Fast algorithms for matching CCD images to a stellar catalogue

Two new algorithms are described for matching two dimensional coordinate lists of point sources that are signifcantly faster than previous methods. By matching rarely occurring triangles (or more complex shapes) in the two lists, and by ordering searches by decreasing probability of success, it is demonstrated that very few candidates need be considered to find a successful match. Moreover, by immediately testing the suitability of a potential match using an efficient mechanism, the need to process the entire candidate set is avoided, yielding considerable performance improvements. Triangles are described by a cosine metric that reduces the density of triangle space, permitting efficient searches. An alternative shape characterization method that reduces computational overhead in the construction phase is discussed. The algorithms are tested on a set of 10 063 wide-field survey images, with fields-of-view up to 4.8 x 3.6 deg, successfully matching 100% of the images in a mean elapsed time of 6 ms (2.4 GHz Athlon CPU). The elapsed time of the searching phase is shown to vary by less than 1 ms for list sizes between 10 and 200 points, demonstrating that fast, robust searches may be completed in nearly constant time, independent of list size.Comment: Accepted for publication in Publications of the Astronomical Society of Australi

Time-frequency analysis of the restricted three-body problem: transport and resonance transitions

A method of time-frequency analysis based on wavelets is applied to the problem of transport between different regions of the solar system, using the model of the circular restricted three-body problem in both the planar and the spatial versions of the problem.. The method is based on the extraction of instantaneous frequencies from the wavelet transform of numerical solutions. Time-varying frequencies provide a good diagnostic tool to discern chaotic trajectories from regular ones, and we can identify resonance islands that greatly affect the dynamics. Good accuracy in the calculation of time-varying frequencies allows us to determine resonance trappings of chaotic trajectories and resonance transitions. We show the relation between resonance transitions and transport in different regions of the phase space

Flat Nonholonomic Matching

In this paper we extend the matching technique to a class of nonholonomic systems with symmetries. Assuming that the momentum equation defines an integrable distribution, we introduce a family of reduced systems. The method of controlled Lagrangians is then applied to these systems resulting in a smooth stabilizing controller

Controlled Lagrangian Methods and Tracking of Accelerated Motions

Matching techniques are applied to the problem of stabilization of uniformly accelerated motions of mechanical systems with symmetry. The theory is illustrated with a simple model-a wheel and pendulum system

The energy–momentum method for the stability of non-holonomic systems

In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top

Nonholonomic Dynamics

Nonholonomic systems are, roughly speaking, mechanical systems with constraints on their velocity that are not derivable from position constraints. They arise, for instance, in mechanical systems that have rolling contact (for example, the rolling of wheels without slipping) or certain kinds of sliding contact (such as the sliding of skates). They are a remarkable generalization of classical Lagrangian and Hamiltonian systems in which one allows position constraints only. There are some fascinating differences between nonholonomic systems and classical Hamiltonian or Lagrangian systems. Among other things: nonholonomic systems are nonvariational—they arise from the Lagrange-d’Alembert principle and not from Hamilton’s principle; while energy is preserved for nonholonomic systems, momentum is not always preserved for systems with symmetry (i.e., there is nontrivial dynamics associated with the nonholonomic generalization of Noether’s theorem); nonholonomic systems are almost Poisson but not Poisson (i.e., there is a bracket that together with the energy on the phase space defines the motion, but the bracket generally does not satisfy the Jacobi identity); and finally, unlike the Hamiltonian setting, volume may not be preserved in the phase space, leading to interesting asymptotic stability in some cases, despite energy conservation. The purpose of this article is to engage the reader’s interest by highlighting some of these differences along with some current research in the area. There has been some confusion in the literature for quite some time over issues such as the variational character of nonholonomic systems, so it is appropriate that we begin with a brief review of the history of the subject

The Lyapunov-Malkin Theorem and Stabilization of the Unicycle with Rider

This paper analyzes stabilization of a nonholonomic system consisting of a unicycle with rider. It is shown that one can achieve stability of slow steady vertical motions by imposing a feedback control force on the rider’s limb

Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System

Matching techniques are developed for discrete mechanical systems with symmetry. We describe new phenomena that arise in the controlled Lagrangian approach for mechanical systems in the discrete context. In particular, one needs to either make an appropriate selection of momentum levels or introduce a new parameter into the controlled Lagrangian to complete the matching procedure. We also discuss digital and model predictive control

Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems

The method of controlled Lagrangians for discrete mechanical systems is extended to include potential shaping in order to achieve complete state-space asymptotic stabilization. New terms in the controlled shape equation that are necessary for matching in the discrete context are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. We also discuss digital and model predictive control.Comment: IEEE Conference on Decision and Control, 2006 6 pages, 4 figure

Matching and stabilization of discrete mechanical systems

Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers