Control of Program Motion of Dynamic Systems Using Relative Motions

Transcription

1 TECHNISCHE MECHANIK, Band 8, Heft 3-4, (8), - 8 Manuskrptengang: 5. Oktber 7 Cntrl f Prgram Mtn f Dynamc Systems Usng Relatve Mtns D Sanh, Dnh Van Phng, D Dang Kha, Tran Duc In ths paper, the prblem f cnstructng ne f the knd f prgram mtn s dscussed. The mtn prgram s a set f cndtns mpsed n the behavr f phase trajectres f the partcles f the system, by whch the pstns and velctes f the system under cnsderatn cannt be arbtrary. In ther wrds, the system f prgram mtn belngs t the systems wth cnstrants. Ths means that t s pssble t treat the prgram as a set f cnstrants that restrct the mtn f the system, n the sense f analytcal mechancs. In ths paper, these cases are dscussed. The passve cntrl system s desgned usng nly relatve mtn. Fr better results, an actve cntrl system can be used. Hwever an nstablty f system culd appear. In such case sme methds f a feedback cntrller system usng sensrs can be used t acheve a stable cntrlled system. Intrductn In the develpment f tday s technlgy, almst all f the engneerng prblems are related t prgram mtn cntrl. As t s cmmnly knwn, a dynamc system perfrms a prgram mtn as t s exerted by apprprate frces frm actuatrs whch are drven by cntrllers. If a dynamc system receves external energy t pwer cntrl actuatrs, t s called an actve cntrl system. Cnversely, a passve cntrl system vares ts energy by changng ts gemetrc prpertes such as the center f mass r the mment f nerta r ts dsspatn and elastcty parameters such as dampng and sprng ceffcents. Frm the energy pnt f vew, an actve cntrl system uses external frces t supplement addtnal energy t perfrm the prescrbed prgram mtn. A passve cntrl system uses nternal frces t vary ts subsystem mtn whch causes t adjust ts energy n rder t carry ut the desred prgram mtn. The advantage f the actve cntrl apprach s creatng fast dynamc respnse and a drect slutn f ths prblem (mtns caused by frces). Hwever, the supplementary energy may cause the system t becme unstable. Mrever, t make actuatrs delver the requred frces smetmes can cause techncally dffcult prblems (e.g. due t actuatrs saturatn) r can cause mpulse mtn (mpact). In cntrast, the advantage f the passve cntrl apprach s t make use f nteractve mtns amng cmpnents f the system t adjust energy smthly, and t elmnate mpulse effects thrugh dampers and sprngs. Hwever, ths apprach hardly creates the requred frces t meet the prgram mtn due t nertal effects and the delay f dynamc respnses. Recently, there appears the hybrd apprach, a sem-actve cntrl methd, whch tres t cmbne the benefts f the tw abve appraches. In the paper, the passve cntrl methd s used t frce the dynamc system t fllw the prgram mtn thrugh the relatve mtn cntrl f a subsystem. In ther wrds, the subsystem s used as a cntrller t frce the mther system t perfrm the desred prgram mtn. The mtn f the subsystem ndrectly changes the mtn f the mther system. Ths knd f prblem nherently embdes the synthess prperty. Let us cnsder a prgram f mtn as a relatn between the tme, crdnates, velctes, and acceleratns, whch mples requrements n the behavr f the slutns f the equatns f mtn f the cnsdered systems. A set f mathematcal expressns descrbng a prgram s called a manfld. Mathematcally, manflds are smlar t mechanc cnstrants. Hwever, mechancal cnstrants are created by physcal nteractns amng bjects, whereas mtn prgrams are smply magnary restrctns n dynamc mtn behavr f the system. Cmmnly, mechancal cnstrants are knwn as physcal cnstrants t dstngush frm mtn prgrams.

2 The Cnstructn f Prgram Mtn Thrugh Relatve Mtns Let s cnsder a dynamc system. Its pstn s lcated by the generalzed crdnates q (=, m ), subject t r deal cnstrants n the frms fq+f =, (-) where f, q and f are matrces, the szes f whch are rxm, mx, and rx, respectvely. The statnary hlnmc cnstrants wth redundant crdnates and statnary lnear nnhlnmc cnstrants are wrtten n the abve mentned frm, where the elements f matrx f are functns f q and the elements f matrx f are functns f q and q. In the case f unstatnary nnlnear nnhhnmc cnstrants f frst rder these elements depend n crdnates, velctes and tme. Fr the am f smplcty we wll nly cnsder the system wth the statnary cnstrants. The purpse f ths paper s t desgn a subsystem muntng n the rgnal dynamc system such that the mtn f the rgnal system can be cntrlled by the mtn f the subsystem t perfrm the prgram mtn n the frm gα (, t q ) = ; α =, s (-) The rgnal system and the subsystem are named as the mther system and the chld system, respectvely. Fr smplcty, the chld system s hlnmc and ts pstn s determned by hlnmc generalzed crdnates u ( α =, s), whch are ndependent t each ther and t the generalzed crdnates q ( =, m). Let s dente α the cntrl frces subject t the subsystem asu γ (γ =, p ), where p depends n the requrements f the cntrl prblem. As mentned abve, the mechancal cnstrans are assumed t be statnary, the knetc energy f the system s f the frms m s m s T = aqq j j + b uu αβ α β + cαqu α, (-3), j= αβ α and we can wrte t n the matrx frms as T T T T = qaq + ubu + qcu, (-4) where A and B are symmetrc quadratc matrces f dmensn f m and s respectvely, whse elements are functns f q and u nly, C s a m s matrx. By means f the Prncple f Cmpatblty, the equatns f mtn f the whle system (the mther-chld system) take the fllwng frms d T T = Qq + Rq ( =, m), (-5) dt q q d T T = Qu α + Ru α ( α =, s), (-6) dt u α uα where R q and R uα are generalzed frces crrespndng t the mechancal cnstrants. Q q and Q uα ( =, m; α =, s ) are generalzed frces f appled frces crrespndng t generalzed crdnates q and u α. ( =, m; α =, s). Cntrl frces U γ (γ =, p ) are ncluded n Q uα (α=, s ). Let s defne the ndependent generalzed crdnates f the mther system as qk,( k, n ; n m r) the cndtn f dealty f the cnstrants, ne frst expresses the generalzed acceleratns terms f the ndependent generalzed acceleratns (α=, s ) by the cnstrants (-). In such a way, we have u α n = =. T buld q (=, m ) n the q k (k =, n ) and the chld system s generalzed acceleratns q = d q +..., (-7) k k k= where the unwrtten terms d nt nclude generalzed acceleratns The matrx D T f sze (m+s) x (n+s) takes the frms quanttes the

3 d d... d, n+ s d d... d T, n+ s D = (-8) dm+ s, dm+ s,... dm+ sn, + s Ntce that sme elements n D T take unt values. As s knwn, the cndtn f dealty f the cnstrants (-) can be expressed n the frm DR = (-9) Obvusly, the generalzed frces crrespndng t the generalzed crdnates absent n (-7) are equal t zer. Accrdngly, ne easly ntces that R u =, (-) where the matrx R u f sze sx takes the frms [ R R ] R u =... s (-) The equatns (-5) and (-6) can be wrtten n the matrx frms as q q q Aq+Cu =Q +G +R (-) u u Cq +Bu=Q +G, (-3) where q u Q, and Q are the matrces f the generalzed frces f the appled frces crrespndng t the generalzed crdnates q and u. q u G, and G are the matrces bult frm the rest part f the equatns. Based n the cndtn f dealty f the cnstrants (-), the equatns (-) and (-3) turn nt the frms Aq+Cu=Q (-4) Cq +Bu=Uº, (-5) where, q A =DA Q D(Q+ G = u ),Uº (Q u +G )C, =DC. (-6) Ntce that the system f equatns (-4), (-) and (-) s a cmplete ne (m+s equatns wth m+s unknwns, q and u α, =, m; α=, s). The cntrl frces are ncluded n the terms f Uº. Nw slve the system f equatns (-), (-) and (-4) wth the fllwng ntal cndtns: we have T qt ( ) = q, qt ( ) = q, ut ( ) = u, ut ( ) = u, (-7) q = q (), t u = u (), t q = qt (), and u = ut () (-8) α α α The cntrl frces U (γ =, p) are determned by substtutng expressns n (-8) nt equatns (-5). The γ prblem s sad t be cmplete f the number f cntrl frces s equal t the number f mtn prgrams. If the number f cntrl frces s greater than that f mtn prgrams, addtnal cnstrants shuld be supplemented nt the mther system. 3 The Cntrller If we use nly relatve mtn u(t) as a cntrl nput, the cntrl system s passve. Fr better perfrmance, we can use actve cntrllers. In ths paper, tw appraches are appled t desgn the cntrller fr the system. The frst ne s an pen lp cntrller. As the llustratn n the next sectn, the cntrl frce F(t) exerts n the slder. By ths apprach, the relatnshp between nput and the resultant state s the equatns f mtn f the system (ncludng addtnal cnstrants). The desred state s btaned by slvng drectly the equatns f mtn (n DAE frm). In cntrast t the passve cntrller, the actve cntrller can cause the system t becme unstable. Hence a clsed lp cntrller s recmmended fr gettng stablty r btanng mre accurate and mre adaptve cntrl. In ths secnd case, bth the cntrl frce F(t) and the mment M(t) vary. One f the nnlnear desgn methds s appled as Feedback Lnearzatn (FBL) methd. The cntrl laws are assumed usng full state feedback (n bservers). The man dea f ths methd s tryng t transfrm a nnlnear system nt a lnear system by crdnate transfrmatn and applyng lnear cntrl desgn methds t the new lnear system (Jean- Jacques E. Sltne; Wepng L, 99). The lkewse pseud nverse methd wll be appled t fnd the cntrl frces U. If the mass matrx M s nvertble, we have: 3

4 q Mq ( ) (( fq,q ) BU) (3-) = + Lnearze the system f equatns (3-) as fllws: q = v, (3-) where v = q d K( q q d ) K( q q d ). Here K, and K are dagnal cnstant matrces that make the fllwng system f equatns stable: e+ Ke + K e = (3-3) where e s the errr vectr, e=q-q d, q are state varables and q d are desred state varables. The cntrl frces are calculated frm equatns (3-) and (3-) as fllws: where B + s the pseud nverse matrx f B. {, } + U=B Mv f(q q) (3-4) 4 An Illustratve Example Cnsder a sngle-wheel vehcle c mvng alng a straght hrzntal D rad under actng the cuple f the B mment M (t) exerted n the wheel b f mass m 3 and radus r rllng wthut slp as sketched n Fgure F 4.. The flrbard s subjected the cuple f mment M (t) n ppste ϕ c ϕ 3 M C drectn. The mass f mtr M attached t the flrbard s c b ϕ A neglected. The ceffcent f rllng F frctn s a. A wrker stands n the x O b M rear f the vehcle, hldng a handle fxed at a heght f L frm the flrbard. The wrker-handle system s mdeled as a massless rd Fgure 4.. The mdel f the sngle-wheel vehcle AB f length L rtated abut jnt A, a partcle B f mass m at the tp f the rd, and a system f a sprng and vscus dashpt wth sprng cnstant c and dashpt cnstant b, respectvely. The system AOC + OD + mtr has the mass m and the nertal mment J abut the cmmn centre f mass f the system at pnt O. The system, supprted by a bearng f dashpt cnstant b and restraned by a trsnal sprng f cnstant c, s cntrlled by a subsystem n rder t keep t n the hrzntal balance. Nte ne f the ends f the trsnal sprng s cnnected t the center O f the wheel and the secnd ne s cnnected t the flrbard AC. The subsystem cnssts f a slder M f mass m and a sprng f cnstant c and a dashpt f cnstant b. The pstn f M s cntrlled by a cntrller devce (nt shwn n the fgure) that drves the frce F(t) actng n the slder n algned drectn wth the flrbard AC and the frce beng equal n ppste t the F(t) n the OD. The generalzed crdnates are x, ϕ, ϕ, ϕ 3 and u whch respectvely are the hrzntal dsplacement f the vehcle, the angular dsplacement f the flrbard frm the hrzntal, the angular dsplacement f rd AB frm the hrzntal, the angular dsplacement f the wheel, and the dsplacement f the slder M wth respect t the flrbard (the relatve dsplacement). The slder M plays a part n the passve cntrl. The cntrl bjectves are t keep the flr AC balanced ( ϕ =, ϕ =,and ϕ = ), the wrker vbrates as lttle as pssble (ϕ π/), the varatn f vehcle s velcty s lttle and the dsplacement u s n the vald range. The fllwng ntatns are used n the paper frm nw n cs ϕ C, sn ϕ S, cs( ϕ + ϕ ) C, sn( ϕ + ϕ ) S j j j The knetc energy and ptental energy f the system are as fllws j T = mx + [J + m ( L + L LLC ) + mu ] ϕ + ml ϕ + J ϕ + mu + [ m ( LS LS ) msu ]x ϕ mls x ϕ + mcxu + m ( L LLC ) ϕϕ 3 3 (4-) 4

5 π Π= mg ( ) (-LS +L S ) + mgus + c( u u ) + cl ( ϕ ) + c ϕ ϕ, (4-) where m =m +m +m +m 3, J 3 =m 3 r / and the centre f mass f the bdy AODC cncdes wth the pnt O n assumptn. The mther system s subject t a hlnmc cnstrant as fllws x + rϕ 3 =, (4-3) r n the frm f (-) as x+ rϕ 3 =. The generalzed frces take the fllwng frms Q =, x Q = mglc mgl C mguc + M aϕ bϕ, 3 π Q = mgl C cl ( ϕ ) bϕ, Q = M + aϕ, 3 3 Q = Ft () mgs c ( u u ) bu. u (4-4) The ndependent generalzed crdnates are x, ϕ, ϕ, and u. The matrx D s n the frm r D = (4-5) In accrdance wth the equatns (-4) and (-5), the equatns f mtn f the system are wrtten as fllws where Mqq ( ) = fq,q ( ) + BU, (4-6) J3 m m( LS LS ) msu mls mc r m( LS LS ) msu J + m( L + L LLC ) + mu m( L LLC ) M(q) =, mls m(l LLC ) ml r mc m [ m( LC LC ) muc] ϕ + mlc ϕϕ + msu 3) ϕ + mlc ϕ + aϕ r mlls ϕϕ muu ϕ mlls ϕ aϕ 3 + mglc mgl C mguc f(q) = π, mlls ϕ mgl C cl ( ϕ ) bϕ - mgs - c( u-u) + mu ϕ bu r M() t B =, U(t) = Ft (). 5

6 5 Smulatn Results The sngle wheeled vehcle s parameters used fr smulatn are as fllws: m = 3 kg, m = 9 kg, m = 6 kg, m 3 = kg, L =.5 m, L =.5 m, J =.75 kgm, c = Nm, c =337 N/m, c =766 N/m, b = Ns/m, b =9 Ns/m, b =6 Ns/m, r=. m, and a=3 Ns/m. The vehcle s steady state speed s mantaned at.77 m/s; the flrbard pstn s kept hrzntally balanced; the rd AB s held vertcally, and the slder s kept arund m frm axs O. The smulatn tme s abut secnds. The ntal cndtns fr the clsed lp case are as fllws x() =, x () =., ϕ () = π /4, ϕ () =, ϕ () = 3 π /4, ϕ () =, ϕ () =, ϕ () =, 3 3 u() =.7, u () =. The ntal pstn f vehcle s nt balanced. Under the effect f the cntrller, the vehcle wuld be gradually drven t the balanced status after a shrt tme. 5. The Open Lp Apprach The smulatn results are shwn as fllws Fgure 5.. The tme hstry f x Fgure 5.. The tme hstry f ϕ Fgure 5.3. The tme hstry f ϕ Fgure 5.4. The tme hstry f ϕ 3 Fgure 5.5. The tme hstry f u Fgure 5.6. The tme hstry f F 6

7 5. The Clsed Lp Apprach The matrces K, K are chsen as fllws: K, = K = The smulatn results are shwn as fllws ph (rad).6.4 dph (rad/s) Fgure 5.7. The angle ϕ Fgure 5.8. The angular velcty ϕ ph (rad)..8 dph (rad/s) Fgure 5.9. The angle ϕ Fgure 5.. The angular velcty ϕ u (m).9 dx (m/s) Fgure 5.. The relatve dsplacement u Fgure 5.. The velcty ẋ 7

8 5 M (Nm) F (N) Fgure 5.3. The mment M (t) Fgure 5.4. The frce F(t) 6 Cnclusn In the develpment f tday s technlgy, mre and mre engneerng prblems are nvlved n cntrl f prgram mtn usng relatve mtn. In the paper, the sngle-wheel vehcle has been kept balanced by usng tw dfferent cntrl methds: ne usng nly the subsystem and the ther usng bth the subsystem and the mther system. Frm cntrl pnt f vew, the cntrl system s actve n bth cases and the cntrl system s passve n case t specfy the relatve mtn. In summary, fr unstable systems (lke the sngle-wheel vehcle), the hybrd methd that cntrls bth the mther system (external frce) and the subsystem (nternal frce) mght acheve the best effect (results). Acknwledgements The publcatn s cmpleted wth fnancal supprt frm the Natnal Basc Research Prgram n Natural Scences. References D Sanh: On the Prncple f Cmpatblty and Equatns f Mtn f Cnstraned Mechancal System, ZAMM 6, Klene Mttelungen, (98), -. Jean- Jacques; E. Sltne; Wepng L: Appled Nnlnear Cntrl, Prentce Hall (99). Stefan Stramgl: Mdelng and IPC Cntrl f Interactve Mechancal Systems: a crdnate free apprach, Sprnger, Lndn (). Rennut, A; Mcaell A; Merlht, X; Andrt, C; Gullaume, F; Chevassus, N; Chablat, D; Chedmal, P.: Passve cntrl archtecture fr vrtual humans. Intellgent Rbts and Systems, (5), Jer-Nan Juang; Mnh Q. Phan: Identfcatn and Cntrl f Mechancal Systems, CAMBRIDGE Unversty Press (). Addresses: Prf. Dr. D Sanh, Prf. Dr. Dnh Van Phng and Tran Duc, Han Unversty f Technlgy, N. - Da C Vet rad - Han Vetnam. D Dang Kha, Unversty f Texas, Austn, USA. emal: dsanhbka@gmal.cm; phng@mal.hut.edu.vn; khaddang.vn@gmal.cm; tranduc.vasys@gmal.cm 8