Date: January 20, 2024

Topic: Kalman Filters

Recall

Notes

Kalman Filter Introduction

• Kalman filters produce continuous and uni-modal distributions during tracking
• Future locations and velocities are predicted based on measurements from past data
• Sensors (e.g., lidars) provide such information for future location predictions

Application in Robotics

• Iterative loop between measurement and motion
• For Kalman filters we iterate between measurement updates and predictions (motion)
• Measurement: Bayes rule (product)
• Motion: LOTP (convolution/sum)

Gaussian distribution is a continuous and unimodal distribution depending on the mean and variance of the graph

The larger the variance, the wider the graph becomes. To get the maximum value of the graph, set $x=\mu$

The certainty of the distribution is shown by the wideness of the curve, with a narrower curve having a more certain distribution

Gaussian Intro

• For Kalman filters, we have a Gaussian distribution (continuous function over the space of locations)
• The area under a Gaussian curve sums up to 1
• We want to localize based on the mean and variance

Gaussian Distribution:

• Gaussian curves have an exponential drop-off at both side and a single peak (uni-modal)

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} $$

• The first term is a normalizing constant, and the second term determines the shape of the graph
• Looking at the second term, $x-\mu$, is being normalized by the variance, $\sigma$
• Hence, the larger the variance $\sigma$, the smaller the exponential term since contribution from $x-\mu$ decreases
• This leads to the function being more spread out
• Maximum value: This is obtained by setting $x=\mu$
• Note: The ends of the Gaussian must tend to 0 as if $x \to \inf$, the exponent term $\to$ 0

Distribution certainties

• The wider curve, the higher the uncertainty (orange graph is most uncertain)

<aside> 📌 SUMMARY: Gaussian distribution helps us to model uncertainties which are unimodal and continuous. The $\mu$ term determines where the center of the graph is, and the $\sigma$ term determines the wideness of the curve

</aside>

Date: January 20, 2024

Topic: Shifting the Mean

Recall

Notes

Gaussian in Motion and Measurement Updates

• When we have 2 different Gaussians, we can multiply them
  • e.g., we have a prior belief (Gaussian 1) and then measure something (Gaussian 2)

Green graph is the result of multiplying the blue and black Gaussians

• With the resultant Gaussian, we have the following:
  • $\mu$ in between the 2 previous Gaussians
  • Higher peak than the 2 previous Gaussian
• The higher peak is because we are actually gaining information from the measurement.

<aside> 📌 SUMMARY: Kalman filters give an approximation of the future location via measuring first then moving

</aside>