In this paper factorization filtering, fusion filtering strategy and related algorithms are presented. Some results of implementation and validation using realistic data are given

Appavu, Raj R

Girija, G

Kashyap, SK

Raol, JR

English

National Aerospace Laboratories Institutional Repository

Tracking filter and multi-sensor data fusionG GIRIJA1, J R RAOL1, R APPAVU RAJ2 and SUDESHKASHYAP11System Identification Laboratory, Flight Mechanics and Control Division,National Aerospace Laboratories, Bangalore 560 017, India2Interim Test Range, Chandipur 756 025, Indiae-mail: jrraol@css.cmmacs.ernet.inAbstract. In this paper factorization filtering, fusion filtering strategy andrelated algorithms are presented. Some results of implementation and validationusing realistic data are given.Keywords. Multi-sensor data fusion; Kalman filtering; tracking.1. IntroductionMulti-sensor data fusion (MSDF) is defined as the process of integrating informationfrom multiple sources to produce the most specific and comprehensive unified data aboutan entity, activity or event. As a technology, MSDF is the integration and application ofmany traditional disciplines and new areas of engineering including communication anddecision theory, uncertainty management, estimation theory, digital signal processing,computer science and artificial intelligence. MSDF is expected to result in robustoperational performance, extended spatial coverage, increased confidence, reduced ambi-guity, improved detection performance, enhanced spatial resolution, improved systemoperational reliability and increased dimensionality (Hall 1992). At a basic level theprocessing operations are dominated by numeric procedures involving linear and nonlinearestimation techniques, pattern recognition processes and various statistical operations. Fora flight test range the tracking of the flight vehicle and sensor fusion are of greatimportance.In the present paper, U-D factorized Kalman filter, state vector fusion filter and a fusionphilosophy are described. The composite scheme is termed UDFF based MSDF strategy.The strategy is validated using post flight data.2. U-D factorization Kalman filterThe problem addressed in this paper is represented by the following set of equations.State model:xj1  j; j1xj  Gwj: 1SaÅdhanaÅ , Vol. 25, Part 2, April 2000, pp. 159±167. # Printed in India159Measurement model:zj  Hxj  vj: 2Here, x is the state vector, w is the process noise with zero mean and covariance matrix Q, zis the measurement vector and v is the measurement noise with zero mean and covariancematrix R, all of appropriate dimensions. The transition matrix of the state equation (1) isgiven as 1 t t2=20 1 t0 0 124 35: 3Here, t is the sampling interval and H  1 0 0. State vector x has position, velocity, andacceleration as its components. The tracking problem is related to estimation of theposition and velocity of a moving object (based on the available measurements) usingKalman filter. This filter is implemented in the factorized form for the present application.It is given in two parts: Time Propagation and Measurement Update algorithms (Bierman1977).Time propagation algorithmState vector evolution (prediction)~xj1  j1; jx^j: 4Covariance update~Pj1  jP^jTj  GQGT : 5With P^  U^D^U^ T , the time update factors ~U and ~D are obtained through modified Gram±Schmidt orthogonalization process (Bierman 1977). The matrix U is an upper triangularmatrix with unit elements on its main diagonal and D is a diagonal matrix. Covariance andgain processing algorithms, operating on U and D factors of state error covariance matrixP, are a technique for implementing "square root filtering" without requiring computationof square roots. The U-D Kalman filtering algorithm is considered efficient, stable andaccurate for real-time applications (Bierman 1977).We define W  U^jG; D  diagD^;Q with WT  w1;w2; . . . ;wn;Here, `T ' denotes transpose of vector/matrix. The U;D factors of ~P  WDWT may becomputed now. For j  n, nÿ 1; . . . ; 2, the following equations are recursively evaluated asshown below.~Dj  hwnÿjj ;wnÿjj iD;~Ui; j  hwnÿjj ;wnÿjj iD=~Dj;wnÿj1i  wnÿji ÿ ~Ui; jwnÿjj ;~D1  hwnÿ11 ;wnÿ11 iD;9>>>=>>>;i  1; . . . ; jÿ 1: 6Here subscript D qualifies the weighted inner product with respect to D.160 G Girija, J R Raol, R Appavu Raj and Sudesh KashyapMeasurement update algorithmThe measurement update in Kalman filtering combines a priori estimate ~x and errorcovariance ~P with scalar observation z  aT x v; aT  H to construct an updated (filteredstate) estimate and covariance as follows:K  ~Pa=;x^  ~x Kzÿ aT~x;  aT ~Pa r;P^  ~Pÿ Ka~P:7Here, r is the measurement noise variance (for scalar data processing). Kalman gain K andupdated covariance factors U^ and D^ can be obtained from the following equations.f  ~UT a; f T   f1: . . . ; fn;v  ~Df ; vi  ~di fi; i  1; 2; . . . ; n;d^i  ~d1r=1;1  r  v1 f1; KT2  v10    0: 8For j  2; . . . ; n recursively the following equations are evaluated:j  jÿ1  vj fj;d^j  ~djjÿ1=j;u^j  ~uj  jKj;Kj1  Kj  vj~uj;9>>=>>; j  ÿ fi=jÿ1: 9where ~U  ~u1; . . . ; ~un, U^  u^1; . . . ; u^n and Kalman gain is given by K  Kn1=n. Here~d  is predicted diagonal element, and d^j is the updated diagonal element of the D matrix.The U-D filter described above is developed in `C' language and implemented on a DECAlpha computer. It has been validated using simulated trajectory data and also real flight data.3. MSDF strategyFigure 1 shows a hierarchical MSDF architecture for a typical range equipped with severaltypes of sensors for tracking of a flight vehicle (Girija et al 1999). It is proposed to havethree sets of fused position x; y; z and corresponding velocity state-information to theDecision Support System (DSS). The data from ground based radars are combined to yieldone set of fused data 1. The second fused data 2 is generated by fusion of INS (inertialnavigation system) and GPS (global positioning system, presently not used) data receivedat the ground station. The angular data from the telemetry (TM) channels which measureonly the azimuth  and elevation  are combined using least squares method togenerate position information. The azimuth and elevation data from EOT (electro-opticaltransducers) are also combined in a similar manner to generate the position information.Although the UDF block is shown explicitly in figure 1 for various sensor channels, onlyone UDF is used and the data are processed sequentially by the same filter. Theseestimates of state vector i.e. position, velocity, acceleration and the error covariancematrices are the input to fusion process to achieve joint or fused state vector estimate basedon multiple sensors.Tracking filter and multi-sensor data fusion 161Figure1.MSDFscheme(PTC-polartocartesian).162 G Girija, J R Raol, R Appavu Raj and Sudesh Kashyap3:1 Fusion algorithmThe fusion algorithm works on the basis of the following equations (Saha 1996):The fused state vector is given byx^F j=j  x^1 j=j  P^1 j=jP^ÿ1E  j=jx^2 j=j ÿ x^1 j=j: 10The covariance of the fused state vector is given byP^F j=j  P^1 j=j ÿ P^1 j=jP^ÿ1E  j=jP^1 j=jT : 11The error covariance P^E j=j given byP^E j=j  P^1 j=j  P^2 j=jT : 12In the present case, the cross covariance between any two-state vectors estimated using datafrom two sensors is neglected for simplicity. The estimates of the state vectors and thecovariance matrices are obtained from the U-D filtering process of each channel data andused in fusion filter of equations (10)±(12) directly, e.g.P1  U1D1UT1 ;because no updating of these factors is needed by the fusion filter. The U-D filteringalgorithm and the fusion filter, (10)±(12), are jointly termed as UDFF in this paper.4. Sensor fusion philosophyIn order to be able to use sensor-channel such as EOT, PCMC, S-Band, two TM, RADAR 1,RADAR 2, INS, and GPS for fusion, it is necessary to develop fusion logic to use theinformation from these sensors. The fusion logic is mainly based upon priority logic andrange limit of individual sensors. Since, for the time being, real data from PCMC, GPS, and2 telemetry (TM) sensors are not available for simulation and validation, the PCMC andGPS data are replaced by INS data (for the sake of the validation process). Also the INS datais used to generate the synthetic trajectory data from two telemetry sensors using simpletriangulation method. The priority logic is decided based on the sensor accuracy within therange of the sensor capability. Based on this priority logic the following sequence could begiven to the sensors (of the first module) within the range limit of, say, RL km. EOT PCMC TM and S-Band fusion S-Band TM Track lossFor range more than RL km PCMC radar tracks the target. For the second module whichcontains the RADARs, the following sequence is followed RADAR 1 and RADAR 2 fusion RADAR 1 RADAR 2 Track lossTracking filter and multi-sensor data fusion 163For the third module which contains the INS and GPS, the following sequence is followed. INS and GPS fusion (GPS data replaced by INS) INS GPS (GPS data replaced by INS) Track lossRule base: The rule base for three modules that depends upon the health of the sensors isgiven. Tables 1±3 cover possible conditions and corresponding actions that can occur in thefusion philosophy.Table 1. Fusion module 1: For range limit<RL (TRAJ 3).EOT S-Band TM Action 1 Action 2 Action 31 1 1 EOT estimation S-band estimation TM estimationand send to DSS1 1 0 EOT estimation S-band estimation TM predictionand sent to DSS1 0 1 EOT estimation S-band prediction TM estimationand sent to DSS1 0 0 EOT estimation S-band prediction TM predictionand sent to DSS0 1 1 EOT prediction Fusion of TM and S-bandSend to DSS0 1 0 EOT prediction Fusion of estimated S-band andpredicted TM and send toDSS for t < m st > m s0 1 0 EOT prediction S-band estimation TM predictionand send to DSS0 0 1 EOT prediction Fusion of predicted S-band andestimated TM and send toDSS for t < m st > m s0 0 1 EOT prediction S-band prediction TM estimationSend to DSS0 0 0 If (ivalpcmc ! 0:0) then PCMC estimation and send toDSS and EOT, S-band, and TM prediction else if(ivalpcmc  0:0) then track loss , EOT, S-band, andTM prediction and EOT send to DSSFor range limit > RLPCMC ACTION1 PCMC estimation and send to DSS0 PCMC prediction and send to DSS(*) m  nt; n is the no. of sampling intervals.164 G Girija, J R Raol, R Appavu Raj and Sudesh Kashyap4:1 Software structureThe software contains the modules and the sub-modules for different fusion channels andfusion levels. The modules are the subroutines of the main program and the sub-modulesare subroutines of modules. At present the fusion is done with maximum two similar sensorchannels. The fusion software reads the sensor data from the files and performs filteringand fusion process for every fusion module to generate TRAJ1, TRAJ2 and TRAJ3. Thesetrajectories are sent to the DSS for further necessary action.Fusion module 1Sub-module 1: Angular telemetry data fusion (triangulation or least square method),filtering and the earth curvature correction if required.Sub-module 2: S-band co-ordinate conversion to telemetry station. Filtering, fusion andearth curvature correction if required.Table 2. Fusion module 2 (TRAJ 1).RADAR 1 RADAR 2 Action 1 Action 21 1 Fusion and send to DSSt < m s1 0 Fusion of estimated radar 1 and predictedradar 2t > m s1 0 RADAR 1 estimation Predicted RADAR 2and send to DSSt < m s0 1 Fusion of estimated radar 2 and predictedradar 1t > m s0 1 RADAR 1 prediction RADAR 2 estimationand send to DSS0 0 RADAR 1 prediction RADAR 2 predictionand send to DSSTable 3. Fusion module 3 (TRAJ 2)INS GPS Action 1 Action 21 1 Fusion and send to DSS1 0 t < m sFusion of estimated INS and predicted GPS1 0 t > m s0 1 INS estimation and Predicted GPSsend to DSSt < m sFusion of estimated GPS and predicted INSt > m s0 1 INS prediction GPS estimation andsend to DSS0 0 INS prediction and GPS predictionsend to DSSTracking filter and multi-sensor data fusion 165Figure 2. Comparison of TRAJ1, TRAJ2, and TRAJ3. Legend: ÐÐÐÐ TRAJ1 ÿ  ÿ  ÿ&ÿ  ÿ  ÿ TRAJ2 ± ±^± ± TRAJ3.166GGirija,JRRaol,RAppavuRajandSudeshKashyapFusion module 2Code for the RADAR 1 and RADAR 2 sensor fusion.Fusion module 3Code for the GPS and INS sensor fusion.4:2 Performance evaluationThe fusion philosophy (figure 1) is tested with one set of post flight data (where data is notavailable, some proxy data are used to carry out the validation). Figure 2 shows thecomparison of TRAJ1, TRAJ2, and TRAJ3. The `VAL' corresponds to validity bit/health ofthe sensor. TRAJ-ID stands for the trajectory identity signifying which module thetrajectory belongs. SEN-ID gives the sensor identity during the trajectory of the target. TheFUS-ID tells whether the trajectory is `filtered only' or fused (after filtering). Thepossibilities given in tables 1±3 are exercised in generating the fused trajectories.5. Concluding remarksThe U-D filtering and sensor fusion filter algorithms have been described and validated. Ahierarchical fusion philosophy has been presented for a typical flight test range and testedfor one set of post flight tracking data. Although the results are encouraging, further up-gradation of the process is under progress.The authors are grateful to Mr Narang, Interim Test Range, Chandipur and Air Cmdre. PBanerjee for technical discussions and support during the course of this work.ReferencesBierman G J 1977 Factorization methods for discrete sequential estimation (New York: AcademicPress)Girija G, Raol J R, Appavu Raj R, Banerjee P 1999 Intelligent multi sensor data fusion. Presented atthe National Seminar on Intelligent and Autonomous Systems, Computer Society of India,MumbaiHall D L 1992 Mathematical techniques in multisensor data fusion (Boston: Artech House)Saha R K 1996 Effect of common process noise on two-track fusion. J. Guidance, Control Dyn. 19:829±835Tracking filter and multi-sensor data fusion 167

Tracking filter and multi-sensor data fusion

NAL-IR

http://nal-ir.nal.res.in/4564/1/Tracking.pdf

Tracking filter and multi-sensor data fusion

Abstract

Similar works

Full text

Available Versions

National Aerospace Laboratories Institutional Repository

NAL-IR