Liam M. Healy

Close conjunction detection on parallel computer

The text should mention that the algorithm described as applied to a single Cartesian coordinate is then successively applied to the other coordinates in the obvious way to select satellites that are within a specified distance of one another.

The Main Problem in Satellite Theory Revisited

Using the elimination of the parallax followed by the Delaunay normalization, we present a procedure for calculating a normal form of the main problem (J2 perturbation only) in satellite theory. This procedure is outlined in such a way that an object-oriented automatic symbolic manipulator based on a hierarchy of algebras can perform this computation. The Hamiltonian after the Delaunay normalization is presented to order six explicitly in closed form, that is, in which there is no expansion in the eccentricity. The corresponding generating function and transformation of coordinates, too lengthy to present here to the same order; the generator is given through order four.

Paint by number: Uncovering phase flows of an integrable dynamical system

Given an integrable dynamical system with one degree of freedom, ‘‘painting’’ the integral over phase space proves to be a powerful technique for uncovering both global and local behavior.This graphical technique avoids numerical integration, employing instead a nonlinear method of assigning contrasting colors to the energy values to distinguish subtle details of the flow.

LIIVe: A Small, Low-Cost Autonomous Inspection Vehicle

The Low-design Impact Inspection Vehicle (LIIVe) is a Naval Research Laboratory research program to design and validate the techniques, sensors, and algorithms required to autonomously inspect a host satellite. LIIVe is envisioned as a self-contained inspection vehicle that is small and inexpensive enough to constitute an off-the-shelf component of a satellite system. Operationally, LIIVe is docked to its host and is released in case of a deployment failure or other on-orbit issue that requires high-quality imagery of the host; it provides imagery of the host to mission controllers while maintaining passively-safe trajectories, and then either re-docks or disposes of itself. In order to facilitate its adoption LIIVe’s guiding design principle is to impose as few design changes on the host as possible. Notably, LIIVe does not require shared communications or power and does not utilize relative navigation aids except for a single optical fiducial attached to the docking port. This paper describes the relative navigation challenges such a system entails and the sensor and algorithmic approaches being taken to address them, including optic flow-based collision detection, real-time passively safe trajectory planning, and low–computational–burden decision making logic.

Applications of Parallel Processing to Astrodynamics

Parallel processing is being used to improve the catalog of earth orbiting satellites and for problems associated with the catalog. Initial efforts centered around using SIMD parallel processors to perform debris conjunction analysis and satellite dynamics studies. More recently, the availability of cheap supercomputing processors and parallel processing software such as PVM have enabled the reutilization of existing astrodynamics software in distributed parallel processing environments, Computations once taking many days with traditional mainframes are now being performed in only a few hours. Efforts underway for the US Naval Space Command include conjunction prediction, uncorrelated target processing and a new space object catalog based on orbit determination and prediction with special perturbations methods.

Trajectory Guidance Using Periodic Relative Orbital Motion

Relative motion of one satellite about another in circular orbit, where the two objects have the same semimajor axis, is periodic in the linearized approximation. A set of orbital elements, the geometric relative orbital elements, which are an exact geometric analogy to the classical orbital elements, can be defined. The relative orbit is manifestly seen to be an ellipse or circle in apocentral coordinates, analogous to perifocal coordinates in inertial motion and different from the local-vertical local-horizontal Cartesian coordinates customarily used for analysis of relative motion problems. These provide a way to do relative motion trajectory design and guidance that stands in contrast to the use of Cartesian coordinates. Algorithms are presented for computing the delta-v and timing for impulsive maneuvers to change a single element: two that change the plane and one that changes the size of the relative orbit. The change in size is a two-maneuver transfer and uses the solution of the three-point perio...

Computation of error effects in nonlinear Hamiltonian systems using Lie algebraic methods

There exist Lie algebraic methods for obtaining transfer maps around any given trajectory of a Hamiltonian system. This paper describes an iterative procedure for finding transfer maps around the same trajectory when the Hamiltonian is perturbed by small linear terms. Such terms often result when an actual system deviates from an ideal one due to errors. Two examples from accelerator physics are worked out. Comparisons with numerical computations, and in simple cases exact analytical calculations, demonstrate the validity of the procedure.

Formation and dynamics of an artificial ring of dust for active orbital debris removal

Recently we suggested a dust-based active debris removal technique to selectively remove small untrackable debris that occupies a very large volume around the Earth. For designing a working system an accurate knowledge of the dynamics of the released dust in orbit is necessary. In this paper we numerically examine the dynamics of non-interacting spherical tungsten dust grains of diameter between 30-60 microns released in a polar low-Earth orbit. We analyze different perturbations due to nonuniform gravity, solar radiation pressure, solar cycles as well as solar and lunar gravity, and dust charging effects, etc., and determine a set of forces adequate to describe the dynamics over the life of the dust in orbit (∼12–15 years). With the resulting force model we analyze the orbits of many dust grains to determine the formation and geometry of the ring. We qualitatively examine the effects of the calculated geometry and dynamics of the dust cloud on the efficiency of the Active Debris Removal scheme.

Orbit Propagation with Lie Transfer Maps in the Perturbed Kepler Problem

The Lie transfer map method may be applied to orbit propagation problems in celestial mechanics. This method, described in another paper, is a perturbation method applicable to Hamiltonian systems. In this paper, it is used to calculate orbits for zonal perturbations to the Kepler (two-body) problem, in both expansion in the eccentricity and closed form. In contrast with a normal form method like that of Deprit, the Lie transformations here are used to effect a propagation of phase space in time, and not to transform one Hamiltonian into another.

Computation of Lie transfer maps for perturbed Hamiltonian systems

Time evolution of a Hamiltonian system can be viewed as a canonical transformation; therefore perturbations, giving rise to near-identity deviations from an unperturbed solution, can be represented by products of Lie transformations, or, together with the unperturbed solution, Lie transfer maps. In this paper I broaden the applicability to all perturbed Hamiltonian systems the method of Dragt and Finn and subsequent co-workers, who developed a representation using a product of Lie transformations factored by phase space variable order. In the present paper, perturbation parameters are no longer necessarily associated with the phase space variables; this method treats both “internal” and “external” perturbations on an equal footing, and a rank is assigned to each variable to reflect the degree of perturbation it represents. With the companion program PGLT, analytic development of the Lie transfer maps is relatively easy for many systems.