] (1) Fuzzy Trajectory Control Design for Underwater Robot. Abstract. 2 Equations of Motion. 1 Introduction

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1 EUSFLAT - LFA 5 Fuzzy Trajectory Control Desgn for Underwater Robot Jerzy Garus Department of Electroncs and Electrcal Engneerng Naval Unversty 8-3 Gdyna, ul. Śmdowcza 69 Poland j.garus@amw.gdyna.pl Zygmunt Ktows Faculty of Mechancal and Electrcal Engneerng Naval Unversty 8-3 Gdyna, ul. Śmdowcza 69 Poland z.tows@amw.gdyna.pl Abstract In the paper applyng of genetc algorthms to desgnng of a fuzzy autoplot for traceepng control of underwater robot s consdered. For the tracng of a reference trajectory, the way-pont lne of sght scheme s ncorporated and three ndependent fuzzy controllers are used to generate command sgnals. Parameters of membershp functons of nput and output are tuned usng genetc algorthms. Qualty of control s concerned wthout and n presence of external dsturbances. Some computer smulatons are provded to demonstrate the effectveness and robustness of the approach. Keywords: Underwater robot, Autoplot, Fuzzy control, Artfcal ntellgence. Introducton Underwater Robotcs has nown an ncreasng nterest n the last years. The man benefts of usage of Underwater Robotc Vehcles (URVs) can be removng a man from the dangers of the undersea envronment and reducton n cost of exploraton of deep seas. Currently, t s common to use the URVs to accomplsh mssons as the nspecton of coastal and offshore structures, cable mantenance, as well as hydrographcal and bologcal surveys. In the mltary feld they are employed n such tass as survellance, ntellgence gatherng, torpedo recovery and mne counter measures. The URV s operated manually wth a joystc or automatcally wth a computer. The automatc control of the underwater robot s a dffcult problem due to ts nonlnear dynamcs. Moreover, the dynamcs can change accordng to the alteraton of confguraton to be suted to the msson. In order to cope wth those dffcultes, the control system should be flexble. An nterestng revew of classcal and modern technques adapted to control the dynamc behavour of unmanned underwater vehcles has been provded n []. Nowadays, fuzzy systems fnd wde practcal applcatons, rangng from soft regulatory control n consumer products to accurate control and modelng of complex nonlnear systems [,6,7,8,9]. In ths paper we desgn the fuzzy logc autoplot to traceepng control of the URV tunng ts parameters by genetc algorthms. Equatons of Moton The general moton of marne vehcle n 6 DOF descrbes the followng vectors [3,4]: η = v = τ = T [ x, y, z, φ, θ, ψ ] T [ u, υ, w, p, q, r [ X, Y, Z, K, M, N ] T ] () η poston and orentaton vector wth coordnates n the earth-fxed frame; v lnear and angular velocty vector wth coordnates n the body-fxed frame; 58

2 EUSFLAT - LFA 5 τ vector of forces and moments actng on a hull n the body-fxed frame. The dynamcal and nematcal equatons of moton can be expressed as follows [3]: ( ) Mv& + C( v) v + D v v + g( η) η& = J( η)v = τ () M nerta matrx (ncludng added mass); C(v) matrx of Corols and centrpetal terms (ncludng added mass); D(v) hydrodynamc dampng and lft matrx; g (η) vector of gravtatonal forces and moments; J velocty transformaton matrx between ( η) Let us assume that the route s composed of some straght lnes defned by turnng ponts P, P, P 3, etc., wth coordnates ( x, y ), ( x,y ), ( x,y ), etc. 3 3 A tracng error of the robot s defned n the reference coordnate system P X Y (n Fg. t s P X Y ) as a perpendcular dstance d of the robot located n poston ( x, y to the predefned ) trajectory. Accordng to ths, the two-component vector of the robot s poston [ x, y n the global ] coordnate system can be expressed as: body and earth-fxed frames. [ ] T y 3 Tracng and Coordnate Systems It s convenent to defne three coordnate systems when analysng tracng control systems for the robot s 3D moton n horzontal plane (see Fg. ) [3,4]:. the global coordnate system OXY (called also the earth-fxed frame);. the local coordnate system O X Y (fxed to the body of the robot); 3. the reference coordnate system P X Y (system s not fxed). ϕ ( ϕ ) sn( ϕ ) x + ( ) ( ) x ϕ cos ϕ y y x cos = (3) y sn x, global coordnates of the pont P ; rotaton of the reference coordnate system wth respect to the global one: y + y ϕ = arctan (4) x+ x [ x, y] T local coordnates of the robot, determned by expresson: x = y x& y& = t cos sn ( ψ ) u sn( ψ ) ( ψ ) u + cos( ψ ) υ υ ψ = ψ ϕ local headng angle defned as the angle between the trac reference lne and the robot s centrelne; ψ, u, υ robot s headng and lnear speeds n nstant t= t; x &, & ncrement of the robot s local [ ] T y t =,,,. coordnates n t= t; tme step; (5) Each tme the robot locaton ( x at the tme ( t), y ( t t satsfes: [ x x ( t) ] + [ y y () t ] (6) ρ )) Fgure : Coordnate systems used to descrpton of trac-eepng control n the horzontal plane: OXY earth-fxed system, O X Y body-fxed system, P X Y reference system where ρ s a crcle of acceptance, the next way-pont should be selected (e.g. n Fg. t s P X Y ) and the robot s poston updated correspondng to the new reference coordnate system. 59

3 EUSFLAT - LFA 5 4 Control System Membershp functons of fuzzy sets of nput varables: error sgnal e() t = η d η() t and derved The man tas of the desgned tracng control change n error e( t) = η ( t) η( t ) as well as system s to mnmze dstance of atttude of the robot s centre of gravty d to the desred trajectory output one (command sgnal) τ () t are shown under assumptons: respectvely n Fg. 4. The notaton s taen as follows: N negatve, Z zero, P postve,. robot can move wth varyng lnear veloctes u, S small, M medum and B bg. υ and angular velocty r;. the robot s poston x, y and headng ψ are measurable; 3. desred trajectory s gven by means of set of η = x, y, ψ ; ponts {( )} d d d 4. command sgnal τ conssts of three components: τ = X and τ = Y forces respectvely n x- and y-axs, τ = N moment about z-axs; 6 5. travel tme s not fxed, thus the navgaton between two ponts s not constraned by tme. A structure of the proposed control system wth the autoplot consstng of three ndependent controllers s depcted n Fg.. d Fgure 3: Fuzzy control structure Presented n Table rules from the Mac Vcar- -Whelan s standard base of rules have been chosen as the control ones [7]. Table : The fuzzy controller s base of rules Error sgnal e NB NM Z PM PB Derved N NB NM NS Z PS change n Z NM NS Z PS PM error e B NS Z PS PM PB Command sgnal τ 4. Tunng of Fuzzy Controllers Fgure : A general structure of the control system 4. Fuzzy Control Law To desgnng of the controllers the fuzzy proportonal dervatve (FPD) controller, adopted from [,8] and worng n confguraton presented n Fg. 3, has been used. The unnown parameters of the proposed fuzzy controllers have been determned usng genetc algorthms (GA), whch are based on Darwn s prncple of reproducton and survval of the fttest [5,9]. In general the GA technques manpulate sets of ndvduals (solutons) by usng genetc operators (selecton, reproducton, crossover and mutaton) n order to propose better ones. Chromosomes represent ndvduals n a populaton. A structure of the chromosome, whch has been used n the tunng procedure, s llustrated n Fg. 5. The chromosome conssts of four values that corresponded to unnown parameters of the membershp functons. Ther tunng range has been defned as follows: 6

4 EUSFLAT - LFA 5 < x e.5,.5 x <, < x M. 5,.5 x. < S < e Due to the objectve s to mnmse the control error, e.g. the tracng error d, a cost functon has been defned n the form: [ d () t + λ τ () t λ τ ( )] J = mn λ t (7) t where λ, λ and λ 3 are constant values. Calculated for the followng confguraton of genetc algorthms:. populaton,. crossover.8, 3. mutaton.5, 4. generaton, the demanded parameters of the membershp functons for three fuzzy controllers of the autoplot are shown n Table. They were obtaned on bass on analyss of the cost functon J durng the robot s movement between two set ponts: o o x B = ( m, m, 9 ), x F = ( 5 m, 5 m, 45 ). A goal of the robot was to reach the pont x F startng from x B. Table : Parameters of membershp functons trajectory n x-axs Controllers trajectory n y-axs course x e x e x S x M Smulaton Study Fgure 4: Membershp functons for fuzzy sets: sgnal e, derved change n error e and command sgnal τ Fgure 5: Chromosome defnton To valdate the performance of the developed nonlnear control law, smulatons results usng the Matlab/Smuln envronment are presented. The model of the UVR s based on a real constructon of a underwater robot called UKWIAL desgned and bult for the Polsh Navy. The URV s an open frame robot controllable n 4 DOF, beng.5 m long and havng a propulson system consstng of sx thrusters. Dsplacement n horzontal plane s done by means of four ones whch generate force up to ±75 N assurng speed up to ±. m/s and ±.6 m/s n x and y drecton, consequently. All parameters of the robot s dynamcs are presented n Appendx A. 6

5 EUSFLAT - LFA 5 Tracng control smulaton results for no added dsturbances and for a sea current dsturbance are shown n Fg. 6 and Fg. 8. The URV has to follow the desred trajectory begnnng from ( m, m), passng target wayponts: (5 m, m), ( m,5 m), ( m, 8 m), (3 m, 8 m), ( m,5 m) and comng bac to start under assumpton that the turnng pont s reached f the robot s nsde of the two-metre crcle of acceptance. In Fg. 7 and Fg. 9 courses of command sgnals are depcted respectvely wthout and wth nteracton of sea dsturbances. Fgure 8: The nfluence of sea current (speed.5 m/s and drecton 35 ) on trac-eepng performance: - desred trajectory, - real trajectory Fgure 6: Smulaton result of trac-eepng wthout envronmental dsturbances: - desred trajectory, - real trajectory Fgure 9: Tme hstores of command sgnals for trac-eepng n presence of the sea current (speed.5 m/s and drecton 35 ) 6 Concluson Fgure 7: Tme hstores of command sgnals for trac-eepng wthout envronmental dsturbances Usng of the fuzzy controllers for underwater robot s trac-eepng has been descrbed. From the results presented, t may be concluded that the proposed approach provdes the autoplot beng robust and havng good performance both wthout and n presence of the sea current dsturbances. Another advantage of the dscussed control system s ts flexblty wth regard to the change of dynamc propertes of the ROV and a performance ndex. Further wors are needed to dentfy the best fuzzy structure of the autoplot and test the robustness of ths approach n the real world. 6

6 EUSFLAT - LFA 5 References [] J. Craven, R. Sutton, R.S. Burns, Control Strateges for Unmanned Underwater Vehcles, J. Navgaton, vol. 5, pp. 79-5, 998. [] D. Dranov, H. Hellendoorn, M. Renfran, An Introducton to Fuzzy Control, Sprnger-Verlag, 993. [3] T.I. Fossen, Marne Control Systems, Marne Cybernetcs AS,. [4] J. Garus, Z. Ktows, Fuzzy Control of Underwater Vehcle s Moton, n Advances n Fuzzy Systems and Evolutonary Computaton, N. E. Mastoras, Ed. WSES Press,, pp. -3. [5] D. Goldberg, Genetc Algorthms n Search, Optmsaton, and Machne learnng, Adson- Wesley, 989. [6] J. Kacprzy, Multstage Fuzzy Control, John Wley and Sons, 997. [7] Z. Mchalewcz, Genetc Algorthms + Data Structures = Evoluton Programs, Sprnger- Verlag, 994. [8] R.R. Yager, D.P. Flev, Essental of Fuzzy Modellng and Control, John Wley and Sons, 994 [9] R.R. Yager, L.A. Zadeh, An Introducton to Fuzzy Logc Applcatons n Intellgent Systems, Kluwer Academc Publshers, 99. Appendx A The URV model The followng parameters of robot s dynamcs were used n the computer smulatons: M = dag { } ( ) = dag{ } D v 7.8 u dag C( v) = 6.w 8.υ g ( η) 6.w 8.5u 45.4υ 3. p 8.υ 8.5u 7.sn( θ ) 7.cos( θ )sn( φ ) 7.cos( θ )cos( φ ) = 79. cos( θ )sn( φ ) w 4. q 6.w 8.υ 5.9r 6.8q ( ) sn( θ ) + cos( θ )cos( φ) 6.w 8.5u 5.9r.3p r 8.υ 8.5u 6.8q.3p 63