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# Track Fitting using Kalman Filter and Smoother - PowerPoint PPT Presentation

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.

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### Track Fitting using Kalman Filter and Smoother

S. Gorbunov DESY Zeuthen

I. KiselKIP, Uni-Heidelberg

E. KryshenSPbSPU, St. Petersburg

KIP

CBM Collaboration Meeting, GSI

March 9-12, 2005

• Analytic formula of track extrapolation

• Kalman Smoother routine

S. Gorbunov, I. Kisel, E. Kryshen

Differential equation of charged particle motion in a magnetic field

Along Z direction :

S. Gorbunov, I. Kisel, E. Kryshen

Coefficients:

Equation of motion:

Extrapolation error:

S. Gorbunov, I. Kisel, E. Kryshen

• Features :

• 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

Example: Third order extrapolation in a homogeneous By field

Small parameter:

S. Gorbunov, I. Kisel, E. Kryshen

Analytic Light

Analytic 1

Analytic 2

Analytic 3

S. Gorbunov, I. Kisel, E. Kryshen

S. Gorbunov, I. Kisel, E. Kryshen

Final result

rn,Cn

Initialization

r0,C0

Filtering

Kk= CkHTk(Vk+HkCkHTk)-1

rk+1= rk+Kk(mk-Hkrk)

Ck+1= (E-KkHk)Ck

Extrapolation

rk = Ak-1 rk-1

Ck = Ak-1Ck-1ATk-1+Qk

rk,Ck

Measurement

mk, Hk, Vk

Noise

Qk

rk,Ck

S. Gorbunov, I. Kisel, E. Kryshen

Final result

r0n,C0n

Extrapolation

r0k= Ak-1r0k+1

C0k= Ak-1C0k+1A-1Tk

Smoothing

Kk= Qk(Ck+Qk)-1

r0k= r0k+Kk(rk-r0k)

C0k= KkCk+(E-Kk)C0k(E-Kk)T

Noise

Qk

r0k+1,C0k+1

S. Gorbunov, I. Kisel, E. Kryshen

• Optimal track parameter estimation at each measurement

• Optimal track parameter estimation between measurements

• Detection and suppression of outliers

S. Gorbunov, I. Kisel, E. Kryshen

• 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