# Input/output linearization

I am studying the book "B. Sicilliano -Robotics. Modelling, Planning and Control" chapter 11, Input/output linearization topic. Where does the formula for theta in 11.75 come from?

In the case of the unicycle, consider the following outputs:

$$y_1 = x + b cos θ;$$

$$y_2 = y + b sin θ$$

with $$b \neq 0$$. They represent the Cartesian coordinates of a point B located along the sagittal axis of the unicycle at a distance |b| from the contact point of the wheel with the ground. How to find the following input-output linearization?

$$\dot{y_1} = u_1;$$

$$\dot{y_2} = u_2;$$

$$\dot{\theta} = \frac{u_2 \cos\theta − u_1\sin\theta}{b}$$

I am not very good at robotics, I would like to get an answer simple to understand.

Thank you for all the help.

• For those who don't have the book, it would be great that you mention what do the variables (b, u_1, u_2) stand for.
– Joe
Feb 8, 2019 at 9:58
• academia.edu/23785978/… Feb 8, 2019 at 16:44

If you observe that $$\omega = \dot{\theta}$$, you can readily derive the equation at hand from:
$$\left[ \begin{array}{c} v \\ \omega \end{array} \right] = \ \left[ \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta/b & \cos\theta/b \ \end{array} \right] \ \left[ \begin{array}{c} u_1 \\ u_2 \end{array} \right].$$