Indoor Autonomous Control of a Two-Wheeled Inverted Pendulum Vehicle Using Ultra Wide Band Technology

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1 snsors Articl Indoor Autonomous Control of a Two-Whld Invrtd Pndulum Vhicl Using Ultra Wid Band Tchnology Dunzhu Xia *, Yanhong Yao and Limi Chng Ky Laboratory of Micro-inrtial Instrumnt and Advancd Navigation Tchnology, Ministry of Education, School of Instrumnt Scinc and Enginring, Southast Univrsity, Nanjing 96, China; 78@suducn (YY); 5675@suducn (LC) * Corrspondnc: xiadz_999@63com; Tl/Fax: Acadmic Editor: Stfano Mariani Rcivd: 6 March 7; Accptd: May 7; Publishd: 5 Jun 7 Abstract: In this papr, w aimd to achiv th indoor tracking control of a two-whld invrtd pndulum (TWIP) vhicl Th attitud data ar acquird from a low cost micro inrtial masurmnt unit (IMU), and th ultra-widband (UWB) tchnology is utilizd to obtain an accurat stimation of th TWIP s position W propos a dual-loop control mthod to raliz th simultanous balanc and trajctory tracking control for th TWIP vhicl A robust adaptiv scond-ordr sliding mod control (-RASMC) mthod basd on an improvd supr-twisting (STW) algorithm is invstigatd to obtain th control laws, followd by svral simulations to vrify its robustnss Th outr loop controllr is dsignd using th ida of backstpping Morovr, thr typical trajctoris, including a circl, a trifolium and a hxagon, hav bn dsignd to prov th adaptability of th control combinations Six diffrnt combinations of innr and outr loop control algorithms hav bn compard, and th charactristics of innr and outr loop algorithm combinations hav bn analyzd Simulation rsults dmonstrat its tracking prformanc and thus vrify th validity of th proposd control mthods Trajctory tracking xprimnts in a ral indoor nvironmnt hav bn prformd using our xprimntal vhicl to furthr validat th fasibility of th proposd algorithm in practic Kywords: two-whld vhicl; indoor; UWB; nonlinar; scond-ordr sliding mod control (-SMC); robust adaptiv Introduction Two-whld invrtd pndulum (TWIP) vhicls ar a popular tst platform to validat th prformanc of advancd control algorithms, such as th control capacity for nonlinar, instability and rapidity situations Th motion of a TWIP vhicl is govrnd by its undr-actuatd configuration, which maks it difficult to control by simply applying traditional robotics approachs [] In rcnt yars, control dsign for th stability and robustnss of TWIP vhicls has attractd grat rsarch intrst In [], a nural ntwork control mthod was proposd for an undr-actuatd whld invrtd pndulum modl with small ovrshoot and quick rspons to changs in th dsird trajctoris In [3], adaptiv backstpping control for a two-whld autonomous robot around an oprating point was invstigatd In [], an adaptation concpt was brought into nural ntwork control, which dmonstratd th yaw angl and pitch angl convrgnc tim was lss than s with an initial valu of 3 rad Th sliding mod control (SMC) mthod was first proposd by Sovit scholars in th 96s Th SMC mthod has som advantags such as robustnss to paramtr uncrtainty, insnsitivity to boundd disturbancs, fast dynamic rspons, a rmarkabl computational simplicity with rspct Snsors 7, 7, ; doi:339/s76 wwwmdpicom/journal/snsors

2 Snsors 7, 7, of 9 8 to othr robust control approachs, and asy implmntation of th controllr [5,6] Howvr, th to chattring othr robust causd control by th approachs, discontinuous and sliding asycontrollr implmntation mor or of lss th limits controllr its application [5,6] Howvr, in th practic chattring [7] causd by th discontinuous sliding controllr mor or lss limits its application in practic Th [7] high-ordr sliding mod control (HOSMC) mthod is th furthr improvmnt of th traditional Th high-ordr sliding mod sliding control, modyilding control lss (HOSMC) chattring mthod and is bttr th furthr convrgnc improvmnt accuracy of whil th traditional prsrving sliding its robustnss mod control, proprtis yilding [8,9] lss Consquntly, chattring and HOSMC bttrmthods convrgnc hav accuracy bn activly whil prsrving invstigatd its in robustnss rcnt yars proprtis for chattring [8,9] attnuation Consquntly, and robust HOSMC control mthods of unknown hav bn uncrtaintis activly invstigatd and prturbations, in rcnt rspctivly yars for chattring Lvant put attnuation forward th and-smc robustmthod control basd of unknown on a diffrntiation uncrtaintis and prturbations, output-fdback rspctivly control systm Lvantin put [], forward and thr -SMC ar many mthod diffrnt basd -SMC on a diffrntiation algorithms that and output-fdback hav bn prsntd control such systm as th in twisting [], and and thr supr-twisting ar many diffrnt ons -SMC put forward algorithms by Lvant, that hav th bn prsntd sub-optimal suchand as global th twisting ons dvlopd and supr-twisting by Bartolini ons and put so on forward by Lvant, th sub-optimal and global Th objctiv ons dvlopd of this papr by Bartolini is sustain and soth on slf-balanc and trajctory tracking of a TWIP vhicl Thin objctiv a GPS dnid of thisnvironmnt papr to sustain Nakano th slf-balanc t al hav proposd and trajctory an stimation/control tracking of a TWIP structur vhicl in a GPS two-whld dnid nvironmnt vhicl whr Nakano both a camra t al hav and proposd a targt objct an stimation/control ar attachd to th structur vhicl [] in a two-whld Th major drawback vhicl whr of both th aapproach camra and basd a targt on objct camra ar attachd is that to it th may vhicl produc [] Th a havy major drawback computational of thload approach and lad basd to a onlow camra ral-tim is that prformanc it may produc Accuracy a havyand computational prcision ar load th and two lad main toprformanc a low ral-tim paramtrs prformanc of th Accuracy indoor andlocalization prcision ar systm th twoportabl main prformanc UWB radio paramtrs ranging of tchnology th indoor was localization introducd systm in [,3] Portabl Th UWB local radiopositioning ranging tchnology systm can was b introducd usd as a standalon in [,3] Th systm UWB or local as a positioning complmntary systm systm can b whr usd asth a standalon GPS is unavailabl systm or as or adnid, complmntary can mt systm high whr ral-tim thrquirmnts GPS is unavailabl and vn or dnid, pntrat can through mt high objcts ral-tim (such as rquirmnts walls, obstacls, andbodis, vn pntrat tc) to a through crtain xtnt objcts (such as walls, obstacls, bodis, tc) to a crtain xtnt In this papr, a prototyp TWIP vhicl (shown in Figur ) was built in in our lab Th TWIP vhicl consists of two indpndnt driv whls on th sam axl, two rotary ncodrs, a car body and othr auxiliary dvics such as a snsor (MPU-65) which has an mbddd 3-axis MEMS gyroscop, a 3-axis MEMS acclromtr (MEMS (MEMS snsors snsors hav hav th th advantags advantags of miniaturization of miniaturization and high-prformanc and high-prformanc []) []) and aand portabl a portabl I-UWB I-UWB locating locating tag tag Y X Z O Figur Th xprimntal two-whld invrtd pndulum vhicl Figur Th xprimntal two-whld invrtd pndulum vhicl Th control systm configuration of our tstbd is illustratd in Figur Th attitud data of th TWIP Th control vhicl systm is masurd configuration via th onboard of our tstbd IMU, th is illustratd location information in Figur of Th th TWIP attitud vhicl data of is th masurd TWIP vhicl by an onboard is masurd UWB via rcivr th onboard and usd IMU, to th control location th information motion of th of vhicl th TWIP In vhicl Figur, is masurd A, A, A by and an A3 onboard ar four UWB UWB rcivr positioning and usd tchnology to control anchors, th motion T of is a th UWB vhicl rcivr In Figur usd, as A, an A, onboard A and tag A3 Th ar four TWIP UWB vhicl positioning can communicat tchnology with anchors, both T cllphons is a UWB rcivr and laptops usd as via an Blutooth onboard tag dvics Th TWIP vhicl can communicat with both cllphons and laptops via Blutooth dvics This This papr papr is is organizd organizd as as follows: follows: in Sction in Sction, th, nonlinar th nonlinar dynamic dynamic modl modl of th TWIP of th vhicl TWIP is vhicl stablishd is stablishd using a Lagrang using a nrgy Lagrang quation nrgy In quation Sction 3, In innr Sction and 3, outr innr loop and controllrs outr loop ar dsignd, controllrs and ar ncssary dsignd, stability and ncssary considrations stability ar considrations discussd Thn ar th discussd fasibility Thn and th ffctivnss fasibility of and th ffctivnss diffrnt control of th mthods diffrnt ar control compard mthods by MATLAB ar compard simulation by MATLAB in Sction simulation, followd in Sction by th, followd by th conclusions in Sction 5 Th configuration of an xprimntal st-up of th TWIP vhicl and th rsults ar givn in Sction 6

3 Snsors 7, 7, 3 of 8 Snsors 7, 7, 3 of 9 conclusions in Sction 5 Th configuration of an xprimntal st-up of th TWIP vhicl and th rsults Snsors 7, ar givn 7, in Sction 6 3 of 9 A Z Y A A Z Y A A A O O T T A3 A3 X Figur Systm architctur Figur Systm architctur Rlatd Work Figur Systm architctur Rlatd A scond-ordr Work adaptiv sliding mod control mthod basd on STW algorithm was proposd for Rlatd th control Work of an undr-actuatd systm in [5], which prformd bttr ffctivnss and A scond-ordr adaptiv sliding mod control mthod basd on STW algorithm was proposd rducd for th A scond-ordr chattring control of an adaptiv compard undr-actuatd sliding with convntional mod systm controlsmc in [5], mthod and which basd ordinary prformd on STW scond-ordr algorithmsmc bttr ffctivnss was proposd Howvr, and for th rducd thadaptiv control chattring of sliding an undr-actuatd mod control compard with systm mthod convntional in [5], is not SMC which vry and prformd ffctiv ordinary bttr for a controllr scond-ordr ffctivnss of SMC and multi-control Howvr, rducd chattring paramtrs th adaptiv compard In [6], sliding with th mod convntional structur of control mthod SMC th and innr/outr is not ordinary vry ffctiv scond-ordr loop control for a SMC systm controllr Howvr, has bn of multi-control th adaptiv usd in sliding simultanous paramtrs mod In control balancing [6], mthodand th structur is not trajctory vry of ffctiv tracking th innr/outr for a controllr for loop control oftwo-whld multi-control systm paramtrs invrtd pndulum has bn usd In [6], in th vhicls, simultanous structur and oftwo balancing thtrajctoris, innr/outrincluding and trajctory loop control a circl tracking systm and control has a trifolium bn for usd hav two-whld inbn simultanous dsignd for invrtd balancing tracking pndulum and For trajctory th purpos vhicls, and tracking of two trajctoris, control controlling for two-whld th TWIP including a invrtd vhicl, circl and pndulum an innr/outr a trifolium vhicls, loop hav bn anddsignd two control trajctoris, structur for tracking including and a For ascond-ordr th circl purpos and a trifolium STW algorithm of controlling hav bn hav th dsignd bn also TWIP vhicl, forusd tracking this an innr/outr For papr th purpos W proposd loop control of controlling a diffrnt structur thadaptiv and TWIPa vhicl, control scond-ordr anmthod innr/outr this STW algorithm looppapr control to hav structur kp convrgnc bn also and usd a scond-ordr of sliding in this papr STWmod W proposd algorithm surfac a havundr diffrnt bn also unknown adaptiv usd indisturbancs control this papr mthod W What s proposd mor, in this papr a diffrnt thr typical to kp adaptiv trajctoris, convrgnc control mthod including of sliding ina this circl, mod papr a trifolium surfac to kpundr convrgnc and a hxagon, unknown of sliding hav bn disturbancs moddsignd surfac What s undr to prov mor, unknown th adaptability thr typical disturbancs of our trajctoris, What s controllr including mor, a thr circl, typical a trifolium trajctoris, and a including hxagon, ahav circl, bn a trifolium dsignd and to prov a hxagon, th adaptability hav bn dsignd of our controllr to prov th adaptability of our controllr 3 Systm Dscription and Modling 3 Systm Dscription and Modling 3 Systm Th Dscription motion of th and TWIP Modling vhicl is dscribd undr an inrtial coordinat systm Th simplifid Th motion schmatic of th TWIP diagram vhicl and isth dscribd two-dimnsional undr an inrtial plan viw coordinat of th systm vhicl Th ar simplifid shown in Th motion of th TWIP vhicl is dscribd undr an inrtial coordinat systm Th schmatic Figur 3 diagram and th two-dimnsional plan viw of th vhicl ar shown in Figur 3 simplifid schmatic diagram and th two-dimnsional plan viw of th vhicl ar shown in Figur 3 X r y l l r x r x r y x x x l x l Figur 3 Th fixd coordinat systm of th vhicl: A simplifid schmatic diagram; Motion Figur 3 Th fixd coordinat systm of th vhicl: A simplifid schmatic diagram; Motion of of th vhicl on th x-y plan th Figur vhicl 3 Th on fixd th x-y coordinat plan systm of th vhicl: A simplifid schmatic diagram; Motion of th vhicl on th x-y plan

4 Snsors 7, 7, of 8 Th origin O of th fixd coordinat systm OXY is th cntr of th axl, th positiv dirction of OX-axis is th forward motion of TWIP, OY-axis is coincidnt with th axl, th positiv dirction of OZ-axis is vrtically upward Th paramtrs and variabls of th systm ar shown in Tabl During th motion procss of our two-whld vhicl, sinc thr ar only two points contacting with th ground, and th body has a motion similar to invrtd pndulum motion, w can mak a fw assumptions in ordr to build dynamic modl for th systm: () Thr xists sufficint friction btwn th whls and th ground, which maks th systm subjct to non-holonomic constraints, i, th assumption of no sliding and skidding holds throughout () Tak th TWIP vhicl as a prfctly rigid body, th distanc from th cntr of mass of th body to th axl is l; (3) Th influnc causd by th dynamic charactristics of DC motors is ignord; () Ignor th ffcts causd by friction and damping forc of barings and rvolut pairs to th motion stat of th systm Tabl Th paramtrs and variabls of two-whld invrtd pndulum vhicl systm Paramtr and Variabl Nam Symbol Valu (Unit) Acclration du to gravity g 98 (m/s ) Mass of th whl M W 3 (kg) Mass of th body m (kg) Radius of th whl r (m) Lngth btwn th whl axl and th cntr of gravity of th body l (m) Rotational inrtia of th whl J W 37 5 (kg m ) Rotational inrtia of th body about th Y-axis J θ 3 (kg m ) Rotational inrtia of th body about th Z-axis J δ (kg m ) Distanc btwn th two whls along th axl cntr D 7 (m) Tilt angl of th vhicl body θ (rad) Th angl whls turnd θ r, θ l (rad) Hading angl of th vhicl δ (rad) Output torqus of lft and right whl DC motors τ r, τ l (Nm) Longitudinal displacmnt along th movmnt dirction X V (m) Longitudinal vlocity of th vhicl v (m/s) Rotational vlocity of th vhicl ω (rad/s) In th papr, w adopt th Lagrang nrgy quation to build th kinmatic modl of th two-whld car In ordr to liminat th ffct causd by th non-holonomic constraint, w choos q = [θ l θ r θ] T as th gnralizd coordinats systm, thn th kinmatic modl of th two-whld car can b writtn in th following form: M(q) q + V ( q, q ) q + G(q) = B(q)u () whr, M(q) R 3 3 dnots th inrtia matrix, V ( q, q ) R 3 3 is th vctor of coriolis and cntrifugal forcs G(q) R 3 is th vctor of gravitational forcs, B(q) R 3 dnots th matrix of control cofficints, u R is th vctor of control inputs Dfining a nw vlocity vctor ξ = [ δ ẋ v] T = [ω v] T, thn th rlationship btwn a nw coordinat vctor ż = [ ξ θ] T = [ω v θ] T and th gnralizd coordinat vctor q = [θ l θ r θ] T can b writtn as follows: q = Hż = r D D r r r ż ()

5 Snsors 7, 7, 5 of 8 whr: Thn th dynamics of th systm can b formulatd as follows: M(z) = whr: α + ml sin θ Qr mlr cos θ mlr cos θ P G(z) = M(z)ż + V( z, ż) z + G(z) = B(z)u (3) mgl sin θ, B(z) =,V ( z, ż) = D r r D r r ml sin θ θ mlr sin θ θ ml sin θ δ, u = P = ml + J θ, Q = M w + J w r + m, ( α = D M w + J ) w r + J δ, β = PQ m l cos θ In ordr to simplify th controllr dsign, th stat-spac quations can b drivd from (3): ω = f (x) + g (x)τ ω () v = f (x) + g (x)τ v (5) θ = f 3 (x) + g 3 (x)τ v (6) f (x) = ml sin θ θ δ α + ml sin θ, g (x) = [ D r ( α + ml sin θ ), f (x) = β ( m gl sin θ m l 3 sin θ cos θ δ + Pml sin θ θ ), g (x) = τ l τ r ], P + mlr cos θ, βr f 3 (x) = β ( m l sin θ θ + Qml sin θ δ Qr + ml cos θ + Qmgl sin θ), g 3 (x) =, βr ( x = δ ω x v v θ θ) T is known as th systm stat u = (τω τ v ) T is th systm control input, and τ ω = τ r τ l, τ v = τ r + τ l Dsign of th Controllr An innr/outr loop controllr nds to b dsignd to mak th vhicl achiv trajctory tracking on th ground whil maintaining its own balanc Th structur of th dsignd innr/outr loop control systm is shown in Figur Th innr loop control consists of two control subsystms: th rotational vlocity control subsystm and th longitudinal vlocity and balanc control subsystm v d, ω v ar dfind as th dsird valu of th longitudinal vlocity v and th rotational vlocity ω rspctivly And th tracking rrors of v and ω ar introducd as v = v v d, ω = ω ω d A rfrnc trajctory is known in prior as x r (t), y r (t), δ r (t), and th control laws of v d and ω d ar givn by th outr loop controllr,

6 Snsors 7, 7, 6 of 8 Snsors 7, 7, 6 of 9 x r y r r x y d v d v v v x y Figur Th structur of th innr/outr loop control systm Figur Th structur of th innr/outr loop control systm Innr Loop Controllr Dsign Innr Loop Controllr Dsign Considring th dsird inclination angl, st of rror stat-spac quations can b Considring th dsird inclination angl θ d =, a st of rror stat-spac quations can b obtaind as follows: obtaind as follows: ω = f f(x) x+ g (x)τx ω ω d (7) (7) d v = f (x) + g (x)τ v v d (8) (8) x x v v d θ = f 3 (x) + g 3 (x)τ v f g (9) (9) x x 3 3 v Th Rotational Vlocity Control It Th is wll Rotational known Vlocity that th Control classical discontinuous sliding mod control provids finit-tim convrgnc for a systm of rlativ dgr on As th rotational vlocity control subsystm (7) It is wll known that th classical discontinuous sliding mod control provids finit-tim has a rlativ dgr qual to two, a sliding surfac is ndd whn using STW algorithm convrgnc for a systm of rlativ dgr on As th rotational vlocity control subsystm (7) has Th tracking rror of δ is introducd as a rlativ dgr qual to two, a sliding surfac δ = δ is ndd r δ And th quilibrium point of systm (7) can whn using STW algorithm b dfind as X Th tracking = [ rror δ ω ] T = [ ] T Dfining a traditional linar sliding surfac s of δ is introducd as δ = δ r - And th quilibrium point as: of systm (7) can T b dfind as X T Dfining s a traditional linar sliding surfac s as: = c δ + ω () s c () whr, c is a positiv constant Diffrntiating () with rspct to tim, it follows that: whr, c is a positiv constant Diffrntiating () with rspct to tim, it follows that: s = c ė δ + ė ω = c ω + f (x) + g (x)τ ω ω d = ϕ (x) + g (x)τ ω () s c c fxgx d xgx () whr, ϕ (x) = c ω + f (x) ω d whr, Th continuous xc fx control law d τ w is constitutd by two trms Th first is dfind by mans of its discontinuous Th tim control drivativ, law τ w whil is constitutd th othr by is atwo continuous trms Th function first is dfind of th availabl by mans sliding of its discontinuous surfac s tim drivativ, whil th othr is a continuous function of th availabl sliding surfac Assum s that thr xists a boundd tim-variabl disturbanc signal d (t) in th systm (7), which satisfis Assum ḋ (t) that L, thr L is xists a Lipschitz a boundd constant tim-variabl According to disturbanc th STW algorithm signal d[], t in th th control systm law(7), of which τ ω can b satisfis obtaind d as t follows: L, L is a Lipschitz constant According to th STW algorithm [], th control law of τ w can b obtaind τ as follows: ω = ( ϕ (x) k s sign(s ) + v )/g (x) v = k ( sign(s x ) + k s ḋ () signsv)/ gx v ksignsd () Thorm According to th convrgnt mthod proposd in [7], whn k >, k > k3 +(k 8)L, th stats of systm (7) can rach s Thorm According = to th ṡ = in finit tim, and convrg to origin convrgnt mthod proposd in [7], whn k >, k > (k ), th stats of systm (7) can rach s =s = in finit tim, and convrg to origin k (k 8) k (k )

7 Snsors 7, 7, 7 of 8 Th Longitudinal Vlocity and Balanc Control W dfin x vd, v d as th xpctd valus of longitudinal displacmnt x v and longitudinal vlocity v rspctivly and considr th tracking rrors of x vd, v d as xv = x v x vd, v = v v d Th quilibrium point of systms (8) and (9) can b dfind as X = [ xv v θ θ] T = [ ] T Th control law of τ v is dsignd to mak th vhicl accomplish th ultimat objctiv, that is, whn t, limθ = as wll as limv = v d Du to its ingnious dsign, simpl algorithm structur and good control ffct, hirarchical sliding mod control thory is widly usd in complx undr-actuatd systms control [8 ] Thrfor, w adopt th hirarchical sliding mod surfac to dsign th control law of τ v In ordr to dsign control stratgy, two traditional linar sliding mod surfacs with known positiv constants c and c 3 ar dfind as: { s = c xv + v s 3 = c 3 θ + θ (3) Diffrntiating s and s 3 with rspct to tim, w can obtain: { ṡ = c v + ė v s 3 = c 3 θ + θ () A control law basd on picwis linar hirarchical sliding mod control mthod nds to b changd occasionally, bringing in svr oscillation during th control procss In viw of th dcoupld sliding-mod control mthod introducd in [], a hirarchical sliding mod control mthod basd on modl analysis is proposd Th first layr sliding mod surfac s and s 3 is usd to construct th scond layr sliding mod surfac S Dfin th scond layr sliding mod surfac S as: Diffrntiating (5) with rspct to tim and thn: S = αs + s 3 with α > (5) S = αc v + c 3 θ + α f (x) + f 3 (x) v d + (αg (x) + g 3 (x))τ v = ϕ (x) + ψ(x)τ v (6) whr ϕ (x) = αc v + c 3 θ + α f (x) + f 3 (x) v d and ψ(x) = αg (x) + g 3 (x) ar Lipschitz continuous Assum that thr xists a boundd tim-variabl disturbanc signal d (t) in th systms (8) and (9), which satisfis ḋ (t) L, L is a Lipschitz constant Similar to th analysis of rotational angl control, th -SMC control law for th longitudinal vlocity and balanc control subsystms (8) and (9) can b dducd as follows: τ v = ( ϕ (x) k 3 S sign(s) + v )/ψ(x) v = k sign(s) + ḋ (7) whr, S is th sliding mod surfac of th control subsystm Thorm According to Thorm, whn k 3 >, k > k3 3 +(k 3 8)L, th stats of subsystms (8) and (9) k 3 (k 3 8) can rach S = Ṡ = and convrg to origin in finit tim with th control law (7) Proof Th procss to prov th subsystms (8) and (9) can rach S = Ṡ = is similar to Thorm s, so for simplicity, th proof is omittd hr Suppos that s > at a momnt bfor th systm

8 Snsors 7, 7, 8 of 8 rturn back to th quilibrium position A ngativ control input τ v is givn to push s to zro, that is, s < At this momnt, undr th sam ngativ control input, s 3 < and ṡ 3 > Sinc α is chosn as a positiv constant, whn S convrgs to, s and s 3 can always satisfy th condition that s >, s 3 <, and gradually convrg to and vic vrsa It can b concludd that subsystms (8) and (9) will convrg to xv = v = θ = θ = in xponntial form 3 Robust Adaptiv -SMC (-RASMC) Considring th rotational vlocity control subsystm (7), whn f (x), g (x) ar unknown functions, dfining a nw coordinat vctor z = s, th systm modl can b writtn as th following form: z = f (x) ω d + c ω + g (x)τ ω = ϕ(x) + g (x)τ ω (8) = ϕ(x) + (g (x) )τ ω + τ ω = ψ(x) + τ ω whr g (x) is assumd strict positivity without loss of gnrality, ϕ(x) = f (x) ω d + c ω and g (x) ar boundd unknown functions, and satisfying th following inqualitis with positiv constants C, Γ m >, Γ M > : ϕ(x) C (9) < Γ m < g (x) < Γ M () As th systm modl contains uncrtain part, th control input τ w can b dsignd as two parts: τ ω = τ ωnom + τ ωcom () whr, τ wnom is th nominal part of τ w, and τ wcom is th compnsation control part of τ w Th nominal systm modl (ψ(x) = ) can b obtaind as: z = τ wnom () According to Thorm, th control systm () is asymptotically stabl and rach z = ż = within finit tim with th following control law of τ wnom (3): { τ wnom = k z sign(z) + v 3 v 3 = k sign(z) k >, k >, (3) In viw of an adaptiv sliding mod control mthod introducd in [9], an adaptiv sliding mod control law of τ wcom is proposd, with which th systm prturbation can b compnsatd Firstly, an auxiliary variabl z aux R is introducd and satisfying ż aux = τ ωcom, thn dfin a sliding mod variabl σ(z) R as: σ(z) = z + z aux () Diffrntiating () with rspct to tim, it is asily obtaind that: σ(z) = ż + ż aux = ϕ(x) + g (x)τ ω τ ωnom = ϕ(x) + (g (x) )τ ω + τ ωnom = a(x, τ ω ) + τ ωcom (5) whr, a(x, τ ω ) = ϕ(x) + (g (x) )τ ω is th total uncrtainty of th systm, and satisfying a(x, τ ω ) < r, whr r is an unknown positiv constant Lt us considr ˆr as th stimation of tru valu r, and consquntly an stimatd rror can b govrnd by r = ˆr r In addition, considring th fact that a(x, τ ω ) is xhibiting slow tim-varying bhaviors, thn, it holds that r ˆr

9 Snsors 7, 7, 9 of 8 Th compnsation control law for th subsystm can b givn by: τ ωcom = ˆr sgn(σ) (6) with th adaptiv law as follows: ˆr = γ σ, γ > (7) Thorm 3 Considr th givn dynamic sliding mod () with th nominal controllr (3) and th compnsation controllr(6), th sliding mod variabl σ = can b guarantd in finit tim Proof Considr a positiv Lyapunov candidat as: V = σ + γ r (8) By diffrntiating V with rspct to tim and considring r ˆr, it holds that: V = σ σ + γ (ˆr r ) ˆr = σ(a(x, τ w ) ˆr sgn(σ)) + (ˆr r ) σ r σ + σ a(x, τ w ) σ (r a(x, τ w ) ) (9) It can b sn that V for t (, + ) By LaSall's invarianc principl [,3] and th Lyapunov stability critrion [], th thorm is provd Similar to th abov analysis, w can dduc th robust adaptiv -SMC control law for th longitudinal vlocity and balanc control subsystms (8) and (9) as follows: τ vnom = k 3 S sign(s) + v v = k sign(s) σ = S + z aux with ż aux = τ vnom τ vcom = ˆr sgn(σ ) with ˆr = γ σ whr, S is th sliding mod surfac (5) of th control subsystm τ v = τ vnom + τ vcom (3), k 3 >, k >, γ > (3) Thorm According to Thorms and 3, whn k 3 >, k >, γ >, th stat of systms (8) and (9) can rach S = Ṡ = with th control law (3), and convrg to origin in finit tim Th proof procss is similar with abov and is omittd hr Comparativ Simulations for th Robust Prformanc In this sction, a comparativ simulation has bn don to vrify th robustnss of th modifid -RASMC control mthod Sinc th -RASMC controllr is basd on th -SMC controllr, thy shar th sam control paramtrs, xcpt two additional paramtrs γ and γ W hav don a lot of simulations to find th appropriat control paramtrs of th dsignd controllr St ω d = 5 rad/s, v d = 5 m/s as th control targt for th -RASMC controllr Thn th initial stat variabls ar supposd as [δ() ω() x v () v() θ() θ()] T = [ ] T Figur 5 dpicts th control prformanc of our controllr for tracking th dscrid ω d and v d with six diffrnt sts of dsignd paramtrs as Tabl shows Figur 5 shows th control prformanc of cas~cas6 without systm uncrtainty It can b concludd that th controllrs hav th sam

10 Control Paramtr c c c 3 k k k 3 k Cas Cas /8 /3 Snsors 7, 7, Cas of 8 Cas /8 /3 Cas control prformanc whn th control paramtrs of th nominal controllrs ar th sam It can b Cas /8 /3 asily obtaind that cas and cas hav th bst control prformanc v rror(m/s) 6 cas5 3 - cas6 - cas3 - cas -6 cas - cas Tilt angl(rad) Tilt angl(rad) cas5 cas6 cas3 cas cas cas cas5 cas6 cas3 cas cas cas w trror(rad/s) - cas5 - cas6 cas3-3 cas - - cas cas Tv(Nm) Tw(Nm) cas cas cas3 cas cas5 cas x cas cas cas3 cas cas5 cas (c) (d) Figur Figur 5 5 Comparison Comparison rsults rsults of of th th controllr controllr with with diffrnt diffrnt control control paramtrs paramtrs v v tracking tracking rror; rror; ωω tracking tracking rror; rror; (c) (c) Tilt Tilt angl angl of of th th TWIP TWIP vhicl; vhicl; (d) (d) Th Th longitudinal longitudinal control control input input τ v and v and th rotational th rotational control control input input τ ω During th procss Tablof Th tuning dsignd th paramtrs control paramtrs, of th -RASMC w controllr hav found th ffcts of th controllr paramtrs on control prformanc Th incrass of c,, k, k, k 3 and k contribut to Control th fastr Paramtr convrgnc c c spd, c 3 and ffmak k ovrshoot k k 3 volum k incras fl fl Howvr, th incras of c contributs Cas to biggr 5 ovrshoot volum 5 and longr sttling 5 tim 3 of 3θ, and biggr Cas /8 /3 Cas Cas /8 /3 Cas Cas /8 /3

11 Snsors 7, 7, of 8 During th procss of tuning th control paramtrs, w hav found th ffcts of th controllr paramtrs on control prformanc Th incrass of c, α, k, k, k 3 and k contribut to th fastr convrgnc spd, and mak ovrshoot volum incras Howvr, th incras of c contributs to biggr Snsors ovrshoot 7, 7, volum and longr sttling tim of θ, and biggr ovrshoot volum and of shortr 9 sttling tim of v Th incras of c 3 maks ovrshoot volum of θ dcras, ovrshoot volum of v incras, ovrshoot and slows volum down and shortr th convrgnc sttling tim spd of v of Th θ incras and v of c 3 maks ovrshoot volum of θ Thn dcras, wovrshoot will considr volum th of v control incras, prformanc and slows down of cas th convrgnc and casspd withof systm θ and v uncrtainty W assumd Thn that w th will mass considr of th body control isprformanc considrd of ascas a variabl and cas valu with in systm th simulation: uncrtainty W assumd that th mass of th body is considrd as a variabl valu in th simulation: { s m =, t s m 5, > s 5, t s Figur Figur 6 shows 6 shows th th simulation rsults of cas and cas v rror(m/s) Tilt angl(rad) Tw(Nm) cas cas x -3 6 (c) 6 8 cas cas () cas cas w rror(rad/s) Tv(Nm) r of cas x -3 - cas -5 cas cas cas (d) (f) Figur 6 Cont

12 Snsors 7, 7, of 9 Snsors 7, 7, of 8 Snsors 35 7, 7, 7 of 9 r of cas r of cas 5 r cas cas (g) (h) (g) (h) Figur 6 Th control prformancs of cas and cas with systm uncrtainty v tracking rror; Figur 6 Th control prformancs of cas and cas with systm uncrtainty v tracking rror; Figur ω 6 tracking Th control rror; prformancs (c) Tilt angl of cas th TWIP and cas vhicl; with (d) systm Th longitudinal uncrtainty control v tracking input rror; v ; () ω tracking rror; (c) Tilt angl of th TWIP vhicl; (d) Th longitudinal control input v ; () Th ω rotational tracking rror; control (c) input Tilt angl Th rotational control input ; of (f) th r of TWIP cas; vhicl; (g) r of (d) cas; Th longitudinal (h) r of cas control and cas input τ ; (f) r of cas; (g) r of cas; (h) r of cas and cas v ; () Th rotational control input τ ω ; (f) r of cas; (g) r of cas; (h) r of cas and cas It can It can b b sn sn that that cas cas has has bttr bttr robustnss robustnss than than cas, cas, so so w w choos choos th sam th sam control control paramtrs paramtrs as cas cas Th Th dsignd paramtrs of of th th innr innr loop loop controllr controllr ar shown ar shown in Tabl in Tabl 3 3 It can b sn that cas has bttr robustnss than cas, so w choos th sam control paramtrs as cas Th dsignd paramtrs of th innr loop controllr ar shown in Tabl 3 Tabl 3 3 Th dsignd paramtrs of of th th innr innr loop loop controllr Control Tabl Paramtr 3 Th dsignd c paramtrs 3 of th k innr k loop k controllr 3 k c c 3 k k k 3 k -SMC -SMC Control Paramtr -RASMC c c c 3 ff 5 k k k 3 5 k/8 /3 fl fl -RASMC /8 /3 -SMC St ω d = -RASMC 5 rad/s, v d = 55m/s as th control targt 5 for our innr loop 5 controllr /8 /3 Thn th St ω initial stat d = 5 rad/s, v variabls ar d = 5 m/s as th control targt for supposd as [ () () () () () our T innr loop controllr Thn th xv v ()] [ ] T In ordr to initial stat variabls ar supposd as [ () () () () () T x vrify th robustnss of th two controllrs, th mass v v ()] [ ] T In ordr to of th body is considrd as a variabl valu St ω d = 5 rad/s, v d = 5 m/s as th control targt for our innr loop controllr Thn th vrify in th simulation: robustnss of th two controllrs, th mass of th body initial stat variabls ar supposd as [δ() ω() x v () v() θ() θ()] is T considrd as a variabl valu th simulation: = [ ] T In ordr to vrify th robustnss of th two controllrs,, t s mth mass of th body is considrd as a variabl valu in th simulation: { 3,, t t s s m, t s Figur 7 shows th simulation rsults m = 3, t s 3, t > s Figur 7 shows th simulation rsults Figur 7 shows th simulation rsults Tilt angl(rad) Tilt angl(rad) SMC -RASMC -SMC -RASMC r v rror(m/s) v rror(m/s) - cas cas SMC - -RASMC SMC -RASMC Figur 7 Cont

13 Snsors Snsors 7, 7, 7, 7, 3 3 of of 8 9 Snsors 7, 7, 3 of 9 w rror(rad/s) Torqu Tv(Nm) - - x - x SMC -5 -SMC -RASMC 3-6 -RASMC (c) (d) (c) (d) 3 3 -SMC x - -SMC x - -RASMC -RASMC SMC SMC 5 -RASMC -5 -RASMC () (f) () (f) Figur 7 Comparison rsults of th two controllrs with systm uncrtainty Tilt angl of th Figur Figur 7 7 Comparison Comparison rsults rsults of th of th two two controllrs controllrs with with systm systm uncrtainty uncrtainty Tilt angl Tilt of angl th TWIP of th TWIP vhicl; v tracking; (c) ω tracking ; (d) Th adaptiv valu of -RASMC; () Th vhicl; TWIP vhicl; v tracking; (c) tracking; ω tracking (c); (d) Th tracking adaptiv ; (d) valu Th ofadaptiv -RASMC; valu () Th of longitudinal -RASMC; control () Th longitudinal control input input longitudinal τ control input v ; (f) Th rotational control input v ; (f) Th rotational control input v ; (f) Th rotational control input τ ω Comparing Figur 7a,b, with Figur 7c,f, it can b concludd that th robust prformanc of Comparing Figur 7a,b, with Figur 7c,f, it can b concludd that th robust prformanc of th control Comparing law Figur of ω is 7a,b, much with bttr Figur than 7c,f, that it can of b th concludd control law thatof thv robust in th prformanc -SMC controllr of th th control law of is much bttr than that of th control law of in th -SMC controllr control Considring law offigur ω is much 7a,b,, bttr it can than b that asily ofsn th control that th lawstat of vvariabls in th -SMC of th controllr subsystms Considring (8) and (9) Considring Figur 7a,b,, it can b asily sn that th stat variabls of th subsystms (8) and (9) Figur in th 7a,b,, -SMC it controllr can b asily oscillats sn that nar th stat th quilibrium variabls of point th subsystms aftr t > (8) and s, whil (9) in th -SMC control in th -SMC controllr oscillats nar th quilibrium point aftr t s, whil th control controllr prformanc oscillats of th -RASMC nar th quilibrium controllr is point almost aftr unaffctd t > s, whil th control prformanc of th prformanc of th -RASMC controllr is almost unaffctd -RASMC By incrasing controllrth is almost valu of unaffctd m, th comparison of th robust prformanc of th two controllrs By incrasing th valu of m, th comparison of th robust prformanc of th two controllrs By incrasing th valu of m, th comparison { of th robust prformanc of th two controllrs,, t s will b mor obvious St th valu of m as m will b mor obvious St th valu of as will b mor obvious St th valu of m as m =, t, thn th simulation rsult is shown as rsult is shown as 5, 5, 5, t t > s s, thn th simulation rsult is shown as Figur Figur Th Th adaptiv valu Torqu Tw(Nm) 3 3 r r r r Tilt Tilt angl(rad) - - -SMC -SMC -RASMC - -RASMC v v rror(m/s) SMC - -SMC - -RASMC -RASMC Figur 8 Cont

14 Snsors 7, 7, of 9 8 w rror(rad/s) Torq Tv(Nm) - - -SMC -RASMC (c) - -SMC -RASMC () Th adaptiv valu Torq Tw(Nm) (d) -SMC -RASMC (f) r r Figur 8 Comparison rsults of th two controllrs with systm uncrtainty Tilt angl of th Figur 8 Comparison rsults of th two controllrs with systm uncrtainty Tilt angl of th TWIP TWIP vhicl; v tracking; (c) ω tracking ; (d) Th adaptiv valu of -RASMC; () Th vhicl; v tracking; (c) ω tracking ; (d) Th adaptiv valu of -RASMC; () Th longitudinal control longitudinal control input v ; (f) Th rotational control input input τ v ; (f) Th rotational control input τ ω From Figur 8, w can s th control prformanc of th -SMC controllr bcoms trribl aftr From t > Figur s, whil 8, w th can -RASMC s th controllr prformanc still maintains of thgood -SMC control controllr prformanc bcomsalthough trribl aftr th dgr t > s, of whil systm thuncrtainty -RASMC controllr is xacrbatd still maintains Combining good Figurs control 7 and prformanc 8, it can b although obsrvd th that dgr th -RASMC of systm uncrtainty controllr has is xacrbatd bttr robustnss Combining Figurs 7 and 8, it can b obsrvd that th -RASMC controllr has bttr robustnss Outr Loop Controllr Dsign Outr Loop Controllr Dsign Th motion stats of th TWIP vhicl on x-y plan ar dscribd by th cntr position of its Th motion stats of th TWIP vhicl on x-y plan ar dscribd by th cntr position of its whl axis (x, y) and th hading angl δ as shown in Figur 9 Dfining xy T, v T whl axis O (x, y) and th hading angl δ as shown in Figur 9 Dfining p = [x y δ] T, q = [v ω] T, th kinmatic quations of th vhicl can b obtaind as: x cos cos δ p y sin q (3) p = xẏ = sin δ q (3) δ Dfining r and δ r as th rfrnc trajctory in th arth fixd coordinat systm, thn th Dfining x r, y r and δ r as th rfrnc trajctory in th arth fixd coordinat systm, thn rfrnc th rfrnc longitudinal longitudinal v r and v rotational vlocitis ω r can b calculatd in prior rspctivly Th r and rotational vlocitis ω r can b calculatd in prior rspctivly position Th position and and orintation orintation rrors rrors ar dfind ar dfind as: as: x cos sin x r x x y cos δ sin δ sin cos x yr y r x (33) y = sin δ cos δ y r y (33) r δ δ r δ Diffrntiating (33) with rspct to tim, th tracking rror dynamics can b obtaind as:

15 Snsors 7, 7, 5 of 8 Snsors Diffrntiating 7, 7, (33) with rspct to tim, th tracking rror dynamics can b obtaind as: 5 of 9 x y ω v + v r cos δ y x = y x ω v + vr v cos r sin δ (3) δ y xω v rsin ω (3) r Backstpping control mthod is oftn usd to dal with undractuatd stabilization Backstpping has good Backstpping tracking prformanc, control mthod fast rspons is oftn timusd without to ovrshoot dal with andundractuatd is suitabl for onlin stabilization control Thrfor, Backstpping w dsign has good thtracking outr-loop prformanc, controllr using fast rspons th ida of tim backstpping without ovrshoot and is suitabl for onlin Tak xcontrol as a virtual Thrfor, control w variabl, dsign th outr-loop nw rror variabl controllr isusing dfind th as: ida of backstpping Tak x as a virtual control variabl, th nw rror variabl is dfind as: x = x λ tanh(ω)y (35) x tanh x y (35) Whn x and δ, according to Equation (35): Whn x and δ, according to Equation (35): y y = x x ω = λ ωtanh(ω)y tanh y,, λ > (36) (36) v r y r r x O r y y v x x r Figur 9 Th postur rrors of th two-whld vhicl Figur 9 Th postur rrors of th two-whld vhicl Lmma For all x R, and th absolut valu of x is boundd, thr is φ(x) = xtanh(x), if and only if Lmma x = th quality For allholds x R, and th absolut valu of x is boundd, thr is φ(x) = xtanh(x), if and only if x = th quality holds So, whn t,, y y xponntially convrgsto to zro, thusx x Adopting backstpping control mthod, th control law is dducd as follows: wd wr w d 3v= ry w cos r + λ 3 v r y sin cos δ + λ sin δ v d = v r cos δ + λ tanh(ω)ωx λ tanh(ω)(v ) r sin δ + λ y ) (37) vd vr cos tanh x tanh vr sin y +λ (37) x λ ( tanh (ω) ωy x tanh y whr, λ, λ, λ 3 and λ ar th dsignd positiv constants whr, λ, λ, λ3 and λ ar th dsignd positiv constants Thorm 5 Th control systm (3) is stabl, and th postur rrors ar asymptotically stabl with th backstpping Thorm 5 control Th control law (37) systm (3) is stabl, and th postur rrors ar asymptotically stabl with th backstpping control law (37) Proof First, dfin a Lyapunov candidat as: Proof First, dfin a Lyapunov candidat as: V = x + y + ( cos δ ) λ 3 cos V x y 3 (38) (38)

16 Snsors 7, 7, 6 of 8 Diffrntiating Equation (38) with rspct to tim: V = x ẋ + y ẏ + λ 3 sin δ δ = x [v r cos δ v λ ( tanh (ω)) ωy λ tanh(ω)(v r sin δ x ω)] λ ωtanh(ω)y + λ 3 sin δ [ω r ω + λ 3 y v r cos δ ] (39) Substituting th control law quations of ω and v (37) into Equation (39), thn: V = λ x λ ωtanh(ω)y λ ( ) sin δ () λ 3 By LaSall s invarianc principl [,3] and th Lyapunov stability critrion [], th stability of th outr-loop controllr is provd 5 Simulation and Discussion In this sction, a lot of comparativ simulations hav bn don to vrify th validity of th proposd control mthods W hav don a lot of simulations and found th appropriat control paramtrs of th dsignd outr loop controllr as shown in Tabl Tabl 5 xhibits svral diffrnt algorithmic approachs combind by divrs control mthods of innr and outr loop controllrs Th simulation framwork for S-B controllr is shown in Figur A of th Appndix A Tabl Th dsign paramtrs of th outr loop controllr Dsign Paramtrs λ λ λ 3 λ Valu 5 Tabl 5 Svral algorithmic approachs of th innr/outr loop control systm Innr Loop Controllr Outr Loop Controllr Dirct Mthod of Lyapunov Function (DMLF) [6] Backstpping PID controllr controllr SMC [9] S-D controllr S-B controllr -RASMC S-D controllr S-B controllr Th control paramtrs of th PID controllr usd for comparison in this papr ar sttld by th Ziglr-Nichols mthod and shown in Tabl 6 Tabl 6 Control paramtrs of th PID controllr Control Paramtr k pω k iω k dω k pv k iv k dv k pθ k iθ k dθ Valu First, a rfrnc circular trajctory with ω r = 5 rad/s, v r = 5 m/s is givn and th radius of th dsird trajctory r is m Furthrmor, w assum that th vhicl start point is [x() y() δ()] T = [3 3 π/3] T, and th initial stat variabls ar supposd as [δ() ω() x v () v() θ() θ()] T = [ ] T Figur shows th practical tracking for th circl mntiond abov with six diffrnt control combinations From Figur a, it is clarly shown that th S-B controllr has th shortst sttling tim of th tilt angl (about 55 s), whil th worst sttling tim of th tilt angl is about 76 s in th controllr Th tracking rror of x dcrass to zro aftr t = 95 s by th S-B controllr Whn it coms to th

17 Snsors 7, 7, 7 of 8 controllr, th tracking rror of x has th longst sttling tim of s (Figur d) Morovr, th ovrshoots of θ and x in th S-B controllr ar smallst among thm Figur d indicats that th DMLF mthod has bttr rspons about th tracking rror of δ, but wors prformancs of x and y than th backstpping mthod Considring ths simulation rsults, th S-B controllr is th most ffctivsnsors control 7, combination 7, stratgy 7 of 9 To furthr vrify th ffctivnss of th proposd controllrs, a mor sophisticatd trifolium prformancs of trajctory is usd as a rfrnc x and y than th backstpping mthod Considring ths simulation rsults, trajctory th S-B controllr is th most ffctiv control combination stratgy A trifolium To furthr trajctory vrify th ffctivnss can b of dsignd th proposd as controllrs, x r (t) a mor = sophisticatd ρ cos(3ω trifolium c t) cos(ω c t), y r (t) = trajctory ρ cos(3ωis c usd t) sin(ω as a rfrnc c t) Hr, trajctory th dsird lngth of ach laf ρ is chosn as ρ = 5, ω c = 5 A trifolium trajctory can b dsignd as x Thn, th rfrnc longitudinal vlocity v r (t) = r (t) = ρcos(3ω ẋr (t) c t)cos(ω + ẏr (t) c t), y and r (t) = ρcos(3ω th rfrnc c t)sin(ω c t) rotational Hr, th dsird lngth of ach laf ρ is chosn as ρ = 5, ω c = 5 Thn, th rfrnc longitudinal vlocity vlocity ω r (t) can b calculatd in prior Furthrmor, w assum that th vhicl start point is [x() y() δ()] T v r t xr t y r t = [53 3 6π] T and th rfrnc rotational vlocity ω r (t) can b calculatd in, and th initial stat variabls ar th sam as that of tracking prior Furthrmor, w assum that th vhicl start point is [ () () ()] a circl T T x y [ ], and th initial stat variabls ar th sam as that of tracking a circl Th systm stats ar divrgnt for tracking trifolium trajctory using th controllr and Th systm stats ar divrgnt for tracking a trifolium trajctory using th controllr and controllr controllr Th rason Th rason why why it divrgs it divrgs is th is th rror rror of th of th longitudinal longitudinal vlocity vlocity v convrgs mor v convrgs slowly than mor th slowly acclration than th of acclration th rfrnc of th longitudinal rfrnc longitudinal vlocity v r vlocity Morovr, th Morovr, simulation th rsults ar shown simulation in th Appndix rsults ar shown A in th Appndix A w tracking(rad/s) Tilt angl(rad) Tilt angl(rad) Tilt angl(rad) w tracking(rad/s) w rror(rad/s) S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B x rror(m) v tracking(m/s) v tracking(m/s) v rror(m/s) y rror(m) Dlta rror(rad) S-B S-B S-D S-D S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B (c) Figur Cont Figur Cont (d)

18 Snsors 7, 7, 8 of 8 Snsors 7, 7, 8 of 9 Torqu Tv(Nm) Torqu Tw(Nm) S-D S-D S-B S-B 6 8 S-D S-D S-B S-B y(m) S-D S-D S-B S-B x(m) () (f) Figur Comparison rsults of control combinations Tilt angl of th TWIP vhicl; v Figur Comparison rsults of control combinations Tilt angl of th TWIP vhicl; v tracking; tracking; (c) ω tracking ; (d) Tracking rrors of x, y and δ; () Longitudinal control input τ v and (c) ω tracking rotational ; (d) control Tracking input rrors τ w ; (f) Tracking of x, y and of th δ; circl () Longitudinal trajctory control input τ v and rotational control input τ w ; (f) Tracking of th circl trajctory Figur shows th comparison information of th four rmaining control combinations, S-D controllr, S-D controllr, S-B controllr and S-B controllr Figur shows th comparison information of th four rmaining control combinations, S-D It is clarly shown in Figur a that th S-D controllr has th shortst sttling tim of th tilt controllr, angl S-D (about controllr, 3 s), whil S-B th controllr longst sttling and S-B tim of controllr th tilt angl is about 37 s in th S-B controllr It In is Figur clarly, shown S-D and in S-D Figur hav a high that rotational th S-D controllr torqu has τ w th up to shortst around sttling Nm, whil timth of th tilt angl (about rotational 3control s), whil torqu th τlongst w in S-B sttling controllr tim is smallr of ththan tilt angl Nm is Th about tracking 37rrors s in th of x S-B and controllr y In Figur ar, much S-D smallr andin S-D th S-B hav controllr high rotational and S-B controllr controlthan torqu thos τof th S-D controllr and S-D w up to around Nm, whil th controllr (Figur d) Morovr, th ovrshoots of x and y in th S-B controllr ar smallst rotational control torqu τ w in S-B controllr is smallr than Nm Th tracking rrors of x and y among thm ar much smallr Finally, a inhxagon th S-Btrajctory, controllr which andcontains S-B controllr an abrupt than variation thos of of longitudinal th S-D controllr vlocity and and S-D controllr rotational (Figur vlocity, d) is Morovr, mor difficult th to track ovrshoots than th trifolium of x and on y Th in hxagon th S-Btrajctory controllr w adopt ar smallst amongis thm dscribd by picwis linar functions Furthrmor, assum that th vhicl start point is [ () () ()] T T Finally, x y a hxagon [37 trajctory, 3 3 /] which, and contains th initial an abrupt stat variation variabls of longitudinal ar st as vlocity and rotational [ () () vlocity, () () () T T xv v is mor ()] difficult [ to track ] than th trifolium on Th hxagon trajctory w adopt is dscribd by picwis linar functions Furthrmor, assum that th vhicl start point is [x() y() δ()] T = [37 3 3π/] T, and th initial stat variabls ar st as [δ() ω() x v () v() θ() θ()] T = [ ] T

19 Snsors 7, 7, 9 of 8 Snsors 7, 7, 9 of 9 Tilt angl(rad) Tilt angl(rad) Tilt angl(rad) w tracking(rad/s) w tracking(rad/s) w rror(rad/s) - S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B (c) v tracking(m/s) v tracking(m/s) v rror(m/s) x rror(m) y rror(m) Dlta rror(rad) S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B (d) Torqu Tv(Nm) S-D S-D S-B -S-B S-D S-D S-B S-B Torqu Tw(Nm) S-D S-D S-B S-B () y(m) x(m) (f) Figur Comparison rsults of control combinations Tilt angl of th TWIP vhicl; v Figur Comparison rsults of control combinations Tilt angl of th TWIP vhicl; v tracking; tracking; (c) ω tracking ; (d) Tracking rrors of x, y and δ; () Longitudinal control input τ v and (c) ω tracking ; (d) Tracking rrors of x, y and δ; () Longitudinal control input τ v and rotational control input rotational τ control input τ w ; (f) Tracking of th trifolium trajctory w ; (f) Tracking of th trifolium trajctory

20 Snsors Snsors 7, 7, 7, 7, of 9 of 8 Similarly, th systm stats ar divrgnt for tracking a hxagon using controllr and controllr Similarly, Morovr, th systm th stats simulation ar divrgnt rsults ar forshown tracking ath hxagon Appndix using A Figur controllr shows and th controllr practical Morovr, tracking prformancs th simulationfor rsults th ar hxagon shownmntiond in th Appndix abov A with Figur four diffrnt shows thcontrol practical tracking combinations prformancs for th hxagon mntiond abov with four diffrnt control combinations w tracking(rad/s) w rror(rad/s) Tilt angl(rad) Tilt angl(rad) Tilt angl(rad) Torqu Tv(Nm) Torqu Tw(Nm) S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B (c) - S-D S-D S-B S-B S-D S-D S-B S-B () v tracking(m/s) v rror(m/s) x rror(m) y rror(m) Dlta rror(rad) y(m) 5 5 S-D S-D S-B S-B S-D S-D S-B S-B 6 8 S-D S-D S-B S-B S-D S-D S-B S-B S-D S-D S-B S-B (d) S-D S-D S-B S-B x(m) (f) Figur Figur Comparison Comparison rsults rsults of of control control combinations combinations v v tracking; tracking; ω ω tracking tracking ; (c) ; (c) Tilt Tilt angl angl of th TWIP vhicl; (d) Tracking rrors of x, y and δ; () Longitudinal control input τ v and of th TWIP vhicl; (d) Tracking rrors of x, y and δ; () Longitudinal control input τ v and rotational rotational control input τ w ; (f) Tracking of th hxagon trajctory control input τ w ; (f) Tracking of th hxagon trajctory

21 Snsors 7, 7, of 8 Figur c shows that th S-B controllr has th shortst sttling tim of th tilt angl (about 3 s), whil th longst sttling tim of th tilt angl is about 33 s in th S-D controllr Th tracking rrors of x and y dcras most quickly to zro in th S-B controllr Whn it coms to th S-D controllr and S-D controllr, th sttling tim of x and y is up to about 7 s (Figur d) Morovr, th ovrshoots of x, y and δ in th S-B controllr ar th smallst Although th hxagon trajctory sms to b trackd ffctivly, th torqu prformanc is actually trribl du to th abrupt variation of th trajctory of longitudinal vlocity and rotational vlocity, and this situation is undsirabl in practic Tabl 7 shows th diffrnt control ffcts of th six control combinations, such as th ovrshoot and th sttling tim of θ, x, y, δ, and a coincidnc factor indicator (CFI) which illustrats th trajctory tracking prformanc Tabl 7 Th ky prformanc indicators of thr diffrnt control combinations Trajctory Circl Trifolium Hxagon Ky Prformanc Indicators DMLF Backstpping PID SMC -SMC PID SMC -SMC Ovrshoot σ (%) θ Sttling tim t s (s) Ovrshoot σ (%) x Sttling tim t s (s) Ovrshoot σ (%) y Sttling tim t s (s) Ovrshoot σ (%) δ Sttling tim t s (s) CFI (%) Ovrshoot σ (%) θ Sttling tim t s (s) Ovrshoot σ (%) x Sttling tim t s (s) Ovrshoot σ (%) y Sttling tim t s (s) Ovrshoot σ (%) δ Sttling tim t s (s) CFI (%) Ovrshoot σ (%) θ Sttling tim t s (s) Ovrshoot σ (%) x Sttling tim t s (s) Ovrshoot σ (%) 5 y Sttling tim t s (s) Ovrshoot σ (%) δ Sttling tim t s (s) CFI (%) W dfin a coincidnc factor indicator (CFI) to dscrib th tracking prformanc Th CFI is obtaind as follows: ( ) xi + xi yi = + yi min max min N ( xi + yi ) i= () CFI = % whr, ()* mans th normalization of xprssions in brackts, min is th minimum of xi + yi from i = to i = N, max is th maximum of xi + yi from i = to i = N, N = T/5 is th numbr of sampls, and T is th total tim of simulation N

22 Snsors 7, 7, of 9 Snsors 7, 7, of 9 whr, ()* mans th normalization of xprssions in brackts, min is th minimum of xi yi whr, ()* mans th normalization of xprssions in brackts, min is th minimum of xi yi from Snsors 7, 7, to N, max is th maximum of xi yi from to N, /5 is of th 8 from i = to i = N, max is th maximum of xi yi from i = to i = N, N T/5 is th numbr of sampls, and is th total tim of simulation numbr of sampls, and T is th total tim of simulation Diffrnt control ffcts ffcts of of th th six six control control combinations combinations ar ar clarly clarly shown shown in Tabl in Tabl 7, whr 7, whr w can w Diffrnt control ffcts of th six control combinations ar clarly shown in Tabl 7, whr w can s that s th that th and th and th controllr controllr hav poor hav adaptability poor adaptability to diffrnt to trajctoris diffrnt trajctoris Th convrgnc Th can s that th and th controllr hav poor adaptability to diffrnt trajctoris Th convrgnc rats of x, rats y and of δ ar x much y and fastr δ by ar using much th fastr backstpping by using th mthod backstpping than using mthod th DMLF than mthod using convrgnc rats of th It can DMLF b clarly mthod snit x, that can y and S-B b clarly δ ar much fastr by using th backstpping mthod than using mthod sn hasthat th S-B bst CFI mthod as wll has asth shortst bst CFI sttling as wll timas ofshortst th DMLF mthod It can b clarly sn that S-B mthod has th bst CFI as wll as shortst th tilt sttling angl fortim all thr of th trajctoris, tilt angl for so itall canthr b concludd trajctoris, that so among it can th b concludd six controlthat combinations, among th S-B six sttling tim of th tilt angl for all thr trajctoris, so it can b concludd that among th six controllr combinations, has th bst control S-B controllr prformanc has for th its bst fastcontrol convrgnc prformanc spd, small for its ovrshoot fast convrgnc combinations, S-B controllr has th bst control prformanc for its fast convrgnc and bst spd, trajctory small coincidnc ovrshoot and bst trajctory coincidnc spd, small ovrshoot and bst trajctory coincidnc 6 Exprimntal Rsults 6 Exprimntal Rsults Sinc th snsor (MPU-65) has an mbddd 3-axis MEMS gyroscop and 3-axis MEMS Sinc th snsor (MPU-65) has an mbddd 3-axis MEMS gyroscop and a 3-axis MEMS acclromtr, w nd to utiliz an information fusion algorithm to obtain rlativly accurat acclromtr, w nd to utiliz an information fusion algorithm to obtain rlativly accurat information Figur 3 shows th fram of th complmntary filtr for information fusion information Figur 3 shows th fram of th complmntary filtr for information fusion Figur 3 Th fram of multi-snsor information fusion algorithm Figur 3 Th fram of multi-snsor information fusion algorithm Th attitud data of th TWIP vhicl is masurd via th onboard IMU, th vlocity data is Th masurd attitud data by two rotary of th ncodrs, TWIP vhicl and is th location masurd via information th of onboard IMU, th TWIP th vhicl vlocity data is masurd is by masurd th UWB positioning by two rotary tchnology ncodrs, and and just th th as shown location in information Figur Th of of th th placmnt TWIP TWIP vhicl vhicl of anchors is masurd is masurd in spac by th by is th shown UWB UWB positioning positioning in Figur tchnology tchnology just asjust shown as shown in Figur in Figur Th placmnt Th placmnt of anchors of in anchors spac is in shown spac in is shown Figur in Figur dis dis dis dis dis dis dis3 dis3 Figur Th placmnt of anchors in spac Figur Th placmnt of anchors in spac Figur Th placmnt of anchors in spac Th position of th Tag can b xprssd as follows: Th position of th Tag can b xprssd as follows: Th position of th Tag can b xprssd as follows: X = dis dis +X X Y = dis dis +Y Y ()

23 Snsors 7, 7, 3 of 9 dis dis X X X Snsors 7, 7, 3 () of 8 dis dis Y Y Y To vrify th tracking prformanc of th proposd dsign, an indoor tracking tst was To vrify th tracking prformanc of th proposd dsign, an indoor tracking tst was implmntd Th dsird trajctory was st to b a circl with ω r = 5 rad/s, v r = 5 m/s implmntd Th dsird trajctory was st to b a circl with ω Th radius of th dsird trajctory r was m Considring th xistnc r = 5 rad/s, v of mismatch r = 5 m/s Th btwn th radius of th dsird trajctory r was m Considring th xistnc of mismatch btwn th ral-tim systm modl and th mathmatical modl, th fdback gains obtaind from simulations ral-tim systm modl and th mathmatical modl, th fdback gains obtaind from simulations may not function wll on th ral-tim platform, thus th control dsign paramtrs nd to b may not function wll on th ral-tim platform, thus th control dsign paramtrs nd to b adjustd through xprimnts on th prototyp In practic, w us th control dsign paramtrs as adjustd through xprimnts on th prototyp In practic, w us th control dsign paramtrs as = 5, c =, c 3 =, α =, k = 5, k =, k 3 =, k = 5, γ = /3, γ = /3, c λ5, = c 8,, λ c = 3 8,, λ, 3 = k, 5, λ k = 5, Figur k3, 5 k shows 5, th / indoor 3, xprimntal / 3, 8, rsults 8, using 3, ths-b 5 Figur controllr 5 shows for tracking th indoor th circl xprimntal rsults using th S-B controllr for tracking th circl (c) (d) () (f) Figur 5 Cont

24 Snsors 7, 7, of 8 Snsors 7, 7, of 9 (g) (h) (i) (j) (k) Figur 5 Exprimntal rsults of tracking for a circl Tilt angl of th TWIP vhicl; (b,c) v and Figur 5 Exprimntal rsults of tracking for a circl Tilt angl of th TWIP vhicl; (b,c) v and ω ω tracking; (d) ω tracking rror; () δ tracking; (f h) Tracking rrors of δ, x, y rspctivly; (i,j) tracking; (d) ω tracking rror; () δ tracking; (f h) Tracking rrors of δ, x, y rspctivly; (i,j) Histograms Histograms of tracking rrors of x, y rspctivly; (k) Practical tracking trajctory of tracking rrors of x, y rspctivly; (k) Practical tracking trajctory Aftr th battry is fully chargd, th TWIP vhicl can run continuously for about 3 min Th tst of tracking a circl lasts about 3 s, and th TWIP vhicl finishd th trajctory tracking for mor than two rounds Figur 5a shows th tilt angl of th TWIP vhicl Figur 5g,h shows th position tracking rrors Bcaus thr is a cm-lvl masur rror in th position masurmnt by UWB tchnology, th position tracking rrors cannot convrg to zro Th maximum position tracking rror is about ±7 m for x-dirction and about ±3 m for y-dirction, but mor than 8% of th all rcordd horizontal points rmain in a circl with a radius of m Th RMS valu of th position rror is 36 m in x-dirction, and 8 m in y-dirction Th sttling tim of th tilt angl is about

25 Snsors 7, 7, 5 of 8 35 s Td th tilt angl is controlld within ±5 This dmonstrats th good tracking prformanc achivd in th indoor nvironmnt 7 Conclusions In this papr, an innr/outr loop control is proposd for th trajctory control of a TWIP vhicl Svral control mthods hav bn utilizd in th innr or outr loop controllr and thn diffrnt control combinations of innr and outr loop control mthods hav bn proposd A lot of simulations hav bn conductd to compar th control prformanc of diffrnt control combinations Thn, th simulation rsults illustrat that th S-B combination has th most ffctiv control prformanc and highly adaptability to diffrnt trajctoris Finally xprimnts hav bn carrid out to vrify th ffctivity of th S-B combination In our futur work, th Kalman filtr and its variants will b considrd to obtain th stat stimation such that th ffcts of snsor noiss can b attnuatd Furthr invstigation will b conductd on path-planning arithmtic for nvironmnt xploration Acknowldgmnts: This work was supportd in part by National Natural Scinc Foundation (Nos 6577, 68), Ky Laboratory of Micro-Inrtial Instrumnt and Advancd Navigation Tchnology, Ministry of Education, China (Projct No KL), Major Projct Guidanc Foundation of Basic Scintific Rsarch Opration Expnss, Southast Univrsity (No 33), Natural Scinc Fund of Jiangsu Provinc (BK395, BK3636), Aronautical Scinc Foundation of China (No 6869) Author Contributions: Dun-Zhu Xia concivd th topic and dvlopd th modl; Yan-Hong Yao and Li-Mi Chng finalizd th simulation; Yan-Hong Yao analyzd th rsults; all authors wrot th papr Conflicts of Intrst: Th authors dclar no conflict of intrst Appndix A Figur Aa d illustrat that th systm stats ar divrgnt for tracking a trifolium trajctory using th controllr and controllr Compard with th circl trajctory, th trifolium trajctory is mor sophisticatd and th tracking vlocity v d changs rapidly at th bginning Figur A3a d illustrat that th systm stats ar divrgnt for tracking a hxagon using th controllr and controllr Comparing with th circl trajctory, th hxagon trajctory is mor sophisticatd and th tracking vlocity v d and w d changs rapidly at th bginning

26 Snsors 7, 7, 6 of 8 Snsors 7, 7, 6 of 9 v d d s du dt du dt s nom com xv v v d v v d d vnom vcom v x v v x u()cos u()sin ; y u()sin u()cos ; u(3); wd wr 3vrycos sin ; v v cos tanh x vr sin y y d r tanh x tanh ; sys() v d ; sys() d ; Figur A Simulation framwork for S-B controllr

27 Snsors 7, 7, 7 of 8 Snsors 7, 7, 7 of 9 Snsors 7, 7, 7 of 9 y(m) y(m) Torqu Tv(Nm) 3 3 idal trajctory idal trajctory x(m) x(m) (c) (c) (d) (d) Figur A Control pformanc of PID&Backstpping controllr and PID&DMLF controllr for Figur A Control pformanc of PID&Backstpping controllr and PID&DMLF controllr for Figur tracking A Control a trifolium, pformanc Tracking of PID&Backstpping trajctoris; Tilt controllr angl of th and TWIP PID&DMLF vhicl; (c,d) controllr Control torqus for tracking tracking trifolium, Tracking trajctoris; Tilt angl of th TWIP vhicl; (c,d) Control torqus a trifolium, τ Tracking trajctoris; Tilt angl of th TWIP vhicl; (c,d) Control torqus τ v and τ w v and τ v and w w Tilt Tilt angl(rad) Torqu Tw(Nm) x x y(m) y(m) Torqu Tv(Nm) idal trajctory idal trajctory x(m) x(m) Tilt Tilt angl(rad) Torqu Tw(Nm) x x (c) (c) (d) (d) Figur A3 Control pformanc of PID&Backstpping controllr for tracking a hxagon, Tracking Figur Figur A3 Control A3 Control pformanc of of PID&Backstpping controllr for for tracking a hxagon, Tracking Tracking trajctoris; Tilt angl of th TWIP vhicl; (c,d) Control torqus τ trajctoris; Tilt angl of th v and τ v w trajctoris; Tilt angl of th TWIP vhicl; (c,d) Control torqus τ v and τ w Rfrncs Isidori, A; Marconi, L; Srrani, A Robust Autonomous Guidanc: An Intrnal Modl Approach; Springr: London, UK, 3; ISBN Yang, C; Li, Z; Cui, R; Xu, B Nural ntwork-basd motion control of an undractuatd whld invrtd pndulum modl IEEE Trans Nural Ntw Larn Syst, 5, 6 [CrossRf] [PubMd]

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