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    AEE3305_Tut3_Sensor_Fusion

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    Glasgow tutorial paper

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    AEE3305_Tut3_Sensor_Fusion

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    aipssyahmi
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    Glasgow tutorial paper

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    Glasgow tutorial paper

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    1 views2 pages

    AEE3305_Tut3_Sensor_Fusion

    Uploaded by

    aipssyahmi
    Glasgow tutorial paper

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
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    Tutorial 3 AEE3305 Flight Systems: Sensor Fusion & Kalman Filtering

    1) One frequently used version of the constant acceleration family of dynamic models used in target tracking is
    the white-noise jerk model given by,
    𝑎̇ (𝑡) = 𝑤(𝑡)
    a) What is the discrete-time state transition matrix for this model?
    b) What is the full state-update equation?

    2) From the definition of error covariance,


    𝑃𝑘 = 𝐸[(𝐱𝑘 − 𝐱̂𝑘 )(𝐱𝑘 − 𝐱̂𝑘 )T ]
    Derive the covariance update equation.

    3) Considering a Kalman Filter tracking the error Δ𝑥 = (𝑥 − 𝑥̂) of estimate 𝑥̂ of the state 𝑥.
    a) Sketch the block diagram for the output-injection state observer.
    b) Show that the Kalman Filter error dynamics for a linear system are given by,
    𝛥𝒙𝑘 = (𝐴 − 𝐾𝐶)𝛥𝒙𝑘−1

    4) It is proposed that a Kalman Filter is designed to observe the states of the following system. Determine if
    such an observer is possible by evaluating the observability matrix.
    𝑥1 1 0 𝑥1 0
    [𝑥 ] = [ ] [𝑥 ] + [ ]𝑢
    2 𝑘 0 1 2 𝑘−1 1
    𝑥1
    𝑦 = [1 0] [𝑥 ]
    2 𝑘

    5) Assuming a stabilised platform, derive a two-state Kalman Filter that estimates the 1D position and velocity
    of a vehicle taking 1D position measurements at 1Hz. Supposing:
    • The initial position and velocity estimates are zero.
    • The initial position and velocity uncertainties have standard deviation (SD) of 10 m and √10
    m/s, respectively, and all covariances between states are initially zero.
    • The first four position measurements are 212, 218, 232 and 241 m.
    • The measurement noise uncertainty is √0.5 m SD and process noise has SD of 1 m and √0.1
    m/s for position and velocity uncertainties.
    What are the estimated position and velocity and their associated error covariance after 2s?

    Tutorial 3 AEE3305 Flight Systems: Sensor Fusion & Kalman Filtering


    University of Glasgow Singapore, last modified: 05/06/2024 1
    6) Assume a Kalman Filter is used to estimate the dynamics of a simple 1D system with position and velocity
    as the states. The matrices for state transition (𝐴), process & measurement noise covariance (𝑄, 𝑅), initial
    conditions of the estimator (𝑥̂0 , 𝑃̂0 ) and measurement matrix (𝐶) are given by,
    1 0.1 0 0
    𝐴=[ ], 𝑄=[ ], 𝑅 = 0.1
    0 1 0 0
    0 1.0 0.1
    𝐱0 = [ ] , 𝑃̂0 = [ ], 𝐶 = [1 0]
    2 0.1 1.0
    Using the equations for the Kalman Filter available in the data sheet,
    a) Calculate the Kalman Gain 𝐾1 on the first iteration 𝑘 = 1.
    b) Calculate the a posteriori state estimate 𝑥̂1 on the first iteration assuming a position measurement of
    𝑦1 = 2.2.
    c) Calculate the a posteriori error covariance matrix 𝑃1 after the first iteration 𝑘 = 1.
    d) Using the Kalman Gain calculated in part (a), derive an equation for the error dynamics and determine
    whether the observer will converge.

    7) Sketch the block diagrams for a loosely coupled open-loop Kalman Filter used to provide an integrated
    INS/GNSS solution.

    Tutorial 3 AEE3305 Flight Systems: Sensor Fusion & Kalman Filtering


    University of Glasgow Singapore, last modified: 05/06/2024 2

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