Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

Highlights: • A numerical calculation method of the Lyapunov exponents in the compass-gait model under OGY control is proposed. • A new linearization method of the impulsive hybrid dynamics around a one-periodic hybrid limit cycle is achieved. • We develop a simple analytical expression of a controlled hybrid Poincaré map. • A dimension reduction of the hybrid Poincaré map is realized. • We describe the numerical computation procedure of the Lyapunov exponents via the designed hybrid Poincaré map. - Abstract: This paper aims at providing a numerical calculation method of the spectrum of Lyapunov exponents in a four-dimensional impulsive hybrid nonlinear dynamics of a passive compass-gait model under the OGY control approach by means of a controlled hybrid Poincaré map. We present a four-dimensional simplified analytical expression of such hybrid map obtained by linearizing the uncontrolled impulsive hybrid nonlinear dynamics around a desired one-periodic passive hybrid limit cycle. In order to compute the spectrum of Lyapunov exponents, a dimension reduction of the controlled hybrid Poincaré map is realized. The numerical calculation of the spectrum of Lyapunov exponents using the reduced-dimension controlled hybrid Poincaré map is given in detail. In order to show the effectiveness of the developed method, the spectrum of Lyapunov exponents is calculated as the slope (bifurcation) parameter varies and hence used to predict the walking dynamics behavior of the compass-gait model under the OGY control.