Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum

Geometric phase can explain the rotation of a dynamical system independent of angular momentum. The canonical example of such is a cat (a non-rigid body with an inbuilt control system), falling from an inverted position, being able to re-orient itself with negligible total angular momentum so as to land on its feet. The system of three bodies moving under mutual gravitation is similarly non-rigid, capable of changing size and shape under the dynamics of that force. Using coordinates that reduce by translations and rotations and simultaneously regularise all binary collisions, which separate shape dynamics from rotational dynamics, we show how certain discrete symmetries (including both reversing and non-reversing symmetries of the equations of motion) can force the geometric phase of motion periodic to vanish. This result is illustrated with periodic orbits discovered in a numerical survey, many of which are heretofore unknown, and the findings of this survey are discussed in detail, including stability, geometric phase, and classification of orbits

Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum

Geometric phase can explain the rotation of a dynamical system independent of angular momentum. The canonical example of such is a cat (a non-rigid body with an inbuilt control system), falling from an inverted position, being able to re-orient itself with negligible total angular momentum so as to land on its feet. The system of three bodies moving under mutual gravitation is similarly non-rigid, capable of changing size and shape under the dynamics of that force. Using coordinates that reduce by translations and rotations and simultaneously regularise all binary collisions, which separate shape dynamics from rotational dynamics, we show how certain discrete symmetries (including both reversing and non-reversing symmetries of the equations of motion) can force the geometric phase of motion periodic to vanish. This result is illustrated with periodic orbits discovered in a numerical survey, many of which are heretofore unknown, and the findings of this survey are discussed in detail, including stability, geometric phase, and classification of orbits

6D physical interaction with a fully actuated aerial robot

This paper presents the design, control, and experimental validation of a novel fully-actuated aerial robot for physically interactive tasks, named Tilt-Hex. We show how the Tilt-Hex, a tilted-propeller hexarotor is able to control the full pose (position and orientation independently) using a geometric control, and to exert a full-wrench (force and torque independently) with a rigidly attached end-effector using an admittance control paradigm. An outer loop control governs the desired admittance behavior and an inner loop based on geometric control ensures pose tracking. The interaction forces are estimated by a momentum based observer. Control and observation are made possible by a precise control and measurement of the speed of each propeller. An extensive experimental campaign shows that the Tilt-Hex is able to outperform the classical underactuated multi-rotors in terms of stability, accuracy and dexterity and represent one of the best choice at date for tasks requiring aerial physical interaction

Distributed feedback control of a fractional diffusion process

International audienceIn this paper, a control law that enforces an output tracking of a fractional diffusion process is developed. The dynamical behavior of the process is described by a space-fractional parabolic equation. The objective is to force a spatial weighted average output to track its specified output by manipulating a control variable assumed to be distributed in the spatial domain. The state feedback is designed in the framework of geometric control using the notion of the characteristic index. Then, under the assumption that the fractional diffusion process is a minimum phase system, it is shown that the developed control law guarantees exponential stability in L2 -norm for the resulting closed loop system. Numerical simulations are performed to show the tracking and disturbance rejection capabilities of the developed controller

Distributed feedback control of a fractional diffusion process

International audienceIn this paper, a control law that enforces an output tracking of a fractional diffusion process is developed. The dynamical behavior of the process is described by a space-fractional parabolic equation. The objective is to force a spatial weighted average output to track its specified output by manipulating a control variable assumed to be distributed in the spatial domain. The state feedback is designed in the framework of geometric control using the notion of the characteristic index. Then, under the assumption that the fractional diffusion process is a minimum phase system, it is shown that the developed control law guarantees exponential stability in L 2-norm for the resulting closed loop system. Numerical simulations are performed to show the tracking and disturbance rejection capabilities of the developed controller

Towards an analytical description of active microswimmers in clean and in surfactant-covered drops

Geometric confinements are frequently encountered in the biological world and strongly affect the stability, topology, and transport properties of active suspensions in viscous flow. Based on a far-field analytical model, the low-Reynolds-number locomotion of a self-propelled microswimmer moving inside a clean viscous drop or a drop covered with a homogeneously distributed surfactant, is theoretically examined. The interfacial viscous stresses induced by the surfactant are described by the well-established Boussinesq-Scriven constitutive rheological model. Moreover, the active agent is represented by a force dipole and the resulting fluid-mediated hydrodynamic couplings between the swimmer and the confining drop are investigated. We find that the presence of the surfactant significantly alters the dynamics of the encapsulated swimmer by enhancing its reorientation. Exact solutions for the velocity images for the Stokeslet and dipolar flow singularities inside the drop are introduced and expressed in terms of infinite series of harmonic components. Our results offer useful insights into guiding principles for the control of confined active matter systems and support the objective of utilizing synthetic microswimmers to drive drops for targeted drug delivery applications.Comment: 19 pages, 7 figures. Regular article contributed to the Topical Issue of the European Physical Journal E entitled "Physics of Motile Active Matter" edited by Gerhard Gompper, Clemens Bechinger, Holger Stark, and Roland G. Winkle

Chemical reactions induced by oscillating external fields in weak thermal environments

Chemical reaction rates must increasingly be determined in systems that evolve under the control of external stimuli. In these systems, when a reactant population is induced to cross an energy barrier through forcing from a temporally varying external field, the transition state that the reaction must pass through during the transformation from reactant to product is no longer a fixed geometric structure, but is instead time-dependent. For a periodically forced model reaction, we develop a recrossing-free dividing surface that is attached to a transition state trajectory [T. Bartsch, R. Hernandez, and T. Uzer, Phys. Rev. Lett. 95, 058301 (2005)]. We have previously shown that for single-mode sinusoidal driving, the stability of the time-varying transition state directly determines the reaction rate [G. T. Craven, T. Bartsch, and R. Hernandez, J. Chem. Phys. 141, 041106 (2014)]. Here, we extend our previous work to the case of multi-mode driving waveforms. Excellent agreement is observed between the rates predicted by stability analysis and rates obtained through numerical calculation of the reactive flux. We also show that the optimal dividing surface and the resulting reaction rate for a reactive system driven by weak thermal noise can be approximated well using the transition state geometry of the underlying deterministic system. This agreement persists as long as the thermal driving strength is less than the order of that of the periodic driving. The power of this result is its simplicity. The surprising accuracy of the time-dependent noise-free geometry for obtaining transition state theory rates in chemical reactions driven by periodic fields reveals the dynamics without requiring the cost of brute-force calculations

ZMP support areas for multi-contact mobility under frictional constraints

We propose a method for checking and enforcing multi-contact stability based on the Zero-tilting Moment Point (ZMP). The key to our development is the generalization of ZMP support areas to take into account (a) frictional constraints and (b) multiple non-coplanar contacts. We introduce and investigate two kinds of ZMP support areas. First, we characterize and provide a fast geometric construction for the support area generated by valid contact forces, with no other constraint on the robot motion. We call this set the full support area. Next, we consider the control of humanoid robots using the Linear Pendulum Mode (LPM). We observe that the constraints stemming from the LPM induce a shrinking of the support area, even for walking on horizontal floors. We propose an algorithm to compute the new area, which we call pendular support area. We show that, in the LPM, having the ZMP in the pendular support area is a necessary and sufficient condition for contact stability. Based on these developments, we implement a whole-body controller and generate feasible multi-contact motions where an HRP-4 humanoid locomotes in challenging multi-contact scenarios.Comment: 14 pages, 10 figure

Geometric Adaptive Control for a Quadrotor UAV with Wind Disturbance Rejection

This paper presents a geometric adaptive control scheme for a quadrotor unmanned aerial vehicle, where the effects of unknown, unstructured disturbances are mitigated by a multilayer neural network that is adjusted online. The stability of the proposed controller is analyzed with Lyapunov stability theory on the special Euclidean group, and it is shown that the tracking errors are uniformly ultimately bounded with an ultimate bound that can be abridged arbitrarily. A mathematical model of wind disturbance on the quadrotor dynamics is presented, and it is shown that the proposed adaptive controller is capable of rejecting the effects of wind disturbances successfully. These are illustrated by numerical examples