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• A design of an explicit analytical expression of Poincaré maps for non-autonomous periodically forced nonlinear systems is achieved. • A linearization of the nonlinear dynamics around a desired period-m limit cycle is performed. • We design three different time-piecewise-constant control laws to stabilize the limit cycle. • An application to the chaotic Duffing oscillator is realized to stabilize period-1 and period-2 limit cycles. In this paper, a design of Poincaré maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period– m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincaré map from the structure of several local maps. An expression of the generalized controlled Poincaré map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT –piecewise–constant control law, a T –piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period– m limit cycle and hence the chaotic motion is controlled. [ABSTRACT FROM AUTHOR]

An analysis of the passive dynamic walking of a compass-gait biped model under the OGY-based control approach using the impulsive hybrid nonlinear dynamics is presented in this paper. We describe our strategy for the development of a simplified analytical expression of a controlled hybrid Poincaré map and then for the design of a state-feedback control. Our control methodology is based mainly on the linearization of the impulsive hybrid nonlinear dynamics around a desired nominal one-periodic hybrid limit cycle. Our analysis of the controlled walking dynamics is achieved by means of bifurcation diagrams. Some interesting nonlinear phenomena are displayed, such as the period-doubling bifurcation, the cyclic-fold bifurcation, the period remerging, the period bubbling and chaos. A comparison between the raised phenomena in the impulsive hybrid nonlinear dynamics and the hybrid Poincaré map under control was also presented. [ABSTRACT FROM AUTHOR]

In our previous work, we have analyzed the passive dynamic walking of the compass-gait biped model under the OGY-based state-feedback control using the impulsive hybrid nonlinear dynamics. Such study was carried out through bifurcation diagrams. It was shown that the controlled bipedal gait exhibits attractive nonlinear phenomena such as the cyclic-fold (saddle-node) bifurcation, the period-doubling (flip) bifurcation and chaos. Moreover, we revealed that, using the controlled continuous-time dynamics, we encountered a problem in finding, identifying and hence following branches of (un)stable solutions in order to characterize local bifurcations. The present paper solves such problem and then provides a further investigation of the controlled bipedal walking dynamics using the developed analytical expression of the controlled hybrid Poincaré map. Thus, we show that analysis via such Poincaré map allows to follow branches of both stable and unstable fixed points in bifurcation diagrams and hence to explore the complete dynamics of the controlled compass-gait biped model. We demonstrate the generation, other than the conventional local bifurcations in bipedal walking, i.e. the flip bifurcation and the saddle-node bifurcation, of a saddle-saddle bifurcation, a subcritical flip bifurcation and a new type of a local bifurcation, the saddle-flip bifurcation. In addition, to further understand the occurrence of the local bifurcations, we present an analysis with a two-parameter bifurcation diagram. Some new hidden walking dynamics are identified. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: The compass-gait walker is a two-degree-of-freedom biped that can walk passively and steadily down an incline without any actuation. The mathematical model of the walking dynamics is represented by an impulsive hybrid nonlinear model. It is capable of displaying cyclic motions and chaos. In this paper, we propose a new approach to controlling chaos cropped up from the passive dynamic walking of the compass-gait model. The proposed technique is to linearize the nonlinear model around a desired passive hybrid limit cycle. Then, we show that the nonlinear model is transformed to an impulsive hybrid linear model with a controlled jump. Basing on the linearized model, we derive an analytical expression of a constrained controlled Poincaré map. We present a method for the numerical simulation of this constrained map where bifurcation diagrams are plotted. Relying on these diagrams, we show that the linear model is fairly close to the nonlinear one. Using the linearized controlled Poincaré map, we design a state feedback controller in order to stabilize the fixed point of the Poincaré map. We show that this controller is very efficient for the control of chaos for the original nonlinear model. [Copyright &y& Elsevier]

Abstract: This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos. [Copyright &y& Elsevier]

Some missing interesting references related to our quoted paper are added. The difference between our analytical approach and that presented in these references is also briefly described. [ABSTRACT FROM AUTHOR]

• A robust feedback control of a 1-DoF impact mechanical oscillator subject to double-side rigid constraints and under bounded-parametric uncertainty is considered. • The control is achieved such that the impact mechanical oscillator, as a slave, tracks a master impact oscillator adopted as a reference model. • The dynamics of the master-slave tracking error is described by a non-autonomous system with impulsive effects. • We use the S-procedure Lemma and the Finsler Lemma to only consider the regions within which the system state evolves and hence to develop BMI stability conditions. • We use the Schur complement Lemma and the Matrix Inversion Lemma to transform these BMIs into LMIs. • A portfolio of numerical simulations is presented to illustrate the master-slave tracking. In this paper, a Linear Matrix Inequality- (an LMI-) based approach for designing a robust state-feedback controller for a 1-DoF, periodically forced, impact mechanical oscillator subject to double-side rigid barriers/constraints and under bounded parametric uncertainties to track another impact oscillator, as a master or reference system, is proposed. The dynamics of such impact oscillator is defined by a hybrid non-autonomous system with impulsive effects, for which the impulsive event occurs when the system state encounters the two barriers and the oscillation motion is limited between them. The main idea in the synthesis of the stability conditions lies in the use of the S-procedure Lemma and the Finsler Lemma in order to only consider the regions inside which the master-slave tracking error evolves. We show that the stability conditions of the tracking error are reformulated by a set of Bilinear Matrix Inequalities (BMIs). Via the Schur complement Lemma and the Matrix Inversion Lemma, a linearization procedure is realized to transform these BMIs into LMIs where the admissible maximum bounds of the parametric uncertainties are maximized. The effectiveness of the proposed feedback controller towards uncertainties is illustrated through simulation results. [ABSTRACT FROM AUTHOR]

• We study the passive dynamic walking of the compass-gait biped model. • A design of an explicit analytical expression of the Poincaré map having the classical form is achieved. • A linearization of the impulsive hybrid nonlinear dynamics around a desired period-1 limit cycle is realized. • We develop a simplified expression of the Poincaré map with a reduced dimension and its Jacobian matrix. • An extensive portfolio of numerical studies, simulations and comparisons is presented showing the validity of the Poincaré map. 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. [ABSTRACT FROM AUTHOR]