Given a non-holonomic dynamical system, \begin{align*} \dot x = v\cos\theta \\ \dot y = v\sin\theta \\ \dot \theta = \omega \end{align*} with constraints $|v| < v_{max}, |\omega| < \omega_{max}$, how can we apply a transformation to this system such that in the new coordinates $(x',y',\theta')$ and with a new time scale $\hat{t}$, \begin{align*} \dot x' = v'\cos\theta' \\ \dot y' = v'\sin\theta' \\ \dot \theta' = \omega' \end{align*} with constraints $|v'| <1, |\omega'| < 1$ ?
The non-holonomic dynamics makes the transformation very non-intuitive.
If we employ a transformation of the form $x' = \frac{\omega_{max}}{v_{max}}x$ , $y' = \frac{\omega_{max}}{v_{max}}y$, $\theta' = \theta$, $\hat{t} = \omega_{max}t$, then the required form is achieved.