Position predictive measurement method for time grating CNC rotary table

Transcription

1 Posiion redicive measuremen mehod for ime graing CC roary able Liu Xiaokang a, Peng Donglin a, Yang Wei a and Fei Yeai b a Engineering Research Cener of Mechanical Tesing Technology and Equimen, Minisry of Educaion, Chongqing Insiue of Technology, Chongqing 45, China b School of Insrumen Science and Oo-Elecronic Engineering, Hefei Universiy of Technology, Hefei, 239 China Tel.: Fax: lxk@cqi.edu.cu Absrac Time graing sensor ransforms sace domain informaion o ime domain informaion and measures saial dislacemen wih ime. To develo high recision ime graing CC roary able and reduce he dynamic osiion feedback error of he able, circular osiion redicive measuremen mehod is roosed for ransforming ime domain informaion back o sace domain informaion based on ime-sace ransformaion echnology. Prediced values are obained by modeling he measured values wih ime series heory, and he las redicion error is correced in real ime using he nex measured values. Modeling mehod and arameer esimaion algorihm are resened. To conform he validiy of he osiion redicion mehod, an exerimenal sysem is designed. The angle dislacemen redicion error of he roary able is 2, and recise redicing is achieved. Keywords: Predicive measuremen, ime-sace ransformaion, ime graing, CC roary able. Inroducion Time graing is a novel dislacemen sensor develoed by he conce of measuring sace wih ime, and angular dislacemen is measured very accuraely wihou he use of a convenional graduaed mechanical scale, which ermis he grealy lowered cos of manufacure []. Time graing ransforms sace domain informaion o ime domain informaion based on ime-sace ransformaion echnology and measures saial osiion every uniform ime. In full closed-loo conrol mode CC sysem requires osiion feeding every uniform sace. To use ime graing as feedback comonen for CC osiioning servo conrol and develo high recision ime graing CC roary able, he new heories and mehods should o be develoed for ransforming ime domain informaion back o sace domain informaion based on ime-sace ransformaion echnology. Predicion is currenly used for indusrial conrol successfully, redicive conrol has become a yical rocess conrol mehod, and redicive conrol heory and algorihm are widely sudied [2]. Predicion conceion was firsly inroduced o measuremen field [3], and his aer will develo circular osiion redicive measuremen mehod o reduce he dynamic osiion feedback error of ime graing CC roary able. 2. Princile of redicive measuremen According o he measuremen rincile of ime graing [, 4], roaing magneic field is used o consruc a moving coordinae sysem S wih consan velociy V, he saial angular dislacemen beween a moving robe P a a random seed v and a fixed robe P b can be measured by deecing he ime difference T ha he roaing magneic field scans he wo robes (Fig. ). I.e. =VT=(36º/scanning eriod) ime difference. In he course of dynamic measuring, he ime inerval of S scans P b is fixed, and he signal of P b riggers daa samling, 2-278

2 hus ime graing ges a measured angle value every scanning eriod T i.e. measuring eriod. v V P a Pb P a P b T ime difference An absolue angle value of CC roary able is measured by ime graing every ime inerval T. The measured angle values θ i-+, θ i-+2,, θ i from ime oin T i-+ o T i can be reaed as a ime series. Time series model is based on he correlaion of he daa, which describes he dynamic characerisic and variaion rule of he series. Thus he fuure values of he ime series can be rediced by using ime series model [5]. The rincile of redicive measuremen mehod is shown in Fig. 2. A curren ime oin T i, he angle dislacemen of O θ i+ θ i θ i- v θ i-+2 θ i-+ T scanning eriod Fig.. Time graing measuremen model. able osiion redicing absolue angle T i measured value T i-+ T i-+2 T i- T i T i+ i rediced value incremenal ulses Fig. 2. Princile of redicive measuremen mehod. he roary able θ i during he nex measuring eriod T (from ime oin T i o T i+ ) can be rediced by modeling he absolue measured angle values θ i-+, θ i-+2,, θ i, hen incremenal ulses are ouued reresening he angle dislacemen θ i during he nex measuring eriod. In his way, discree absolue measured angle values can be ransformed o coninuous incremenal ulses ha are fed o CC sysem. The angle dislacemen of he roary able θ k during he kh measuring eriod (from ime oin T k- o T k ) is:. k k k () Using ime series heory he rediced angle dislacemen value of he roary able ˆi during he nex measuring eriod (from ime oin T i o T i+ ) can be obained as: ˆ L (,,, ). (2) i i i i i The number of ulse ouued in PWM (Pulse-Widh Modulaion) mode during he nex measuring eriod (from ime oin T i o T i+ ) is: 2-279

3 P ( ˆ e) Q, (3) i i i where Q is he ulse equivalen and e i is he redicion error of he las measuring eriod (from ime oin T i- o T i ). Time graing will ge a new measured angle value θ i+ when ime oin T i+ come. θ i+ can be used o calculae he redicion error beween he rediced value and he measured value. Eq. (3) shows ha he redicion error of he las measuring eriod is correced when curren redicing is conduced. And he redicion error will no be cumulaed, which assures he aforemenioned mehod of high recision. 3. Algorihm for osiion redicion Predicion model is based on saionary daa series. Firsly i is necessary o rerocess he dynamic daa measured by ime graing, remove he deerminisic rends and ransform non-saionary daa series o saionariy [5, 6]. And we aly saionary daa series o modeling, analysis and redicion for he residual daa hen finish modeling and analysis for curren and as angle measured values of ime graing ogeher wih he deerminisic rends. Finally redicing angle dislacemen of he CC roary able is achieved. The exression of h order auoregressive model AR() for ime series {X } can be described as: X a X,, (4) j j j where { } is whie noise W(, 2 ), real number a=(a, a 2,, a ) T is he auoregressive coefficien and a. X n+ can be successively rediced wih daa X n, X n-,, X n-+. The exression for oimal linear redicion is: n n n n n j n j j Xˆ L( X X, X,, X ) a X. (5) Parameer esimaion algorihm is as follows for auoregressive coefficien a. Firsly we need o rerocess he observed daa X, X 2,, X o obain a new ime series {Y } wih zero mean value: Y X X,,2,, X j X j. (6) Then we model he {Y } wih AR(). The esimaion for auocovariance funcion of he rerocessed daa is: k y y, k,,,. (7) k j jk j The square esimaion for auoregressive coefficien and whie noise T ( a 2, a2, a ), are deermined by Yule-Walker equaion a 2 2 a2 2 a (8) 2-28

4 angular acceleraion(/ms 2 ) angular acceleraion(/ms 2 ) angle() angular velociy(/ms) and 2 ( a a2 2 a ). (9) 4. Exerimen seu and resuls Exerimen seu is shown in Fig. 3. An AC servomoor drives a roary able insalled wih a.8 ime graing sensor acing as osiion feedback comonen. Siemens 82D digial CC servo sysem works wih SMC3 encoder inerface module for full closed-loo osiion conrol. The CC sysem receives he feedback signals from ime graing and hen Fig. 3. Exerimen seu. recisely conrols he osiion of he roary able. HEIDEHAI angular encoder ROD88 wih 36 line coun and accuracy of is mouned for esing he dynamic redicion error and he osiioning accuracy of he roary able. Inerolaion and digiizing elecronics IBV66B is alied for convering he ROD88 sinusoidal signals o TTL square-wave ulse rains as well as 4-fold subdivision. Consequenly he measuring se of he esing sysem 角度 ime(s) Fig. 4. Tesed osiion curve of roary able ime(s) Fig. 5. Velociy curve of roary able ime(s) Fig. 6. Acceleraion curve of roary able measured value rediced value redicion error ime(s) Fig. 7. Predicion resuls of negaive acceleraion moion. 2-28

5 is.9. A couner card is develoed o rocess he ouued signals from IBV66B and he incremenal ulses from ime graing and can simulaneously lock he wo couning resuls for synchronous dislacemen comarison. The card is conneced o a comuer via a RS-232C or, and he comuer can auomaically acquire he wo locked daa. The comuer can also ge he measured angle value of ime graing direcly wih anoher RS-232C or. Figure 4 shows a esed osiion curve of roary able. The CC sysem drove he able roaing from o wihin 7 seconds. The comuer samled he osiion of he roary able en imes er second. There were four differen moion saus including res, osiive acceleraion moion, aroximae uniform moion and negaive acceleraion moion (Fig. 5). The angular acceleraion curve of roary able is shown in Fig. 6. Figure 7 illusraes he redicion resuls of negaive acceleraion moion (he fourh moion course denoed in Fig. 5) using AR(3) model. The acceleraion redicion error varies form -.23/ms 2 o.9/ms 2. And he redicion error goes bigger along wih he larger acceleraion variaion. According o he relaionshi beween dislacemen and acceleraion, he angle dislacemen redicion error of he roary able is Conclusion This aer briefly inroduced he rincile of ime graing. Circular osiion redicive measuremen mehod was roosed o reduce he dynamic osiion feedback error of ime graing CC roary able. Prediced values were calculaed by modeling he measured values wih auoregressive model, and he redicion coefficien comuing algorihm was resened. The las redicion error was correced in real ime using he nex measured values. Discree absolue measured angle values were ransformed o coninuous incremenal ulses ha were fed o CC sysem. Exerimenal resuls have shown ha recise osiion redicing for ime graing roary able was achieved. Furher research will focus on coninuous redicing for comosie moion. 6. Acknowledgemen This rojec is funded by aional aural Science Foundaion of China (o. 5855). References. X.K. Liu, D.L Peng, X.H. Zhang and X.H. Chen. Solid Sae Phenomena: Mecharonic Sysems and Maerials. 26, vol. 3, S. Joe Qin and Thomas A. Badgwell. Conrol Engineering Pracice. 23, vol., X.K. Liu, Y. Fei, D. Peng and X. Wang. Measuremen Technology and Inelligen Insrumens VIII: Key Engineering Maerials. 28, vols , Liu Xiaokang, Peng Donglin, Zhu Ge, e al. Chinese Journal of Mechanical Engineering. 28, vol. 2, Shuyuan He. Alicaions for Time Series Analysis. Beijing Universiy Press. China S. Franisek. Predicions in Time Series Using Regression Models. Sringer-Verlag, ew York