The Unscented Kalman Filter for Nonlinear Estimation Young Ki Baik

References An Introduction to the Kalman Filter –Greg Welch and Gary Bishop (TR 2004) A New Extension of the Kalman Filter to Nonlinear –Simon J. Julier Jefferey K. Uhlmann (1997) The Unscented Kalman Filter for Nonlinear Estimation –Eric A. Wan et al. (2001)

Contents Kalman Filter Extended Kalman Filter Unscented Kalman Filter Conclusion

The Kalman Filter Example (Simple Gaussian form) –Assumption All error form Gaussian noise –Estimated value –Measurement value

The Kalman Filter Example (Simple Gaussian form) –Optimal variance –Optimal mean Kalman gain Innovation

The Kalman Filter The process to be estimated –Process –Measurement Process noise Measurement noise State Measurement

The Kalman Filter The process to be estimated –Priori estimate error –Posteriori estimate error Priori state Posteriori state

The Kalman Filter The process to be estimated –Kalman gain –Posteriori state estimate Innovation or residual

The Kalman Filter Overall process –Time Update (“predict”) –Measurement Update (“correct”)

Estimation in nonlinear systems Kalman filter –Linear system (O) –Non-linear system (X) Extended Kalman filter –Linearized version of Kalman filter –Non-linear system (O)

The Extended Kalman Filter The Transformation of Uncertainty – is a random variable with mean covariance –We wish to calculated the mean covariance Consistency condition

The Extended Kalman Filter The Taylor series expansion (Linearization)

The Extended Kalman Filter Overall process –Time Update (“predict”) –Measurement Update (“correct”)

The Extended Kalman Filter Example –Beacons location –Real location (0,1) –Accuracy Range (2cm standard deviation) Angle (15degree standard deviation) sensor r

The Extended Kalman Filter Example –True : Monte Carlo simulation with extremely large number of samples Linearization error

The Unscented Kalman Filter The Basic Idea of Unscented Transform Sigma point

The Unscented Kalman Filter The Unscented Transform –The n-dimensional random variable with mean covariance is approximated by 2n+1 weighted point by

The Unscented Kalman Filter The Unscented Transform –Instantiate each point through the function to yield the set of transformed sigma points –The mean and covariance are given by the weighted average and the weighted outer product of the transformed points,

The Unscented Kalman Filter Example =2 nd order accuracy

The Unscented Kalman Filter Overall process –Time Update (“predict”)

The Unscented Kalman Filter Overall process –Measurement Update (“correct”)

The Unscented Kalman Filter

Summary The Extended Kalman Filter –Linearized version of Kalman filter –Non-linear system (O) –Linearization error problem The Unscented Kalman Filter –Approximates the distribution –Accurate to at least the 2 nd order. –No Jacobians or Hessian are calculated. –Efficient “sampling” approach. –The UKF consistently achieves a better level of accuracy than the EKF at a comparable level of complexity.