Application of Kalman Filter in Track Prediction of Shuttlecock

Transcription

1 Proceedngs of the 9 IEEE Internatonal Conference on Robotcs and Boetcs Deceber 9-3, 9, Guln, Chna Applcaton of Kalan Flter n Trac Predcton of Shuttlecoc Man Yongu, Zhao Lang, Hu Jngxn School of Inforaton Scence and Engneerng, Northeastern Unversty, Chna Abstract - Ths paper deals wth the applcaton of Kalan flter for optzng and flterng the poston sgnal of shuttlecoc obtaned by the vson servo syste of Shuttlecoc Robot []. Non-unfor ass dstrbuton and ar resstance effect can ae uch nose not only n vson recognton but also n neatc odel analyss of shuttlecoc. The Kalan flter algorth s used to flter the shuttlecoc poston sgnal by tang the error of easureent and the error of shuttlecoc oton odel nto account. Besdes, by consderng the requreent of fast ovng control, we reduce densons of state vector by decoposton of shuttlecoc oton to shorten the executve cycle. The sulaton results show ts affectvty on provng the accuracy of trac predcton. It can also accoplsh trac predcton fast and accurately when appled on Shuttlecoc Robot. Keywords: least squares, Kalan flter, ar resstance, Shuttlecoc Robot I. INTRODUCTION In odern feld of robot, sport robots have been reported a lot and a great any researches have been done n ths feld, but there s lttle report on the research of shuttlecoc robots. Shuttle Robot s a new nd of sport robot. Shuttlecoc Robot s a new research area of vson servo syste applcaton and syste control strategy research. Generally, the syste of Shuttlecoc Robot s coposed of three core parts, vson subsyste, servo subsyste and achnery subsyste [], whch s shown n Fg. [].Vson subsyste s used to obtan the poston sgnal of shuttlecoc, servo subsyste and achnery subsyste are the executve devces wth three degree freedo. When the robot s n operaton, the two otors are controlled to poston the robot leg to the predcted fall-pont of shuttlecoc, and the thrd otor wll ht the shuttlecoc at the requred te, so the accuracy of trac predcton s so portant that t wll decde the executng effect of executve devce. However, trac predcton of rregular-shaped object such as shuttlecoc based on nforaton obtaned by vson syste also has several probles whch are dffcult to be solved by ost of current ethods: Frstly, the shape of shuttlecoc s rregular, and ts centrod s too dffcult to be recognzed well by vson syste when flyng n the ar randoly. So t s possble to obtan the poston sgnal of shuttlecoc by vson syste Fg. Real syste of shuttlecoc robot wthout any errors and nose even by the vson recognton algorth n [] and [4] or ost of other age processng algorths proposed untl now. Secondly, the neatc odel of shuttlecoc s a specal nd of projecton oton odel. A odel of projecton oton s proposed n [3]. However, as shuttlecoc s a nd of rregular-shaped object whch s affected by ar resstance ntensely when rotatng and flyng n the ar, so the oton of shuttlecoc s too coplex to be descrbed /9/$5. 9 IEEE. 5

2 wthout any error. Thrdly, because the accuracy of trac predcton s assocated wth the saplng frequency of vson syste, t s portant to shorten the executve cycle [7]. However, havng too any densons of the state vector s a unversal proble n any Kalan flters such as the one proposed n [], whch s dsadvantage to shorten the executve te. Consderng the above-entoned probles, ths paper at frst analyzes the neatc odel of shuttlecoc, and then the algorth of trac predcton based on the odel s proposed. In order to optze and flter the poston sgnal of shuttlecoc before trac fttng, Kalan flter s added. By the te the sutable values of paraeters R (easureent error atrx) and Q (odel error atrx) s set, the vson recognton errors and odel errors are taen nto account. However, because the oton of shuttlecoc s n three-densonal space, all varables of poston, velocty, and acceleraton n three x, y and z densonal space are needed to descrbe the oton and to calculate n Kalan flter, but f at frst the neatc odel s decoposed nto x, y and z three axes, then Kalan flter can be appled n three axes separately. As every flter only has a three-denson process state vector, the total te spent on atrx operatons n Kalan flter algorth can be uch less [6]. II. GENERALLY INTRODUCTION OF ALGORITHM OF TRACK PREDICTION AND FALL POINT FORECASTING In order to get accurate and adequate nforaton of shuttlecoc oton, the neatc trac equaton of shuttlecoc s obtaned by trac fttng through seres of poston nforaton of shuttlecoc, then several portant nforaton such as the poston of fall pont, and fall te of shuttlecoc can be solved by the equatons. As an proveent, as shown n Fg.,the poston nforaton of shuttlecoc s optzed by the applcaton of Kalan flter before the trac of shuttlecoc oton s ftted so that not only the poston nforaton of shuttlecoc s obtaned ore accurately, but also the velocty and acceleraton nforaton can be acqured, whch contrbute to ore effectve control of Shuttlecoc Robot. Fg. Flowchart dagra of trac predcton and fall pont forecastng III. ANALYSIS OF KINEMATIC MODEL OF SHUTTLECOCK MOTION ) Analyss on Kneatcs By analyzng the oton of shuttlecoc on neatcs, we now t s a nd of oton wth a varable acceleraton: s( v( Kalan flter on x axs Obtan the poston of shuttlecoc by vson syste Next cycle N v( dt + s a( dt + v Where s ( s the poston, v( s the velocty and a( s the acceleraton of shuttlecoc whle s s the ntal poston and v s the ntal velocty. ) Analyss of Ar Resstance Model a) Descrpton of the Ar Resstance Based on General Ar Resstance Model v v Start Kneatc odel of shuttlecoc Kalan flter on y axs Trac predcton by least squares Calculate the fall te Calculate the poston of fall pont Reach the fall pont? Y Over Kalan flter on z axs Servo syste () f () Where and are nherent paraeters whch are related to the volue and shape n the absence of varaton of atospherc densty. Consderng () and (), we regard the state varables of Kalan flter as a vector whch has ore than 9 densons. 6

3 Besdes, there are couplng relatonshps of x, y and z because of quadratc ter n (). Because of several operatons of 9 9 atrx, especally nverse operatons of atrx, the executable cycle s too long to satsfy the request of hgh speed []. Therefore the practcal ethod s descrbed as below. b) Splfcaton of Ar Resstance Model Based on Theoretcal Analyss and Measureent The ar resstance odel of shuttlecoc s analyzed agan. It s a fact that the range of the heght of shuttlecoc oton s wthn eter, so the axu speed of the shuttlecoc ust be wthn 5/s whch s uch saller than the velocty of low-speed bullet that we can use the ar resstance odel of low-speed objects. Besdes, through the easureent of the paraeters of ar resstance odel of shuttlecoc we now that <<.All n all, the ar resstance odel of shuttlecoc s splfed as below f v (3) Where s the paraeter of ar resstance of shuttlecoc. Consderng the gravty, we get f + G a (4) 3) Kneatc Equatons of Shuttlecoc Moton The acceleraton of shuttlecoc oton s decoposed nto x, y and z three densons based on (3) and Fg.3. ax f x f cosθ snϕ v cosθ snϕ vx a y f y f snθ snϕ v snθ snϕ v y az f z g f cosϕ g v cosϕ g vz g Fg.3 The decoposton of velocty and acceleraton of shuttlecoc oton (5) Fro (5) and (3), we have: v t x v x x( e + + x v y t v y y( e + + y (6) g v t gt g + v z( ( + ) e + + x So the oton of shuttlecoc s decoposed nto x, y and z three densons. IV. TRACK FITTING OF SHUTTLECOCK MOTION WITH THE KINEMATIC MODEL AND POSITION OF SHUTTLECOCK The equaton of x-axs s used for exaple: v t x v x Frstly x( e + + x s transfored to: y a x + (7) a t ( e ) Where x s equal to. Then the trac s ftted by least squares [8], and (8) s obtaned as below: ( y ( y a x a x a ) x a ) By solvng the equatons, we can get the values of a and a. Slarly, the trac equatons of y-axs and z-axs are solved n the sae way and then the trac equatons of shuttlecoc oton are obtaned. V. OPTIMIZATION OF THE POSITION INFORMATION OF SHUTTLECOCK MOTION BY KALMAN FILTER ) Dscrete State Transton Equatons of Shuttlecoc Moton In reference to equatons of shuttlecoc oton and te dscretzaton of vson syste, the dscrete for of equatons of shuttlecoc oton s obtaned. As the saplng cycle s very sall (alost /3 second) and the value of velocty of shuttlecoc can not change sharply because of the ntal effect, the velocty and acceleraton (see (5)) of shuttlecoc oton are consdered as constant n the perod of a cycle. Wth (5) and equatons of unforly accelerated oton, we have the state transton equatons of shuttlecoc oton: (8) 7

4 xt x( x( ) + x ( ) + ( ) x ( x ( ) + x( ) T (9) x( x ( [ x ( ) + x( ) T ] ) Descrptons of Kalan Flter [3] [5] [9] a) Coputaton of state transton The process fro one state to another s governed by (). X ( ) A X ( ) + X_Y_OR_Z B U ( () Where x( X ( x ( x( [ g] s state atrx of the syste. U ( s the control atrx and g represents acceleraton of gravty. X_Y_OR_Z s a Boolean data whch s whle the axs s x or y and whle axs s z. The 3 atrx A relates the state at the prevous te step X ( ) to the state at the current step X ( ) under the crcustance of not consderng control atrx [4]. Accordng to (9), we have A T T T. T b) Coputaton of error covarance P ( ) A P( ) A + Q Where P () s state vector error covarance atrx and Q s process nose covarance. c) Updatng of the estate wth easureent Z () X ( X ( ) + Kg( [ Z( H X ( )] () P( ) H Kg( (3) H P( ) H + R Where Kg( s Kalan gan and Z ( s the easureent vectors. H [ ] s a 3 atrx gvng the deal(nose-free) connecton between the easureent and state vector whle H s transpose of t. R s easureent error covarance. d) Coputaton of error covarance for updated estate. P ( ( I Kg( H ) P( ) (4) Where I s a 33 unt atrx. VI. SIMULATION RESULTS In order to evaluate results of Applyng Kalan flter n trac predcton of shuttlecoc oton, several experents are done on Matlab. Frstly, the easured values obtaned by vson syste are sulated by nputtng whte nose whose sgnal-to-nose rato s 5 nto actual values. Besdes, we set the process nose covarance Q as.4 whle easureent error covarance R as.4. ) The easured values, the fltered easured values by Kalan flter and the actual values of poston of shuttlecoc are copared on x-axs, y-axs and z-axs, as shown n Fg.4 (a), Fg.4 (b) and Fg.4(c). X axs coponent of poston of shuttlecoc() Y axs coponent of poston of shuttlecoc() Z axs coponent of poston of shuttlecoc() Actual value Measured value Measured value after flterng by Kalan flter Te(s) (a) Actual value Measured value Measured value after flterng by Kalan flter Te(s) (b) Actual value Measured value Measured value after flter by Kalan flter Te(s) (c) 8

5 Fg.4 (a) Coparson of the easured values, the fltered easured values by Kalan flter and the actual values of shuttlecoc poston on x-axs, (b) Coparson of the easured values, the fltered easured values by Kalan flter and the actual values of shuttlecoc poston on y-axs, (c) Coparson of the easured values, the fltered easured values by Kalan flter and the actual values of shuttlecoc poston on z-axs. Results analyss: Through Fg.4 (a), Fg.4 (b) and Fg.4 (c) soe conclusons are reached. Frstly, the errors of easureent are nzed by the use of Kalan flter. Secondly, velocty and acceleraton of shuttlecoc oton are obtaned whch are useful to servo syste. Thrdly, the curves of the fltered easured values by Kalan flter converge to the curves of actual values. ) Coparson of the convergence of fall-pont poston predcton between usng Kalan flter and not usng Kalan flter Dstance between actual fall pont and predct fall pont() Wthout Kalan flter Wth Kalan flter Te(s) Fg.5 Coparson of the convergence of fall-pont poston predcton Results analyss: As shown n Fg.5, after Kalan flter s added, the fluctuaton of errors of trac predcton s saller, the convergence of errors of trac predcton s faster, and the steady state value of errors of trac predcton s uch saller than before, whch are very portant for controllng shuttlecoc robot qucly, steadly and accurately. VII. SUMMARY AND CONCLUSIONS Ths paper concerns about the accuracy of recognton and trac predcton of rregular-shaped object by bnocular vson syste. In reference to the sulaton results, soe conclusons are acqured: By usng Kalan flter, the nose of recognton s reduced, the velocty and acceleraton of shuttlecoc are obtaned, the accuracy of trac predcton s proved, and the fall-pont of shuttlecoc s predcted faster and ore accurately. These proveents entoned above are portant for us to control Shuttlecoc Robot well. REFERENCES [] Man Yongu, Hu Jngxn, Tang We, Zhao Lang. A Shuttlecoc Robot Based on Vson Servo Syste. MCGM 9. [] Yongu Man and Jngxn Hu, Shuttlecoc robot and control ethod[p], Chna Patent, 999, [3] R. E. Kalan, "A new approach to lnear flterng and predcton probles", Transactons of the ASME, Journal of Basc Engneerng, 96. [4] Sulaton of Bearngs-only Passve Target Localzaton Based on TLS and Kalan Flter. SU WeWANG Guo-hongZHONG Pe-lnXIU Jan-juan. Naval Aeronautcal and Astronautcal Unversty,9 [5] Lee S.A Kalan flter approach for accurate 3-D oton estaton fro a sequence of stereo ages. CVGIP:Iage Understandng,99,9:44~58. [6] Atya S, Hager G D.Real-Te Vson-Based Robot Locaton. IEEE Transacton on Robotcs And Autoaton 993, 9(6): [7] Heen A W, Segers A J. Modelng and predcton of envronental data n space and te usng Kalan flter [J ]. Stochastc Envronental Research and Rc Assessent,, (6) :54. [8] Xaoyong Ja and Chuansheng Xu, The nventon and way of thnng on least squares[j], Journal of Northwest Unversty (Natural Scence Edton), Vol 36, No 3, Jun, 6. [9] Greg Welch, Gary Bshop. An Introducton to the Kalan Flter. Unversty of North Carolna at Chapel Hll. [] Adly A. Grgs,R. Grover Brown. IEEE. APPLICATION OF KALMAN FILTERING IN COMPUTER RELAYING. IEEE Transactons on Power Apparatus and Systes, Vol. PAS-, No. 7 July 98. [] Jn Guo, Xanyong Lu, Xaonng Chen. Autoatc Measureent of the Mechancal Part Based on Bnocular 9

6 Stereo Vson. Proceedngs of the 7thWorld Congress on Intellgent Control and Autoaton.June 5-7, 8. [] R Clare, J Waddngton and J N Wallace. THE APPLICATION OF KALMAN FILTERING TO THE LOAD/PRESSURE CONTROL OF COAL-FIRED BOILERS. [3] MICHAEL ÖSTERLUND and ÖRJAN NILSSON. PROJECTILE MOTION IN REAL LIFE. [4] Yng Ne, Tao Wu, Xangjng An, and Hangen He. Fast Bnocular Vson wth Flters. Proceedngs of 8 3rd Internatonal Conference on Intellgent Syste and Knowledge Engneerng.