Speed and Direction Angle Control of Four Wheel Drive Skid-Steered Mobile Robot by Using Fractional Order PI Controller

Transcription

1 hp://dxdoiorg/5755/jeie56337 ELETONIA I ELETOTECHNIA ISSN 39-5 VOL NO 5 6 Speed and Direcion Angle Conrol of Four Wheel Drive Skid-Seered Mobile obo by Using Fracional Order Conroller ail Oran Abdullah Basci Adnan Derdiyok 3 Deparen of Elecronics and Auoaion Erzincan Universiy Vocaional High School Erzincan Faculy of Engineering Aaürk Universiy Elecrical and Elecronics Engineering Erzuru 3 Faculy of Technology Sakarya Universiy Elecrical and Elecronics Engineering Sakarya abasci@aauniedur Absrac In his paper speed and direcion angle conrol of four-wheel drive skid-seered obile robo (4WD SSM) is realized by Fracional-Order Proporional Inegral () conroller Speed and direcion angle of he obile robo are calculaed by using angular velociy of each oors conroller produces he orques of each oor of obile robo for rajecory racking and sabilizaion in he desired posiion A well-uned convenional conroller is also applied o obile robo for coparison wih he Experienal resuls prove ha he shows beer rajecory racking perforance han conroller in ers of rajecory racking accuracy and error levels Index Ters 4WD SSM; fracional order ; BLDC oor; rajecory racking I INTODUCTION In he pas few years researchers have focused on he auoaed guided vehicle (AGV) rajecory racking probles and differen approaches have been discussed Especially he nonholonoic consrains of AGV have been aken ino consideraion Norey-ico e al [] have proposed a pah racking conroller based on a robus D algorih Their ehod uses a siple linearized odel of he obile robo coposed of an inegraor and a delay They used an easy synhesis procedure and he obained rules were siilar o he Ziegler Nichols ehod for D conrollers Zhao e al [] were developed a suiable reference odel based on he vehicle paraeers for Auonoous vehicle by siplified bicycle odel of an auoobile In addiion hey used he adapive D conrol syse for adapabiliy and sabiliy Taogna and ar [3] have proposed a conrol srucure ha akes possible he inegraion of a kineaic conroller and an adapive fuzzy conroller for rajecory racking conrol of nonholonoic obile robos In heir ehod adapive conroller used a Fuzzy Logic Syse for esiaing he nonlinear robo funcions involving unknown robo paraeers for racking Manuscrip received 9 April 6; acceped 5 Augus 6 4 conrol of wheeled obile robos Huang e al [4] suggesed a high-gain observer based adapive oupu feedback racking conrol design schee for nonholonoic obile robos They used observers o esiae he unknown linear and angular velociies respecively Peng e al [5] have discussed in he obile device wo subsyse ha is nonholonoic obile plafor and holonoic anipulaor subsyse They derived a kineaic conroller for wo subsyse o obain a desired velociy by Lyapunov funcions Also hey proposed a robus adapive racking conroller and according o he Lyapunov sabiliy heory he racking errors and adapive coefficien errors are all bounded Alhough fracional order calculaion has an old hisory besides all hese developens oday i can be considered a new issue Is onse is based on GW Leibnz (695) and L Euler (73) I is presened basic ah of fracional calculus [6] soluions of fracional order differenial equaions [7] [8] soe provisions in he engineering of hese syses [9] Thus conrollers are increasingly becoing popular and have been gained any working area especially in conrol applicaions [] [4] Sudies have shown ha he fracional order syses can produce uch beer resuls han ineger order syse The conrol is robus in syses including uncerainy Therefore i is necessary o deerine he opiu paraeers For soluion i is presened he elecroagneis and evoluionary opiizaion algorihs [5] fracional-order conroller design using geneic algorihs [6] paricle swar opiizaion algorih [7] an ieraive opiizaion ehod according o nonlinear funcion iniizaion [8] an auo-uning ehod for he fracional order λ D μ conroller using he relay es [9] These proposed ehods allow direc selecion of he paraeers of he conroller hrough he knowledge of he agniude and phase of he plan a he frequency of ineres obained wih he relay es and a pracical and syseaic uning procedure [] In his paper a conroller and a well-uned convenional conroller are applied o experienal seup

2 ELETONIA I ELETOTECHNIA ISSN 39-5 VOL XX NO X 6 of four-wheel skid-seering obile robo for speed and direcion angle conrol respecively The deerinaion of he and conroller paraeers are achieved by rial and error The experienal resuls showed ha he has given beer resuls han he conroller in ers of rajecory racking accuracy and error levels II 4WD SSM DESIGN Four-wheel skid-seering obile robo (oving on wo diensional plane wih inerial coordinae frae) is depiced in Fig The kineaic equaions of he skid-seered obile robo are derived as follows [] [3] Fig ineaics of 4WD SSM The vehicle has four fixed wheels and wheels are skidseer oion and each wheel is driven by a Brushless DC oor hrough a gear box as shown in Fig vy No slip consrain sin( ) r r v l sin( ) Pure rolling consrain: (6) y c cos( ) x c sin( ) (7) x c cos( ) y c sin( ) L r r (8) x c cos( ) y c sin( ) L r l (9) where r is radius of wheels L is he disance beween he lef and righ wheels r and l are righ and lef wheels angular velociy v and are speed and direcion angle of 4WD SSM vx and vy are speed coponen of cenre of graviy of he 4WD SSM (P c) for x and direcions The acual posiion of he 4WD SSM is represened by generalized coordinaes Pc ( x c yc ) A Obaining he Transfer Funcion of BLDC Moors Firsly consider he equaion of he DC oor o obain he ransfer funcion of he BLDC oor The volage equaion and ransfer funcion of DC oor can be expressed as [4] di V L i E () y where: d E () e di d L V i e () d d i J B (3) Fig Drive syse of each wheel vx r r x r y l r r L L () r v r l () r r l L (3) r r l l (4) cos( ) r r v l cos( ) (5) The ransfer funcion is obained using () and (3) in s- for G( s) LJ s J B L s B ( ) e (4) We can siplify () considering he following assupions: B is sall and in ha case is ends o so J BL 3 e B If he negligible values are eliinaed (4) wrien as G( s) LJ s J s e by uliplying op and boo of (5) by (5) 5

3 ELETONIA I ELETOTECHNIA ISSN 39-5 VOL XX NO X 6 Transfer funcion (6) e G( s) e J L J s s e e (7) where is he orque consan (N/A) e is elecrical orque (Vsec/rad) L is Terinal inducance phase o phase (H) B is fricional coefficien of oor and load (N/ (rad/sec)) J is he oor ineria ( g ) is Terinal resisance phase o phase (Ω) The echanical ie consan The elecrical ie consan Subsiuing (8) and (9) ino (7) J (8) e L e (9) G( s) e es s () so (8) and (9) indicae he difference beween DC and Brushless DC oors The echanical and elecrical consans are very iporan pars of oor odel paraeers Given he syerical srucure he echanical ie consan (8) becoes [5] The elecrical ie consan J J () L e e e L () There is a syerical arrangeen a hree phases Therefore he echanical and elecrical consans becoe: J3 e L e 3 (3) Also using he elecrical power and echanical power equaions; he relaionship beween e and is expressed as follows: 3 E I N T 6 (4) E T N I 6 3 (5) e (6) B Maxon Moor (EC 3) Brushless DC Moor TABLE I SPECIFICATION OF MAXON BLDC MOTO (8889) S No Paraeer(uni) Specificaion Noinal volage (V) 4 No load speed (rp) 3 No load curren (A) 86 4 Noinal speed (rp) 95 5 Noinal orque (N) Noinal curren (A) Sall orque (N) Saring curren (A) 73 9 Maxiu efficiency 76 % TABLE IICHAACTEISTICS OF MAXON BLDC MOTO (8889) S No Paraeer(uni) Specificaion Terinal resisance phase o phase (Ω) 39 Terinal inducance phase o phase (H) 6 3 Torque consan (N/A) 5 4 Speed consan (rp/v) Speed/orque gradien (rp/n) 35 6 Mechanical ie consan (s) oor ineria (gc ) The aheaical odel of he BLDC oor is odelled based on he paraeers fro able - he values for e and e need o calculaed: 39 ; 6 J gc g ; 3 5 N / A 5 N / A ; 659 s 659 s 3 L 6 6 e (7) J3 659 [s] e (8) 6 J3 339 e (9) e 673 [V sec/rad] (3) Using () G(s) becoes: G( s) s 659 s 69 G( s) s 659 s (3) (3) 6

4 ELETONIA I ELETOTECHNIA ISSN 39-5 VOL XX NO X 6 III FACTIONAL ODE CONTOLLE The fracional-order differeniaor can be denoed by a p general fundaenal operaor a D where a and are he liis of operaions The fracional-order differeniaor and inegral are defined as follows a D p p d p r p -p (d ) p a (33) where p is he fracional order which can be a coplex nuber however he consan p is relaed o iniial condiions There are several aheaical definiions o describe he fracional derivaives and inegrals [6] [7] Beween hese definiions here are wo coonly used ones ie he Grünwald Lenikov (GL) and he ieann Liouville (L) The GL definiion is a li h p p j p a D f ( ) h ( ) f ( jh) (34) h j j where [] eans he ineger par while he L definiion is given as n p d f ( ) a D f ( ) d (n-p) n (35) pn a ( ) Torque of he lef fron and back wheels uv u l (38) 4WD SSM speed and direcion angle is calculaed fro he equaion () and (3) given for lef and righ oors Considering (5) and (6) uv and u signals are generaed by he conroller o calculae he righ and lef reference orque Fig 3 Block diagra of he 4WD SSM conrol syse V EXPEIMENTAL ESULT In his secion he perforance of he conroller is achieved by coparing he speed and direcion angle of 4WD SSM wih well-uned conroller resuls The boh conrollers are execued on he 4WD SSM shown in Fig 4 The 4WD SSM is equipped wih four fas response Brushless DC oors wih increenal encoders couning 5 pulses/urn and speed reducions gear boxes and an indusrial PC wih DAQ for (n p n) () is he Euler s gaa funcion a is he iniial ie and paraeer is used when he differenial and inegral are aken The general for of he fracional order conroller is he and is general ransfer funcion is given as C( s) i p s (36) where is fracional order p and i are he proporional gain and inegraion consan respecively The opiizaion of he ree paraeers p i and akes designing of conroller ore challenging han ineger order conroller Several ehods are proposed for opiizaion In his paper he deerinaion of he ree paraeers is achieved by rial and error IV CONTOL OF 4WD SSM The conrol ehod is explained as shown in Fig 3 The required orque for each oor is copued by uv and u ha are defined as oupus of conroller Torque of he righ fron and back wheels uv u r (37) Fig 4 4WD SSM plafor: acual view In order o show perforance of he proposed conroller hree experiens have been conduced and resuls are shown in Fig 5 Fig 7 In he firs experien a sinusoidal speed and sinusoidal direcion angle references are chosen for esing and conrollers As shown in Fig 5 he boh conroller have siilar responses and he axiu percenage errors of sine wave reference for is 4 % and for is 59 % respecively Due o chosen a reference signal which slowly changing over ie he percenage errors are close o each oher As i can be seen fro figure a he sar of he obile plafor oveen he direcion angle error of boh conrollers is higher han speed error This causes by slippage beween he wheel and he ground floor If we disregard he error occurred a he saring poin of sine wave he axiu percenage errors of sine wave reference for is 43 % and for 7 % respecively 7

5 ELETONIA I ELETOTECHNIA ISSN 39-5 VOL XX NO X 6 Speed (/s) Direcion Angle (rad) Error (/s) a) b) Fig 5 Experienal resuls for he sinusoidal speed (a) and sinusoidal direcion angle (b) references Error (rad) 5 5 Speed (/s) Direcion Angle (rad) Error (/s) Error (rad) a) b) Fig 6 Experienal resuls for he square wave speed (a) and sinusoidal direcion angle (b) references Speed (/s) Direcion Angle (rad) Error (/s) Error (rad) a) b) Fig 7 Experienal resuls for he sawooh speed (a) and sinusoidal direcion angle (b) references 8

6 ELETONIA I ELETOTECHNIA ISSN 39-5 VOL XX NO X 6 In he second experien speed reference is a square wave and direcion angle is chosen as sine funcion for conrol of 4WD SSM The square wave reference is iporan o es he perforance of he conrollers for sep changes I can be seen fro Fig 6 ha when he square wave reference changes gives 6 sec rise ie wih having 453 % overshoo while gives 9 sec rise ie wih having 48 % overshoo which is obviously uch beer Addiionally when he speed is fixed a /s boh conrollers displays sae seady sae perforance On he oher hand if we disregard he error occurred a he saring poin of sine wave he axiu percenage errors of sine wave reference for is 5 % and for 6 % respecively Finally a sawooh wave speed and sinusoidal direcion angle references are chosen In Fig 7 i is observed ha when he sawooh wave reference sudden changes gives 3 sec rise ie wih having 444 % overshoo while gives 4 sec rise ie wih having 384 % overshoo which is saller han In a siilar way he axiu percenage of errors of he sinusoidal direcion angle reference for is 47 % and for % respecively VI CONCLUSIONS In his paper an experienal sudy on he applicaion of conroller o a four-wheel skid-seering obile plafor under he differen references was presened In order o show he effeciveness of he a well-uned convenional conroller is also applied The experienal resuls show ha he conroller shows beer seady sae perforance wih having less overshoo and saller speed error when i copared o he responses of On he oher hand gives fas rise ie han and direcion angle errors of boh conroller is siilar because of using sae reference and slippage beween he wheel and he ground floor To conclude he applied conroller resuls in beer responses han conroller o conrol he speed and direcion angle of he vehicle under changing references EFEENCES [] J E Norey-ico e al Mobile robo pah racking using a robus D conroller Conrol Engineering Pracice vol 9 no pp 9 4 [Online] Available: hp://dxdoiorg/6/ S967-66()66- [] P 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