Bicycle model for a car

Even if you are not into robotics, the bicycle model will make your driver's license test easier. (I hope so :)

The content is taken from the text book Corke, Peter I., Witold Jachimczyk, and Remo Pillat. Robotics, vision and control: fundamental algorithms in MATLAB. Vol. 73. Berlin: Springer, 2011.

I (and some labmates)had been stuck with the derivative of $\alpha$ for some time. Thanks for my co-supervisor, Lionel Birglen, who points out that it is the derivative of $\arctan$ that confused us. So here it goes:

Assume the goal position is fixed.

Remember, $x$ and $y$ are the x-axis and y-axis position of the robot, so we have $\dot{x} = - \dot{\Delta x}$and $\dot{y} = - \dot{\Delta y}$. Then:

since $\dot{x} = v \cos \theta$ and $\dot{y} = v \sin\theta$

With $\Delta x = \rho \cos(\alpha+\theta)$and $\Delta y = \rho \sin(\alpha+\theta)$, comes: