Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of
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ORIGINAL PAPER
Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of the controlled Poincaré map Wafa Znegui · Hassène Gritli
· Safya Belghith
Received: 20 January 2020 / Accepted: 27 July 2020 © Springer Nature B.V. 2020
Abstract The compass-gait biped robot is a two-DoF legged mechanical system that has been known by its passive dynamic walking. This kind of passive biped robot is modeled by an impulsive hybrid nonlinear system that exhibits complex behaviors. This paper is concerned with the stabilization of the passive dynamic walking of the compass-gait biped robot by designing an analytical expression of the controlled Poincaré map. The suggested analytical method starts with the time-piecewise linearization of the impulsive system augmented with the controller, around a desired oneperiodic passive hybrid limit cycle. By virtue of the first-order Taylor series, we design an explicit expression of the controlled Poincaré map. We present also a simplified expression of the controlled Poincaré map having a reduced dimension. In order to accomplish our goal concerned with the stabilization of the passive walking dynamics of the compass-gait biped robot, we develop first the linearized Poincaré map around the W. Znegui · H. Gritli (B) Institut Supérieur des Technologies de l’Information et de la Communication, Université de Carthage, 1164, Borj Cedria, Tunis, Tunisia e-mail: [email protected] W. Znegui e-mail: [email protected] W. Znegui · H. Gritli · S. Belghith Laboratoire Robotique, Informatique et Systèmes Complexes (RISC, LR16ES07), Ecole Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, BP. 37, Le Belvédère, 1002 Tunis, Tunisia e-mail: [email protected]
period-1 fixed point of the Poincaré map and we adopt a state-feedback control law to stabilize it. Furthermore, in order to enhance the effectiveness and robustness of the stabilization process, we adopt an LMI-based optimization approach by considering the controlled Poincaré map to design the feedback gain. In the end of this work, we provide some numerical and graphical simulation results, to show the validity of the designed control law by means of the controlled Poincaré map in the stabilization of the passive dynamic walking of the compass-gait biped robot. Keywords Compass-gait biped robot · Impulsive hybrid nonlinear dynamics · Time-t map · Controlled Poincaré map · Analytical method · Stabilization · LMI approach
1 Introduction 1.1 Background Researchers in humanoid robots’ fields have fulfilled interesting results and achieve significant improvements and progress in last decades leading to design several prototypes for multiple purposes [1]. One of these fields that we will focus on is the legged walking biped robots [2,3]. Its particularity consists on real-life matching with human locomotive mechanism. The nature is full of plenty examples of legged biped walking features, where the human being certainly
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belongs to them. Such kind
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