15.2 Error Propagation

Consider the distance traveled by. differential wheeled robot given the rotations of its wheels.

We have the expected (commanded) Displacement + unexpected slips, which we can split into radial and tangenet displacement.

As $t$ grows, the uncertainty of the position grows as well. Error will propagate faster orthogonally to the robots direction of motion.

How about using a sensor to correct this? those are noisy too! But lets do it anyway.

The variance of each componenet in a random variable has a weight proportional to how much it contributes to the final value. Time for a partial derivative!

let $y=f(x)$ be a function that maps $x$ (a sensor reading) to a random variable $y$. ${\sigma }_x$ is the standard deviation of $x$

${\sigma }_y=(\frac{\delta f}{\delta x}){\sigma }_x$ and the variance:

${\sigma }_y^2=(\frac{\delta f}{\delta x}{)}^2{\sigma }_x^2$

if $y=f(x)$ is multi-variable:

${\Sigma }^Y=J{\Sigma }^X{J}^T$ where ${\Sigma }^Y$ and ${\Sigma }^X$ are $n\times n$ and $m\times m$ matrices holding the variances of the input and output and $J$ is a Jacobian Matrix.

This chapter explains some math for Line Fitting and Odometry and its pretty interesting, but I’m skipping it because if I really need to reference this later, I’m better off looking at the textbook.