Hassène Gritli

INTERMITTENCY AND INTERIOR CRISIS AS ROUTE TO CHAOS IN DYNAMIC WALKING OF TWO BIPED ROBOTS

Recently, passive and semi-passive dynamic walking has been noticed in researches of biped walking robots. Such biped robots are well-known that they demonstrate only a period-doubling route to chaos while walking down sloped surfaces. In previous researches, such route was shown with respect to a continuous change in some parameter of the biped robot. In this paper, two biped robots are introduced: the passive compass-gait biped robot and the semi-passive torso-driven biped robot. The period-doubling scenario route to chaos is revisited for the first biped as the ground slope changes. Furthermore, we will show through bifurcation diagram that the torso-driven biped exhibits also such route to chaos when the slope angle is varied. For such biped, a modified semi-passive control law is introduced in order to stabilize the torso at some desired position. In this work, we will show through bifurcation diagrams that the dynamic walking of the two biped robots reveals two other routes to chaos namely the intermittency route and the interior crisis route. We will stress that the intermittency is generated via a saddle-node bifurcation where an unstable periodic orbit is created. We will highlight that such event takes place for a Type-I intermittency. However, we will emphasize that the interior crisis event occurs when a collision of the unstable periodic orbit with a weak chaotic attractor happens giving rise to a strong chaotic attractor. In addition, we will explore the intermittent step series induced by the interior crisis and also by the Type-I intermittency. In this study, our analysis on chaos and the routes to chaos will be based, beside bifurcation diagrams, on Lyapunov exponents and fractal (Lyapunov) dimension. These two tools are plotted in the parameter space to classify attractors observed in bifurcation diagrams.

CYCLIC-FOLD BIFURCATION AND BOUNDARY CRISIS IN DYNAMIC WALKING OF BIPED ROBOTS

The aim of this paper is to show that the onset/destruction of bipedal chaos in the dynamic walking of a passive compass-gait biped robot and a semi-passive torso-driven biped robot walking down a slope can occur via a transition mechanism known as boundary crisis. It is known that such biped robots exhibit a scenario of period-doubling bifurcations route to chaos as one of their geometrical or inertial parameters changes. In this paper, we show that a cyclic-fold bifurcation is the key of the occurrence of a double boundary crisis. We demonstrate through bifurcation diagrams how the same period-three unstable periodic orbit generated from the cyclic-fold bifurcation causes the sudden birth/death of the bipedal chaos in the dynamic walking of the two biped robots. We stress that a double boundary crisis is responsible for the fall of each biped robot while walking down an inclined surface and as some bifurcation parameter varies. Stability of the cyclic-fold bifurcation under small perturbations is also discussed.

Cyclic-fold bifurcation in passive bipedal walking of a compass-gait biped robot with leg length discrepancy

We report on the analysis of passive bipedal walking patterns generated by a compass-gait biped robot having leg length discrepancy. Such two-degrees-of-freedom biped robot but with equal leg length is known to walk passively and steadily down sloped surfaces without any source of actuation. It is known also that the passive walk exhibits only a period-doubling bifurcation leading to chaos in response to a change in some inertial or geometric parameter. In this paper, we show through bifurcation diagrams that the compass-gait biped with leg length discrepancy reveals also cyclic-fold bifurcations in its passive dynamics walking pattern. We show also that such bifurcations occur in pair giving rise to the coexistence of two distinct attractors which can be either periodic or chaotic. Furthermore, we stress that a cyclic-fold bifurcation is responsible on the fall down of the biped robot and also on the generation of another new walking patterns. In this paper, the hybrid model of the compass gait model is given. The whole dynamic model is normalized and is written with respect to a normalized parameter expressing discrepancy percentage in leg length.

A period-three passive gait tracking control for bipedal walking of a compass-gait biped robot

In this paper we interest on the tracking control subject of a period-three passive bipedal walking for a planar compass-gait biped robot. This biped robot is an impulsive mechanical system of two-degrees-of-freedom known to possess passive periodic bipedal walking patterns found to be reminiscent of human walking. It is well known that such compass-gait biped can walk steadily down an inclined slope without any control. The behavior of its gait depends closely on the ground slope as well as on its inertial parameters. We showed recently that such compass-gait robot can possess two distinct periodic stable gaits for the same ground slope: a period-1 gait and another period-3 gait which has not been fully studied yet. This paper deals with the period-3 passive stable gait. Some properties of this typical gait are discussed in this paper revealing its typical symmetry, efficiency and its importance for the control subject. We reveal that the compass-gait biped robot presents an elegant and attractive manner to walk with the period-three passive gait which can be employed in some desirable objectives for the passive bipedal walking. We show in addition that the period-3 gait is very sensitive to small perturbations and becomes unstable where the passive walk of the compass biped is found to converge to the period-1 gait. This conducts to the fact that the basin of attraction of such period-three gait is very restricted. Then, in order to stabilize the passive period-three gait, we propose in this paper an energy tracking control law. Furthermore, we apply this same control law to track the period-three passive gait when the biped robot walks down a different sloped surface.

From Hopf Bifurcation to Limit Cycles Control in Underactuated Mechanical Systems

This paper deals with the problem of obtaining stable and robust oscillations of underactuated mechanical systems. It is concerned with the Hopf bifurcation analysis of a Controlled Inertia Wheel Inverted Pendulum (C-IWIP). Firstly, the stabilization was achieved with a control law based on the Interconnection, Damping, Assignment Passive Based Control method (IDA-PBC). Interestingly, the considered closed-loop system exhibits both supercritical and subcritical Hopf bifurcation for certain gains of the control law. Secondly, we used the center manifold theorem and the normal form technique to study the stability and instability of limit cycles emerging from the Hopf bifurcation. Finally, numerical simulations were conducted to validate the analytical results in order to prove that with IDA-PBC we can control not only the unstable equilibrium but also some trajectories such as limit cycles.

Master-slave controlled synchronization to control chaos in an impact mechanical oscillator

This work aims at controlling chaos exhibited in the impulse hybrid dynamics of a 1DOF impact mechanical oscillator by achieving a master-slave controlled synchronization. Our objective is to synchronize the motion of a chaotic slave impact oscillator with that of a periodic master impact oscillator via an external control input. The master-slave synchronization problem is reformulated as the stabilization of the synchronization error by means of a state-feedback controller. For the design of the control input, we deal only with the linear dynamics of the two systems during their oscillation phase. Our fundamental approach hinges mainly on the use of the S-procedure in order to reduce the conservatism of the classical Lyapunov approach. We employ also the Schur complement and the Matrix Inversion Lemma in order to transform a BMI into a LMI. We show the effectiveness of the proposed method for the control of chaos by applying the designed control input to the chaotic impact oscillator.