Motion Planning and Posture Control of the General 3-Trailer System K. Raghuwaiya, B. Sharma, J. Vanualailai

Transcription

1 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 Moton Plannng an Poture Control of the eneral -Traler Sytem K. Raghuwaya B. Sharma J. Vanualala Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ Abtract Th paper preent a et of artfcal potental fel functon that mprove upon n general the moton plannng an poture control wth theoretcally guarantee pont an poture tablte convergence an collon avoance properte of the general-traler ytem n a pror known envronment. We bacally egn an nect two new concept; ghot wall an the tance optmzaton technque (DOT to trengthen pont an poture tablte n the ene of yapunov of our ynamcal moel. Th new combnaton of technque emerge a a convenent mechanm for obtanng feable orentaton at the target poton wth an overall reucton n the complety of the navgaton law. Smulaton are prove to emontrate the effectvene of the control law. Keywor Artfcal potental fel -traler ytem moton plannng poture. M I. INTRODUCTION OTION plannng for tractor-traler moble robot currently one of the mot actve area of moble robot technologe an ha attracte the attenton of many cholar n the fel of robotc an control engneerng. Followng the appearance of varou new metho an new technologe reearcher to ate have more n-ght nto the moton plannng problem. The nonholonomc moton plannng problem nvolve fnng a feable path from ome ntal confguraton to ome ere fnal confguraton for a ytem wth nonholonomc velocty contrant. Thee nonntegrable contrant are from the conton of non-lppage on the wheel n rollng contact wth another rg boy. Some eample of thee type of nonholonomc ytem nclue moble robot tractor-traler vehcle an moble manpulator. A we range of problem n varou robotc applcaton have been olve by utlzng the artfcal potental fel metho. It maor avantage nclue eaer analytc repreentaton of ytem ngularte an nequalte t mplcty an proceng pee. The unerlyng prncple of th metho to attach attractve fel to the target an repulve fel to the obtacle. The robot' workpace then flle wth potve an negatve fel n whch the robot attracte to t egnate target an K. Raghuwaya wth the School of Computng Informaton an Mathematcal Scence at the Unverty of the South Pacfc F Ilan (phone: ; fa: ; e-mal: raghuwaya_k B. Sharma wth the Faculty of Scence Technology an Engneerng at the Unverty of the South Pacfc F Ilan (phone: ; fa: ; e-mal: J. Vanualala wth the Reearch Offce at the Unverty of the South Pacfc F Ilan (phone: ; fa: ; e-mal: vanualala@up.ac.f. repule away from the obtacle. The poneer work on moton plannng an control of robot va the artfcal potental fel wa one by Khatb n []. Tractor-traler ytem conttute a generalzaton of the moble robot. They are bacally compoe of a moble robot an everal traler pulle by the moble robot whch atfy nonholonomc retrcton a well. Tractor traler moble robot have great applcaton n labor ntenve work or n area whch poe a hgh rk to human health. Thee artculate robot ytem are foun n a varety of place uch a arport factore ralway taton nuclear power plant mne n multple-traler truck a they ncreae tranportaton effcency. Reearcher are currently egnng varou control algorthm for moton plannng of thee mult-boy vehcle that are capable of performng a we range of tak n varou envronment. Many reearcher have tue tractor-traler ytem wth on ale htchng but only a few have focue on the off-ale htche traler ytem ue to t knematc tructure whch hghly complcate. A uch analy an controller egn become ffcult epecally wth rear wheel rven prme mover. In [] the author ue a front wheel rve tractor to erve the knematc moel for the general n-traler to olve the nonholonomc moton plannng problem. In [] Bolzernet al. propoe control law for the off-ale htche traler ytem bae on lnearzaton of a vrtual on-ale vehcle whch hare ome properte wth the actual one. In [4] ee et al. preente epermental ata for the egn an control of pave multple traler ytem both off an on-ale. Moton plannng an collon avoance cheme were conere by mnmzng the traectory trackng error wth the reference traectory mplyng the traectory of the towng vehcle. In [5]- [9] the author conere moton plannng an poture control an formaton type of the tanar an general - traler robot where pont to pont moton were controlle ung a yapunov bae control cheme. hot wall an the tance optmzaton technque (DOT were utlze to orchetrate "near perfect" fnal orentaton of every ol boy of the artculate robot ne a egnate parkng bay. Th paper make ue of yapunov technque a a tool for themoton plannng of tractor-traler robot. Specfcally theauthor eal wth the general -traler ytem. Themultboy robot navgate t way towar the target n acontrane workpace populate wth fe obtacle. Here thewall of the boune workpace an the tatc obtacle aretreate a ghot obtacle. To avo thee obtacle we utlze Khatb' collon avoance cheme to propoe potental fel toafely travere n the workpace towar the 6

2 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ target poton anattan the ere fnal poture. The paper organze a follow. In Secton II the vehcle moel efne. In Secton III IV an V moton plannng carre out. Thecontructon of tablzng control law preente n SectonVI whle Secton VII contan ome mulaton reult. The paperen wth ome conclung remark n Secton VIII. II. VEHICE MODE Two fferent traler ytem can be tnguhe from lterature; tanar an the general traler ytem groupe nto two fferent categore bae upon ther fferent hookng cheme. Bacally thee ytem cont of a tractor towng an arbtrary number of traler whch motly are pave n orer to reuce the cot ofmplementaton. The author wll coner a rear wheel rvencar-lke vehcle an an off-ale (general ytem htche two-wheele pave traler n Euclan plane. The tractor robot utlze heren bacallyperform moton mlar to that of a car-lke robot wth front-wheel teerng an ecree the path of the attache traler. In th reearch the general-traler ytem emboe a carlke tractor robot an three off-ale htche two-wheele pave traler. That the traler not attache eactly to the mpont of the rear ale of the tractor robot but here the couplng ont locate beyon the center pont of the rear ale.e. at a potve tance > (ee Fg. }. Eentally a kngpn on the two ol boe wth > an for... a potve length from the mpont of the rear ale of the two vehcle. Fg. Knematc moel of the general -traler robot Wth reference to Fg. ( y repreent the Cartean coornate an gve the reference pont of the th ol boy of the artculate robot whle θ gve t orentaton wth repect to the z a. Alo the tance between the two ale of the tractor robot an l the length of each ale. The connecton between any two boe gve re to the followng holonomc contrant on th ytem: + + θ θ θ co co co y y θ θ θ n n n + for.... We efne : ε + a where a a mall offet for the th vehcle (ee Fg.. Thee contrant wll reuce the menonof the confguraton pace nce the y can be epree completely n term of poton ( ( y θ for.... o Fg. Schematc agram of a general -traler ytem an the ghot wall If we let m be the ma of the full robot F the force along the a of the tractor robot Γ the torque about a vertcal a y an I the moment of nerta of the tractor robot then at o the ynamc moel of a general -tralerytem etene from [5] gven n (. The electon of the reference pont ue to the mplcty n the contructon of the potental fel functon. It mportant to note that when the knoynamcal moel for the off-ale htchng eactly the ame a that of the nale htchng prove we efne where the S S correponng length for the tanar traler ytem. A. Mnmzng C-Space To enure that the entre vehcle afely teer pa an obtaclethe planar vehcle can be repreente a a mpler fe-hapeobect uch a a crcle a polygon or a convehull []. Th repreentaton facltate wth thenherent vew of mnmzng the obtacle pace n the workpace. 7

3 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ ɺ vcoθ ω n θ yɺ vnθ + ω co θ ɺ v θ tan φ : ω ɺ θ n ( θ θ co ( θ θ + + n( θ θ co( θ θ + n ( θ θ n( θ θ n ( θ θ k k ( n( θ θ co( θ θ co( θ θ v + co( co θ θ ( θ θ + k + n ( θ θ + co ( θk θk + n ( θ θ co( θ θ n( θ θ n( θ θ n( θ θ co ( θ θ ω v σ : F ɺ m ɺ ω σ : Γ I In threearch gven the clearance parameter ε an ε the author encloe the artculate vehclewthn eparate protectve crcular regon (a een nfg..e. a protectve regon for each olboy whch bacally reuce the unneceary growth of thec-pace na [8] an ubequently preent a greateret of opton. Hence crcular regon C centere at ( y for... wth rau + V ( + ε + ( + ε r l ( ( + rv a + l + ε an If we let + for... then rv rv. Alo wth the choce of the reference pont an therau of the crcular regon of the vehcle we have for... III. ATTRACTIVE POTENTIA FIED FUNCTIONS Th ecton formulate collon free traectore of the robot ytem uner knoynamc contrant n a fe an boune workpace. It aume that the car-lke robot have pror knowlege of the whole workpace. We want to egn the acceleraton controller σ an σ o that the moble robot move afely towar ttarget.. A. Attracton to Target A target agne for the robot to reach after ome tme t. For the th boy of the tractor traler ytem we efne a target {( R } : T z z z p + z p rt wth center ( p p an rau rt. For the attracton to t egnate target we coner an attractve potental functon V ( ( p + ( y p + v + ω ( B. Aulary Functon To guarantee the convergence of the moble robot to t egnate target we egn an aulary functon efne a: ( + ( + ρ ( θ p y p p ( where p the ere fnal orentaton of the th boy of the artculate robot. Thee potental functonare then multple to the repulve potental functon to be egne n the followng ecton. IV. REPUSIVE POTENTIA FIED FUNCTIONS We ere the th boy of the moble robot to avo all tatonary obtacle nterectng ther path. For th we contruct the obtacle avoance functon that merely meaure the tance between each boy an the obtacle n the workpace. To obtan the ere avoance thee potental functon appear n the enomnator of the repulve potental fel functon. Th create a repulve fel aroun the obtacle. A. Fe Obtacle n the Workpace et u f w ol obtacle wthn the workpace an aume that the qth obtacle crcular wth center ( oq oq an rau ro q. For the thboy wth a crcular avoance regon of rau r V to avo the lth obtacle we aopt FO ( q ( oq + ( y oq ( roq + rv (4 for... an q... w. B. Workpace mtaton We ere to etup a framework for the workpace of our robot. Our workpace a fe cloe an bounerectangular regon efne for ome η > r for k wth r r a WS ( z z R : z η z η. V { } k 8

4 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ We requre the robot to tay wthn the rectangular regon at alltme t. Therefore we mpoe the followng bounaryconton: z z : z eft Bounary: Upper Bounary: Rght Bounary: ( : ower Bounary: z z : z η z z z η z z : z. In our yapunov-bae control cheme thee bounare areconere a fe obtacle. For the th boy of each robot to avo thee we efne the followng potental functon for the left upper rght an lower bounare repectvely: W rv (5 W η ( y + rv (6 W η ( + rv (7 W y r (8 4 V for... Now nce ηk > rv for k each of the functon potve n WS.Embeng thee functon nto the control law wll contan themoton of the tractor-traler robot wthn the pecfebounare of the workpace an wll prevent t from crong overthe bounare. C. Orentaton One ffculty that et wth contnuou tmenvarantcontroller that although the fnal poton reachable t vrtually mpoble to get eact orentaton at the equlbrum pont of th pecal cla of ynamcal ytem arect reult of Brockett' Theorem [9]. In th paper we contruct ghot wall along the e of thetarget parallel to the ere fnal orentaton of the robot ana thr ghot wall erecte n-front of the target. Th technquereuce the poble entry route to a ngle openng a the otherentry route are blocke by the ghot wall.net we utlze an ea npre by the work carre out bykhatb n [] for the avoance of thee ghot walln orer to force the ere orentaton. The technque we uecalculate the mnmum tance from the robot to a ghot wall anavo the reultant pont on that ghot wall. Avong thecloet pont on any lne bacally affrm that the moble robotavo the whole wall. Th algorthm help greatly mplfy thenavgaton law. z z -plane Now let u coner the kth ghot wall n the ( from the pont ( ak bk to the pont ( k k the pont ( a b. We aume that y cloet to t at the tangent lne whch pae through the pont.from geometry t known that f y thepont of nterecton of th tangent then k k λ a + λ a a y b + b b k k k k k k k k k k where k ( ak k ( y bk rk ( ak ak bk bk k rk ( a a + ( b b ( a a + ( b b λ + an k k k k k k k k. λ then we let λ f λ then we let λ k k otherwe we accept the value of λ k between an n whch cae there a perpencular lne to the pont y of th ( y on the ghot wall from the center k k boy of the artculate vehcle at every tme t.for the th boy of the robot to avo the cloet pont of eachof the kth lne egment we coner a potve potental felfuncton: S ( ( y y r k k + k V for... an k... m. V. DYNAMIC CONSTRAINTS Practcally the teerng an benng angle of themoble robot are lmte ue to mechancal ngularte whle the tranlatonal pee retrcte ue to afety reaon. Subequently we have; ( v vma where vma the mamal π pee of the tractor; φ φma < k k If (9 where φ the ma π mamal teerng angle an ( θ θ θma < where θma the mamum benng angle of the traler wth repect to the orentaton of the tractor. The traler can freely rotate wthn π π about ther lnkng pont wth the tractor. Conerng thee contrant a artfcal obtacle we have the followng potental fel functon: ( ( ( + ( U vma v vma v v ma v ma U ( ω ω + ρ mn ρ ( mn ( ma ( ( ma + ( DC θ θ θ θ θ θ ( Thee potental functon guarantee the aherence to the above retrcton place upon the tranlatonal velocty v teerng angle φ an the rotaton θ for the th traler. VI. CONTRO AWS Combnng all the potental functon ( an ntroucng contant enote a the control parameter k β ζ γ q κ > k q N we efne a canate yapunov functon 9

5 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 V + + S 4 k W k k ζ w γ κ q q DC q FO U β ( Clearly locally potve an contnuou on the { 8 : oman D R W > Sk > FOq > DC > U ( > }. We efne e : ( p p p an equlbrum pont of ytem(. Thu we have ( e. n σ ( co n δ ω + g θ f θ + m + zɺ n + (4 f nθ g co θ z + h ɺ n + f n h θ m z + m ɺ for n where δ δ > are contant commonly known a convergence parameter an n. Proof: The tme ervatve of our yapunov functon along a partcular traectory of ytem ( then: Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ Fg. The total potental The total potental a n Fg. aregenerate for target attracton an avoance of two tatonary k-hape obtacle. For better vualzaton the target of the leaer locate at( t t ( 55 an the k are fe at ( o o ( 9 ( o o ( 9 wth ra of ro ro. whle l l. Alo the velocty an angular component of the robot have been treate a contant uch that v.5 ω an θ. To etract the control law we fferentate the varou component of eparately an carry out the neceary ubttuton from (. The nonlnear control law for ytem ( wll be egne ung yapunov' Drect Metho. The proce begn wth the followng theorem: Theorem: The equlbrum pont of ytem( table n the ene of yapunov prove n n σ ( co n δ v + f θ + g θ + m + uɺ n + ( f nθ + h m u ɺ an n + f n g co θ u ɺ θ e n ( ( δv δ ω ɺ + for all D an ɺ ( e where the functon f g h g m for... n an are efne a (upon uppreng : β β f + ( p W W w γ o FO q k k q q k k k k q a a S ( ( k k k ( k + b b y y S ( θ h p q k k q q k k k k k k k k β4 β g + ( y p W 4 W w γ y o FO m q b b r y y S ( ( k k k ( k + a a r S ζ k k k k ( θ θ DC where an + κ U β ζ γ w k q k Sk W DC q FOq U κ

6 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ uɺ n( θ θ co( θ θ + + n ( θ θ co ( θ θ + n( θ θ n ( θ θ n ( θ θ n ( θ θ co( θ θ co ( θ θ zɺ co( θ θ co ( θ θ + k + n ( θ θ + co ( θk θk k k ( θ θ ( θ θ ( θ θ + n co n n ( θ θ n( θ θ co ( θ θ. A careful crutny of the properte of our calar functon reveal that e an equlbrum pont of ytem ( n the ene of yapunov an a legtmate yapunov functon guaranteeng tablty. Th n no contracton wth Brockett reult [] a we have not proven aymptotc tablty. VII. SIMUATION To llutrate the effectvene of the propoe controller we preent a cenaro ee Fg. 4of where the car-lke robot an t pave traler move towar t egnate goal whle avong fe obtacle n t workpace. The ue of the ghot wall help n attanng the ere poture of the tractor an the traler robot. Fg. 5 Orentaton of the tractor an t traler TABE I NUMERICA VAUES OF INITIA STATES CONSTRAINTS AND PARAMETERS FOR SCENARIO Intal Confguraton Intal Conton y 5 v.5 ( ω π π ( θ θ θ θ 4 ( 4 4 Fnal Confguraton ( p p p p p p ( 4.5 Fe Obtacle ( o o ( o o Phycal mtaton Dmenon of Robot Workpace Bounare Clearance Parameter Safety Parameter hot Wall Fe Obtacle Dynamc Contrant Workpace Retrcton Convergence Parameter Control Parameter 9 9 ro ro. v ma 5 π ma φ ρ mn l.5 c.5 η η 4 ε ε. ξ ξ.4. k γ 8 γ γ γ q q q q ξ ξ ξ κ β δ δ ω. raph n Fg. 5 how the orentaton of the tractor an t off-ale traler.the correponng ntal an fnal tate an other etal for the mulaton are lte n Table I (aumng that approprate unt have been taken nto account. Fg. 4 The reultng table traectory of the general -traler robot VIII. CONCUSION Th paper preent a et of artfcal fel functon erve ung yapunov rect metho that mprove upon n general the poture control wth theoretcally guarantee pont an poture tablte an convergence an collon avoance of a general -traler moble robot. We have a

7 Worl Acaemy of Scence Engneerng an Technology Internatonal Journal of Mathematcal Computatonal Scence an Engneerng Vol:8 No: 4 Internatonal Scence Ine Vol:8 No: 4 waet.org/publcaton/ centralze traectory plannng algorthm whch to ome etent emontrate autonomy an multtakng capablte of human. The new algorthm prove u wth a utable an fttng platform to harvet collon-free traectore from ntal to ere tate an generate maneuver that culmnate to practcally reaonable poture wthn a contrane envronment whlt atfyng the nonholonomc contrant of the ytem. The propoe controller tablze the confguraton coornate of the vehcle to an arbtrary mallneghborhoo of the target. We note here that convergence only guarantee from a number of ntal tate of the ytem. The erve controller prouce feable traectore an enure a nce convergence of the ytem to t equlbrum tate whle atfyng the neceary knematc an ynamc contrant. We note here that convergence only guarantee from a number of ntal tate of the ytem. Future reearch wll are a warm of thegeneral - tralermoble robot. REFERENCES [] O. Khatb "Real-tme obtacle avoance for manpulator an moble robot" Internatonal Journal of Robotc Reearch 5 (: 9 98 March 986. [] A. W. Dvelb an J. T. Wen. Traectory trackng control of a cartraler ytem IEEE Tranacton on Control Sytem Technology 5(: [] P. Bolzern R.M. DeSant an A.ocatell. An nput-output lnearaton approach to the control of an n boy artculate vehcle Journal of Dynamc Sytem Meaurement an Control(:9-6. [4] J. ee W. Chung M. Km C. ee an Jeabok-Song A pave multpletraler ytem for noor ervce robot n IEEE/RSJ Int. Conf.Intellgent Robot an Sytem Mau HI pp [5] Sharma. B Vanualala. J Raghuwaya. K an Praa. A (8 New Potental Fel Functon for Moton Plannng an Poture Control of - Traler Sytem Internatonal Journal of Mathematc an Computng Scence (: [6] Sharma.B Vanualala. J an Chan. U (9 Flockng of Multagent n Contrane Envronment European Journal of Pure an Apple Mathematc (: [7] K. Raghuwaya S. Sngh B. Sharma an J. Vanualala Autonomou Control of a Flock of -Traler Moble robot Proc of the Internatonal Conference on Scentfc Computng a Vega USA pp [8] K. Raghuwaya S. Sngh B. Sharma. ngam Formaton Type of a Flock of -Traler Moble Robot Proc of The 7th IMT-T Internatonal Conference on Mathematc Stattc an t Applcaton Bangkok Thalan pp [9] B. Sharma New Drecton n the Applcaton of the yapunov-bae Control Scheme to the Fnpath Problem PhD Dertaton Unverty of the South Pacfc F July 8. [] P. C-Y. Sheu an Q. Xue Intellgent Robotc Plannng Sytem Worl Scentfc Sngapore 99. [] R. W. Brockett Dfferental eometry Control Theory chapter Aymptotc Stablty an Feeback Stablaton page 8-9. Sprnger-Verlag (98.