Kalman Filteroptimal Estimator- A Tracker Development Approach

IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 3, March 2015. www.ijiset.com

ISSN 2348 – 7968

53

Kalman FilterOptimal Estimator- A Tracker Development Approach

Student Lovely Professional University India Department of Electronics and Communication

Abstract-The prediction of position and velocity of an object accurately is the biggest challenge faced by the communication system these days. The algorithm developed fulfils the need to accurately measure and predict the correct position and velocity. The approach used includes using an initial estimated position and then recursively calling the Kalman Filter equations to reduce the error in position and velocity and correctly predict the position and velocity. After successive recursions the predicted and actual position are found to be nearly the same with a very low margin for error

Keywords— LMS, RMS

I. INTRODUCTION

In Kalman filter, filtering means actually estimating the state vector at the present time which is based upon the past observed data. On the other hand prediction is estimating the state vector at the future time [2]. Kalman filter forms the basis of most state estimation algorithms. We use this state estimation just because it is the key for obtaining the best possible navigation solution from the various measurements available with us. Kalman filter uses all the measurement information that is input to it over time, not just the most recent set of measurements. It is actually an estimation algorithm, rather than a filter. It also maintains real time estimates of a number of parameters. Estimates are updated using a series of measurements that are subject to noise. It uses knowledge of the deterministic and random properties of system parameters and the measurements for obtaining the optimal estimates of the information available. It is also called Bayesian estimation technique. [1] The Kalman filter is also called a tool for estimating the variables of a wide range of processes. In mathematical terms we can say that a Kalman filter estimates the states of a linear system. The Kalman filter not only works well in practice, but it is theoretically attractive too because it can be shown that of all possible filters, it is the one that minimizes the variance of the estimation error.

Kalman filters are often implemented in embedded control systems because in order to control a process, we first need an accurate estimate of the process variables [1].

II. ELEMENTSOFKALMANFILTER

1.State vector: It is a set of parameters which describe a system, known as states, which the Kalman filter estimates. Each state may be constant or may be time varying. For many navigation applications, the state includes the components of position or position error. Velocity, altitude and navigation sensor error states may also be estimated.Along with the state vector there is an error covariance matrix which represents the uncertainties in the Kalman filter’s state estimate and degree of correlation between errors in those estimates. This correlation information within the error covariance matrix is important for the following 3 reasons.

(a) It enables the error distribution of the state estimates to be completely represented.

(b) There is not always sufficient information from the measurement to estimate the Kalman filter states independently. The correlation information enables estimates of linear combinations of these states to be maintained while awaiting further measurement information.

(c) Correlations between errors can build up over the integral between measurements. Modeling this can enable us to determine one from the another.

2. System model:This model is also called the process model or time propagation model which describes how Kalman filter states and error covariance matrix vary with time.

will unc erro mod on k 3. M mea func mea cov nois new regu inte

4. M mea stat [2]. equ com stat of t vari pres eve is u

5. K esti con as t part in t filte time equ nex forw cov equ pos mea I

l increase w certainty, the ors will bec

del is determ known prope Measuremen asurements ctions of asurement variance mat

se on the m w measurem ular interval erval between

Measureme asurement ve te vector in The Kalm uations wh mputational

te of a proce the squared ious aspects sent, and ev n when the p unknown to u

Kalman Fi mates a pr ntrol like for

the process ticular point the form of er equations

e update e uations. To o xt time step ward (in ti variance est uations get th

terior estim asurement in

IJISET - Internatio

with time as e position come more ministic for

erties of the

nt vector: It of propertie state vec vector is trix that de measurement. ment informa ls. But in so n measurem

ent model:I ector varies the absence man filter is

hich provid (recursive) ss, in a way

error. The f s: it suppor ven future s precise natur us.

ilter Algori rocess simp rm. The ope

is estimate of time and f noisy mea can be divi quations an obtain the a

the time u ime) the c timates. Th he feedback mate which nto the a prio

onal Journal of Inn

the integra and velocit correlated. the states, a system.

t is a set of es of system

tor. Along a measure escribes the

. For many ation is inpu ome other ca ments can be

It describes with the fun e of measur a set of m des us a

means to that minimi filter is very rts estimatio states, and i re of the mo

ithm:The K ply by usin eration may ed by the fi d the feedbac asurements.

ided into tw nd measurem a priori estim update equa current state he measurem

to obtain an h incorpora ori estimate.

novative Science, E www

al of velocity ty estimation The system as it is base

simultaneou m which ar g with th ement nois statistics o applications ut to K.F a ases the tim

irregular.

how th nction of tru rement nois mathematica an efficien estimate th izes the mean y powerful in ons of past it can do s odeled system

Kalman Filte ng feedback

be describe ilter at som ck is obtaine The Kalman wo categories

ment updat mates for th ations projec

e and erro ment updat n improved ates a new

[3]

Engineering & Tec w.ijiset.com y n m d us re he se of s, at me he ue se al nt he n n t, o m er k d me d n s: te he ct or te a w Fig.1 Ope Theappl 1. T h 2. C 3. O 4. N 5. S 6. W 7. R In the signals a which c surround Though problem converge some ot fulfill th the corre is not dir devices. For exam tunnel.  N f a s i  S R h

chnology, Vol. 2 I

eration of Kalm

III.APPLICA

ications of K Tracking obj heads) Channel estim Orbit determ Navigation Structural he Weather fore Radar Tracke IV multi sensi are obtained corrupts the dings. But w there are so m but they h

ence rate, ther techniq he need to a

ect position rectly access

mple if the

Now-a-days facing a lo accurate po stationary increasing co Some estim RMS have high rate of

ssue 3, March 201

man Filter

ATIONS OF KALM

Kalman Filte jects (e.g., mation. mination ealth monitor ecasting er etc. V.MOTIVATION ing environ d along with data becau we need to many algor have some computation ques should

accurately m of the objec sible to the G

object is p

communic ot of proble osition of a or non-st ongestion, n mation tech computation convergence 15. ISSN MAN FILTER er includes: balls, faces ring N nment, som h the desire use of the c obtain corre rithms to com

problems li nal complex

be develop measure and

ct even if th GPS or other

passing thro

cation syste ems in gett any object

tationary d

noise etc. hniques like

nal complex e.

2348 – 7968

The 1.T the trac algo corr filte of o 2. app is u prev in m Filt dete the vers

In r cha upd Dis repr exp whe j=ti Fig.2 I

e research is

rack object help of Ka cked. The orithm of Ka rect equatio er we can ab object.

Track obje proach Kalm

used for bet vious one. T my base pa ter will rema

ector will be SURF featu sion of SIFT

V. research me rts for initi date of Kalm

A. crete time resented in pression:

ere; =state

me variable

2 System mode

IJISET - Internatio

based on fo

t in real tim alman Filter algorithm alman Filter on and recu ble to estima

ect with fe man Filter alo tter tracking This is the im aper. The al

ain same on e added. The ure detector T.

. RESEARCH ME

ethodology, ialization of man filter equ

.SYSTEM MOD e linear s

a state varia

(scalar); a,b .

el of Kalman F

onal Journal of Inn

llowing obje

me: In this ap object is e used here . By using it ursively call ate and track

eature dete ong with fea g of the obj mprovement lgorithm of nly the addit e feature det

r which is t

ETHODOLOGY

system mod f Kalman m uation is desc

DEL systems ar able format + b=constant; Filter [3]

novative Science, E www

ectives:

pproach with stimated an

is the basi ts predict an ling Kalman k the position

ector:In thi ature detecto ject than th I want to d the Kalman tional featur

ector may b the advance

del and flow matrices an cribed.

e generally given by th

=input;

Engineering & Tec w.ijiset.com h d ic d n n is or he o n re be d w d y he B. FLO The firs position the initia Transitio Filter E Vector, the Proc initialize estimate calculati

Fig. 3 Flow

C.U

Here w Kalman Innovati initialize with the

chnology, Vol. 2 I

OWCHART FO

st or the b of any obje alization of t on Matrix,

rror Co-Var the Measure cess Noise C

ed. These ed guess of

ing the error

w Chart for Ini

UPDATE OF K

we calculate n Gain, we c

ion. The va ed in the fi e Filter Error

ssue 3, March 201

OR INITIALIZA MATRICES basic step f

ect using th the Kalman the Obser riance Matr ement Noise Co – Varianc

allow the the position r in the next r

itialization of K

KALMAN FILT

the Kalma an calculate alue of the irst step is r Co-Varianc

15.

ISSN

ATION OF KAL

for predictin he Kalman F variables. T rvation Mat rix, the Filt e Co- Varia ce Matrix ha filter to m n and hence recursion.

Kalman Matric

TER EQUATIO

an Gain. Us e the Residua Filter State now update ce Matrix

2348 – 7968

55 LMAN

ng the Filters is The State trix, the ter State ance and ave to be make an

e allows

ces

ONS

IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 3, March 2015. www.ijiset.com

ISSN 2348 – 7968

56 Fig. 4 Flowchart for Kalman filter Update Equation

VI.CONCLUSION

By the use of the algorithm applied in base paper, the trajectory can be estimated with an accuracy of 96.5%. But accuracy may be altered by changing some of the parameters involved in the algorithm for e.g. by changing the pre-defined model of filtering to a more accurate model of the object. If an initial estimation of the position will be taken, it will help in increasing the accuracy.

The algorithm used here can be modified for objects having accelerated trajectories which is

generally the case in practical scenario. The values of all the matrices would be differently initialized for that purpose and also a very different pre-defined model of object would have to be used. The accuracy of the algorithm in that case would highly depend on the predefined model.

ACKNOWLEDGMENT

The paper has been written with the kind assistance, guidance and active support of my department who have helped me in this work. I would like to thank all the individuals whose encouragement and support has made the completion of this work possible.

REFERENCES I. Books

[1] Adaptive Digital Filters (Second edition) by Maurice. [2] Signal Detection and Estimation “MauradBarkat”. [3] An introduction to Kalman Filter, G Welch and G Bishop. [4]Handbook of Digital Signal Processing, D Elliot ed, Academic Press, 19986

[5] Digital and Kalmanfiltering:an introduction to discrete-time filtering and optimum linear estimator, SM Bozic, Halsted Press 1994.

[6] An Engineering Approach to Optimal Control and Estimation Theory, GM Siouris, John Wiley and Sons, 1996. [7] Statistical and Adaptive Signal Processing, DG Manolakis, VK Ingle, SM Kogon, McGraw Hill, 2000.

II. Research Papers

[8]SangHyun Chang, RangoliSharan, Michael Wolf, Naoki Mitsumoto, Joel W. Burdick. Int J Robot(2010), “People Tracking with UWB Radar Using a Multiple-Hypothesis Tracking of Clusters (MHTC) Method”, 2:3-18. DOI 10.1007/s1269-009-00399-x.

[9] Arghya Das, “Implementation of Kalman Filter in Radar Tracking using MATLAB”, Int C and IEEE.

[10]Ki Hwan Eom, SeungJoon Lee, Yeo Sun Kyung, Chang Won Lee, Min Chul Kim and Kyung Kwon Jung, “Improved Kalman Filter Method for Measurement Noise Reduction in Multi sensor RFID systems”, Sensors 2011, 11, 10266-10282;doi:10.3390/s111110266.

[11]Dr. C. Nicholas Pryor, “An Adaptive Kalman Filter Tracker for Multi-Mode Range/Doppler Sonar”, Aspects of signal processing, NATO Advanced study institutes series volume 33-1, 1977, ppp265-278.

[12]Sheldon Xu and Anthony Chang, Robust Object Tracking Using Kalman Filters with Dynamic Covarience”, Cornell University

IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 3, March 2015. www.ijiset.com

ISSN 2348 – 7968

57 2008, pp 495-512.

[14] UtpalBhattacharjeeandPranab Das, “Performance Evaluation of Wiener Filter and Kalman Filter Combined with Spectral Subtraction in Speaker verification System”, Int J of Innovative Technology and Exploring Engineering (IJITEE), ISSN:2278-3075, Volume-2, Issue-2, January 2013. [15] Vadehi.V, S.Vasuhi, K.Sri Ganesh, C. Theanammai,.NareshBabu N T, N Uthiravel, P. Balamuralidhar, Grish Chandra, “Person tracking Using Kalman Filter in Wireless Sensor Network”, Parallel Processing Lab, Department of Electrical Engineering, Madras Institute of Technology, Anna University, TCS, Banglore.