Nonholonomic conditions - Rolling disk

Question

Suppose there is a rolling disk subject to non-slipping condition. The kinematic model follow below.

\begin{equation} \begin{cases} \dot x = v \, cos(\theta) \\ \dot y = v \, sin(\theta) \\ \dot \theta = \omega \end{cases} \end{equation}

If I know the development $(x(t), y(t))$, then $\theta(t) = atan2(\frac{\dot x}{\dot y})$, $\omega(t) = \frac{d}{dt} \theta(t)$ and $v^2 = \dot{x}^2 + \dot{y}^2$. Is it possible to find the signal for linear velocity $v$ from the equations above?

Answer 1

I found an easy way to compute it:

$ sign(v) = \begin{cases} sign\left(\frac{\dot{x}}{\cos{\theta}}\right) & \cos{\theta} \neq 0 \\ sign\left(\frac{\dot{y}}{\sin{\theta}}\right) & otherwise \end{cases} $