ROBUST STABILIZATION FOR UNCERTAIN NONHOLONOMIC DYNAMIC MOBILE ROBOTS WITH MONOCULAR CAMERA

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1 IJRRAS 8 () August pdf ROBUST STABILIZATION FOR UNCERTAIN NONHOLONOMIC DYNAMIC MOBILE ROBOTS WITH MONOCULAR CAMERA Yanan Du & Zhnying Liang * School of Scinc, Shandong Univrsity of Tchnology, Zio 55, China ABSTRACT In this papr, a nw kind of sooth ti-varying controllr was proposd for th roust stailiation prol of nonholonoic kinatics oil roots with uncrtain cara paratrs asd on visual fdack. And a torqu controllr was also dsignd y using th thod of coputd-torqu for th dynaic odl associatd with uncrtain disturancs. This torqu controllr can lt th vlocity rror convrg to ro and lt th actual vlocity track th dsird on in finit ti. Siulation rsults donstrat th ffctivnss of th proposd thods. Kywords: Moil root, Nonholonoic syst, Dynaic, Stailiation.. INTRODUCTION Th roust stailiation prol of nonholonoic whld oil roots (WMR) has rcivd a grat dal of attntion in th past svral dcads [,,3,4,5,6,7]. WMR is a typical nonholonoic chanical syst. It can't us sooth or ti invariant stat fdack to staili th nonholonoic chanical syst caus it dosn t t th Brocktt sooth stailiation conditions [3]. Many scholars took th chaind transfor [4], synovial control [5], discontinuous fdack [6] and ti-varying stat fdack [7] and so on to rali th control of oil root. In th control of nonholonoic WMR, it is usually assud that th root stats ar availal y using snsor asurnts. In practic, thr ar uncrtain disturancs of kinatics odl. In ordr to ovrco this prol, visual fdack is an iportant approach to iprov th control prforanc of oil roots. Visual fdack is usually lack of dpth inforation [8,9,]. Thrfor, th cara is fixd on th ciling. In [9], th fdack fro uncaliratd and fixd (ciling-ountd) caras wr usd. In [], th adaptiv tracking controllrs wr dvlopd to achivs asyptotic tracking. Control stratgy was proposd asd on th iag y controlling th stailiation syst for oil root in polar gotry pol in []. Th dynaic roust stailiation of oil roots with uncrtain cara paratrs was invstigatd in []. In [3], a nw thod asd on tivarying fdack was dvlopd to staili th tractor-trailr whld root around th origin. In [4], a nw sooth ti-varying fdack controllrs was proposd to xponntially staili th uncrtain chaind syst y using stat-scaling and control thoris for two cass. Th contriution of this papr is that a kinatics controllr is proposd to staili a class of dynaic odls of nonholonoic oil roots in th prsnc of uncrtain disturanc with unknown paratrs xponntially in a or gnral situation. And a torqu is proposd y using a coputd-torqu thod to ak th actual vlocity track th dsird on infinit ti. Th rst papr is organid as follows. Sction introducs th kinatic odl and dynaic odl of WMR. In sction 3, th controllrs ar dsignd for WMR and giv th proof rigorously. In sction 4, th siulation rsults donstrat th ffctivnss of th proposd stratgis. Finally, th ajor contriutions of th papr ar suarid in sction 5.. PROBLEM STATEMENT.. Cara visual odl In Figur, th oil root is shown. It is assud that a pinhol cara is fixd to th ciling, th typ (,) oil root is undr th cara. It is assud that th cara plan runs paralll to th oil root apac. Thr ar thr coordinat fras, naly th inrtial fra X-Y-Z, th cara fra x-y-, and th iag fra u-o -v. Assu that th plan of th cara fra is th idntical on with th plan of th iag coordinat plan. Its coordinat rlativ to X-Y plan is (c x,c y), th coordinat of th original point of th cara fra with rspct to th iag fra is dfind y (O c,o c). Suppos that (x,y ) is th coordinat of (x,y) rlativ to th iag fra. 84

2 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara Figur. Cara-root syst configuration Pinhol cara odl yilds [9] x y cos sin sin x c cos y c x y O O c c () whr and ar positiv constants which ar dpndnt on th dpth inforation, focal lngth. dnots th angl twn u axis and X axis with a positiv anticlockwis orintation... Kinatic odl For th typ of (,) oil root, assu that th gotric cntr point and th ass cntr point of th root ar th sa. Th kinatic odl can dducd y th following quations: x v cos y vsin () Considring () and (), th kinatic odl can dducd y th following quation in th iag fra: x v cos( -) y v sin ( -) (3).3. Dynaic odl According to th Eulr-Lagrangian forulation, th nonholonoic dynaic odl of th oil root can dscrid as [5] whr M( q V( q ) q G( A ( (4) nn n M( R is a sytric, positiv dfinit inrtial atrix, q R is gnralid coordinats, nn V( q ) R is th cntriptal and Coriolis atrix, G( R is th gravitational vctor, R is an r n input transforation atrix, R is th torqu applid to th right and lft whls, A( R is th atrix associatd with th constraints, R is a Lagrang ultiplir which xprsss th constraint forc. In ordr to siplify th analysis, th root is assud to do plan otion and r n. Hnc, G (. Th nonholonoic constraints of oil root can writtn as n nr 85

3 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara A( q (5) q is givn y q S( (6) nr whr S( R is coposd of a group asis in th solution spac of A (, [,, r ] is th corrsponding stat varials. Diffrntiating oth sids of (6) and coining (4), th dynaic syst can rwrittn as M( S( [ M( S ( V( q ) S( ] A ( (7) Pr-ultiplying oth sids of (7) y S (, on otains S ( M( S( S ( [ M( S ( V( q ) S( ] S ( (8) Pr-ultiplying oth sids of (8) y [ S ( ], on otains H( F( ) (9) whr H ( ( S B) S MS, F( ) ( S B) S [ MS VS]. For th typ (,) oil root, th atrics of quation (6) ar givn q [ x, y, ], [ v, ]. x, y and hav n givn th physical aning in th syst configuration. v is th straight lin vlocity and is th angular vlocity of th root. 3. CONTROLLER DESIGN In this sction, a nw kind of sooth ti-varying controllr and a torqu controllr will dsignd for nonholonoic uncrtain syst (3) undr th Assuptions and Las prsntd low. Th roust stailiation of syst (3) will discussd. Assuption. is known., ar unknown, thr xist four positiv known constraints,,, such,. that Assuption. [5] Th dynaic dscription of th oil root with uncrtain disturancs can dscrid as H ( F( ) () d whr d H( f is oundd unknown disturancs of root, f [ f, f], fi fli, i,, [, ] is th input torqu applid to th right and lft whls. La. [5] Suppos thr xists a continuous function V : D R such that th following conditions hold: a. V is positiv dfinit.. Thr xist ral nurs c and (,) and V cv () Thn thr xist a finit ti point T which satisfis T V (). And if t T, w hav V ( t). c( ) 86

4 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara La. [6] Considr a ti-varying linar syst dfind y x ( A t)) x () If RR A R is a Hurwit atrix and for vry lnt in B (t) which satisfis th following conditions: a. li t) t. B ( t) dt Thn, th syst () is xponntially stal. La 3. [7] If A R, th charactristic polynoial of atrix is dnotd as follows I A a a. A is Hurwit atrix if and only if a a., Basd on Assuption and rplacing y in syst (3), w hav x v cos y v sin (3) For (3), tak rvrsil transforation as follows: x cos( ) sin( ) x x sin( ) cos( ) y (4) Diffrntiating oth sids of (4), on otains x x x v cos v sin x x v sin cos v sin cos (5) For syst (5), tak stat-scaling transforation y x y x, (6) t ( whr t) (, ). Thn, diffrntiating oth sids of (6), on otains y y v cos v sin y v( )sin cos y y For syst (7), w dsign th dsird vlocity as follows [7], (7) v d k y k y, d k (8) 87

5 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara whr k. k, k and k ar indtrinat paratrs to dsignd. Dnot th vlocity rror signal as follows: v v, (9) c v d c W know that th cofficints of H (, F( ) d in dynaic quation () ar only dpndnt on angl and hav nothing to do with x and y []. Hnc, H (, F( ) ar known as long as is otaind fro cara. Th torqu controllr can dsignd as follows y using th thod of coputd-torqu: H( d F( ) H( u () whr [ vd, d ], u( t) [ u, u d ] ar corrsponding control input. Th following txt is to analy th stailiation of syst. Dfin th control input u (t) as follows: l u sgn ( v )( f v ), u sgn( )( f ) () whr, and sgn( ) () l Thor. For uncrtain syst (3), choosing th controllr () and k, k and k satisfy th conditions low undr Assuption and. k, h /( k ) k h ( k kh) k kh (3) Thn, vry stat in syst (3) convrgs to ro xponntially as t gos to infinity. And naly li x( t), y( t), ( t). t Proof. Considring () and (), on otains f u (4) d Sustituting (9) into (4) and coining (), w hav v sgn( v)( fl v sgn( )( fl f f (5) To prov Thor, w considr a non-ngativ scalar function V v : v v V (6) Th drivativ of V v is 88

6 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara V v { sgn( v )( f v ) f } v ( f v sgn( v ) f ) v (7) v l l It is clarly sn that V v Vv and v whn t T fro La. In th sa way, whn t T whr T and T ar dfind as V () () v V T, T (8) ( ) ( ) It is clarly sn that V v() and V () ar oundd. Hnc, v and whn t ax { T, T}. Considring (8) and (9), w hav v k y k y, k (9) Sustituting (9) into th first colun in (7), on otains k (3) Th solution of (3) is ( t) ( () k k ) t k t (3) whr () is th initial valu of.fro (9) and (3), it is sn that and convrg to ro xponntially as t gos to infinity. Fro (7), (9) and (3), w hav y k y k y k ( )sin y k( )sin y ( ) y y sin cos sin cos y [ hk ( )] y [ hk ( )] y [( h) k ( ) ( )] y [( h)( ) k ] y (3) whr k h and h is dfind y (3). Syst (3) can writtn as Y ( A t)) Y (33) [ whr Y y, y ] and k k A, t) hk( ) hk( ) k ( )sin, k ( )sin, sin cos ( h) k( ) sin cos ( h) k( ) Th charactristic polynoial of atrix A is I A a a whr 89

7 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara a [ k kh( )], a k k h W can s that a, a whn k, k and k satisfy (3). Thn, A is Hurwit atrix y th argunt in La 3. convrgs to ro xponntially as t gos to infinity y (3). Thrfor, sin approxiat to and cos approxiat to /. Thn, w hav / h. Hnc, sin cos ( ) h. k Considring / k / fro (9), w hav / and vry lnt ij ( i, j,) convrg to ro xponntially as t gos to infinity. Syst (7) is stailid xponntially whn t ax { T, T} y La. Fro ( t), w can s that y and y convrg to ro xponntially. To su up, choosing torqu controllr (), th syst (3) convrgs to dsird postur whn t ax { T, T}. Hnc, th syst (3) is xponntial stailiation. Thor is provd. 4. SIMULATION For th uncrtain chaind syst (3), choos initial stat [ x (), y (), (), v(), ()] [.3,-.3,,,]. Th 4 paratrs ar chosn as,,, 4, f, f,.5,.5, k, k, k 3 l l. Uncrtain disturancs ar rando ral nur twn [-,]. All th trajctoris of stats of oil root ar plottd in Figur -5. Figur. Trajctory of th root in th iag fra Figur 3. Stats of th root with rspct to ti Figur 4. Actual vlocity with rspct to ti Figur 5. Vlocity rrors with rspct to ti 5. CONCLUSION In this papr, w discuss th roust stailiation prol of nonholonoic dynaic oil roots with uncrtain cara paratrs in a or gnral situation. For th kinatic oil of th root with a onocular cara, a stat scaling transforation is applid to its original syst and a dsird vlocity is dsignd to staili th oil root xponntially undr is known and and ar unknown. Th thod of coputd-torqu controllr 9

8 IJRRAS 8 () August 6 Du & Liang Moil Roots with Monocular Cara which can lt th actual vlocity track th dsird on in finit ti. Siulation rsults donstrat th ffctivnss of th thods proposd in this papr. 6. REFERENCES [] G. Capion, G. Bastin, B. D Andréa-Novl. Structural proprtis and classification of kinatic and dynaic odls of whld oil roots[j]. IEEE Transactions on Rootics and Autoation, 996, (): [] B.L. Ma, S.K. Tso. Roust discontinuous xponntial rgulation of dynaic nonholonoic whld oil roots with paratr uncrtaintis[j]. Intrnational Journal of Roust Nonlinar Control, 8, 8(9): [3] R.W. Brocktt. Asyptotic staility and fdack stailisation[j]. Divrntial Gotry Control Thrapy, 983, 8: 8-9. [4] Z.P. Jiang, H. Nijijr. A rcursiv tchniqu for tracking control of nonholonoic systs in chaind for[j]. IEEE Transactions on Autoatic Control, 999, 44(): [5] J.M. Yang, J.H. Ki. Sliding od control for trajctory tracking of nonholonoic whld oil roots[j]. IEEE Transactions on Rootics & Autoation, 999, 5(3): [6] A. Astolfi. Discontinuous control of nonholonoic syst[j]. Systs & Control Lttrs, 996, 7(): [7] Y.P. Tian, S.H. Li. Exponntial stailiation of nonholonoic dynaic systs y sooth ti-varying control[j]. Autoatica,, 38(7): [8] H.S. Wang, Y.H. Liu, D.X. Zhou. Adaptiv visual srvoing using point and lin faturs with an uncaliratd y-in-hand cara[j]. IEEE Transactions on Rootics, 8, 4(4): [9] W.E. Dixon, D.M. Dawson, E. Zrgroglu. Adaptiv tracking control of a whld oil root via an uncaliratd cara syst[j]. IEEE Transactions on Systs Man & Cyrntics Part B,, 3(3): [] Y.C. Fang, W.E. Dixon, D.M. Dawson, P. Chawda. Hoography-asd visual srvo rgulation of oil roots[j]. IEEE Transactions on Systs Man & Cyrntics Part B, 5, 35(5): 4-5. [] G.L. Mariottini, G. Oriolo, D. Prattichio. Iag-Basd Visual Srvoing for Nonholonoic Moil Roots Using Epipolar Gotry[J]. IEEE Transactions on Rootics, 7, 3(): 87-. [] G. Wang, C.L. Wang, Q.H. Du, Y.F Ji. Dynaic roust rgulation of oil roots with unknown cara paratrs[j]. Journal of Univrsity of Shanghai for Scinc and Tchnology, 5, 37(4): [3] A.K. Khalaji, S.A.A. Moosavian. Stailiation of a tractor-trailr whld root[j]. Journal of Mchanical Scinc & Tchnology, 6, 3(): [4] Z.Y. Liang, C.L. Wang. Th xponntial stailiation of uncrtain chaind for systs of oil roots asd on visual srvoing[j]. Journal of Systs Scinc & Coplxity, 5, -. [5] S.P. Bhat, D.S. Brnstin. Finit-Ti Staility of Continuous Autonoous Systs[J]. Sia Journal on Control & Optiiation,, 38(3): [6] J. Slotin, W.P. Li. Applid nonlinar control, Englwood Cliffs[M]. Nw Jrsy: Prntic Hall, 99. [7] R.C. Dorf, R.H. Bishop. Modrn control syst[m]. Nw Jrsy: Prntic Hall,