Modelling of the precise movement of a ship at slow speed to minimize the trajectory deviation risk

Transcription

1 Computatonal Methods and Expermental Measurements XIV 29 Modellng of the precse movement of a shp at slow speed to mnmze the trajectory devaton rsk J. Maleck Polsh Naval Academy, Poland Faculty of Mechancs and Electrcal Engneerng, Poland Abstract In ths paper, soft computng technques appled to desgnng an autoplot enablng the control of the precse movement of a hydrographc shp n a horzontal plane along the desred trajectory have been consdered. The reference trajectory s determned by partcular way-ponts, and command sgnals are generated by ndependent fuzzy controllers. The mnmzaton of the rsk of devatng from the trajectory has been consdered as a functon of control qualty n an envronment wth, and wthout, dsturbances. Wth the am of presentng the tests effectveness and robustness, selected results of the executed smulaton have been presented. Keywords: precse control, slow speed of shp, mnmsaton of devaton rsk, artfcal ntellgence. 1 Introducton Modellng of a safe shp s movement wth the ad of soft computng technques, and consderng all real condtons of operaton, s a complcated process due to problems wth the evaluaton of coeffcents of dynamcs, the descrpton of external forces and sea envronmental dsturbances. A hydrographc shp has been consdered as a control plant [6]. Contemporary shps are often equpped wth control systems allowng for complex manoeuvres and operaton executon. The process of a shp s automatc control s a dffcult task due to ts nonlnear dynamcs. Moreover, the dynamcs can change accordng to the modfcaton of the drve s confguraton, whch s suted to the executed task of shp [2]. do:1.249/cmem9271

2 296 Computatonal Methods and Expermental Measurements XIV Nowadays fuzzy systems n control of specal shp motons fnd wde practcal applcatons, rangng from soft regulatory control n consumer products to accurate control and modellng of complex nonlnear systems. In ths work use of the fuzzy autoplot n control of a shp movement has been proposed [4]. The paper has been organzed n the followng way: t starts wth a bref descrpton of the dynamcal and knematcal equatons of shp moton. Then the problem of moton along predefned trajectory has been dscussed. Next, fuzzy control law has been descrbed and the results of smulaton tests have been nserted. Fnally, selected conclusons have been presented. 2 Modellng of precse shp movement - equatons of moton eneral movement of a shp n 6 degrees of freedom (DOF) s descrbed wth the ad of the followng vectors [2, 3, 6]: T η = [ x, y,z, Φ, Θ, Ψ] T υ = u,v,w, p,q,r (1) where: η τ = [ ] [ X,Y,Z,K,M,N] T vector of poston and orentaton n the earth-fxed frame; x, y, z coordnates of poston; Φ, Θ, Ψ coordnates of orentaton; υ vector of lnear and angular veloctes n the body-fxed frame; u, υ, w lnear veloctes along longtudnal, transversal and vertcal axes; p, q, r angular veloctes about longtudnal, transversal and vertcal axes; τ vector of forces and moments actng on the shp n the body-fxed frame; X, Y, Z forces along longtudnal, transversal and vertcal axes; K, M, N moments about longtudnal, transversal and vertcal axes. Smplfed equatons of shp s precse moton are descrbed n followng form [6]: m u& 2 2 vr + wq x q + r + y pq r& + z pr + q& = X m [ ( ) ( ) ( )] [ v& 2 2 wp + ur y ( r + p ) + z ( qr p& ) + x ( qp + r& )] I z r& + ( I I ) pq ( q& rp ) I + ( q p ) I + ( rq p& ) y x + m = Y I zx [ x ( v& wp + ur ) y ( u& vr + wq )] = N yz xy (2)

3 Computatonal Methods and Expermental Measurements XIV 297 In general dynamcal and knematcal equatons of moton [1, 2] can be expressed n followng form [3,, 6]: where: η& J() η v ( v) M v& + C ( v ) v + D v + g ( η ) = M nerta matrx ncludng a rgd-body and an added mass nerta matrx; C(v) matrx of Corols and centrpetal terms ncludng a rgd-body and an added mass Corols and centrpetal matrxes; D(v) hydrodynamc dampng matrx; g ( η ) restorng forces and moments matrx; J ( η) velocty transformaton matrx transformng parameters between body-fxed and earth-fxed coordnate systems. 3 Precse control of a shp n a horzontal plane For the am of analyss and mathematcal descrpton of a shp moton n the horzontal plane, three coordnate systems (fg. 1) have been defned [3,, 6]: global coordnate system OXY, also called the earth-fxed coordnate system; local coordnate system O X Y, fxed to the body of a shp; reference coordnate system WP X Y ths system s not fxed. Precse control system of the shp has been desgned under the followng assumptons: the shp moves wth varyng lnear veloctes u, v and the angular velocty r; coordnates of poston x, y and the headng Ψ are measurable; the desred trajectory s a broken lne defned by set of way-ponts WP 1, WP 2, WP 3, etc. wth coordnates respectvely (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ), etc.; the command sgnal conssts of three components τ=[x, Y, N]. It has been assumed that X-axs n the reference coordnate system WP X Y covers wth straght lne passng through ponts ( x, y ) and ( x, + 1 y+1 ). The reference system can be obtaned from the global one as a result of the transformaton consstng of movement along the vector O WP and rotaton about Z-axs around the angle φ. A man goal of the precse steerng system of the shp moton n the horzontal plane s to mnmze mean square devatons y and Ψ measured n the system WP X Y, (fg. 1): y s a perpendcular dstance of the shp s centre of gravty to the predefned trajectory; Ψ s a local headng angle defned as the angle between the track reference lne and the shp s centrelne. = τ (3)

4 298 Computatonal Methods and Expermental Measurements XIV X ρ Φ +1 X x +1 ρ WP +1 X +1 Φ Y +1 x x WP y O Ψ v Y Ψ u X Y O y y +1 y Y Fgure 1: Three coordnate systems used n the descrpton of a shp s moton n the horzontal plane: O X Y earth-fxed system, OXY body-fxed system, and WP X Y reference system. The structure of the above control system descrbed s presented n fg. 2. The form of an assumed cost functon J cost descrbes the followng expresson: where: Jcost 2 2 ( y y ( t) + λ Ψ ( t) ) = mn λ (4) Ψ t () t = sn( )( x ( t) x ) + cos( φ )( y ( t) y ) y φ ()

5 Computatonal Methods and Expermental Measurements XIV 299 Route Planner x +1, y +1 Trajectory Regulator X-axs Y-axs Ψ-course X Y N Shp u v r Fgure 2: eneral structure of the control system of the shp. ( t) y ( t) x, nstantaneous coordnates of the shp s centre of gravty n the global system O X Y ; x, coordnates of the pont WP n the system O X Y ; φ y angle of rotaton of the reference coordnate system wth respect to the global one. Ths angle has been descrbed n followng form: () t = Ψ( t) () t Ψ φ, y + 1 y φ = arctan (6) x+ 1 x Ψ nstantaneous course of the vehcle n the system O X Y ; t tme; λ y, λ Ψ constant coeffcents. Each tme the shp locaton ( x ( t ), y( t )) [ x x () t ] + [ y y () t ] ρ here (see fg. 1): ρ +1 at the tme t satsfes: a crcle of acceptance, the next way-pont should be selected, the reference coordnate system WP X Y changed nto WP +1 X +1 Y +1 ; φ an angle of rotaton, calculated accordng to [6]. New shp s poston (x +1, y +1 ) s updated correspondng to new reference coordnate system. 4 Fuzzy control law For the am of desgnng of the fuzzy proportonal dervatve controller (FPD), controller adopted from [6, 7] has been used (fg. 3). (7)

6 3 Computatonal Methods and Expermental Measurements XIV η dk - e k e k Fuzzy Regulator τ k Shp η k Fgure 3: Fuzzy control structure. Membershp functons of fuzzy sets of nput varables are: error sgnal ek () t = ηdk ηk () t and derved change n error ek ( t) = η k ( t) η k ( t 1) and output one (command sgnal) τ k ( t). Here k s equal to the number of DOF. The notaton s taken as follows: N negatve, Z zero, P postve, S small, M medum and B bg. Table 1 rules presented from the Mac Vcar-Whelan s standard base of rules have been chosen as the control ones [1, 7]. The unknown parameters of the proposed fuzzy controllers have been determned wth the ad of fuzzy base of knowledge. Table 1: The fuzzy controller s base of rules. Error sgnal e k NB NM Z PM PB Derved N NB NM NS Z PS change n Z NM NS Z PS PM error e k B NS Z PS PM PB Command sgnal τ k Smulaton study Fgures 4 and contan selected results of smulaton of studes consdered n ths artcle. Smulatons of the automatc steerng system operaton are based on data executed on a real hydrographc shp. Her mathematcal model was used to smulate studes n ths artcle [, 6]. The smulated studes of the worked out automatc steerng system were executed n the Matlab envronment. Forces of thrust generated by mathematcal model of the man propulson (2 screw) have been presented n fg. 4. Dsturbances of sea envronment: wnd B, wave and sea current 1 m/s nfluence on the shp [6]. Fuzzy proportonal-dervatve controller (PD) was used as a regulator n automatc steerng system. The same forces of thrust generated by the man propulson and bow thruster wthout nfluence of dsturbances are llustrated n fg.. In ths case classcal proportonal-ntegral-dervatve regulator (PID) was used n automatc steerng system.

7 Computatonal Methods and Expermental Measurements XIV 31 2 Thrust force F n12 [kn] F n1 F n tme 1 Thrust force F ss [kn] 1 F ss tme [s] Fgure 4: Automatc steerng system of a shp wth a fuzzy proportonaldervatve regulator (PD). Thrust forces the man propulson F s1, F s2 (upper plot) and thrust forces the bow thruster F ss (lower plot).

8 32 Computatonal Methods and Expermental Measurements XIV Thrust force F n12 [kn] F n1 2 F n tme [s] Thrust force F ss [kn] tme [s] Fgure : Automatc steerng system of shp wth a classcal proportonalntegral-dervatve regulator (PID). Thrust forces the man propulson F s1, F s2 (upper plot) and thrust forces the bow thruster F ss (lower plot).

9 Computatonal Methods and Expermental Measurements XIV 33 6 Concluson The present state of knowledge assocated wth problems of desgnng the fuzzy automatc steerng system s formalzed n a lttle degree. Ths state decdes about the effectveness of workng of these types of systems. The effectveness of ther acton usually s based on experence and ntuton of desgner and experence and practcal knowledge of the helmsmen and navgators. In the large measure they defne the base of the fuzzy algorthm of steerng and they decde about the way the control law wll descrbe acton of the steered process [3, 6]. In ths work fuzzy regulators ncludng varables wth - 7 fuzzy sets were used. Too small a quantty of these sets does not allow the essence of the steerng object, however too large a quantty causes the consderable and nconvenent growth of the number of parameters descrbng the algorthms of steerng. Conducted research allows one to state that the use of fuzzy logc to the precse steerng of shp movement enables executon of hydrographc tasks wth large precson. In ths paper, the use of fuzzy controllers n precse control along a desred trajectory has been descrbed. Results presented of numercal researches prove that an appled approach n desgnng of autoplots gves postve effects provdng the sutable performance of the proposed control system. Another advantage of the control system dscussed s ts flexblty wth regard to the change of dynamc propertes of the shps and a performance ndex. Further works are needed to dentfy the best fuzzy structure of the autoplot and to test the robustness of ths approach n presence of envronmental dsturbances. Acknowledgement The present work was partally supported by State Commttee for Scentfc Research n Poland under rant no. N4 - O References [1] Drankov, D., Hellendoorn, H., & Renfrank, M., An ntroducton to fuzzy control, Sprnger-Verlag, [2] Fossen, T.I., Marne control systems, Marne Cybernetcs AS, New York, 22. [3] arus, J., Usng of softcomputng technques to modellng of moton of underwater robot under condtons of envronmental dsturbances, Polsh Journal of envronmental studes, vol. 16, no 4B, pp , 27 [4] Kacprzyk, J., Multstage fuzzy control, John Wley and Sons, New York, [] Maleck, J., Mathematcal model for control of shp at slow speed, Polsh Journal of envronmental studes, vol. 16, no 4B, pp , 27

10 34 Computatonal Methods and Expermental Measurements XIV [6] Maleck, J., Mathematcal model of shp as object of precse steerng, Scentfc Magazne Polsh Naval Academy, no 169/K, pp , dyna, 27 (n Polsh). [7] Yager, R.R., & Flev, D.P., Essental of fuzzy modellng and control, John Wley and Sons, 1994.