Biped

We consider a simple model of biped walker, seen as a hybrid oscillator. The figure opposite shows a schematic diagram of the model, where \(l\) is the length of the legs; \(M\) and \(m\) are the masses of the hip and the foot, respectively; \(\phi_1\) and \(\phi_2\) specify the angles of the swing and support legs and \(\gamma\) is the angle of the slope. This model exhibits a stable limit-cycle oscillation for appropriate parameter values that corresponds to periodic movements of the legs. The model has a unique mode (\(U=\{1\}\)) and a continuous state variable \(\textbf{x}(t) = (\phi_1(t), \overset{.}{\phi_1}(t), \phi_2 (t), \overset{.}{\phi_2} (t))^\top\) . The dynamics is described by

\begin{equation} \textit{f}(\textbf{x}) = \begin{pmatrix} \overset{.}{\phi_1} \\ sin(\phi_1-\gamma)\\ \overset{.}{\phi_2}\\ sin(\phi_1 - \gamma) + \overset{.}{\phi_1^{2}} sin \phi_2 - cos (\phi_1 - \gamma) sin \phi_2 \end{pmatrix} \end{equation} \begin{equation} Reset(\textbf{x}) = \begin{pmatrix} -\phi_1\\ \overset{.}{\phi_1}sin(2\phi_1)\\ -2\phi_1\\ \overset{.}{\phi_1 }cos 2\phi_1 ( 1 - cos 2\phi_1 ) \end{pmatrix} \end{equation} \begin{equation} Guard(\textbf{x})=0 \equiv (2\phi_1 - \phi_2 =0\ \wedge \phi_2<-\delta) , \label{Biped_pi} \end{equation}