17. Extended Kalman Filter

How to Perform a Taylor Expansion

The general form of a Taylor series expansion of an equation, $f(x)$ , at point $\mu$ is as follows:

$f(x) \approx f(\mu) + \frac{\partial f(\mu)}{\partial x} ( x - \mu)$

Simply replace $f(x)$ with a given equation, find the partial derivative, and plug in the value $\mu$ to find the Taylor expansion at that value of $\mu$ .

See if you can find the Taylor expansion of $arctan(x)$ .

Let’s say we have a predicted state density described by

$\mu = 0$ and $\sigma = 3$ .

The function that projects the predicted state, $x$ , to the measurement space $z$ is

$h(x) = arctan(x)$ .

and its partial derivative is

$\partial h = 1/(1+ x^2)$ .

I want you to use the first order Taylor expansion to construct a linear approximation of $h(x)$ to find the equation of the line that linearizes the function $h(x)$ at the mean location $\mu$ .

A) $h(x) \approx x$

B) $h(x) \approx 1/(1+x^2)$

C) $h(x) \approx x + arctan(x)$

D) $h(x) \approx 3 + x$