# track fitting using kalman filter and smoother

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Track Fitting using Kalman Filter and Smoother. S. Gorbunov DESY Zeuthen I. Kisel KIP , Uni-Heidelberg E. Kryshen SPbSPU, St. Petersburg. KIP. CBM Collaboration Meeting, GSI March 9-12, 2005. Contents. Analytic formula of track extrapolation Kalman Smoother routine. - PowerPoint PPT Presentation

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• Track Fitting using Kalman Filter and SmootherS. Gorbunov DESY Zeuthen I. Kisel KIP, Uni-HeidelbergE. Kryshen SPbSPU, St. PetersburgCBM Collaboration Meeting, GSIMarch 9-12, 2005KIP

S. Gorbunov, I. Kisel, E. Kryshen

• Contents Analytic formula of track extrapolation

Kalman Smoother routine

S. Gorbunov, I. Kisel, E. Kryshen

• Extrapolation in Magnetic FieldDifferential equation of charged particle motion in a magnetic field Along Z direction :

S. Gorbunov, I. Kisel, E. Kryshen

• Analytic ExpressionExtrapolation error:Coefficients:Equation of motion:

S. Gorbunov, I. Kisel, E. Kryshen

• Analytic Expression FeaturesFeatures :

The precision of extrapolation does not depend on a shape of the magnetic field.

One can cut off the higher-order terms in the series.

S. Gorbunov, I. Kisel, E. Kryshen

• Analytic 3Example: Third order extrapolation in a homogeneous By fieldSmall parameter:

S. Gorbunov, I. Kisel, E. Kryshen

• Coefficients in the Analytic FormulaAnalytic 1Analytic 2Analytic 3Analytic Light

cx= 1.13e-04cy= 5.64e-04cz= 3.09e-04cxx= 3.72e-07cxy= 2.57e-06cxz= 1.25e-06cyx= 3.95e-06cyy= 2.39e-04cyz= 3.04e-05czx= 5.21e-07czy= 4.64e-06czz= 3.74e-06cxxx = 5.15e-09cxxy= 2.47e-08cxxz= 8.73e-09cxyx= 3.36e-08cxyy= 3.83e-07cxyz= 6.57e-08cxzx= 5.28e-09cxzy= 3.20e-08cxzz= 1.78e-08cyxx= 4.81e-08cyxy= 5.46e-07Cyxz= 9.34e-08cyyx= 7.67e-07cyyy= 5.85e-05Cyyz= 3.75e-06cyzx= 1.30e-07cyzy= 1.35e-06Cyzz= 4.03e-07czxx= 8.33e-09czxy= 3.35e-08Czxz= 1.47e-08czyx= 4.76e-08czyy= 8.63e-07Czyz= 1.33e-07czzx= 1.59e-08czzy= 9.91e-08Czzz= 5.49e-08

S. Gorbunov, I. Kisel, E. Kryshen

• Performance of Extrapolation

MethodResidualsPullsp/pxytxtyq/pxytxtyRunge-Kutta 40.6427241.51.51.171.051.011.021.00Analytic 30.6427241.51.51.181.051.001.021.00Analytic 20.6827241.51.51.301.081.011.031.00Analytic 10.9430251.51.51.901.371.031.101.02Analytic Light0.6427241.51.51.191.051.001.021.00Analytic Central2.4938251.71.53.772.231.031.331.00

S. Gorbunov, I. Kisel, E. Kryshen

• Kalman Filter SchemeInitializationr0,C0Extrapolationrk = Ak-1 rk-1Ck = Ak-1Ck-1ATk-1+Qk

FilteringKk= CkHTk(Vk+HkCkHTk)-1rk+1= rk+Kk(mk-Hkrk)Ck+1= (E-KkHk)CkFinal resultrn,CnNoiseQkMeasurementmk, Hk, Vkrk,Ckrk,Ck

S. Gorbunov, I. Kisel, E. Kryshen

• Kalman Smoother SchemeFinal resultr0n,C0nExtrapolationr0k= Ak-1r0k+1C0k= Ak-1C0k+1A-1Tk

SmoothingKk= Qk(Ck+Qk)-1r0k= r0k+Kk(rk-r0k)C0k= KkCk+(E-Kk)C0k(E-Kk)TNoiseQkr0k+1,C0k+1

S. Gorbunov, I. Kisel, E. Kryshen

• Kalman Smoother Optimal track parameter estimation at each measurement Optimal track parameter estimation between measurements Detection and suppression of outliers

MethodResidualsPullsp/pxytxtyq/pxytxtyAnalytic Light 0.6427241.51.51.191.051.001.021.00+ Smoother 0.6427241.51.51.191.051.001.021.00

S. Gorbunov, I. Kisel, E. Kryshen

• Summary An analytic formula of extrapolation in magnetic field has been derived

A simplified formula is used in the KF fitter and CA track finder Kalman smoother is ready for track fitting in presence of noise hits

Tracking is well tested and available

S. Gorbunov, I. Kisel, E. Kryshen

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