On-line Parameter Estimation in Sliding-mode Control of Pioneer 3-DX Wheeled Mobile Robot

Transcription

1 roceedngs of the 7th WSES Internatonal Conference on Systems Theory and Scentfc Computaton, thens, Greece, ugust -, 7 7 On-lne arameter Estmaton n Sldng-mode Control of oneer 3-DX Wheeled Moble Robot. FILIESCU, L. STNCU, S.FILIESCU, G.STMTESCU Department of utomaton and Industral Informatcs Unversty Dunarea de Jos of Galat Domneasca, 7, Galat, 88 Faculty of utomaton and Computers olytechnc Unversty of Bucharest Splaul Independente, 33, Bucharest, ROMNI dran.flpescu@ugal.ro bstract: - arameter dentfcaton scheme and dscrete-tme adaptve sldng-mode controller appled to oneer 3-DX wheeled moble robot (WMR are presented n ths paper. The dynamcal model for moble robot wth one par of actve wheels, tme varyng mass and moment of nerta have been used n sldngmode control. Two sldng-mode controllers correspondng to angular and poston moton have been desgned. Two closed-loop, on-lne parameter estmators have been used n order to acheve robustness aganst parameter uncertantes (robot mass and moment of nerta. Closed-loop crcular trajectory trackng oneer 3- DX control results are presented. Key-Words: - Dscrete-tme oneer 3-DX model, sldng-mode control, on-lne parameter estmaton. Introducton Dfferent approaches have been proposed n the lterature for output trackng of one par of actve wheels moble robots (WMR, [ and [3. The control problem of non-holoc systems when there are model uncertantes has been wdely addressed. Relatvely few results have been presented about the robustness of WMR control concernng model uncertantes and eternal dsturbances. The structural (parameter and/or unstructural uncertantes n the model of the MIMO non-lnear systems and the dffcultes n parameter dentfcaton make necessary the desgn of the controller such that the closed loop robustness s acheved. It s well known that the robustness to structural, un-structural uncertantes and eternal dsturbances of the WMR closed loop can be acheved wth a varable structure controller, [, [ and [. Mantanng the system on a sldng surface weakens the nfluence of the uncertantes n the closed loop and quckly leads to an equlbrum pont. The man advantage of the dscrete-tme sldng mode control s wth the drect and easy realtme mplementaton. Snce the sldng mode control s orgnal from contnuous tme, t s more dffcult to choose a synthess n dscrete-tme. The dscretetme sldng mode control, [, s qute dfferent of performng the control desgn n the contnuous-tme doman. Dscrete-tme sldng-mode controller desgn s usually based on an appromate sldngmode system evoluton due to the non unque attractveness condton and appromate evoluton on sldng surface, [, [. The robust trajectory trackng problem has been addressed n [3 usng a contnuous tme sldng-mode control. The performng control desgn, usng the knematcal model of the vehcle does not eplctly take nto account parameters varaton (robot mass and moment of nerta and eternal dsturbances (frctons and vscous forces, [. The controller desgn usng the WMR dynamcal model, where uncertantes n the robot physcal parameters can be eplctly taken nto account, tends to nterest actual researches on ths feld. In ths paper, the trajectory trackng problem for oneer 3-DX one par of actve wheels type WMR, n the presence of uncertantes (tme-varyng mass and moment of nerta, has been solved by dscrete-tme sldngmode controllers based on the dscrete-tme WMR dynamcal model. Two closed loop, on-lne parameter estmators have been used aganst parameter uncertantes. The paper s organzed as follows. In Secton the dynamcal model of one par of actve wheels oneer 3-DX moble robot s presented. lso, the dscrete-tme state space model, ts uncertantes, non-holoc constrant and the output trackng errors of oneer 3-DX are presented. Secton 3 descrbes on-lne parameter

2 roceedngs of the 7th WSES Internatonal Conference on Systems Theory and Scentfc Computaton, thens, Greece, ugust -, 7 73 estmators correspondng to angular and poston moton. The sldng adaptve controllers, assocated to angular and poston moton, are desgned n Secton and. oneer 3-DX sldng-mode closed loop smulaton results are presented n Secton and conclusons remarks n Secton 7. Contnuous and dscrete-tme oneer 3-DX dynamc model ssumpton: The WMR moton s supposed to be pure rollng, wthout of any slppng. Fgures and show oneer 3-DX wth oneer - DOF manpulator and the schema of a WMR, respectvely. X Y s a moble frame attached to the uncycle and XY defnes an nertal erence system. The vehcle dynamcs s fully descrbed by a three dmensonal vector of generalzed coordnates q ( t consttuted by the coordnates (( ( t, y( t of the mdpont between the two drvng wheels, and by the orentaton angle Φ ( t. The velocty constrant (non-holoc constrant of vehcle moton s & sn Φ y& cos Φ. Defne by τ r and τ l the torques provded by DC motors to the rght and left wheel, respectvely. The vehcle s descrbed by the followng dynamcal model as n τ r + τ l m && my& Φ & + cos Φ r τ r + τ l my && m& Φ & + sn Φ r IΦ && D r ( τ τ r l ( where mkg, I, Dcmm, r9.cmm are the robot mass, moment of nerta, dstance between wheels and wheels radus, respectvely. The real mass of the WMR s supposed to be tme-varyng wth bounded uncertanty wth known nal mass. Due to the tme-varyng mass, the moment of nerta becomes tme-dependng wth bounded uncertanty. ssumpton: Even f the moment of nerta s consdered tme-varyng, the robotc mass s supposed to be unformly dstrbuted all the tme. Let defne two parameters correspondng to the angular and poston moton, such as: ( t D ( I( t r, π ( t ( m( t r. The real values of the parameters are tme-varyng wth upper bounded uncertantes π real real ( t ( t ; ( t π π ( t ; ma π π ma ( Fg.. oneer 3-DX wth -DOF rm Fg.. WMR confguraton varables for angular and poston moton. Let R be the state vector, whose elements are, y, 3 Φ (3 & & Φ&, y, Defne the control nput correspondng to angular, u τ r τ l and poston moton, u τ r + τ l, respectvely. The state space representaton of WMR and the nonholoc constrant wll be dscretzed wth the samplng perod T, replacng the dervatve by a fnte dfference and usng a zero-order-hold for the control nputs k + k + T k ( ( ( ( k + ( k + T ( k T ( k + T + Tπ cos( 3 u ( k + + T + Tπ sn( 3 u ( k + + T u 3 ( sn( 3 cos( 3 ( k (

3 roceedngs of the 7th WSES Internatonal Conference on Systems Theory and Scentfc Computaton, thens, Greece, ugust -, 7 7 k beng the k th tme nterval where the correspondng varable s evaluated ( t kt. Let e R be, the vector of output errors: e where ;, L, tracked., s the trajectory to be 3 ngular and poston moton on-lne parameter estmaton The closed loop structure, shown n fgure, s proposed. For each robot moton, angular and poston, respectvely, an on-lne parameter estmator and a sldng controller have been ntroduced. Due to the tme-varyng of the oneer 3-DX mass, the control nput parameters ( t and π ( t are on-lne updated n order to be used n the correspondng sldng mode control nput. The robustness aganst mass uncertanty wll be assured. The mamum bounds of control nput parameters correspondng to angular and lnear moton wll be used n the attractveness condton of approprate sldng surface. s wll be shown n the net sectons, the attractveness condton of the correspondng sldng surface only on certan nterval s satsfed. Outsde of t, on-lne parameter estmates wll be used to compute the control nput. Moreover, n dscrete-tme, the sldng condton wth some appromaton s satsfed. When the system s nsde of the sldng sector or n the neghborhood of sldng surface, the parameter updatng law can provde convergent estmates. Let S and S be two sldng surfaces correspondng to the control nput for angular and poston moton, respectvely. s parameter updatng law, the recursve least squares method s used. The control nput for angular moton has two terms: the frst one, denoted compensaton part u comp (k, has to compensate the rotatonal dynamcs; the second one, denoted sldng mode u k, corresponds to system evoluton nsde part, ( of sldng surface neghborhood. The whole control nput for angular moton s comp u u u + ( The calculus and the steps for gettng both components of the angular moton control nput are gven n Secton. Epressng the estmated value for angular moton control nput parameter, ˆ ˆ, the net sequence, correspondng to recursve least squares method, [8 and [9, can be used to provde an estmaton of the uncertanty scalar term ( k u ( k L + [ u ( k ( k ( k L u ( k ( k ˆ ˆ ( k ˆ ( k u ( k + ( ( + L u k S k T at the k th step (7 (8 (9 Remark: Snce for each robot moton just one parameter s estmated, the gan L and the covarance are scalars. k, has The control nput for poston moton, u ( only sldng-mode part, u u correspondng parameter, ˆ π π ˆ π. For the, smlar updatng law s used π ( k u ( k L π ( + u k k [ ( π ( π ( k L π uπ ( k π ( k ˆ π ˆ π ( k T ˆ π ( k u ( k + L π ( k + π u ( k + S S ( k where L π, π prevously and wll be defned later. π ( S ( have the same meanng as Remark: For both parameter updatng laws, (9 and (, the epresson n brackets s vald when the system evolutes n the neghborhood of the correspondng sldng surface. ngular moton sldng-mode control nput synthess The followng stable sldng surface has been chosen, n order to desgn the control nput for angular moton S ( k ( k + µ ( k (3 where ( ( ( ( k δ e k k 3 k arctg ( ( ( k δe k wth: µ (, δ, δ. arameter µ T and the poston errors, e, e establsh the dynamcs of sldng surface. The nterval set of δ and δ assures the stablty of poston errors. If

4 roceedngs of the 7th WSES Internatonal Conference on Systems Theory and Scentfc Computaton, thens, Greece, ugust -, 7 7 the non-holoc constrant correspondng to the erence trajectory ( 3 arctg ( s taken nto account, then the angular error e 3 vansh when e, e tend to zero. 3 Remark: The sldng surface defned n (3 has been chosen such as whenever a sldng mode s acheved on t and e, e vansh, the orentaton angle Φ tends to ts erence value. For computng the control nput, the followng attractveness condton, as n [7 and [, has been used: S ( k ( k + < ( k + ( where ( k + S ( k + S (7 n appromate sldng-mode evoluton can be assured on the surface (3. If for the compensaton part of the control nput the epresson ( ( k + δ e k + comp arctg u ( T ( ( ( ( ( k + δe k k + T k µ k + (8 s chosen, then, after replacng (, (3 and ( n (7, one obtans + ( u ( k + T comp u S ( k Wth (9, ( becomes T [ u + T + T [ comp [ u u comp [ u k [ S < [ ( (9 ( Introducng the upper bound of the angular moton parameter uncertanty, the above second degree nequalty can be wrtten n the compact form ma ( u T + ma comp u [ S < If u ma comp > and S T u ( >, then the sldng-mode part of the control nput can be epressed as u ma comp ma < S T u ( ( u <, the nequalty ( s satsfed for When ma comp T u S u > (3 ma 3 Remark: Both epressons of the sldng-mode part, ( and (3, can be wrtten compactly S ma comp ρ u ( k T u ( ma ρ. where ( ma comp When S T u, the attractveness condton ( can not be satsfed. The sldng mode part of the control nput stll can be computed by usng estmates of parameter. The recursve least square method used to compute ˆ, gven by (7, (8 and (9, s convergent only when the system evolves n the neghborhood of sldng surface. Theore, an appromate sldng S k + T. mode condton s satsfed ( [ ˆ comp u + ˆ u ( Ths appromate s used n order to compute the control nput for angular moton comp u ˆ u ( ˆ ( Remark: Usng (, the updatng law (9 can be rewrtten as ˆ k ˆ k + L ( ( [ ˆ ( k u ( ( ( ( k comp + ˆ k u k S k T ( oston moton sldng-mode control nput synthess The followng sldng surface s proposed S ([ + [ [ e ( k + [ δ e ( k δ (7 Startng wth the thrd equaton of model (, usng a trgoetrc equalty and the non-holoc constrant (, the followng equalty holds tg ( T ( k + ( k + + ( k + ( k + (8 Moreover, ntroducng the epressons of the state varables, from state model (, and usng the constrant (, the above equalty becomes

5 roceedngs of the 7th WSES Internatonal Conference on Systems Theory and Scentfc Computaton, thens, Greece, ugust -, 7 7 tg ( + Tπ u ( T [ [ T ( [ + [ Let defne S k cos T ( [ ( ([ + [ [ ( ( k + e k + [ ( k + δ e (9 δ (3 The sldng moton on the surface (7 concerns the reduced order system of the robotc model, wthout of 3 rd and th equaton. The same attractveness condton, as n [, for computng the poston moton control nput has been consdered S ( k + < ( k + (3 ( k + S ( k + S (3 n appromate sldng mode evoluton on the surface (7 can be assured. Consequently of sldng-mode evoluton on (3, the angular state ( T δ e ( k 3 tends to hold the followng epressons 3 ( [ δ e ( k + [ δ e ( k cos ( T3 ( δe ( k [ δe ( k + [ δ e ( k sn (33 (3 Usng (8, the followng epresson can be obtaned [ ( k + + [ ( k + [ ( T cos ([ ( [ ( ( ( ( k + k T π π k u k (3 Wth (3 and (9, ( and (3 become S k + S k T cos T k π π k u k ( ( ( ( ( T cos ( k + S S ( T π π ( u ( ( (3 (37 Usng (3, (37 and upper bound of poston moton uncertanty, from (, the second degree nequalty can be wrtten ma ( π π cos ( T T u + S [ S < (38 If u > and S S >, then the sldng control nput for poston moton s S S u ρ (39 ma T[ cos( T ( π π S k S k, the where ρ (. When ( ( attractveness condton (3 can not be satsfed. The control nput stll can be computed usng onlne estmates for π. Remark: The recursve least square method used to compute ˆ π, gven by (, ( and (, s convergent only when the system evolves n the neghborhood of sldng surface. Theore, the appromate sldng mode condton s satsfed, S ( k +,.e. T[ cos( T ( π ˆ π u + S ( From above, the control nput can be epressed as u S T[ cos( T ( π ˆ π ( Remark: s result of (, ( can be rewrtten as ˆ π ( k ˆ π ( k T[ cos( T ( [ ( ( ( ( ( + k π ˆ π k L π k u k + S k S k ( When the system evolves n sldng-mode on the surface (7, can epress the followngs e ( k k k e k ( ( ( δ (3 δ ( Theore, output trackng error dynamcs assocated to the reduced order system can be epressed as: e k + e k δ Te k ( ( ( ( ( k + e Te ( k e δ ( For δ, δ stable., the above dynamcs errors are T oneer 3-DX sldng-mode closed loop control For testng the proposed dscrete-tme sldng-mode adaptve controller oneer 3-DX wth on board C and wreless adapter has been used n crcular trajectory trackng. The rugged 3-DX s cm 38cm cm alumnum body wth.cm da drve wheels. The two motors use 38.3: gear ratos and contan -tck encoders. Ths dfferental drve platform s hghly holoc and can rotate n place

6 roceedngs of the 7th WSES Internatonal Conference on Systems Theory and Scentfc Computaton, thens, Greece, ugust -, 7 77 movng both wheels, or t can swng around a statonery wheel n a crcle of 3cm radus. rear caster balances the robot. The followng parameters of model (3 were used: mkg, Dcm, I, kgm, T.3s. The moment of nerta has been computed assumng the mass unformly dstrbuted. lnear-tme varyng mass addtonally to the nal one has been consdered. More precsely, the robotc tme-varyng mass has been ncreased lnearly from kg to kg. The crcle trajectory trackng, shown n fgures 3, was obtaned for ma., π ma. 33. The followng values have been chosen for the constants: µ., ρ ρ. 99, δ δ 3. 33, ( π (. y(t 3 poston trajectory - 3 (t 8 erence robot angular trajectory erence robot 8 t [sec - control nput - 8 t [sec trackng error angular poston poston angular - 8 t [sec Fg. 3. WMR closed loop response for crcular erence and ntal condtons (33; (33; 3( π /7; (-.; (.; (.. Concluson Dscrete-tme, sldng-mode adaptve controllers and parameter estmators for trajectory trackng appled to control angular and poston moton of oneer 3- DX one par of actve wheels moble robot, have presented n ths paper. The tme-varyng mass and moment of nerta dynamcal state space model have been undertaken n order to desgn the controllers. Even f as parameter uncertantes, only the robotc mass and moment of nerta have been consdered, the proposed controllers assure closed loop robustness to a wde typology of parameter and model uncertantes and eternal dsturbances. Two sldng-mode adaptve controllers have been desgned, for angular and poston moton, respectvely. The robustness s guaranteed by sldng-mode controllers and by on-lne parameter estmators. Controllers parameters, on-lne updated, assure an appromate sldng-mode evoluton even f the attractveness condton s not satsfed and contrbute to an ncreased robustness. References: [ L. E. ghlar., T.Hamel, and. Soueres, Robust path-followng control for wheeled robots va sldng mode technques. roceedngs of the 997 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems, Vol. 3, 997, pp [ C. Canudas de Wt, and O. J.Sordalen,. Eponental stablsaton of moble robots wth nonholoc constrants. IEEE Transactons on utomatc Control, 37, 99, pp [3 C. Canudas de Wt, B. Sclano, and K.. Valavans, Trends n moble robot and vehcle control, n Control roblems n Robotca, Lecture Notes n Control and Informaton Scence, Sprnger- Verlag, vol.3, 998, pp.-7. [ R. Ferro, and F.L. Lews, Robust practcal pont stablzaton of a nonholoc robot usng neural network, Journal of ntellgent and Robotc Systems,. 997, pp [. Flpescu, U. Nunes, S. Stamatescu, Dscretetme, sldng-mode wmr control based on parameter dentfcaton. reprnts of the th IFC World Congress, July, rague, Czech Republc. [ K. Furuta, Sldng mode control of a dscrete system. System & Control Letters, vol., 99, pp.-. [7 T. Leo, and G. Orlando,. Dscrete-tme sldng mode control of a nonholoc moble robot. roceedngs of the Nonlnear Control Systems desgn Symposum, NOLCOS, 998, pp.7-8. [8 L. Ljung, System Identfcaton ( nd edton. rentce Upper Saddle Rver, 999, NJ. [9. Stoca, and. hgren, Eact ntalsaton of the recursve least squares algorthm. Internatonal Journal of daptve Control and Sgnal rocessng, vol.,,, pp [ V. I. Utkn, Sldng mode n control optmzaton, Berln, Sprnger Verlag, 99. [ J. M. Yang and J. H. Km, Sldng mode control for trajectory trackng of nonholoc wheeled moble robots, IEEE transacton on Robotcs and utomaton, vol., no. 3, 999, pp [ X. Yu, and J. X. Xu, Varable Structures Systems: Towards st Century, Sprnger_Verlag Berln Hedelberg,.