3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Joe Feb 8 at 9:58
  • $\begingroup$ academia.edu/23785978/… $\endgroup$ – jsotola Feb 8 at 16:44
2
$\begingroup$

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]. $

$\endgroup$
  • 1
    $\begingroup$ Thank you. Can't believe I missed that! $\endgroup$ – Leon Rai Feb 8 at 11:00

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.