Kalman Filter I've got it! Sequential Bayes Filtering is a - - PowerPoint PPT Presentation
Kalman Filter I've got it! Sequential Bayes Filtering is a general approach to state estimation that gets used all over the place. But, implementations like histogram filters or Kalman filters are computationally complex. Is it
approach to state estimation that gets used all over the place.
Kalman filters are computationally complex.
simple?
I've got it!
Kalman filters are particularly well suited for tracking moving objects In general: state estimation
Image: Thrun et al., Probabilistic Robotics Image: UBC, Kevin Murphy Matlab toolbox
Process update: Measurement update:
Image: Thrun et al., Probabilistic Robotics
Recall the state transition function: – this probability can be expressed as a table: Current state Next state Can also be expressed as a function:
A linear system is any system where:
(technically, this is a linear Gaussian system)
Also written:
How get it into this form? Equation of motion:
How get it into this form? Equation of motion: Integrate forward one timestep:
A linear system is any system where:
(technically, this is a linear Gaussian system)
Also written: Also, assume that the observation function is linear Gaussian:
update initial position x y x y prediction x y measurement x y
Image: Thrun et al., CS233B course notes
Image: Thrun et al., CS233B course notes
prior Measurement evidence posterior
Image: Thrun et al., CS233B course notes
Univariate Gaussian: Multivariate Gaussian:
Suppose: Calculate:
x y x y
Suppose: Then:
Image: Thrun et al., CS233B course notes
Suppose: Then: Marginal distribution
Process update (discrete): Process update (continuous):
Process update (discrete): Process update (continuous): prior Transition dynamics
Process update (discrete): Process update (continuous): prior Transition dynamics
Observation update: Where:
Observation update: Where:
Observation update: Where:
But we need:
Suppose: Calculate:
But we need:
Prior: Process update: Measurement update: System:
Prior: Process update: Measurement update: System:
Prior: Process update: Measurement update: This factor is often called the “Kalman gain”
Process update: Measurement update: – covariance update is independent of observation – Kalman is only optimal for linear-Gaussian systems – the distribution “stays” Gaussian through this update – the error term can be thought of as the different between the
Image: Thrun et al., CS233B course notes
prior Measurement evidence posterior
Image: Thrun et al., CS233B course notes
Process update: Measurement update: System:
Image: Thrun et al., CS233B course notes
initial position prediction measurement
x
x
x
next prediciton
x
update
x
Image: Thrun et al., CS233B course notes
past measurements prediction
Level of tank Fill rate Process: Observation:
What should I do?
What should I do? Well, there are some options...
What should I do? Well, there are some options... But none of them are great.
What should I do? Well, there are some options... But none of them are great. Here's one: the Extended Kalman Filter
Take a Taylor expansion: Where: Where:
Take a Taylor expansion: Where: Where: Then use the same equations...
Prior: Process update: Measurement update:
Image: Thrun et al., CS233B course notes
Image: Thrun et al., CS233B course notes