2,904 research outputs found
Combination of Lyapunov Functions and Density Functions for Stability of Rotational Motion
Lyapunov methods and density functions provide dual characterizations of the solutions of a nonlinear dynamic system. This work exploits the idea of combining both techniques, to yield stability results that are valid for almost all the solutions of the system. Based on the combination of Lyapunov and density functions, analysis methods are proposed for the derivation of almost input-to-state stability, and of almost global stability in nonlinear systems. The techniques are illustrated for an inertial attitude observer, where angular velocity readings are corrupted by non-idealities
Lyapunov instability of fluids composed of rigid diatomic molecules
We study the Lyapunov instability of a two-dimensional fluid composed of
rigid diatomic molecules, with two interaction sites each, and interacting with
a WCA site-site potential. We compute full spectra of Lyapunov exponents for
such a molecular system. These exponents characterize the rate at which
neighboring trajectories diverge or converge exponentially in phase space.
Quam. These exponents characterize the rate at which neighboring trajectories
diverge or converge exponentially in phase space. Qualitative different degrees
of freedom -- such as rotation and translation -- affect the Lyapunov spectrum
differently. We study this phenomenon by systematically varying the molecular
shape and the density. We define and evaluate ``rotation numbers'' measuring
the time averaged modulus of the angular velocities for vectors connecting
perturbed satellite trajectories with an unperturbed reference trajectory in
phase space. For reasons of comparison, various time correlation functions for
translation and rotation are computed. The relative dynamics of perturbed
trajectories is also studied in certain subspaces of the phase space associated
with center-of-mass and orientational molecular motion.Comment: RevTeX 14 pages, 7 PostScript figures. Accepted for publication in
Phys. Rev.
Nonlinear Feedback Control of Axisymmetric Aerial Vehicles
We investigate the use of simple aerodynamic models for the feedback control
of aerial vehicles with large flight envelopes. Thrust-propelled vehicles with
a body shape symmetric with respect to the thrust axis are considered. Upon a
condition on the aerodynamic characteristics of the vehicle, we show that the
equilibrium orientation can be explicitly determined as a function of the
desired flight velocity. This allows for the adaptation of previously proposed
control design approaches based on the thrust direction control paradigm.
Simulation results conducted by using measured aerodynamic characteristics of
quasi-axisymmetric bodies illustrate the soundness of the proposed approach
On mathematical modelling of insect flight dynamics in the context of micro air vehicles
This paper discusses several aspects of mathematical modelling relevant to the flight
dynamics of insect flight in the context of insect-like flapping wing micro air vehicles (MAVs).
MAVs are defined as flying vehicles ca six inch in size (hand-held) and are developed to
reconnoitre in confined spaces (inside buildings, tunnels etc). This requires power-efficient,
highly-manoeuvrable, low-speed flight with stable hover. All of these attributes are present in
insect flight and hence the focus of reproducing the functionality of insect flight by engineering
means. This can only be achieved if qualitative insight is accompanied by appropriate
quantitative analysis, especially in the context of flight dynamics, as flight dynamics underpin
the desirable manoeuvrability.
We consider two aspects of mathematical modelling for insect flight dynamics.
The first one is theoretical (computational), as opposed to empirical, generation of the
aerodynamic data required for the six-degrees-of-freedom equations of motion. For these
purposes we first explain insect wing kinematics and the salient features of the corresponding
flow. In this context, we show that aerodynamic modelling is a feasible option for certain flight
regimes, focussing on a successful example of modelling hover. Such modelling progresses
from first principles of fluid mechanics, but relies on simplifications justified by the known
flow phenomenology and/or geometric and kinematic symmetries. In particular, this is relevant
to six types of fundamental manoeuvres, which we define as those steady flight conditions for
which only one component of both the translational and rotational body velocities is non-zero
(and constant).
The second aspect of mathematical modelling for insect flight dynamics addressed here
deals with the periodic character of the aerodynamic force and moment production. This
leads to consideration of the types of solutions of nonlinear equations forced by nonlinear
oscillations. In particular, the existence of non-periodic solutions of equations of motion is of
practical interest, since this allows steady recitilinear flight.
Progress in both aspects of mathematical modelling for insect flight will require further
advances in aerodynamics of insect-like flapping. Improved aerodynamic modelling and
computational fluid dynamics (CFD) calculations are required. These theoretical advances
must be accompanied by further flow visualisation and measurement to validate both the
aerodynamic modelling and CFD predictions
Nonlinear Dynamics Analysis and Control of Space Vehicles with Flexible Structures
Space vehicles that implement hardware such as antennas, solar panels, and other extended appendages necessary for their respective missions must consider the nonlinear rotational and vibrational dynamics of these flexible structures. Formulation and analysis of these flexible structures must account for the rigid-flexible coupling present in the system dynamics for stability analysis and control design. The system model is represented by a flexible appendage attached to a central rigid body, where the flexible appendage is modeled as a cantilevered Euler-Bernoulli beam. Discretization techniques, such as the assumed modes method and the finite element method, are used to model the coupled dynamics by transforming the partial differential equations of motion into a finite set of differential equations. State feedback control laws are designed to achieve stability and desired motion in the presence of rigid-flexible coupling. An optimal control law in the form of a linear quadratic regulator is presented and compared with a Lyapunov-based control law that guarantees asymptotic stability. Conventional and adaptive sliding mode control laws are also presented to account for any uncertainties in the linearized system model. Full-order and reduced-order observers are included in the control system to account for lack of velocity state measurements that are generally unavailable in real world applications
The dynamics of spin stabilized spacecraft with movable appendages, part 1
The motion and stability of spin stabilized spacecraft with movable external appendages are treated both analytically and numerically. The two basic types of appendages considered are: (1) a telescoping type of varying length and (2) a hinged type of fixed length whose orientation with respect to the main part of the spacecraft can vary. Two classes of telescoping appendages are considered: (a) where an end mass is mounted at the end of an (assumed) massless boom; and (b) where the appendage is assumed to consist of a uniformly distributed homogeneous mass throughout its length. For the telescoping system Eulerian equations of motion are developed. During all deployment sequences it is assumed that the transverse component of angular momentum is much smaller than the component along the major spin axis. Closed form analytical solutions for the time response of the transverse components of angular velocities are obtained when the spacecraft hub has a nearly spherical mass distribution
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