Uniform Circular robot motion

Question

Here is the initial diagram:

The robot is moving counter-clockwise around the circle.

I am having trouble seeing how they derived the centers of the circles. They have defined the center of a circle as:

$\ x_c = x - (v/\omega)sin(\theta) $

$\ y_c = y + (v/\omega)cos(\theta) $

I always thought that the center of a circle was:

$\ x_c = x - (v/\omega)cos(\theta) $

$\ y_c = y - (v/\omega)sin(\theta) $

So how did they derive their equations for the center of a circle?

Answer 1

This is simply because you are used to the regulardefinition of $\theta$ as being measured from the positive direction of the x-axis, as opposed to its definition here.