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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

  • 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

  • Acceleration
  • Why is it not enough?
  • 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)


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


A gyroscope measures a 3D angular velocity plus an offset and white measurement noise in the sensor co-ordinate frame

  • The spectrum of the gyroscope offset has a low cutoff frequency in comparison with the bandwidth of the kinematic signals that are to be measured

A 3D accelerometer measures acceleration minus gravity and a white noise component, all in the sensor co-ordinate frame

  • The acceleration of a body segment in the global system can be described as low pass filtered white noise
inclination estimated from gyroscope
Inclination estimated from gyroscope

Remove offset

Strapdown integration



inclination estimated from accelerometer
Inclination estimated from accelerometer

Remove body acceleration





what assumptions can we make
What assumptions can we make?
  • 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