RamseteController

Ramsete controller for tracking tank trajectories with unit-less gains.

The standard Ramsete control law from equation \((5.12)\) of this paper is \[ \begin{pmatrix}v\\ \omega\end{pmatrix} = \begin{pmatrix} v_d \cos (\theta_d - \theta) + 2 \zeta \sqrt{\omega_d^2 + b v_d^2} \Big\lbrack (x_d - x) \cos \theta + (y_d - y) \sin \theta \Big\rbrack\\ \omega_d + b v_d \frac{\sin(\theta_d - \theta)}{\theta_d - \theta} \Big\lbrack (x_d - x) \cos \theta - (y_d - y) \sin \theta \Big\rbrack + 2 \zeta \sqrt{\omega_d^2 + b v_d^2} (\theta_d - \theta) \end{pmatrix} \] where \(\zeta \in (0, 1)\) and \(b \gt 0\). To rid the gains of units, let \(b = \frac{\bar{b}}{l^2}\) where \(l\) is the track width.