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Karman filter and attitude estimation

This paper presents a comprehensive approach to attitude estimation using the Kalman filter, focusing on yaw, pitch, and roll angles. It discusses the representation of orientation within the sensor coordinate system and the limitations of accelerometers and gyroscopes in isolation. The Kalman filter's role is emphasized as a method for recursively estimating the internal state of a dynamic system based on noisy measurements. Key concepts include state transition models, measurement noise, and optimal Kalman gain, offering valuable insights for sensor fusion and motion tracking applications.

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Karman filter and attitude estimation

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  1. Karman filter and attitude estimation Lin Zhong ELEC424, Fall 2010

  2. Yaw, pitch, and roll angles

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

  4. Orientation (attitude) • How xyz is represented in the sensor coordinate XYZ: GSR=[ Sx, Sy, Sz]T Ga = GSR·Sa

  5. Accelerometer • Acceleration • Why is it not enough?

  6. Gyroscope • Angular velocity • Why is it not enough?

  7. 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)

  8. 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

  9. 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)

  10. 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)

  11. Recursive estimation • At time 0, and are known • Given them at k-1, Predict and Update & Measurement residual Residual covariance Optimal Kalman gain

  12. 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

  13. 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

  14. Inclination estimated from gyroscope Remove offset Strapdown integration GSRt GSRt-1

  15. Inclination estimated from accelerometer Remove body acceleration SzA=Sgt/|Sgt|= Predict Rotate GSR

  16. 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

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