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Publication# Local Vulnerabilities and Global Robustness of Coupled Dynamical Systems on Complex Networks

Abstract

Coupled dynamical systems are omnipresent in everyday life. In general, interactions between individual elements composing the system are captured by complex networks. The latter greatly impact the way coupled systems are functioning and evolving in time. An important task in such a context, is to identify the most fragile components of a system in a fast and efficient manner. It is also highly desirable to have bounds on the amplitude and duration of perturbations that could potentially drive the system through a transition from one equi- librium to another. A paradigmatic model of coupled dynamical system is that of oscillatory networks. In these systems, a phenomenon known as synchronization where the individual elements start to behave coherently may occur if couplings are strong enough. We propose frameworks to assess vulnerabilities of such synchronous states to external perturbations. We consider transient excursions for both small-signal response and larger perturbations that can potentially drive the system out of its initial basin of attraction. In the first part of this thesis, we investigate the robustness of complex network-coupled oscillators. We consider transient excursions following external perturbations. For ensemble averaged perturbations, quite remarkably we find that robustness of a network is given by a family of network descriptors that we called generalized Kirchhoff indices and which are defined from extensions of the resistance distance to arbitrary powers of the Laplacian matrix of the system. These indices allow an efficient and accurate assessment of the overall vulnera- bility of an oscillatory network and can be used to compare robustness of different networks. Moreover, a network can be made more robust by minimizing its Kirchhoff indices. Then for specific local perturbations, we show that local vulnerabilities are captured by generalized resistance centralities also defined from extensions of the resistance distance. Most fragile nodes are therefore identified as the least central according to resistance centralities. Based on the latter, rankings of the nodes from most to least vulnerable can be established. In summary, we find that both local vulnerabilities and global robustness are accurately evaluated with resistance centralities and Kirchhoff indices. Moreover, the framework that we define is rather general and may be useful to analyze other coupled dynamical systems. In the second part, we focus on the effect of larger perturbations that eventually lead the sys- tem to an escape from its initial basin of attraction. We consider coupled oscillators subjected to noise with various amplitudes and correlation in time. To predict desynchronization and transitions between synchronous states, we propose a simple heuristic criterion based on the distance between the initial stable fixed point and the closest saddle point. Surprisingly, we find numerically that our criterion leads to rather accurate estimates for the survival probability and first escape time. Our criterion is general and may be applied to other dynamical systems.

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Accomplishment of many interactive tasks hinges on the compliance of humans. Humans demonstrate an impressive capability of complying their behavior and more particularly their motions with the environment in everyday life. In humans, compliance emerges from different facets. For example, many daily activities involve reaching for grabbing tasks, where compliance appears in a form of coordination. Humans comply their handsâ motions with each other and with that of the object not only to establish a stable contact and to control the impact force but also to overcome sensorimotor imprecisions. Even though compliance has been studied from different aspects in humans, it is primarily related to impedance control in robotics. In this thesis, we leverage the properties of autonomous dynamical systems (DS) for immediate re-planning and introduce active complaint motion generators for controlling robots in three different scenarios, where compliance does not necessarily mean impedance and hence it is not directly related to control in the force/velocity domain. In the first part of the thesis, we propose an active compliant strategy for catching objects in flight, which is less sensitive to the timely control of the interception. The soft catching strategy consists in having the robot following the object for a short period of time. This leaves more time for the fingers to close on the object at the interception and offers more robustness than a âhardâ catching method in which the hand waits for the object at the chosen interception point. We show theoretically that the resulting DS will intercept the object at the intercept point, at the right time with the desired velocity direction. Stability and convergence of the approach are assessed through Lyapunov stability theory. In the second part, we propose a unified compliant control architecture for coordinately reaching for grabbing a moving object by a multi-arm robotic system. Due to the complexity of the task and of the system, each arm complies not only with the objectâs motion but also with the motion of other arms, in both task and joint spaces. At the task-space level, we propose a unified dynamical system that endows the multi-arm system with both synchronous and asynchronous behaviors and with the capability of smoothly transitioning between the two modes. At the joint space level, the compliance between the arms is achieved by introducing a centralized inverse kinematics (IK) solver under self-collision avoidance constraints; formulated as a quadratic programming problem (QP) and solved in real-time. In the last part, we propose a compliant dynamical system for stably transitioning from free motions to contacts. In this part, by modulating the robot's velocity in three regions, we show theoretically and empirically that the robot can (I) stably touch the contact surface (II) at a desired location, and (III) leave the surface or stop on the surface at a desired point.

In this thesis we describe a strategy to control robotic knees and ankles. A dynamical system is used to generate a position trajectory to control a servo motor replacing the missing joint. The dynamical system consists in a pool of coupled oscillators modeling a central pattern generator (CPG). As a first step, anthropometric trajectories of the knee and ankle are learned by the system through the convergence of the oscillators to the specific frequencies, corresponding amplitudes and phase relations. The same system is then used to play back these trajectories. As a sensory feedback to trigger the playback we use one adaptive frequency oscillator to synchronized with the acceleration from the thigh. We use a bipedal model in a physics-based robot simulation environment to test the proposed system. Finally we present a simple hardware implementation of this system on the Agonist-Antagonist Active Knee prototype.

2010Accomplishment of many interactive tasks hinges on the compliance of humans. Humans demonstrate an impressive capability of complying their behavior and more particularly their motions with the environment in everyday life. In humans, compliance emerges from different facets. For example, many daily activities involve reaching for grabbing tasks, where compliance appears in a form of coordination. Humans comply their handsâ motions with each other and with that of the object not only to establish a stable contact and to control the impact force but also to overcome sensorimotor imprecisions. Even though compliance has been studied from different aspects in humans, it is primarily related to impedance control in robotics. In this thesis, we leverage the properties of autonomous dynamical systems (DS) for immediate re-planning and introduce active complaint motion generators for controlling robots in three different scenarios, where compliance does not necessarily mean impedance and hence it is not directly related to control in the force/velocity domain. In the first part of the thesis, we propose an active compliant strategy for catching objects in flight, which is less sensitive to the timely control of the interception. The soft catching strategy consists in having the robot following the object for a short period of time. This leaves more time for the fingers to close on the object at the interception and offers more robustness than a âhardâ catching method in which the hand waits for the object at the chosen interception point. We show theoretically that the resulting DS will intercept the object at the intercept point, at the right time with the desired velocity direction. Stability and convergence of the approach are assessed through Lyapunov stability theory. In the second part, we propose a unified compliant control architecture for coordinately reaching for grabbing a moving object by a multi-arm robotic system. Due to the complexity of the task and of the system, each arm complies not only with the objectâs motion but also with the motion of other arms, in both task and joint spaces. At the task-space level, we propose a unified dynamical system that endows the multi-arm system with both synchronous and asynchronous behaviors and with the capability of smoothly transitioning between the two modes. At the joint space level, the compliance between the arms is achieved by introducing a centralized inverse kinematics (IK) solver under self-collision avoidance constraints; formulated as a quadratic programming problem (QP) and solved in real-time. In the last part, we propose a compliant dynamical system for stably transitioning from free motions to contacts. In this part, by modulating the robot's velocity in three regions, we show theoretically and empirically that the robot can (I) stably touch the contact surface (II) at a desired location, and (III) leave the surface or stop on the surface at a desired point.