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Universit¨ at Hamburg MIN-Fakult¨ at Fachbereich Informatik Kalman Filters Kalman Filters Jonas Haeling and Matthis Hauschild Universit¨ at Hamburg Fakult¨ at f¨ ur Mathematik, Informatik und Naturwissenschaften Fachbereich Informatik Technische Aspekte Multimodaler Systeme November 9, 2014 J. Haeling and M. Hauschild - Kalman Filters 1

Kalman Filters - tams.informatik.uni-hamburg.de · Kalman Filters Kalman Filters Jonas Haeling and Matthis Hauschild Universit at Hamburg Fakult at fur Mathematik, Informatik und

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  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Kalman Filters

    Kalman Filters

    Jonas Haeling and Matthis Hauschild

    Universität HamburgFakultät für Mathematik, Informatik und NaturwissenschaftenFachbereich Informatik

    Technische Aspekte Multimodaler Systeme

    November 9, 2014

    J. Haeling and M. Hauschild - Kalman Filters 1

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Kalman Filters

    Table of Contents

    1. Motivation

    2. The Discrete Kalman Filter

    3. Model Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    4. Extended Kalman Filter

    5. Conclusion

    J. Haeling and M. Hauschild - Kalman Filters 2

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Table of Contents

    1. Motivation

    2. The Discrete Kalman Filter

    3. Model Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    4. Extended Kalman Filter

    5. Conclusion

    J. Haeling and M. Hauschild - Kalman Filters 3

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Robot localization scenario

    I A robot drives along a one dimensional roadI It localizes itself using

    I OdometryI Sonar sensor

    J. Haeling and M. Hauschild - Kalman Filters 4

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Current estimation of position

    J. Haeling and M. Hauschild - Kalman Filters 5

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Current estimation of position

    J. Haeling and M. Hauschild - Kalman Filters 6

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Current estimation of position

    J. Haeling and M. Hauschild - Kalman Filters 7

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Current estimation of position

    J. Haeling and M. Hauschild - Kalman Filters 8

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Current estimation of position

    J. Haeling and M. Hauschild - Kalman Filters 9

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    History of the Kalman Filter[5]

    The Kalman Filter is a linear filter producing an optimal estimateof the system state using noisy input data

    I Named after Rudolf Emil KálmánI Born 1930 in BudapestI Hungarian-US-American electrical engineer & mathematician

    I Invented in 1960 (with assistance from Richard Bucy)

    I First use: trajectory estimation in the Apollo program

    I Special case of non-linear filter by Stratonovich invented ealier

    J. Haeling and M. Hauschild - Kalman Filters 10

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Motivation Kalman Filters

    Applications of the Kalman Filter

    I Generally position estimationI Robotics: robot localization, (moving) object or human trackingI Military: navigation of missiles, submarinesI Aeronautics: position of a plane, attitude control of the ISS

    I Electronics: phase-locked loop

    I Computer graphics: stabilizing depth measurements, fittingBezier patches

    J. Haeling and M. Hauschild - Kalman Filters 11

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    The Discrete Kalman Filter Kalman Filters

    Table of Contents

    1. Motivation

    2. The Discrete Kalman Filter

    3. Model Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    4. Extended Kalman Filter

    5. Conclusion

    J. Haeling and M. Hauschild - Kalman Filters 12

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    MIN-FakultätFachbereich Informatik

    The Discrete Kalman Filter Kalman Filters

    The Discrete Kalman Filter[3][1]

    I Tries to estimate the state x ∈ Rn of a discrete-time controlledprocess

    I Is an optimal linear filterI Incorporates all available dataI Produces a statistically minimized error

    I Assumes white gaussian noise both for process prediction andmeasurement

    J. Haeling and M. Hauschild - Kalman Filters 13

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    The Discrete Kalman Filter Kalman Filters

    The Discrete Kalman Filter[3]

    I Time update projects the current state estimate ahead in time

    I Measurement update adjusts the projected estimate by anactual measurement at that time

    J. Haeling and M. Hauschild - Kalman Filters 14

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    The Discrete Kalman Filter Kalman Filters

    Time Update (“Predict”) - Step 1/2[3]

    State estimation

    x̂−k = A · x̂k−1 + B · uk−1

    I x̂−k : The observed state at timestep k

    I A: Relates the state at timestep k − 1 to the state at kI uk−1: Control input at timestep k − 1I B: Relates optional control input to state x

    J. Haeling and M. Hauschild - Kalman Filters 15

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    The Discrete Kalman Filter Kalman Filters

    Time Update (“Predict”) - Step 2/2[3]

    Error covariance projection

    P−k = A · Pk−1 · AT + Q

    I P−k : A priori estimate error covariance

    I Pk : A posteriori estimate error covariance

    I A: Relates the state at timestep k − 1 to the state at kI Q: Process noise covariance

    J. Haeling and M. Hauschild - Kalman Filters 16

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    MIN-FakultätFachbereich Informatik

    The Discrete Kalman Filter Kalman Filters

    Time Update (“Predict”) Recap[3]

    J. Haeling and M. Hauschild - Kalman Filters 17

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    The Discrete Kalman Filter Kalman Filters

    Measurement Update (“Correct”) - Step 1/3[3]

    Kalman Gain computation

    Kk = P−k · H

    T · (H · P−k · HT + R)−1

    I Kk : Controls the influence of the measurement on the aposteriori state estimation at timestep k

    I P−k : A priori estimate error covariance

    I H: Relates measurement to state

    I R: Measurement noise covariance

    J. Haeling and M. Hauschild - Kalman Filters 18

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    The Discrete Kalman Filter Kalman Filters

    Measurement Update (“Correct”) - Step 2/3[3]

    State estimation update with measurement

    x̂k = x̂−k + Kk(zk − H · x̂

    −k )

    I x̂k : A posteriori state estimate

    I x̂−k : The observed state at timestep k

    I Kk : Controls the influence of the measurement on the aposteriori state estimation at timestep k

    I zk : Measurement at timestep k

    I H: Relates measurement to state

    J. Haeling and M. Hauschild - Kalman Filters 19

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    The Discrete Kalman Filter Kalman Filters

    Measurement Update (“Correct”) - Step 3/3[3]

    Error covariance update

    Pk = (I − Kk · H) · P−k

    I Pk : A posteriori estimate error covariance

    I I: Identity matrix

    I Kk : Controls the influence of the measurement on the aposteriori state estimation at timestep k

    I H: Relates measurement to state

    I P−k : A priori estimate error covariance

    J. Haeling and M. Hauschild - Kalman Filters 20

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    The Discrete Kalman Filter Kalman Filters

    Measurement Update (“Correct”) Recap[3]

    J. Haeling and M. Hauschild - Kalman Filters 21

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    The Discrete Kalman Filter Kalman Filters

    Operation of the Kalman Filter[3]

    J. Haeling and M. Hauschild - Kalman Filters 22

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    The Discrete Kalman Filter Kalman Filters

    Summary: The Discrete Kalman Filter

    I Is used for combining noisy data

    I Is an optimal filter

    I Has a cyclic recursive approach

    I Assumes white gaussian noise

    I Predicts an estimate of the current state x̂ with a measurementscaled through the Kalman gain K

    J. Haeling and M. Hauschild - Kalman Filters 23

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    Model Process of a Kalman Filter Kalman Filters

    Table of Contents

    1. Motivation

    2. The Discrete Kalman Filter

    3. Model Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    4. Extended Kalman Filter

    5. Conclusion

    J. Haeling and M. Hauschild - Kalman Filters 24

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    Model Process of a Kalman Filter Kalman Filters

    Model Definition Process[2]

    The Kalman Filter removes noise by assuming a pre-defined modelof a system.

    1. Understand the situation

    2. Model the state process

    3. Model the measurement process

    4. Model the noise

    5. Test the filter

    6. Refine filter

    J. Haeling and M. Hauschild - Kalman Filters 25

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    1. Understand the situation[2]

    I Task: Measure the level of water in a tank

    I Measurements obtained via floating device

    I Average water level could be changing or static

    I Water could be sloshing or stagnant

    J. Haeling and M. Hauschild - Kalman Filters 26

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    2. Model the state process[2]

    I Water level L is constant

    I State x̂k is the estimate of LI Constant model:

    I Ak is 1 for any k ≥ 0I Control variables B and u are 0

    Reminder: Time Update

    x̂−k = A · x̂k−1 + B · uk−1P−k = A · Pk−1 · A

    T + Q

    J. Haeling and M. Hauschild - Kalman Filters 27

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    3. Model the measurement process[2]

    I Float gives us the measurement zkI Measurement scale is the same scale as state estimate

    → H = 1

    Reminder: Measurement update

    Kk = P−k · H

    T · (H · P−k · HT + R)−1

    x̂k = x̂−k + Kk(zk − H · x̂

    −k )

    Pk = (I − Kk · H) · P−k

    J. Haeling and M. Hauschild - Kalman Filters 28

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    4. Model the noise[2]

    I Error due to process→ Process variance matrix Q = q

    I Noise from the measurement→ Measurement variance matrix R = r

    I Noise from the estimation→ State variance matrix Pk = p (scalar)

    J. Haeling and M. Hauschild - Kalman Filters 29

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    5. Test the filter[2]

    I Simplified equations:

    Predict

    x̂−k = x̂k−1P−k = Pk−1 + q

    Update

    Kk = P−k · (P

    −k + r)

    −1

    x̂k = x̂−k + Kk(zk − x̂

    −k )

    Pk = (1− Kk) · P−k

    I Filter is completely defined, let’s test it!

    J. Haeling and M. Hauschild - Kalman Filters 30

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    MIN-FakultätFachbereich Informatik

    Model Process of a Kalman Filter - Constant Model Kalman Filters

    5. Test the filter[2]

    I True water level L = 1I Start state x0 arbitrarly initialized to 0I Start variance P0 is 1000, system noise q = 0.0001,

    measurement noise r = 0.1 (z1 = 0.9)

    Predict

    x̂−1 = 0P−1 = 1000 + 0.0001 = 1000.0001

    Update

    K1 = 1000.0001 ∗ (1000.0001 + 0.1)−1 = 0.9999x̂1 = 0 + 0.9999 ∗ (0.9− 0) = 0.8999P1 = (1− 0.9999) ∗ 1000.0001 = 0.1000

    J. Haeling and M. Hauschild - Kalman Filters 31

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    5. Test the filter[2]

    I Another step:

    Predict

    x̂−2 = 0.8999P−2 = 0.1000 + 0.0001 = 0.1001

    I Hypothetical measurement of z2 = 0.8

    Update

    K2 = 0.1001 ∗ (0.1001 + 0.1)−1 = 0.5002x̂2 = 0.8999 + 0.5002 ∗ (0.8− 0.8999) = 0.8499P2 = (1− 0.5002) ∗ 0.1001 = 0.0500

    J. Haeling and M. Hauschild - Kalman Filters 32

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    5. Test the filter[2]

    t x̂−k P−k zk Kk x̂k Pk

    3 0.8499 0.0501 1.1 0.3339 0.9334 0.03344 0.9334 0.0335 1 0.2509 0.9501 0.02515 0.9501 0.0252 0.95 0.2012 0.9501 0.02016 0.9501 0.0202 1.05 0.1682 0.9669 0.01687 0.9669 0.0169 1.2 0.1447 1.0006 0.01458 1.0006 0.0146 0.9 0.1272 0.9878 0.01279 0.9878 0.0128 0.85 0.1136 0.9722 0.0114

    10 0.9722 0.0115 1.15 0.1028 0.9905 0.0103

    J. Haeling and M. Hauschild - Kalman Filters 33

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    Model Process of a Kalman Filter - Constant Model Kalman Filters

    5. Test the filter[2]

    J. Haeling and M. Hauschild - Kalman Filters 34

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    Model Process of a Kalman Filter - Filling Tank Kalman Filters

    Filling Tank Model[2]

    I A 20 % error produced a 5 % inaccuracy

    I But what if the true situation is not static?→ Static model, but the tank is filling at a constant rate

    I Tank level at time k: Lk = Lk−1 + f

    I Filling rate f = 0.1 per time step

    I Tank level starts at L0 = 0

    I Measurement and process noise remains the same

    I Let’s see what happens!

    J. Haeling and M. Hauschild - Kalman Filters 35

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    Model Process of a Kalman Filter - Filling Tank Kalman Filters

    Filling Tank model with q = 0.0001 and r = 0.1[2]

    t x̂−k P−k zk Kk x̂k Pk L

    3 - - - - 0 1000 01 0.0000 1000.0001 0.11 0.9999 0.1175 0.1000 0.12 0.1175 0.1001 0.29 0.5002 0.2048 0.0500 0.23 0.2048 0.0501 0.32 0.3339 0.2452 0.0334 0.34 0.2452 0.0335 0.50 0.2509 0.3096 0.0251 0.45 0.3096 0.0252 0.58 0.2012 0.3642 0.0201 0.56 0.3642 0.0202 0.54 0.1682 0.3945 0.0168 0.6

    J. Haeling and M. Hauschild - Kalman Filters 36

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    Model Process of a Kalman Filter - Filling Tank Kalman Filters

    Filling Tank model with q = 0.0001 and r = 0.1[2]

    J. Haeling and M. Hauschild - Kalman Filters 37

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    Model Process of a Kalman Filter - Filling Tank Kalman Filters

    Filling Tank model with q = 0.01 and r = 0.1[2]

    J. Haeling and M. Hauschild - Kalman Filters 38

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    Model Process of a Kalman Filter - Filling Tank Kalman Filters

    Filling Tank model with q = 0.1 and r = 0.1[2]

    J. Haeling and M. Hauschild - Kalman Filters 39

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    Model Process of a Kalman Filter - Filling Tank Kalman Filters

    Filling Tank model with q = 1 and r = 0.1[2]

    J. Haeling and M. Hauschild - Kalman Filters 40

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    Model Process of a Kalman Filter - Filling Model Kalman Filters

    A Filling Model[2]

    I You can relax a model by increasing your estimated error

    I But a bad model will not give good estimates!

    2. Model the state process

    I State x = (xl , xf )T where xl is the estimated level and xf the

    estimated filling rate

    I Ak =

    (1 ∆k0 1

    )represents the filling tank with timestep k

    I B and u still ignored

    J. Haeling and M. Hauschild - Kalman Filters 41

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    Model Process of a Kalman Filter - Filling Model Kalman Filters

    A Filling Model[2]

    I Cannot measure filling rate

    I But noisy measurement of L

    3. Model the measurement process

    I Scaling remains the same: H =(1, 0

    )I z =

    (z , 0

    )T

    J. Haeling and M. Hauschild - Kalman Filters 42

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    Model Process of a Kalman Filter - Filling Model Kalman Filters

    A Filling Model[2]

    4. Model the noiseI Measurement process is unchanged: R = r

    I State process is changed:

    I Estimate error covariance no longer scalar: P =

    (pl plfplf pf

    )I Discrete noise model: Q =

    (qf /3 qf /2qf /2 qf

    )with filling noise qf

    (Q derived from the continuous Q, skipped here)

    J. Haeling and M. Hauschild - Kalman Filters 43

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    Model Process of a Kalman Filter - Filling Model Kalman Filters

    A Filling Model[2]

    5. Test the modelI Measurement noise r = 0.1

    I Process noise of qf = 0.0001, which is quite accurate

    I Initial state x0 =(0, 0

    )TI Initial variance P0 =

    (1000 0

    0 1000

    )I True filling rate f = 0.1 per timestep

    J. Haeling and M. Hauschild - Kalman Filters 44

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    Model Process of a Kalman Filter - Filling Model Kalman Filters

    A Filling Model - example[2]

    J. Haeling and M. Hauschild - Kalman Filters 45

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    Model Process of a Kalman Filter - Filling Model Kalman Filters

    A Filling Model with a constant level[2]

    J. Haeling and M. Hauschild - Kalman Filters 46

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    Model Process of a Kalman Filter - Non-linear Model Kalman Filters

    Constant, but sloshing model[2]

    I Another model: Water level is constant, but it is sloshing

    I Sloshing modeled as a sine wave:

    L = c ∗ sin(2 ∗ π ∗ r ∗∆k) + lc : scales the amplituder : cycle ratel : average level

    I We use c = 0.5, r = 0.05, l = 1

    I What do you notice?

    J. Haeling and M. Hauschild - Kalman Filters 47

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    Model Process of a Kalman Filter - Non-linear Model Kalman Filters

    Constant, but sloshing model - example[2]

    J. Haeling and M. Hauschild - Kalman Filters 48

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    Model Process of a Kalman Filter - Summary Kalman Filters

    Summary of the Three Tank Examples[2]

    Six steps for defining a Kalman Filter model:

    1. Understand the situation 2. Model the state process3. Model the measurement process 4. Model the noise5. Test the filter 6. Refine filter

    I Filter will fit measurements to provided model

    I May not always be desirable (sloshing could be just noise)

    I Initialization and noise components affect the results

    I Think of the outcome of your filter (linear model works, butlags)

    I An Extended Kalman filter is required to model non-linearitycorrectly

    J. Haeling and M. Hauschild - Kalman Filters 49

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    Extended Kalman Filter Kalman Filters

    Table of Contents

    1. Motivation

    2. The Discrete Kalman Filter

    3. Model Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    4. Extended Kalman Filter

    5. Conclusion

    J. Haeling and M. Hauschild - Kalman Filters 50

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    Extended Kalman Filter Kalman Filters

    Extended Kalman Filter[3]

    I Previously: linear stochastic difference equation

    I Process or measurement may be non-linear

    I KF that linearizes about the current mean and covariance iscalled an Extended Kalman Filter

    I Uses partial derivatives of the process and measurementfunction

    Process with state x ∈ Rn

    xk = f (xk−1, uk−1,wk−1)

    Measurement with z ∈ Rm

    zk = h(xk , vk)

    J. Haeling and M. Hauschild - Kalman Filters 51

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    Extended Kalman Filter Kalman Filters

    EKF Time Update Equations[3]

    State estimation

    x̂−k = f (x̂k−1, uk−1, 0)

    Error covariance projection

    P−k = Ak · Pk−1 · ATk + Wk · Qk−1 ·W Tk

    I Jacobian matrix of partial derivatives of f with respect to x

    A[i ,j] =δf[i ]δx[j]

    (x̂k−1, uk−1, 0)

    I Jacobian matrix of partial derivatives of f with respect to w

    W[i ,j] =δf[i ]δw[j]

    (x̂k−1, uk−1, 0)

    J. Haeling and M. Hauschild - Kalman Filters 52

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    Extended Kalman Filter Kalman Filters

    EKF Measurement Update Equations[3]

    Kalman Gain Computation

    Kk = P−k · H

    TK · (Hk · P

    −k · H

    Tk + Vk · Rk · V Tk )−1

    State estimation update with measurement

    x̂k = x̂−k + Kk · (zk − h(x̂

    −k , 0))

    Error covariance update

    Pk = (I − KkHk) · P−k

    H[i ,j] =δh[i ]δx[j]

    (x̃k , 0) and V[i ,j] =δh[i ]δv[j]

    (x̃k , 0)

    x̃ : approximate state, w : process noise, v : measurement noiseJ. Haeling and M. Hauschild - Kalman Filters 53

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    Extended Kalman Filter Kalman Filters

    Predict-Correct-Cycle of EKF[3]

    J. Haeling and M. Hauschild - Kalman Filters 54

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    Extended Kalman Filter Kalman Filters

    Summary - Extended Kalman Filter[4][3]

    I EKF are needed when you either have a non-linear process ormeasurement relationship

    I Uses function f for difference equation and function h for themeasurement equation

    I No longer optimal estimator (only in linear cases)

    I Considered by some as the de facto standard for non-linearstate estimation

    I Heavily used in navigation systems and GPS

    J. Haeling and M. Hauschild - Kalman Filters 55

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    Conclusion Kalman Filters

    Table of Contents

    1. Motivation

    2. The Discrete Kalman Filter

    3. Model Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    4. Extended Kalman Filter

    5. Conclusion

    J. Haeling and M. Hauschild - Kalman Filters 56

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    Conclusion Kalman Filters

    Advantages and Disadvantages

    I AdvantagesI Optimal if you have a linear system with Gaussian noiseI Recursive ⇒ Real-time capableI EKF can handle non-linearityI Relatively easy to useI Wide use in practice speaks for itself

    I DisadvantagesI Loses optimality in non-linear systemsI Unimodal because of Gaussians ⇒ Only one hypothesisI Models may be too complex⇒ Sensivity analysis required because of imprecisions⇒ Or Kalman Filters may not be useful at all

    J. Haeling and M. Hauschild - Kalman Filters 57

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    Conclusion Kalman Filters

    Comparison to other filters

    Particle Filter or Sequential Monte Carlo methods

    I Estimate density represented with particles ⇒ MultimodalI Does not require Gaussian noise

    I Often used in complex non-linear models

    I Large state space dimensionality requires lots of particles

    Hybrid

    I Particle Filter superior for dealing with multi-modal data

    I EKF superior for dealing with updates with little noise

    I Use PF until variance is below a certain level, switch to KF

    J. Haeling and M. Hauschild - Kalman Filters 58

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    Conclusion Kalman Filters

    Conclusion

    I The Kalman Filter is a good and easy to use filter to get morereliable output from your sensors

    I The recursive approach makes it usuable for real-time purposessuch as in robots

    I But: you have to be able to describe the underlying modelproperly

    J. Haeling and M. Hauschild - Kalman Filters 59

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Conclusion Kalman Filters

    Thank you for your attention!

    Jonas Haeling and Matthis [email protected] and

    [email protected]

    Universität HamburgFakultät für Mathematik, Informatik und NaturwissenschaftenFachbereich Informatik

    Technische Aspekte Multimodaler Systeme

    J. Haeling and M. Hauschild - Kalman Filters 60

  • Universität Hamburg

    MIN-FakultätFachbereich Informatik

    Conclusion Kalman Filters

    Bibliography

    [1] Peter Maybeck. Stochastic models, estimation, and control. AirForce Institute of Technology, 1979.

    [2] Ashutosh Saxena. Kalman Filter Applications. CornellUniversity, 2008. http://www.cs.cornell.edu/courses/cs4758/2012sp/materials/mi63slides.pdf.

    [3] G. Welch and G. Bishop. An Introduction to the Kalman Filter.University of North Carolina at Chapel Hill, 2006.

    [4] Wikipedia. Extended Kalman filter, 2014. http://en.wikipedia.org/wiki/Extended_Kalman_filter.

    [5] Wikipedia. Kalman filter, 2014.http://en.wikipedia.org/wiki/Kalman_filter.

    J. Haeling and M. Hauschild - Kalman Filters 61

    http://www.cs.cornell.edu/courses/cs4758/2012sp/materials/mi63slides.pdfhttp://www.cs.cornell.edu/courses/cs4758/2012sp/materials/mi63slides.pdfhttp://en.wikipedia.org/wiki/Extended_Kalman_filterhttp://en.wikipedia.org/wiki/Extended_Kalman_filterhttp://en.wikipedia.org/wiki/Kalman_filter

    MotivationThe Discrete Kalman FilterModel Process of a Kalman FilterConstant ModelFilling TankNon-linear ModelSummary

    Extended Kalman FilterConclusion