Unscented Kalman Filter

As we've seen in the previous chapter, when the State Transition model \( \boldsymbol{f}(\boldsymbol{x}) \) and Observation model \( \boldsymbol{h}(\boldsymbol{x}) \) are close to linear, the EKF performance is satisfying. However, when \( \boldsymbol{f}(\boldsymbol{x}) \) or \( \boldsymbol{h}(\boldsymbol{x}) \) models are highly non-linear, the linearization error can cause estimations that are significantly different from the true value of the state and estimation uncertainties that don't capture the true uncertainties in the state.

The Unscented Kalman Filter is an alternative approach to linearization. While Extended Kalman Filter treats the non-linearity using analytical linearization, the Unscented Kalman Filter performs statistical linearization based on a set of rules.

What is the meaning of the name "unscented"?

A running joke was made that "unscented" is a contrast to "scented," meaning the EKF performance is "stinky."

UKF creator Jeffrey Uhlmann explained that "unscented" was an arbitrary name he adopted to avoid being referred to as the "Uhlmann Filter."

"Initially, I only referred to it as the "new filter." Needing a more specific name, people in my lab began referring to it as the "Uhlmann filter," which obviously isn't a name that I could use, so I had to come up with an official term. One evening everyone else in the lab was at the Royal Opera House, and as I was working, I noticed someone's deodorant on a desk. The word "unscented" caught my eye as the perfect technical term."

Jeffrey Uhlmann also says:

"What was most striking to people about the UT was not the accuracy so much as the ease with which it could be implemented. There was no longer a need to derive a linearized approximation which would then have to be coded up for use in the filter."

The Unscented Transform (UT)

The Unscented Transform is a method for calculating the statistics of a random variable that undergoes a non-linear transformation.

The Unscented Transform includes four steps:

  • Step 1 - Select a set of points from the input distribution.
  • Step 2 - Propagate each selected point through the non-linear function, producing a new set of points belonging to the output distribution.
  • Step 3 - Compute sigma points weights.
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