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Ph.D. Thesis:
Title: Analysis and Control of Chaos in Robotics: Case of Planar Biped Robots.
Titre : Analyse et Contrôle du Chaos en Robotique : Cas des Robots Bipèdes Planaires.
Abstract:
Bipedal robots are in fact a particularly interesting class of mobile mechatronic systems, because of their adaptability to various walking surfaces allowing them to move in very constrainedin size or unstructured environments. The complexity and sophistication underlying human locomotion make it difficult translating walking, its multitude of shapes and rapid adaptation to changes in walking surfaces in terms of control objectives for the biped robot.
In my PhD Thesis, I was interested in the analysis and control of chaos in dynamic walking of planar biped robots and especially the passive dynamic walking (PDW) of a compass-type biped robot and the semi-passive dynamic walking (SPDW) of a torso-driven biped robot. I studied also the PDW of a compass-gait biped robot with leg length discrepancy. The (semi-)passive dynamic walking of the two models is described by an impulsive hybrid nonlinear dynamics, which is known very complex in terms of analysis and control, and very complex from point-of-view of various nonlinear behaviors that can be displayed such as bifurcations and chaos.
A Compass-Gait Biped Robot
Bifurcation Diagram showing all PDW of the Compass-Gait Biped Robot
A Torso-Driven Biped Robot
Bifurcation Diagram showing all SPDW of the Torso-Driven Biped Robot
A Compass-Gait Biped Robot with Leg Length Discrepancy
Bifurcation Diagram showing all PDW of the Torso-Driven Biped Robot with Leg length Discrepancy
The analysis concerned the determination of limit cycles, the study of bifurcations and the calculation of the Lyapunov exponents and the fractal dimension.
My main contributions in the analysis of chaos and bifurcations concern:
Firstly, the demonstration of a Type-I intermittency and an interior crisis as two new routes to chaos as well as of a boundary crisis,
Secondly, the search for new periodic cycles of the dynamic walking based on the concept of grazing bifurcation and the iterative scheme of Davidchack-Lai, and
Thirdly, the demonstration of a cyclic-fold bifurcation giving rise to period-3 stable passive cycles leading to chaos. An energetic control has been introduced to stabilize/track the period-3 passive cycle.
Fourthly, the demonstration of a cyclic-fold bifurcation and a hysteresis phenomenon in the passive dynamic walking of a compass-gait bped robot with leg length discrepancy (Unequal Leg Length).
In addition, an approach based on the OGY method to control chaos generated in the passive dynamic walking of the compass-gait biped robot has been developed. In this context, a reduced model with an impulsive hybrid linear dynamics has been developed from the linearization around a desired limit cycle. Thus, a particular expression of a constrained Poincaré map has been obtained. A state feedback control for the stabilization of the fixed point of the Poincaré map and the control of chaos was then been proposed.
Publications List: ==>
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Master Thesis:
Title: Master-Slave Synchronization of Chaos: Application to a Mechanical System.
Titre : Synchronisation Maître-Esclave du Chaos : Application à un Système Mécanique.
Abstract:
During my research in Master, I was mainly interested in learning of new techniques from the principle of reducing the conservatism of the analysis compared to the classical approach of Lyapunov stability in the context of polyhedral piecewise affine systems (PWA). These systems are an important class of hybrid dynamical systems that are very complex from analysis and control point of view. Thus, I was interested to realize the synchronization of two identical PWA mechanical systems (master and slave) showing chaotic behavior for very specific system parameters.
Two Machanical PWA Systems
Two main contributions were made:
Using the conservatism principle for the development of a control law for the master-slave synchronization of two PWA systems using the:
S-procedure.
Transformation of a Bilinear Matrix Inequality (BMI) in a Linear Matrix Inequality (LMI) using the:
Schur complement,
Matrix Inversion Lemma.
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