd ]=[0.7560,0.0085,1.3340], J = 7269.0

u = 8kn [k p,k j,k d ]=[0.2278,0.0161,0.4788], J = 7170.9

Table II shows the dynamic performance indexs of GAbased PID for heading control with current disturb.

The simulation results show that the control effect is alsogood within the current disturbs. However, the adjusting timeis obvious longer than the control without current disturbs .

60 80 100 120 140 160Time(s)

60 80 100 120 140 160Time(s)

40

40

IIII·1:1::\

!I

!:I{i-..-.= = - - - - - - - - - - ­

JI',l20

rr7"'=-----------'===~~

jf!J::1

'I~ 1il)j!Il

20o

-5

o

5

25

35

30

70

10

20

50

60

-oa.{j 40

J~ 30

Fig. 4 Rudder angle control curves without current

Table 1 shows the dynamic performance indexs of GAbased PID for heading control without current disturb.

According to the simulation results and Table I, it isshowed that in the ideal condition (no ocean current), thecontrol of GA based PID on the design operating point iseffective and have good dynamic characteristics. With the

C. Simulation Results

Fig (3) is the step response curves of heading control onsurge velocity operating point (2kn, 4kn, 8kn). The headinginstruction is step changed from °degree to 60 degree atsimulation time 20s. There is no ocean current disturbs in thesimulation. Fig (4) is the rudder angle control curves and therudder angle rang is restricted in [-35" 35"].The obtained PID

parameters and cost function value are:u = 2kn [kp,k j,kd ]=[2.4143,0.0216,5.6728], J =6838.8

u = 4kn [k p,k j,kd ]=[0.7698,0.0110,2.1280], J =6706.4

u = 8kn [k p,k j,kd ]=[0.3108,0.0168,0.5291], J =6655.6

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TABLE IIHEADING CONTROL PERFORMANCE INDEX WITHCURRENT

operating point (u) u~2kn u=4kn u= 8kn

rise time tr /s 7.5 6.5 6

adjustin g time ts /s 26.5 16.5 16

overshoot CY % 5.2 4.1 3

70

60 /50

If

~ 40 Ifl ,

~ 30P=1

20

10

--Im pr oved GA b as ed P ID- - GA b as ed PID

current disturbs with velocity I.Okn and north direction. Theseresults show that the improved GA based PID has much bettercontrol effect with faster adjusting time and smaller overshoot.

V. CONCLUSION

The single PID control is not effective for AUV motioncontrol because the parameters are adjusted difficultly and theparameters can't suit for nonlinear and diverse system state.The GA based PID can using the good high-speed parameteroptimization ability to realize the automatic parameterselecting and nonlinear control. Although, there are manyproblems , such as early-maturing, poor real-time, and so on,more and more methods are raised to improve GA based PID.This paper using GA and improved GA based PID to realizethe motion control of AUV, which obtains a good effect. Thesimulation results show that it has a good dynamicperformance and definite effect to restrain the exotericdisturbs.

R EFERENCES

Fig. 7 Compring of heading control curves without current using GA basedPID and improved GA based PID

Fig. 8 Compring of heading control curves with current using GA based PIDand improved GA based PID

Fig.(7) and Fig.(8) are the step response curves of headingcontrol using GA based PID and improved GA based PIDrespectively with the surge velocity operating point 2kn. InFig.(7), there is not current disturbs. But in Fig.(8), there is

--Im proved GA based PID- - GA b ase d PID

70

60

50

20

10

o

o

20

IIIJ

IIII!I1

20

40

40

60 80 100Tim efs)

60 80 100T ime fs)

120

120

140

140

160

160

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