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Design of an explicit expression of the Poincaré map for the passive dynamic walking of the compass-gait biped model

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  • Znegui, Wafa
  • Gritli, Hassène
  • Belghith, Safya

Abstract

This paper presents a design method of an explicit analytical classical expression of the Poincaré map for the passive dynamic walking (PDW) of the planar compass-gait biped model. Our methodology is based chiefly on a time-piecewise linearization of the impulsive hybrid nonlinear dynamics of the PDW around a desired one-periodic hybrid limit cycle. We start by linearizing the continuous dynamics of the swing phase and also the discrete dynamics of the impact phase. Thus, we obtain a simplified impulsive hybrid linear dynamics. By means of the first-order Taylor approximation, we design an explicit expression of the Poincaré map. Moreover, we provide a simplified expression of the Poincaré map with a reduced dimension. The expression of the Jacobian matrix of this reduced Poincaré map is also developed. Finally, some numerical results and graphical simulations are presented in order to compare between the impulsive hybrid nonlinear dynamics and the developed Poincaré map. These results show the similarity between the two models and then the efficiency and the validity of the designed Poincaré map in the analysis of the PDW of the compass-gait biped robot.

Suggested Citation

  • Znegui, Wafa & Gritli, Hassène & Belghith, Safya, 2020. "Design of an explicit expression of the Poincaré map for the passive dynamic walking of the compass-gait biped model," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    • Handle: RePEc:eee:chsofr:v:130:y:2020:i:c:s0960077919303820
    DOI: 10.1016/j.chaos.2019.109436
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    References listed on IDEAS

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    1. Gritli, Hassène & Belghith, Safya, 2018. "Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark–Sacker bifurcation," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 158-168.
    2. Gritli, Hassène & Belghith, Safya, 2017. "Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid Poincaré map," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 72-87.
    3. Gritli, Hassène, 2019. "Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 127-145.
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    Cited by:

    1. Rao, XiaoBo & Gao, JianShe & Ding, ShunLiang & Liang, Jie & Zhang, Jiangang, 2023. "Multistability of gaits, the basin of attraction and its external topology in the simplest passive walking model on stairs," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    2. Jiang, Bo & Jiang, Hui & Liu, Qihuai & Jiang, Guirong, 2024. "Periodic gait classification and control of a biped model with telescopic legs and pulse thrust," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).

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    1. Gritli, Hassène, 2019. "Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 127-145.
    2. Gritli, Hassène & Belghith, Safya, 2018. "Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark–Sacker bifurcation," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 158-168.
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