Biped:

We consider a simple model of biped walker [1], seen as a hybrid oscillator. The figure below 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) , \end{equation} We consider: \(\gamma=0.009\), \(\delta=0.1\) and initial condition \(x(0) = (0.009, -0.05869, -0.0009629, -0.3432) \) (See [2]).

Results:

We consider a system with uncertainty (\(w=0.0001\)), set of initial conditions \(B(x_0,\varepsilon)\) with \(\varepsilon=0.001\), the time-step used in Euler's method is \(\tau =2\cdot 10^{-5}\), and we take \(T=k\tau\) with \(k=194129\) as an approximate period.
Using the shown results, we check that:
  • \(B((i_0+1)T)\subset B(i_0T)\) for \(i_0=4\)
  • besides, for the state \(\phi_1\), the minimum \(m^1_+=-0.1856\) (represented by a small cyan ball) of the upper green curve \(\tilde{u}_1(t)+\delta_{{\cal W}}(t)\) is less than the maximum \(M^1_-=0.1856\) (represented by a small gray ball) of the lower green curve \(\tilde{u}_1(t)-\delta_{{\cal W}}(t)\)
  • and \(\Sigma_{i=1}^k\lambda_i\approx\)\(-3010 < 0\).
Then, we can conclude that the system converges towards an attractive LC contained in \([B(4T),B(5T)]\).

The figures below show respectively the simulation of \(\phi_1(t), \overset{.}{\phi_1}, \phi_2 (t), \overset{.}{\phi_2} (t)\) and \(\delta_{{\cal W}}(t)\) with perturbation (w=0.0001) over 5 periods (5T=19.41) for dt=0.00002.
In the figures \(\phi_1(t), \overset{.}{\phi_1}, \phi_2 (t)\) and \(\overset{.}{\phi_2} (t)\), the red curves represent the Euler approximation, the green curves correspond to the borders of tube Bw. The black vertical lines delimit the portion of the tube between t=i0*T0 and t=(i0+1)T_0. The cyan point represents the minimum m+^1 of the upper green curve and the gray point shows the maximum M-^1 of the lower green curve.

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References:

[1] SHIRASAKA, Sho, KUREBAYASHI, Wataru, and NAKAO, Hiroya. Phase reduction theory for hybrid nonlinear oscillators. Physical Review E, 2017, vol. 95, no 1, p. 012212.
[2] JERRAY, Jawher, FRIBOURG, Laurent, and ANDRÉ, Étienne. Guaranteed phase synchronization of hybrid oscillators using symbolic Euler's method (verification challenge). EPiC Series in Computing, 2020, vol. 74, p. 197-208.