Showing papers in "Journal of The Astronautical Sciences in 2005"

A Numerical Implementation of Spherical Object Collision Probability

Abstract: Collision probability analysis for spherical objects exhibiting linear relative motion is accomplished by combining covariances and physical object dimensions at the point of closest approach. The resulting covariance ellipsoid and hardbody can be projected onto the plane perpendicular to relative velocity when the relative motion is assumed linear. Collision potential is determined from the object footprint on the projected, two-dimensional, co-variance ellipse. The resulting double integral can be reduced to a single integral by various methods. This work addresses the numerical computation of this single integral using Simpson’s one-third rule to achieve at least two significant figures of accuracy over a wide range of parameters.

Relating Position Uncertainty to Maximum Conjunction Probability

Abstract: The effects of positional uncertainty on the Gaussian probability computation for orbit conjunction are examined and an upper bound determined. Relative motion between two objects is assumed linear for a given encounter with time-invariant position covariance. A method is developed to assess the maximum probability for various satellite sizes, encounter geometries, and covariance sizes and shapes. The associated standard deviation then defines the boundary of probability dilution. The assertion is made that orbit positions should be sufficiently accurate to avoid this dilution region. This work shows how to calculate the upper bounds of probability by assuming worst-case covariance orientation and size. Power series approximations are developed for aspect ratios ranging from 1 to 50 to capture 99% of all conjunction possibilities. An analytical approximation is also given for an infinite aspect ratio to capture all possibilities. These expressions can be used as a simple pre-filter or to determine worst-case scenarios. Although desired, the actual covariances are not needed. What is needed is the ratio of major-to-minor axes of the projected combined covariance ellipse, the object sizes, and the relative distance at the point of closest approach.

Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem

Abstract: In this paper, a technique for the analysis and the design of low-energy interplanetary transfers, exploiting the invariant manifolds of the restricted three-body problem, is presented. This approach decomposes the full four-body problem describing the dynamics of an interplanetary transfer between two planets, in two three-body problems each one having the Sun and one of the planets as primaries; then the transit orbits associated to the invariant manifolds of the Lyapunov orbits are generated for each Sun-planet system and linked by means of a Lambert’s arc defined in an intermediate heliocentric two-body system. The search for optimal transit orbits is performed by means of a dynamical Poincare section of the manifolds. A merit function, defined on the Poincare section, is used to optimally generate a transfer trajectory given the two sections of the manifolds. Due to the high multimodality of the resulting optimization problem, an evolutionary algorithm is used to find a first guess solution which is then refined, in a further step, using a gradient method. In this way all the parameters influencing the transfer are optimized by blending together dynamical system theory and optimization techniques. The proposed patched conic-manifold method exploits the gravitational attractions of the two planets in order to change the two-body energy level of the spacecraft and to perform a ballistic capture and a ballistic repulsion. The effectiveness of this approach is demonstrated by a set of solutions found for transfers from Earth to Venus and to Mars.

Stable Constellations of Frozen Elliptical Inclined Lunar Orbits

Abstract: Higher altitude orbits (in the 500 to 20,000 km range) at the Moon are dominated by Earth perturbations and result in motions that do not ascribe to the standard notions of orbits dominated by nonspherical gravity effects (such as from oblateness). This fact complicates orbit design of lunar orbiter constellations that require specific and persistent coverage over a selected lunar region. Using a combination of analytical theory and numerical simulation, a technique is developed for designing a lunar constellation of three spacecraft where two spacecraft are always in view from the lunar surface for the polar regions.

Navigation accuracy guidelines for orbital formation flying

Abstract: Some simple guidelines based on the accuracy in determining a satellite formation s semi-major axis differences are useful in making preliminary assessments of the navigation accuracy needed to support such missions. These guidelines are valid for any elliptical orbit, regardless of eccentricity. Although maneuvers required for formation establishment, reconfiguration, and station-keeping require accurate prediction of the state estimate to the maneuver time, and hence are directly affected by errors in all the orbital elements, experience has shown that determination of orbit plane orientation and orbit shape to acceptable levels is less challenging than the determination of orbital period or semi-major axis. Furthermore, any differences among the member s semi-major axes are undesirable for a satellite formation, since it will lead to differential along-track drift due to period differences. Since inevitable navigation errors prevent these differences from ever being zero, one may use the guidelines this paper presents to determine how much drift will result from a given relative navigation accuracy, or conversely what navigation accuracy is required to limit drift to a given rate. Since the guidelines do not account for non-two-body perturbations, they may be viewed as useful preliminary design tools, rather than as the basis for mission navigation requirements, which should be based on detailed analysis of the mission configuration, including all relevant sources of uncertainty.

Orbital design of earth-oriented tethered satellite formations

Abstract: A formation of distributed sensors has great potential to enhance Earth surveillance. This paper investigates the dynamics of tethering several subsatellites together in a three-dimensional configuration. To keep the system oriented toward Earth, the Likins-Pringle rigid body equilibria are used as a baseline design. A flexible lumped-mass model is used to assess the stability of the tethered system. Three parameters related to the formation size, masses and spin rate are varied in order to find designs that demonstrate desirable dynamic behavior. While none of the designs are Lyapunov stable, formations with a spin axis near the local vertical are well-behaved over time spans of several orbits.

Deflection of Earth-Crossing Asteroids/Comets Using Rendezvous Spacecraft and Laser Ablation

Abstract: Space missions are presented to deflect four fictitious Earth impacting objects by using an advanced magnetoplasma spacecraft designed to deliver a laser ablation payload. The laser energy required to provide sufficient change in velocity is estimated for one long-period comet and three asteroids, and an optimal rendezvous trajectory is provided for each threat scenario. The end-to-end simulations provide an overall concept for solving the deflection problem. These analyses illustrate that the optimal deflection strategy is highly dependent on the size and the orbital elements of the impacting object, as well as the amount of warning time. A rendezvous spacecraft with a multi-megawatt laser ablation payload could be available by the year 2050. This approach could provide a capable and robust orbit modification approach for altering the orbits of Earth-crossing objects with relatively small size or long warning time. Significant technological advances, multiple spacecraft, or alternative deflection techniques are required for a feasible scenario to protect Earth from an impacting celestial body with large size and short warning time.

An Analytical Solution for Relative Motion with an Elliptic Reference Orbit

Abstract: A method for the determination of accurate analytical expressions for the time dependence of the relative motion of two noninteracting masses under the gravitational influence of a spherical primary mass is presented and discussed. The case of the elliptic reference orbit is considered and the case of a circular reference orbit is analyzed as a special case. The approach also leads to accurate initial conditions for the elimination of drift away from periodic relative motion. The solution for the relative motion is in closed-form in terms of the eccentric anomalies of the target and chaser orbits, while the eccentric anomalies themselves are expressed in terms of the orbits’ respective eccentricities, using an iterative method.

Dynamics of Lagrangian Point Multi-Tethered Satellite Systems

Abstract: This paper examines the dynamics of a tether-connected multi-spacecraft system, arranged in a wheel-spoke configuration, in the vicinity of the L2 Lagrangian point of the Sun-Earth system. Equilibrium configurations of the system are determined and small motions about one of these configurations are analyzed. A simple feedback controller is also developed to stabilize the motion of the system. Numerical simulation of the free and controlled motion of the parent mass and tether librations is carried out for a three-mass case.

Modeling of Image Formation in Multi-Spacecraft Interferometric Imaging Systems

Abstract: In this paper, we develop a detailed model of the process of image formation in Multi- Spacecraft Interferometric Imaging Systems (MSIIS). We show that the Modulation Trans- fer Function of, and the noise corrupting, the synthesized optical instrument are dependent on the trajectories of the constituent spacecraft and obtain these explicit functional rela- tionships. We show that "good" imaging using MSIIS is equivalent to painting a "large disk" with smaller "paintbrushes" while maintaining a minimum thickness of paint, given that the goal of imaging is the correct classification of the formed images. This implies that the trajectories of the constituent spacecraft have to be "dense" enough in a given region, while making sure that they are "slow" enough. This is illustrated through an example. In this paper, we model the process of image formation in an MSIIS. We also model the noise inherent in such systems. We show that both the Modulation Transfer function (MTF) of the synthesized optical instrument and the noise corrupting the image formed by such an optical instrument are dependent on the trajectories of the constituent spacecraft. Further, if we formulate the goal of imaging as the correct classifi- cation of the formed images, we show that satisfactory imaging by an MSIIS is analogous to the "painting" of a large resolution disk with smaller "coverage" disks or "paintbrushes" while maintaining a minimum thickness of paint. The problem of design of MSIIS is related to the fields of synthetic aperture optics and formation flying. The relationship of our work to these topics is discussed next. The topic of long baseline interferometry falls under the category of synthetic aperture optics, 4 that was first developed in the context of synthetic aperture radars (SAR). 5 The method consists of emulating a large optical instrument by a number of smaller ones and combining their contributions in a proper way to obtain an image that has resolution comparable to that of the large optical instrument. For a discussion of the various metrics used in the optimization of these systems, please refer to 6 and the references therein. All the abovementioned designs optimize the locations of the constituent telescopes such that some metric of image quality is maximized. Thus, these correspond to static optimization problems. However, for an MSIIS, due to the high resolution requirements, the "design variables" are the trajectories of the constituent spacecraft. In fact, we show the explicit dependence of the MTF on the trajectories of the constituent spacecraft. Further, we show that the noise corrupting the image in an MSIIS is a function of the spacecraft trajectories and the rate of arrival of photons on the observation plane. Given that the goal of imaging is the correct classification of images, the design of an MSIIS reduces

Zero Relative Radial Acceleration Cones and Controlled Motions Suitable for Formation Flying

Abstract: Assume that a constellation of satellites is required to remain close to a given nominal trajectory and that there is some freedom in the selection of the geometry of the constellation. If we are interested in avoiding large variations of the mutual distances between the spacecraft, we can consider the possible existence of regions of zero relative radial acceleration with respect to the nominal trajectory. The motion along these regions will reduce the expansion or contraction of the constellation. The goal of this paper is the study of these regions and the controlled motions between them.

Sliding Mode Control and Optimization for Six DOF Satellite Formation Flying Considering Saturation

Abstract: This paper considers the problem of controlling both position and attitude states for a formation flying system using only thrusters. A coupled six degree of freedom position and attitude model is derived, where the coupling comes from the gravity gradient torque. Furthermore, the J2 perturbation, mass variation, thruster model, and thruster layout are considered. A globally stable chattering free sliding mode robust controller is designed with respect to functional bounded uncertainties. Based on the studies of open-loop responses, the range of control parameters is properly selected to avoid thruster saturation. A genetic algorithm is then applied to select the controller parameters. Both position and attitude states in a formation are precisely and continuously controlled for more than two years without recharging, using the selected electronic ion engine.

QUEST and The Anti-QUEST: Good and Evil Attitude Estimation

Abstract: An attitude determination algorithm, proposed 25 years ago as a back-up algorithm in case the QUEST algorithm failed to function properly on its maiden mission, is finally compared with the QUEST algorithm. A comparison of the attitude determination accuracies of both algorithms is carried out using the QUEST measurement model. The back-up algorithm, which we have chosen to call “The Anti-QUEST,” while it lacks the special features that have made QUEST so popular today and is rather clumsy, slow and deficient in many ways, works well under the almost ideal conditions of the Magsat mission, but not generally. Further comparisons of QUEST and The Anti-QUEST provide useful insights into the different behaviors of optimal and deterministic attitude estimators. This work also presents useful practical techniques for the covariance analysis of nonoptimal algorithms.

Automatic Generation and Integration of Equations of Motion for Flexible Multibody Dynamical Systems

Abstract: In this paper, we consider a new direction for generating and simultaneously solving equations of motion for dynamical systems, using automatic differentiation. We overview the current computational approaches for solving multibody dynamics problems and discuss several choices for equation of motion formulation. We present an operator-overloading method for generating equations of motion automatically via Lagrange’s Equations and solve them in a direct fashion. Several numerical examples are presented to demonstrate the accuracy and efficiency of the method for simulating the motion of multibody dynamical systems.

Investigating the Effects of Cross-Link Delays on Spacecraft Formation Control

Abstract: In this paper, the ability to perform tight formation control in the presence of communication and measurement delays is investigated. The paper discusses cross-link requirements for formation control and compares centralized and decentralized control strategies. A simple two degree-of-freedom model with communication delays is introduced for analysis. A typical approach for systems with small time delays is to base the design on a nominal system model that does not contain the time delays. The limitations of this approach on stability and closed-loop bandwidth are discussed. Finally, the time delay is considered in the design of two control structures: state-feedback and state-feedback with integral feedback. The effects of the time delay on stability, stability margins, and bandwidth are investigated. An example illustrating these effects is presented.

The Orbital Perturbation Environment for the COBRA and COBRA Teardrop Elliptical Constellations

Abstract: Elliptical, non-geostationary satellite orbit (NGSO) satellite constellations have been proposed as a solution to the problem of the shortage of slots in the Geostationary Ring for new communications satellites. These elliptical constellations have the potential to greatly multiply the available space real estate without interfering with the existing GEO satellites. These constellations employ eight-hour “leaning” elliptical orbits, operating at approximately the critical inclination of 63.4 degrees. The three active arcs of a single satellite have a ground trace that resembles a coiled cobra, hence the acronym name COBRA. Due to the exact repeat ground-track, with three orbital revolutions for each rotation of the Earth, these COBRA orbits experience deep tesseral resonance perturbations with the 3rd, 6th, and other (multiples of three) order geopotential harmonic coefficient pairs. These resonant perturbations are dependent on the RAAN of the particular ground-track. These orbits also experience strong perturbations due to the lunar-solar point masses. The semi-analytical satellite theory DSST is used to explore both the tesseral resonance and lunar-solar point mass perturbations.

Large Aperture Space Telescopes in Formation: Modeling, Metrology, and Control

Abstract: In this paper we summarize a concept definition and feasibility analysis for formation flying a representative large space telescope composed of separated optical elements. A virtual-structure construct (an equivalent rigid body) is created by unique metrology and control that combines both centralized and decentralized methods. The formation may be in orbit at GEO for super-resolution Earth observation or it may be in an Earth-trailing orbit for astrophysics. Extended applications are envisioned for exo-solar planet interferometric imaging by a formation of very large separated optics telescopes. Space telescopes, with such large apertures and optics with focal length f/10 to f/100, are not feasible if connected by massive metering structures. Instead, the new virtual-structure paradigm of information and control connectivity between the formation elements provides the necessary spatial rigidity and alignment precision for the telescope.

Low -Thrust Autonomous Control for Maintaining Formation and Constellation Orbits

Abstract: Control strategies to maintain spacecraft formation flying and constellation station keeping orbits are presented in this paper. For small or nanosatellites, the potential control thruster used in orbital control could be of the low-thrust type. The control methods that can adapt to different thrust levels without compromising the system performance and the mission task are investigated. Autonomous algorithms (analytic solutions that can be easily executed in real-time) that can be easily implemented on board are considered at the same time to reduce the overall cost of the mission.

Minimum-Parameter Representations of N-Dimensional Principal Rotations

Abstract: Classic techniques have been established to characterize N × N proper orthogonal matrices using the N-dimensional Euler’s theorem and the Cayley transform. These techniques provide separate descriptions of N-dimensional orientation in terms of the constituent principal rotations or a minimum-parameter representation. The two descriptions can be linked by the canonical form of the extended Rodrigues parameters. This form is developed into a new minimum-parameter representation that directly links to the principal rotations. The new representation is solved using analytic and geometric approaches for N = 3 and N = 4, and numerical solutions are found for N= 5. In fact multiple solutions, which are related geometrically by different coordinatizations of the principal planes, have been found. The new parameters represent a projection of the principal rotations onto the planes formed by the body coordinates.

Keeping a Spacecraft on the Sun-Earth Line

Abstract: Measurements of Earth's atmosphere as it occults sunlight can be obtained advantageously from a spacecraft placed in the proximity of the Sun-Earth Lagrange point L2. Maintaining the condition of continuous solar occultation by all parts of the atmospheric disk requires that the displacement of the spacecraft perpendicular to the Sun-Earth line remains less than 200 km. However, the gravitational force exerted by the Earth s moon must be negated by propulsion in order to meet this rather tight constraint. We provide an estimate of propulsive force needed to keep the spacecraft coincident with L2, as well as estimates of velocity increments needed to maintain various trajectories in the close vicinity of L2.

Effects of Orbit Variations and J2 Perturbations on a Class of Earth-Orbiting Interferometric Observatories

Abstract: This paper revisits a class of Earth-orbiting interferometric observatories introduced previously and reviews the general procedures to achieve wave-number plane coverage. The effect of eccentric spacecraft trajectories and gravity field J2 perturbations on wave-number plane coverage are considered. Conditions for complete wave-number plane coverage are found for certain classes of orbit perturbations. This analysis leads to design criteria for interferometric observatories that ensure wave-number plane coverage as a function of perturbation strength.

Long-Term Evolution of the KOMPSAT-1 Orbit

Abstract: A long-term orbital evolution of the Korea Multi-Purpose Satellite-1 is monitored and analyzed in this paper. Four years of orbital evolution is divided into four phases in terms of the operational status of the satellite. Orbit determination results, which are obtained from the satellite’s GPS navigation data, are used to convert the osculating orbit into the mean orbit. Variation of the mean altitude by orbit maneuvers and safe-hold mode operations is also analyzed. Decay of the mean altitude by natural atmospheric drag is correlated with the solar flux variation. Variation of the mean inclination is analyzed in terms of the shift of the local time of the ascending node crossing. The evolution from a near frozen orbit is monitored and analyzed in a phase plot of the mean argument of perigee and mean eccentricity.

The Potential of Remote Sensing for Neutral Atmospheric Density Estimation in a Data Assimilation System

Abstract: New data assimilation techniques have improved time-dependent estimates of the neutral atmospheric density, making it possible to better estimate the drag perturbation on low-Earth-orbiting satellites. This study looks at the potential for using satellite remote sensing from space as an effective density observation source in a data assimilation system. Changes in the neutral density can occur on a minute-to-minute basis, particularly during geomagnetic storms. Although coverage from only a few (two) satellites may be limited, remote sensing provides observations with a high temporal and spatial resolution. To quantify the effectiveness of the observing platform, a simulated “truth” neutral atmosphere is created using a physical model. This “truth” neutral atmosphere is sampled according to the mechanics of the remote sensing platform, and the results are statistically evaluated. With the resolution afforded by remote sensing, results show that two remote sensing satellites provide a stable solution of degree 4 (5 × 5) every ten minutes. Although coverage from two remote sensing satellites is limited, the coverage is sufficient to provide a pattern correlation coefficient consistently higher than 0.92.

A Simple Derivation of the Gauss-Bonet Theorem

Abstract: A simple derivation of the Gauss-Bonet theorem is presented based on the representation of spherical polygons by Euler angles and Rodrigues transposition theorem. This leads to a derivation of the theorem which avoids completely the explicit evaluation of rotation matrices.

MixTeam Scheduling Algorithms: Application to Space-Based Interferometric Observatories

Abstract: Long-baseline interferometric observatories comprising multiple space telescopes in the same circular Earth orbit have been proposed for high-resolution imaging of exosolar planets. In this paper, we describe the novel class of MixTeam scheduling algorithms for such observatories. A range of schedulers within this class was evaluated through simulation. The results indicate that observation schedules can be shortened by up to 21% by applying well-designed learning mechanisms. The influence of constellation geometry on observation scheduling was also studied and is characterized in terms of tradeoffs among coverage, capacity and utilization of the observatory.

Velocity Perturbations Analysis of the Spot 1 Ariane Rocket Fragmentation

Abstract: The breakup of the Spot 1 Ariane third stage rocket was one of the most prolific and puzzling satellite fragmentation events ever. The cause of the breakup was long listed as unknown, with several studies pointing to different conclusions. This study re-examines the breakup by studying the velocity distribution patterns of the fragments. The presence of six fragments in high energy orbits suggests collision as a possible cause of the breakup. The existence of butterfly patterns in the spread velocities of the fragments in planes between the vertical and the downrange directions also points to fragmentation by impact. The orientations of the debris concentrations flanking the high energy ricochet uncovers the possible path of the projectile that could have struck the Ariane rocket—the projectile could have come from an angle of 11° or less above the horizontal and about 6° from the left of the Ariane’s path. The velocity perturbations distributions in the radial and downrange components show that fragments lying outside the Gaussian envelopes comprised the high energy ricochet and ejecta which emanated from the contact area. Both of the Gaussian envelopes exhibited displacements opposite from the ricochet and ejecta, thus betraying recoil effects. The cross-range velocity distribution pattern was severely distorted with the ejecta forming concentrations on either side of the Gaussian envelope—a further sign of possible collision. Finally, a cluster analysis of the fragments indicates that the vast majority of the fragments suffered only small velocity changes while the high energy ricochet and ejecta formed their own distinct clusters, which is yet another indication of possible fragmentation by impact. The possible path of the projectile would place itself among the members of the Sun-synchronous orbiting group.