Role of Kalman Filters in Probabilistic Algorithm

Transcription

1 Volume 118 No , 5-10 ISSN: (prined version); ISSN: (on-line version) url: hp:// doi: /ijpam.v118i11.2 ijpam.eu Role of Kalman Filers in Probabilisic Algorihm K.Aniha Deparmen of Mahemaics, S.A.Engineering College, Chennai, India. Absrac: Probabilisic Algorihms provide an Informaion hrough probabiliy disribuions over a whole space of possible hypohesis which will give degree of belief in mahemaical way o conclude he problem of uncerainy. his paper describes he basics of Kalman filer algorihm and is applicaion in localizaion of Roboics. Keywords: Belief disribuion, Localizaion, Kalman filer. 1. Inroducion Probabilisic Roboics is he process of esimaing he posiion of Robos from sensor daa which canno be observed direcly bu ha can be inferred. Generally Robos has gahered he informaion from nearby obsacles hrough sensors. Bu sensors carry only parial informaion due o noise. Probabilisic algorihms minimize his noise and will give he sae esimaion hrough belief disribuion over he possible sae space. he main objecive of his paper is o inroduce he mahemaical ool for esimaing sensor daa hrough Exended Kalman Filer algorihm for non-linear sysems. Robos can influence he environmen hrough is acuaors and gaher he informaion from camera image, range scan hrough sensors. his percepual ineracion is called Measuremen. Measuremen daa a ime will be denoed by M. he se of all measuremen acquired from ime 1 o 2 is given by he following equaion M M, M, M,..., M (1) 1: Conrol acions change he esimae sae a any ime. Even hough Robos do no play any acion iself i will execue he conrol acions. Hence we should consider he conrol daa. I carries he informaion abou he change of sae of he environmen and i will be denoed by C and defined he change of sae beween -1 and. he following expression denoes he sequence of conrol daa 5

2 C C, C, C,..., C (2) 1: Belief Disribuion he probabilisic law characerizing he evoluion of sae will be denoed by P( x x M C ) P( x x,c ) (3) 0: 1, 1: 1, 1: 1 he above expression is known as condiional independence and i will give he sae ransiion probabiliy. I specifies how environmens are sochasic and evolves over a ime as a conrol funcion C A belief disribuion assigns a probabiliy o each possible hypohesis in erms of is rue sae and i will reflec he inernal knowledge of robo abou he environmen sae. he belief value over a sae x is given by bel( x ) p( x / M, C ) (4) 1: 1: he poserior probabiliy afer Equ(4), such ha before incorporaing M., jus afer execuing C will be denoed as follows bel( x ) p( x / M, C ) (5) 1: 1 1: Equ(5) predics he sae a ime based on he previous sae poserior, before incorporaing he measuremen a ime. Calculaion of (4) and (5) are called Measuremen updae 3. he Kalman Filer he powerful mahemaical ool used o esimae he sochasic environmen from noisy sensor measuremen is Kalman Filer which was invened by Rudolph Emil Kalman in 1950s as a echnique for linear mean square error filering and predicion in linear sysems which implemens belief compuaion for coninuous saes. For he given addiive combinaion of 6

3 signals and noise i will minimise he mean square error and give he separaion of signals from noise. A ime he belief is represened by he (i) (ii) mean Covariance and he following assumpions have been made o consruc he Kalman Filer Algorihm (i) he nex sae probabiliy P( x / C, x 1) Mus be linear funcion in is argumens wih added Gaussian Noise which is expressed in he following equaion x A x 1 BC (6) Here x, x -1 are sae vecors and C hey are of he form x1, c1, x2, c2, x & C : : x c n, m, A is a square marix of order n and B is a marix of order n X m he mean poserior sae is given A x 1 Bu & Covariance is Ramun Schmid and Guido gave he noise suppression algorihm using Weighed Overlap Add (WOLA) decomposiion mehod. Gelb e al gave he descripion for exended Kalman Filer algorihm for non-linear sysem. he Ieraed Exended Kalman Filer algorihm was given by Bruyninckx and Suherland. Lefebvre described he properies of Linear Regression Kalman Filer. his filer was comprised by Cenral difference filer (Schie 1997), Firs order divided 7

4 difference filer (Ravn 2000) and Unscened Kalman Filer (Uhlman and Durran-Whye 2000). his paper gives he Exended Kalman Filer algorihm for non-linear sysem. 4. Exended Kalman Filer for Non-linear sysems he following expression gives he belief value from he belief one ime sep earlier. bel ( x ) P ( x x C ) bel ( x ) dx (7) 1, bel( x ) exp( ( x A x 1 B C ) 2 ( x A x B C ) exp( ( x ) ( x ) dx (8) he above equ (8) can be rewrien as bel( x ) exp( L ) dx (9) 1 o solve his inegral in closed form we should decompose L in he following form bel( x ) exp{ L ( x )} (11) he resuling disribuion over x will be enirely defined hrough L (x ) such as seing he firs derivaive of L o zero gives he mean and he Kalman gain Exended Kalman algorihm () i A B C 1 ( ii) A 1A ( iii) K Z ( Z Z Q ) 1 ( iv) K ( M Z ) ( v) ( I K Z ) ( vi) Re urn, 8

5 5. Conclusion he above Exended Kalman algorihm gives he opimal noise reducion in non-linear sysem which will be implemened in GPS racking sysem, Localizaion of Robos ec REFERENCE [1] Bar-Shalom, Y., and Li, X., 1993, Esimaion and racking: principles, echniques and sofware (Boson, London: Arech House). [2] Bayes,., 1763, An essay owards solving a problem in he docrine of chances. Philosophical ransacions of he Royal Sociey of London, 53, Reprined in Biomerika, 45, , [3] Gelb, A., Kasper, J., Nash, R., Price, C., and Suherland, A., 1974, Applied opimal esimaion (Cambridge, MA: MI Press). [4] Julier, S., Uhlman, J., and Durran-Whye, H., 2000, A new mehod for he ransformaion of means and covariances in filers and esimaors. IEEE ransacions on Auomaic Conrol, 45(3), [5] Mehra, R., 1972, Approaches o adapive filering. IEEE ransacions on Auomaic Conrol, 17(5), [6] Nørgaard, M., Poulsen, N., and Ravn, O., 2000b, New developmens in sae esimaions for nonlinear sysems. Auomaica, 36(11), [7] Ramun Schmid,Guido M. Schuser, A Kalman filer based noise suppression algorihm using speech and noise models derived from spaial informaion, IEEESignal Processing Conference, h European,ISSN : [8] Schei,., 1997, A finie-difference mehod for linearisaion in nonlinear esimaion algorihms. Auomaica, 33(11), [9] anizaki, H., 1996, Nonlinear filers: esimaion and applicaions second, revised and enlarged ediion (Berlin-Heidelberg: Springer-Verlag) [10] Uhlmann, J., 1995, Dynamic map building and localizaion: new heoreical foundaions. PhD hesis, Universiy of Oxford, UK 9

6 10