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Karman filter and attitude estimation. Lin Zhong ELEC424, Fall 2010. Yaw, pitch, and roll angles. Inclination. How z is represented in the sensor coordinate XYZ : S z. Orientation (attitude).

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Karman filter and attitude estimation

Karman filter and attitude estimation

Lin Zhong

ELEC424, Fall 2010



Inclination
Inclination

  • How z is represented in the sensor coordinate XYZ: Sz


Orientation attitude
Orientation (attitude)

  • How xyz is represented in the sensor coordinate XYZ: GSR=[ Sx, Sy, Sz]T

Ga = GSR·Sa


Accelerometer
Accelerometer

  • Acceleration

  • Why is it not enough?


Gyroscope
Gyroscope

  • Angular velocity

  • Why is it not enough?


Kalman filter
Kalman filter

  • What does it do?

    • Estimate the internal state x of a linear dynamic system from noisy measurements

  • How does it estimate it?

    • Linearity of the system

    • Statistical properties of the system and measurement

    • Recursive (dynamic programming)


The target system
The target system

  • The system evolves in discrete time steps

    • Fk is the state transition model

    • Bk is the control-input model

    • wk is the process noise, assumed to a zero mean multivariable normal distribution wk~N(0, Qk)

http://en.wikipedia.org/wiki/Kalman_filter


The measurement
The measurement

  • The measurement (observation) of the state xk is a linear function of xk

    • Hk is the measurement model

    • vk is the measurement noise, a zero mean multivariable normal distribution vk~N(0, Rk)


Discrete kalman filter
Discrete Kalman filter

  • Two variables are updated at each stage (k)

    • : state estimation given measurements up to and including time k

    • : error covariance matrix (how accurate is)


Recursive estimation
Recursive estimation

  • At time 0, and are known

  • Given them at k-1, Predict and Update &

Measurement residual

Residual covariance

Optimal Kalman gain




Inclination estimated from gyroscope
Inclination estimated from gyroscope white noise component, all in the sensor co-ordinate frame

Remove offset

Strapdown integration

GSRt

GSRt-1


Inclination estimated from accelerometer
Inclination estimated from accelerometer white noise component, all in the sensor co-ordinate frame

Remove body acceleration

SzA=Sgt/|Sgt|=

Predict

Rotate

GSR


What assumptions can we make
What assumptions can we make? white noise component, all in the sensor co-ordinate frame

  • Offset of gyroscope and accelerometer can be calibrated, automatically.

  • Roll and pitch are small

  • Body acceleration is small (if engines are controlled properly)

  • Goal

    • Yaw should be constant

    • Roll and pitch should be small


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