State variable of a mobile robot, operating in planar environment, is its x$-$y location and its global

heading direction $.$ Suppose we know x and y with high certainty, but the orientation $$ is unknown.

This reflected by our initial estimate:

$$

$0\mathrm{.}0100$

$000$ and $00\mathrm{.}010$

$0010000$

$$

$$

$==$

$$

a$)$ Draw, graphically, the posterior over the robot pose after robot moves d$=1$ units forward. For

this exercise the expected location of the robot after motion:

cos

sin

x x

y y

$$

$$

$$

$+$

$=+$

$$

b$)$ Now develop this motion into a prediction step for EKF. $($for that you have to generate a

state transition function and linearize it$.)$ Then generate Gaussian estimate of the robot pose

using the linearized model.

c$)$ Draw the uncertainty ellipse of the Gaussian.