Phase Synchronization Control of Robotic Networks on Periodic Ellipses with Adaptive Network Topologies

This paper presents a novel formation control method for a large number of robots or vehicles described by Euler-Lagrange (EL) systems moving in elliptical orbits. A new coordinate transformation method for phase synchronization of networked EL systems in elliptical trajectories is introduced to define desired formation patterns. The proposed phase synchronization controller synchronizes the motions of agents, thereby yielding a smaller synchronization error than an uncoupled control law in the presence of bounded disturbances. A complex time-varying and switching network topology, constructed by the adaptive graph Laplacian matrix, relaxes the standard requirement of consensus stability, even permitting stabilization on an arbitrary unbalanced graph. The proofs of stability are constructed by robust contraction analysis, a relatively new nonlinear stability tool. An example of reconfiguring swarms of spacecraft in Low Earth Orbit shows the effectiveness of the proposed phase synchronization controller for a large number of complex EL systems moving in elliptical orbits

Floquet Stability Analysis of Ott-Grebogi-Yorke and Difference Control

Stabilization of instable periodic orbits of nonlinear dynamical systems has been a widely explored field theoretically and in applications. The techniques can be grouped in time-continuous control schemes based on Pyragas, and the two Poincar\'e-based chaos control schemes, Ott-Gebogi-Yorke (OGY) and difference control. Here a new stability analysis of these two Poincar\'e-based chaos control schemes is given by means of Floquet theory. This approach allows to calculate exactly the stability restrictions occuring for small measurement delays and for an impulse length shorter than the length of the orbit. This is of practical experimental relevance; to avoid a selection of the relative impulse length by trial and error, it is advised to investigate whether the used control scheme itself shows systematic limitations on the choice of the impulse length. To investigate this point, a Floquet analysis is performed. For OGY control the influence of the impulse length is marginal. As an unexpected result, difference control fails when the impulse length is taken longer than a maximal value that is approximately one half of the orbit length for small Ljapunov numbers and decreases with the Ljapunov number.Comment: 13 pages. To appear in New Journal of Physic

Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems

We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constant-dimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincare map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior

Modeling of subharmonics and chaos in DC motor drives

In this paper, the nonlinear dynamics of both voltage-mode and current-mode controlled dc motor drive systems are presented. The investigation is based on the derivation of the discrete mappings that describe their system subharmonics and chaos in the continuous conduction mode of operation. It illustrates that different bifurcation diagrams can be obtained by using different modes of control while varying the same system parameters. A unified modeling approach for the period-1 and hence the period-p orbits as well as their stability analysis during both voltage-mode and current-mode of control is proposed and verified.published_or_final_versio

Optimal Control of Electrodynamic Tethers

Low thrust propulsion systems such as electrodynamic tethers offer a fuel-efficient means to maneuver satellites to new orbits, however they can only perform such maneuvers when they are continuously operated for a long time. Such long-term maneuvers occur over many orbits often rendering short time scale trajectory optimization methods ineffective. An approach to multi-revolution, long time scale optimal control of an electrodynamic tether is investigated for a tethered satellite system in Low Earth Orbit with atmospheric drag. Control is assumed to be periodic over several orbits since under the assumptions of a nearly circular orbit, periodic control yields the only solution that significantly contributes to secular changes in the orbital parameters. The optimal control problem is constructed in such a way as to maneuver the satellite to a new orbit while minimizing a cost function subject to the constraints of the time-averaged equations of motion by controlling current in the tether. To accurately capture the tether orbital dynamics, libration is modeled and controlled over long time scales in a similar manner to the orbital states. Libration is addressed in two parts; equilibrium and stability analysis, and control. Libration equations of motion are derived and analyzed to provide equilibrium and stability criteria that define the constraints of the design. A new libration mean square state is introduced and constrained to maintain libration within an acceptable envelope throughout a given maneuver. A multiple time scale approach is used to capture the effects of the Earth’s rotating tilted magnetic field. Optimal control solutions are achieved using a pseudospectral method to maneuver an electrodynamic tether to new orbits over long time scales while managing librational motion using only the current in the tether wire

Phase Synchronization Control of Robotic Networks on Periodic Ellipses with Adaptive Network Topologies

This paper presents a novel formation control method for a large number of robots or vehicles described by Euler-Lagrange (EL) systems moving in elliptical orbits. A new coordinate transformation method for phase synchronization of networked EL systems in elliptical trajectories is introduced to define desired formation patterns. The proposed phase synchronization controller synchronizes the motions of agents, thereby yielding a smaller synchronization error than an uncoupled control law in the presence of bounded disturbances. A complex time-varying and switching network topology, constructed by the adaptive graph Laplacian matrix, relaxes the standard requirement of consensus stability, even permitting stabilization on an arbitrary unbalanced graph. The proofs of stability are constructed by robust contraction analysis, a relatively new nonlinear stability tool. An example of reconfiguring swarms of spacecraft in Low Earth Orbit shows the effectiveness of the proposed phase synchronization controller for a large number of complex EL systems moving in elliptical orbits

Path controller implementation for airborne wind energy systems

The implementation of a path controller to a two-line kite model is presented. Within the first chapter, an introduction to Airborne Wind Energy systems and the discussion of some typical control methods can be found. The following chapter deals with the mathematical model of a two line kite. This model considers a kite-surf size kite that can be controlled via two equal tethers. Some thoughts and explanations on the model are included. Thereafter, an open loop control law capable of allowing figure of eight trajectories is defined. Accordingly, an analytical expression for such figure of eight orbits is presented. Some insight on Floquet theory is required in order to properly understand the physics behind periodic orbits. A general purpose predictor-corrector algorithm for periodic orbit propagation determines a set of feasible initial conditions that yield a periodic orbit for a given control law. By means of this tool, it is possible to obtain a periodic orbit applying the control law that has been previously defined. A discussion on such orbit is included, together with its stability analysis. At this point, it is of interest to perform a parametric analysis with the aim of understanding how the stability and the trajectory respond to variations in the control law. Finally the path controller scheme is presented in the form of an optimal control problem. The latter selection was triggered by the failure in implementing a proportional-derivative runtime controller. The results of the project are a deep understanding on the kite sensitivity to variation of tether lengths, i.e. their controls, together with a controller capable of determining optimal control laws for any given desired target path.Ingeniería Aeroespacial (Plan 2010

A SIMPLE APPROACH TO CALCULATION AND CONTROL OF UNSTABLE PERIODIC ORBITS IN CHAOTIC PIECEWISE-LINEAR SYSTEMS

This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor; and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given

Autonomous Spacecraft Control During Close-Proximity Near-Earth Object Operations

A control scheme is proposed for a satellite orbit controller around a small, irregularly shaped near-Earth object (NEO) combining classical control theory and orbital mechanics into a continuous hybrid control system that achieves and maintains a circular orbit in a perturbed environment. NEOs are asteroids and comets that approach Earth\u27s orbit around the Sun. They are currently being studied for resource allocation and threat mitigation, while providing unique opportunities for control systems. The NEO environment consists of a weak and complex gravity field, as well as other perturbations such as solar radiation pressure (SRP) and third-body gravitational disturbances. This project focuses on the gravity field of the NEO and characterizes orbital stability within the NEO\u27s gravity field. A three-term Proportional, Integral, and Derivative (PID) controller is utilized in order to achieve and maintain a circular orbit in close-proximity to the NEO 25143 Itokawa. The proposed control scheme merges a simple controller with orbital mechanics to maximize the effectiveness and efficiency of the thrusters. It uses the PID controller to thrust in the radial direction in order to maintain the proper orbital radius, which is found to be an effective method of correcting perturbed orbits in the NEO environment. This is followed by a change in the orbital velocity of the spacecraft in order to match the specific mechanical energy for the desired circular orbit, which is typically the most efficient method of correcting perturbed orbits. Systems Tool Kit (STK) is used to run the simulation and a MATLAB-STK interface was developed that allows for sophisticated orbit control development. Using the STK simulation software allows for the ability to test multiple orbit parameters for stability. This was applied in studying the interaction between the complex gravity model and its effect on the satellite using a harmonic excitation analysis. It was found that when the ratio of the excitation frequency to the natural frequency (ω/ωn) is greater than seven, the orbit is stable. This thesis provides methods for simulating and predicting satellite orbit control as well as providing guidelines for regions of stability for NEO missions