Dynamic Data Processing with Kalman Filter

Adaptive information processing methods are widely used for data processing. Most often they are used for two or more sensor information complex processing. In navigation GPS information is most often complex treated with inertial sensor information using adaptive filtering techniques [2,3]. If navigation system has only one source of information, adaptive filtering can be used [1, 4]. Widely known are following algorithms: Least Mean Square (LMS) algorithm, Recursive Least Squares (RLS) algorithm and Kalman Filtering (KF) algorithm. Results of research Least Square Method (LSM) for one source information filtering with sliding window are shown in [1]. This work describes results of Kalman filter modeling and optimization and usage of adaptive Kalman filter for GPS information processing. Comparison of data processing results with adaptive Kalman filter and Least Square Method (LSM) with sliding window is shown.


Introduction
Adaptive information processing methods are widely used for data processing.Most often they are used for two or more sensor information complex processing.In navigation GPS information is most often complex treated with inertial sensor information using adaptive filtering techniques [2,3].If navigation system has only one source of information, adaptive filtering can be used [1,4].Widely known are following algorithms: Least Mean Square (LMS) algorithm, Recursive Least Squares (RLS) algorithm and Kalman Filtering (KF) algorithm.Results of research Least Square Method (LSM) for one source information filtering with sliding window are shown in [1].This work describes results of Kalman filter modeling and optimization and usage of adaptive Kalman filter for GPS information processing.Comparison of data processing results with adaptive Kalman filter and Least Square Method (LSM) with sliding window is shown.

Modeling of data processing with two component Kalman filter
Trajectory of mobile object movement is supposed to be linear with radical change in one point.We modeling true coordinate in point i or in time t i using expression (1).All modeling values are relative, velocity is a relative coordinate change in the time step between two points where x 0 is coordinate in starting point, but V x is velocity of x coordinate change.
Maximal object movement dynamic is if velocity changes from positive to negative or contrary.In Fig. 1   Measured coordinate xm i and velocity of coordinate change are estimated using two measuring devices: first determinate coordinate, for example positioning system GPS and second determinate velocity, for example using accelerometers.In this case Kalman filtering algorithms must be use for estimation of mobile object movement trajectory x e : , and velocity V e : , where Sx and SV spectral density of object trajectory and velocity fluctuation, Rx and RV -covariance of the coordinate and velocity estimation error, Rax and RaV predict covariance of the coordinate and velocity estimation error, Kx and KV Kalman filter transfer function, Modeling results of trajectory estimation using Kalman filter, when Sx = SV =0,  e =2, Rx 0 =4 and RV 0 =0.3 are shown in Fig. 2  How it is seen from Fig. 2 estimation results is good, if coordinate change is linear.When velocity changes (Fig. 3) estimation error has very high value beginning with velocity changing point.To decrease dynamic errors often use Kalman filtering with some artificial object fluctuation [4].So we modeling filtering process using Sx=0.02 and SV=0.02.Results are shown in Fig. 4. How it can be seen dynamic error decrease, but modeling results also show that the estimation error in this case grow, for example in Fig. 4 it is two times more.Value of artificial object fluctuation can be chosen for every dynamical object individual, using optimization method of static and dynamic errors.
Another method is method witch we use for adaptive filtering with sliding window [1].In this method when estimated and measure trajectory value difference is more than same constant, the parameters of Kalman filter is change.We research Kalman filter adaptation when measure value of trajectory Y i coordinate and estimated coordinate xe i difference is more than When equation ( 4) is true then filter transfer functions are set K i =KV i =1 and filter values are so V i =0 and xe i =xm i .
Results of modeling so adaptive Kalman filter with two measuring devices and velocity change in point i=20 from +0.5 to -0.5, Sx = SV =0 is shown in Fig. 5. Filter adaptation is done in point i=25, when difference between estimation and measuring error is more than  e =2.This method can be used also in case when velocity measuring device have offset In this case Kalman filtering give growing estimation error as it is shown in Fig. 6.When adaptive Kalman filter with condition (4) for parameter correction is used offset effect can be decrease Fig. 7.
Described method for offset correction can be used, but better is employ third equation in Kalman filter for determination of the offset.

Modeling of data processing with Kalman filter if one measuring device is used for trajectory and velocity measuring
When one device is used, velocity can be processing from coordinate measuring results using Kalman filter or some another filter.If Kalman filter is used, velocity V e processing equations are following: . In this case also dynamic error is very large and we must use adaptive Kalman filtering.If filter adaptation with condition (4) is used, modeling error decrease (Fig. 9).

GPS data filtering with adaptive Kalman filter
Next step is to use the adaptive Kalman filter for some fragment of GPS data processing.There are 4 turns in this data fragment [1].Fig. 10 shows measured data (xm) and estimated data (xe) processing with Kalman filter.Comparison results of GPS date filtering with LMS algorithm [1] shows that Kalman adaptive filter gives better results for filtering dates in short time interval.In this work research results of dynamic data processing with adaptive Kalman filter are presented.In filtering process parameters of filter are change on results of difference between measuring and expected coordinate.Modeling results of adaptive Kalman filtering are comparing with classic Kalman filter.Modeling results using adaptive Kalman filtering for coordinate estimation shows that estimation error decrease in points when parameters of movement change very rapidly.Results of adaptive Kalman filter modeling and filter use for GPS receiver information processing are described.Ill.11, bibl.4 (in English; abstracts in English and Lithuanian).Pateikiami Kalmano filtrų dinaminio apdorojimo rezultatai gauti jų parametrus keičiant matuojamos ir apskaičiuotos koordinatės skirtumo dydžiu.Klasikinio filtro duomenys lyginami su adaptyviųjų filtrų modeliavimo duomenimis.Nustatyta, kad paklaida gerokai sumažėja greitai keičiantis judesio parametrams.Pateikiami realaus GPS imtuvo tyrimo duomenys.Il. 11, bibl.4 (anglų kalba; santraukos anglų ir lietuvių k.).
is shown true coordinate time function, if linear movement and radical change of 180 0 are used for mobile object trajectory modeling.True trajectory measurement errors e i are modeling as normal process with mean value M e equal zero and root square value - e .In Fig. 1 are shown true trajectory with error Y i = xt i + e i modeling when  e =2.Y i is measured trajectory Y i.

Fig. 1 .
Fig. 1.True coordinate xt i changing model if x 0 =0,V x =0.5 and changes to V x = -0.5 and measure value Y i when errors are modeling with M e =0,  e =2

Fig. 4 Fig. 5 .
Fig.4 Trajectory estimation results with two measuring devices and Kalman filtering and velocity change in point i=20 from +0.5 to -0.5, Sx = SV =0.02

Fig. 6 .
Fig. 6.Trajectory estimation results with two measuring devices and Kalman filtering when velocity measuring device have offset V=0.04

Fig. 7 .
Fig. 7. Trajectory estimation results with two measuring devices and Kalman filtering when velocity measuring device have offset V=0.04 and in point i=16 is make filter parameter correction

Fig. 8 .Fig. 8 .
Fig.8.shows modeling results when one device measurements are process with Kalman filter using (2) and (6) and mobile object trajectory change in point i=20.

Fig. 9 .
Fig. 9. Trajectory estimation results with one measuring devices and Kalman filtering and velocity change in point i=20 from +0.5 to -0.5, Sx = SV =0, in point i=25 filter parameters are corrected

Fig. 10 .
Fig. 10.GPS data processing results with Kalman filter.Filtering gives quite significant error in turns.Estimation results using adaptive Kalman filter are shown in Fig. 11.As shown in the Fig. 11 result of data filtering is much better.

Fig. 11 .
Fig. 11.GPS data processing results with adaptive Kalman filter