Periodic orbits in Hill’s restricted three-body problem have been thoroughly studied because of their wide applicability. However, collision trajectories, in particular transient-collision (TC) trajectories, received less attention, despite their potential relevance to exploration missions. This study presents a methodology for identifying TC trajectories with respect to the secondary body within the framework of Hill’s problem with an oblate primary (HPO), including a specific application to the Mars-Deimos system. The Hamiltonian dynamics are regularized using the Levi-Civita method, followed by a systematic numerical grid search over three independent parameters, based on an integration of the regularized equations backward in time from a collision state. A trajectory suitable for encounter with Deimos is identified using the highest value of the Jacobi constant, which consistently yields collision solutions. Subsequently, a k-nearest neighbors-based method is used to characterize the robustness of the trajectory to variations in initial conditions. The solution is then validated using high-fidelity simulations, which indicate that the HPO model is a useful preliminary design tool for exploration missions to minor celestial bodies.

Safe and reliable spacecraft rendezvous maneuvers are crucial for numerous space missions. However, missed-thrust events (MTEs), characterized by a sudden loss of thrust, present significant challenges that can jeopardize mission success. In this research, rendezvous strategies are designed to mitigate the impact of MTE on spacecraft equipped with electric propulsion. A resilient trajectory shaping and control methodology is developed to mitigate the effect of thrust failures. The new approach introduces a missed-thrust sensitivity grade to generate trajectories with reduced sensitivity to MTE, incorporating a finite-horizon linear quadratic regulator and zero-effort miss/ zero-effort velocity guidance, together with an optimization of the final time. The MTE-resilient rendezvous is designed while accounting for a realistic thruster operation mode, including conversion from continuous to pulsed commands. The results indicate that the proposed methodology significantly enhances the resilience of rendezvous maneuvers to MTE. Rendezvous plans designed using the MTSG approach demonstrate improved performance, with reduced variations in miss distances and miss velocities, as well as minimized variations in maneuver timing and approach direction, both in low Earth orbits and geostationary orbits.

This paper presents a new semianalytical orbit propagation method designed to account for the J2 oblateness perturbation. Unlike common existing methods, such as those developed by Brouwer and Kozai, which typically rely on averaging methods to separate satellite motion into secular, short-periodic, and long-periodic terms, the new approach is based on the development of the Fourier series expansion of the Lagrange planetary equations. It provides an approximate solution that directly yields the short-periodic corrections of the classical orbital elements under the J2 perturbation. A key contribution is the detailed, explicit derivation of the Fourier series form of the J2 perturbing potential. An analysis of simulation data highlights the new method’s particular suitability and advantage in accuracy for orbits with low eccentricities and low inclinations, especially in the in-plane directions, compared to Kozai’s short-periodic solution. On the other hand, at high eccentricities and inclinations, the proposed method does not have an advantage in terms of accuracy. The utility of this method is demonstrated by propagating a two-satellite relative motion in low Earth orbit, where it results in significantly smaller in-plane errors.

Spacecraft rendezvous missions have become increasingly important, with new applications such as debris removal, in-orbit servicing and cooperative orbital control. However, accurate autonomous rendezvous with a target moving on an elliptic orbit is a challenging problem, characterized by time-varying relative orbital dynamics. Whereas the state-of-the-art includes an abundance of rendezvous algorithms, little attention has been given to developing simple closed-loop elliptic-rendezvous algorithms that are both implementable on realistic thruster configurations and are able to accommodate a required approach direction. The current work develops a feedback controller for elliptic rendezvous based on a simple finite-horizon linear quadratic regulator and a concomitant Lyapunov-based pulse-generation scheme. This methodology is implemented on a triaxial thruster configuration and on a vernier-assisted configuration. It is shown how the proposed feedback control can be supplemented to yield R/V/H-bar directional rendezvous, using ideas borrowed from sliding-mode control. Simulations indicate that the new pulsed-feedback strategy is effective for rendezvous with targets on eccentric orbits under practical low-thrust constraints.

Satellite orbit control is essential for numerous space missions, requiring both fuel efficiency and accuracy in the presence of orbital perturbations. Thus far, however, little attention has been given to developing low-thrust closed-loop orbit control with guaranteed stability, accuracy, and performance, while accounting for the constant-magnitude thrust constraint typical to electric propulsion (EP) systems. This research develops a new closed-loop Lyapunov-based control law for low-thrust orbital transfers using the Milankovitch vectorial elements, with a particular emphasis on constant-magnitude thrust systems. Although EP systems are highly fuel-efficient, their constant-magnitude thrust attribute introduces challenges that have been scarcely addressed in previous works. This work suggests a new method to solve these challenges by adopting an integrative approach, wherein the closed-loop orbital transfer algorithms combine a Lyapunov-based controller with linear quadratic regulation and sliding-mode control. Simulations indicate that, compared to existing methods, the newly-proposed integrative approach improves transfer accuracy while guaranteeing stability, eliminating chattering and reducing propellant consumption for transfers to near-circular target orbits.

Far-range space rendezvous is a complex process, designed to achieve terminal conditions required for the subsequent close-range rendezvous phase. In far-range rendezvous, the relative dynamics between the chaser spacecraft and the target are nonlinear, thus challenging the simple orbit control methods used in close-range rendezvous. In this paper, a new nonlinear closed-loop low-thrust far-range rendezvous law applicable to both spacecraft rendezvous and minor celestial body rendezvous is developed by means of angular momentum and eccentricity vector matching, which steers the chaser into the target's orbital plane. This matching is realized by a nonlinear Lyapunov-based feedback control, in which the chaser and target angular momentum and eccentricity vectors are expressed in target-fixed local-vertical local-horizontal coordinates. In order to reduce the resulting along-track offset, the targeted semimajor axis is appended with an along-track-dependent bias, such that the along-track relative distance and speed are ultimately nullified. It is proven that the newly-developed rendezvous law remains invariant under the gravitational acceleration exerted by a minor celestial body, and is, therefore, applicable to minor celestial body rendezvous. Simulation results for both low Earth orbit spacecraft rendezvous and minor celestial body rendezvous indicate that the proposed far-range rendezvous algorithm is effective, steering the chaser spacecraft to close proximity of the target, thus enabling to initiate the close-range rendezvous phase.

Earth’s oblateness effects on the long-term evolution of satellite relative motion have been extensively studied in recent decades. However, much less attention has been given to the effect of sectoral and tesseral harmonics on the inter-satellite distance growth. This subtle effect becomes increasingly important in modern missions, involving large constellations of satellites, characterized by strict safety and fuel-use constraints. In this context, the current work provides a comprehensive modeling of short-periodic (SP) terms, emanating from the tesseral and sectoral harmonics, and their effect on the averaged inter-satellite distance. The averaged distance is utilized as a metric for quantifying the long-term effects of the gravitational harmonics on the relative drift. It is proven that differential sectoral and tesseral perturbations induce a secular inter-satellite drift due to SP terms related to the semimajor axis. These SP terms can be utilized as periodic corrections, mapping between mean and osculating orbital elements, thus mitigating the anticipated drift. A method that formulates the differential drift effects for a geopotential of arbitrary order is developed, and the dominant drivers of inter-satellite drift are detected. These drivers are then modeled in numerical simulations, showing that incorporating SP corrections of the semimajor axes can significantly mitigate the inter-satellite distance drift, thereby facilitating safe and efficient operation in, e.g., large satellite constellations.

Feedback control for close-range orbital rendezvous, which permits choosing the direction of the final approach, is beneficial for numerous space missions. However, developing such a feedback controller for satellites operating with constant-magnitude low thrust without introducing chattering is nontrivial due to the underactuated nature of the dynamics. This paper develops a feedback control law for rendezvous with a target on a nearly circular orbit, assuming that the chaser satellite utilizes constant-magnitude low thrust. Based on the Clohessy–Wiltshire (CW) model, a sliding surface is chosen on which the states approach the origin. A feedback law driving given initial states to the sliding surface is developed first. Close to the origin, a bang-bang type controller is then employed, which drives the state errors to zero in finite time while allowing to choose the final approach direction along the R-, V-, or H-bar. Solutions to the state trajectories are obtained in closed form, except for the case of the R-bar approach, which is proven to be finite-time stable. It is shown that the higher-order dynamics neglected in the CW equations can be handled through dynamic inversion combined with a hopping maneuver.

Utilizing broadband low Earth orbit satellite signals in an opportunistic manner for navigation is becoming increasingly popular. This paper deals with a particularly useful approach for navigation based on satellite signals of opportunity, which uses carrier Doppler-shift observables. We provide analytically derived and simplified formulas for the Jacobian involved in the numerical computation of the navigation solution and derive a global navigation satellite system-like dilution-of-precision metric that can be used to assess accuracy. A numerical study provides preliminary computational results.

Smart dust devices are tiny systems-on-a-chip platforms capable of sensing, storing and transmitting data wirelessly as part of a large network with distributed capabilities. Previous works investigated the long-term orbital evolution of smart dust in space by studying the combined effect of gravitational perturbations, solar radiation pressure (SRP) and atmospheric drag. In the current work, the problem of finding long-term orbital equilibria conditions for smart dust is recast and extended to include Poynting–Robertson and Solar Wind (PRSW) drag. By including the PRSW effects and defining new equilibrium conditions on the orbital orientation, some additional partial equilibrium solutions are found. Moreover, it is shown that even though PRSW is not dominant compared to SRP or J₂, it still influences the evolution of the relative Sun-orbit orientation. For orbits with higher initial perigee altitudes, where drag and J₂ effects subside, it is shown that PRSW influences long-term orbital behavior, and should be considered in the orbit design scheme of smart dust devices.

The study of libration-point location in the Mars-Phobos and Mars-Deimos three-body systems is a relatively new endeavour. Although the study of libration points is centuries old, calculating the locations of libration points and their dynamical substitutes commenced only after advances in perturbation theories have been made. Thus far, most of the studies on libration-point locations were mainly theoretical, usually without treating a specific planetary system. In particular, only a handful of studies included calculations of the libration-point location in a realistic Martian dynamical environment; all of them focused only on the Mars-Phobos system. In this paper, the locations of the L₁ and L₂ libration points in the vicinity of Phobos and Deimos are calculated, while considering the oblateness of Mars, the non-spherical gravity fields of Phobos and Deimos, the orbital eccentricity of Phobos, and the gravitational perturbation due to the Sun. The averaged effect of the Sun on the libration-point locations is also studied. The resulting libration-point locations differ from their classical values by up to 0.88 km in the case of Phobos and 0.45 km in the case of Deimos. It is shown that the displacement of libration points L₁ and L₂ in the vicinity of Deimos is mostly in the xz plane of the rotating frame, and that the libration-point motion under the gravitational perturbation of the Sun creates a three-dimensional figure-eight-shaped curve that collapses into an ellipse-like shape every Martian equinox. When the averaged gravitational effect of the Sun is considered, the libration-point locations are calculated for multiple dates along the Martian year.

Spacecraft rendezvous is a crucial aspect of space exploration and satellite servicing; hence, it is one of the most widely-studied problems in astronautics. The first spacecraft rendezvous missions date back to the dawn of the space age. In this paper, we focus on the problem of spacecraft rendezvous using constant-magnitude thrust, because, this setup is more realistic for small spacecraft operating electric propulsion systems. Considering the fact that the only optimal solutions for the general constant-magnitude-Thrust rendezvous reported in the literature were found by means of direct optimization, we propose a simple state transformation, which leads to new solutions for rendezvous with constant-magnitude thrust. First, we show that by expressing the rendezvous dynamics as a driftless dynamical system, the solution of the TPBVP emanating from the minimum-Time rendezvous problem becomes simpler; second, the same driftless form can be used to synthesize a feedback rendezvous law considering the constant thrust-magnitude constraint.

Position and velocity estimation using Global Navigation Satellite Systems (GNSS) has been widely studied and implemented. In contrast to existing GNSS, the idea of using low Earth orbit (LEO) satellites for position and velocity determination is relatively new. On one hand, the launch to LEO is more affordable compared to GNSS orbits. On the other hand, LEO satellites provide reduced coverage and suffer from orbit determination uncertainties. In this article, we study position and velocity estimation for an aerial platform using signals from a LEO satellite constellation, designed to produce a relatively long coverage duration, while minimizing the geometric dilution of precision. We determine the receiver’s position by using the trilateration method and the velocity by using Doppler estimation, and improve the accuracy thereof by utilizing an Extended Kalman Filter (EKF). We suggest a solution for the trilateration initialization problem, which arises for LEO navigation satellites, which relies on averaging the Earth projection of all the satellites within sight. We examine two scenarios, one wherein the EKF’s dynamical model matches the reference dynamical model, and another with a model mismatch. When the dynamical model is approximated, the EKF reduces the position and velocity errors considerably. When the dynamical model is known, the position and velocity errors can be reduced by an order of magnitude.

Time difference of arrival (TDOA)-based geolocation refers to position estimation of a terrestrial emitter by using at least three receivers. This type of geolocation is used by a variety of systems. Existing systems, however, do not provide continuous coverage of a given target area. In this study, satellite constellations that can provide continuous geolocation services in a specific target region are designed and optimized. The problem of TDOA-based geolocation is formulated, and the lower bounds of the estimation errors are expressed in the form of the Cramér–Rao lower bound, and the position dilution of precision (PDOP). Geolocation constellations in low Earth orbits are designed using global optimization techniques, aimed at minimizing the PDOP.

We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

The parameterised post-Newtonian (PPN) formalism is a weak-field and slow-motion approximation for both General Relativity (GR) and some of its viable generalisations. Within this formalism, the motion can be approached using various parameterisations, among which are the Lagrange-type and Gauss-type orbital equations. Often, these equations are developed under the premise of the Lagrange constraint. This constraint makes the evolving orbital elements parameterise the instantaneous conics always tangent to the actual trajectory. Arbitrary mathematically, this choice of a constraint is convenient under perturbations dependent only on positions. However, under perturbations dependent also on velocities (like in the relativistic celestial mechanics) the Lagrange constraint unnecessarily complicates solutions that can be simplified by relaxing the constraint and introducing a freedom in the orbit parameterisation, which is analogous to the gauge freedom in electrodynamics and gauge field theories. Geometrically, this freedom is the freedom of non-osculation, i.e., of the degree to which the instantaneous conics are permitted to be non-tangent to the actual orbit. Under the same perturbation, all solutions with different degrees of non-osculation look mathematically different, though describe the same physical orbit. While non-intuitive, the modeling of an orbit with a sequence of nontangent instantaneous conics can at times simplify calculations. The appropriately generalised (“gauge-generalised”) Lagrange-type equations, and their applications, appeared in the literature heretofore. We, in this paper, derive the gauge-generalised Gauss-type equations and apply them to the PPN two-body problem. Fixing the gauge freedom in three different ways (i.e., modeling an orbit with non-osculating elements of three different types), we find three parameterisations of the PPN two-body dynamics. These parameterisations render orbits with either a fixed non-osculating semimajor axis, or with a fixed non-osculating eccentricity, or with a fixed non-osculating argument of periastron. We also develop a transformation from non-osculating to classical osculating orbital elements, and illustrate the new solutions using numerical simulations.

Satellite formation establishment may consume considerable fuel if cross-track distance (CTD) is a mission requirement. This work proposes a strategy for significantly reducing the required fuel. Instead of direct out-of-plane maneuvers, it is suggested to use in-plane maneuvers, while using the nodal precession caused by the J₂ perturbation. The CTD dynamics are analyzed, and representative formation geometries are considered. Analytical expressions for the CTD and related establishment timing are derived for the case of impulsive thrust. For most cases, the expected CTD establishment duration is reasonable, and the fuel saving with the J₂-assisted CTD establishment strategy is considerable. For the case of J₂-assisted CTD establishment with continuous thrust, optimal control methods are used. The proposed strategy is proven to be optimal for two cases: limited and unlimited thrust magnitude. Finally, the cases of J₂-assisted CTD establishment with impulsive and with continuous thrust are compared. It is shown that continuous thrust is preferable in terms of fuel, for similar CTD establishment durations.

CubeSats usually carry no propulsion system because of difficulties in scaling down existing propulsion technologies to meet the stringent size, volume, and power limits. The most challenging case in this context is the one-unit CubeSat: the use of which has been rapidly growing. To increase CubeSat functionality by allowing small orbit corrections, particularly due to orbit injection errors, this paper develops a new propulsion method relying on an ejection of masses from a reaction wheel. Using the reaction wheel’s kinetic energy, propellant masses are released from the wheel’s outer circumference using an impulsive application of an electric current. This paper models the dynamics of the satellite, the modified reaction wheel, and the ejected masses; evaluates the pros and cons of the proposed method; and provides a methodology for constructing a model of a mass-ejecting reaction wheel designed according to given specifications of a desired velocity change and satellite attitude stability. The system is useful for small orbit corrections, which can be used (for example) for CubeSat formation establishment after orbital injection.

With the onset of autonomous spacecraft formation flying missions, the ability of satellites to autonomously navigate relatively to other space objects has become essential. To implement spacecraft relative navigation, relative measurements should be taken, and processed using relative state estimation. An efficient way to generate such information is by using vision-based measurements. Cameras are passive, low-energy, and information-rich sensors that do not actively interact with other space objects. However, pointing cameras with a conventional field-of-view to other space objects requires much a-priori initialization data; in particular, dedicated attitude maneuvers are needed, which may interfere with the satellite's main mission. One way to overcome these difficulties is to use an omnidirectional vision sensor, which has a 360-degree horizontal field of view. In this work, we present the development of an omnidirectional vision sensor for satellites, which can be used for spacecraft relative navigation, formation flying, and space situational awareness. The study includes the development of the measurement equations, dynamical models, and state estimation algorithms, as well as a numerical study, an experimental investigation, and a space scalability analysis.

Satellite relative motion around the Earth has been thoroughly studied during the last two decades. However, considerably less attention has been given to the study of satellite relative motion around Mars. As the cost of space technologies decreases and more space missions are within reach, formation flying missions around Mars have the potential to benefit future exploration missions launched to the Red Planet. A key parameter in such missions will be the frequency at which the spacecraft need to perform formation-keeping maneuvers to compensate for unwanted drifts due to differential perturbations. The Martian J₃ and J₄ gravitational harmonics are significant enough to warrant a dedicated investigation of bounded satellite relative motion configurations. In this study, we derive conditions for bounded satellite relative motion in non-critical inclinations around Mars, while considering its gravitational harmonics up to J₄. We first introduce a family of stable frozen orbits facilitating the implementation of formation flying and then apply differential nodal precession negation and differential periapsis rotation negation methods while considering the gravitational harmonics up to J₄. Using this procedure, we demonstrate how the secular growth of the relative distance can be arrested during long time intervals.

Constellation design theory has been studied extensively. However, analysis of longitude-dependent perturbation effects on inter-satellite distance drift has not received much attention. In addition to oblateness-related perturbations, sectoral and tesseral perturbations are non-negligible for low Earth orbits, due to their effect on inter-satellite distance evolution. This paper introduces the idea of reducing the tesseral/sectoral relative drift by including these perturbations in the short-periodic correction of the mean elements. An analytical expression of the correction is derived based on the Lie transform theory. Different from previous works, the ratio between the Earth rotation rate and the satellite’s mean motion is chosen as the small parameter in the Lie theory formulation. The independent variables of the generating-function-related partial differential equations can be reduced to a single variable when using this small parameter. Numerical simulations validate that the sectoral and tesseral effects on the inter-satellite distance drift in satellite constellations can be mitigated by using the proposed correction of the mean elements.

The effects of third-body perturbations on natural and artificial satellite orbits have been studied extensively. However, much less attention has been given to the case considering the orbital inclination of the third body. In this paper, it is shown that a perturbed orbit, even with a small initial inclination—much less than the critical Kozai inclination of 39.23 degrees—may significantly increase its inclination due to the inclination of the third-body’s orbit. To study this phenomenon, a new method for finding the inclination of the perturbed body is proposed based on a coordinate transformation. This enables the derivation of an analytical solution for the orbital inclination, predicting the long-term evolution thereof when the third body’s orbit is inclined. It is found that the amplitude of the inclination depends on the inclination of the third body, and on the relative angle between the orbital planes of the perturbed orbiter and the perturbing body. Numerical simulations illustrate the accuracy of the proposed methodology for predicting the long-term evolution of the orbital inclination.

Low-thrust orbit control has been widely studied, mostly by using direct or indirect optimization to detect time-optimal and fuel-optimal orbit transfers. This paper proposes an alternative approach for designing efficient low-thrust orbit transfers. The main idea is to create an artificial resonance between the period of the control accelerations and a characteristic period of the orbital dynamics. To that end, Gauss’s variational equations are written using a Fourier series expansion in the mean anomaly. The secular changes of the orbital elements are determined based on averaging theory. It is shown that an artificially created low-thrust resonance can save fuel compared with piecewise-constant thrust. A general resonant-control candidate for changing all mean orbital elements efficiently is proposed. Four simple decoupling control laws are designed for changing the semimajor axis, eccentricity, inclination, and right ascension of the ascending node separately. In addition, periodic corrections transforming between mean elements and osculating elements are derived. The proposed decoupling control laws are applied to a formation-keeping problem, illustrating the potential merit of the new control law compared with a fixed-thrust-magnitude feedback control.

Third-body perturbations have been extensively studied in recent years. Almost all previous works, however, assumed that the perturbations are caused by a third body that orbits the primary on a radius larger than the semimajor axis of the perturbed object. This assumption is justified as long as the primary is not accompanied by a third body in close orbit. In this work, we present an analytic model for the dynamics of a perturbed object that orbits the primary on an orbit with a semimajor axis larger than the semimajor axis of the third body. Such a third body is referred to as an inner third body. An analysis of the long-term evolution of the orbital elements is presented, followed by simulation results, which demonstrate the validity of the model. A more generalized model is then developed, which includes a nonzero eccentricity for the orbit of the third body. An analogy between the J₂ problem and the inner third-body perturbation is indicated as well.

Using differential solar radiation pressure (DSRP) for satellite formation keeping is not a new idea. However, there is no existing design methodology of DSRP-based controllers for lunar satellite formation keeping. Lunar orbits are different from low Earth orbits, because the third-body effects are dominant. This paper develops a DSRP-based formation keeping controller, using mean orbital element feedback, for regulating relative distances between satellites in lunar orbits. It is shown that the most effective way to mitigate the intersatellite drift is to adjust the semimajor axes. This is achieved by modifying the cross-sectional areas of the satellites. The stability of the closed-loop system is proven based on the finite time stability theory. Numerical simulation results illustrate that the new DSRP-based controller is able to arrest the relative distance drift in lunar orbits for several years.

This paper is dedicated to developing a new approach for solving the low-thrust trajectory optimization problem of spacecraft in low Earth orbits. The unknown control function is parameterized by its Fourier coefficients up to a given order, thus transforming the optimal control problem into a nonlinear parameter optimization problem. An adequate choice of the period and the maximum degree of the truncated Fourier expansion leads to the creation of a resonance between the natural frequency of the orbital dynamics and the control function. Resonance is a well-known phenomenon; however, the novelty of the presented approach is to create the resonance artificially, in order to increase the effect of the control acceleration on the satellite trajectory evolution. This artificial resonance contributes to creating a rapid change of the cross-track orbital elements, using the low-thrust control acceleration. The proposed technique is applied to low Earth orbits constellation optimization, involving the minimization of the geometric dilution of precision of an Earth coverage problem.

Using differential solar radiation pressure (DSRP) for formation keeping is not new. However, there is no design of a DSRP-based controller for lunar satellite formation keeping, where the C₂₂ sectorial harmonic and third-body effects are large, other than the J₂zonal harmonic. This paper develops a DSRP-based controller utilizing mean orbital elements feedback for controlling satellites in a bounded region based on the astrodynamical analysis. It is shown that the most effective way to mitigates the drifts for using SRP is to adjust the semi-major axis. Numerical simulation results illustrate the new DSRP-based controller has the ability to arrest the relative distance drifts for serval years.

A general third-body perturbation problem, considering the perturbing body in an elliptic and inclined orbit, is investigated by using a semi-analytical theory. Previous works have contributed to deriving the averaged third-body-perturbed dynamics, but did not provide a transformation between osculating and mean elements in the general case. In this paper, an analytical transformation between osculating and mean elements is developed explicitly using von Zeipel’s method, in addition to developing the long-term dynamical equations. The resulting dynamical model is improved, because the disturbing function is averaged as a whole, instead of separating the disturbing function into many terms and averaging them independently. The simulation results indicate that the new singly averaged dynamical model behaves much better than the doubly averaged dynamics in propagating the long-term evolution of the orbital elements. Moreover, it is shown that the perturbing body’s inclination and eccentricity have a vital influence on the evolution of the satellite’s inclination and eccentricity.

Satellite navigation constellations orbit the Earth in medium and geosynchronous orbits. Their high altitude provides wide coverage, which may be redundant if only regional coverage is needed. In this paper, a design scheme for regional navigation satellite constellations in low-Earth orbit is proposed. The design yields long coverage durations, which are characterized by a minimized Geometric Dilution of Precision (GDOP), with respect to a predefined mid-latitude receiver. The proposed constellations are defined for various combinations of the following parameters: Repeat Ground-Track commensurability, receiver latitude, number of satellites, and internal satellite arrangement. Optimal constellations, with respect to the GDOP integral, are searched, design guidelines are set up, and a mathematical model is fitted. The proposed constellations provide GDOP-optimized coverage durations, recurring twice each day. The results include numerical solutions for prograde near-polar constellations in altitudes ranging between 550 and 870 km, with receivers positioned at latitudes from 30° to 60°, and number of satellites ranging from 13 to 36. The average coverage duration varies from 20 to 80 min, with a mean GDOP ranging from 2.5 to 3.

Finding relative orbits for satellite formations flying around the Earth, which are long-term bounded under various perturbations, has been a vibrant field of study. However, much less attention has been given to detecting such orbits for circumlunar formation flying missions. As opposed to low-Earth orbits, in circumlunar missions the third-body effect is large, and the magnitude of the C ₂₂ sectorial harmonic has the same order as the J ₂ zonal harmonic. This renders the analysis of bounded relative orbits more challenging. In this paper, we detect a new family of long-term bounded lunar relative orbits, which can be used for lunar formation flying missions. The main idea is to find an analytical solution for the mean relative distance among the formation satellites, and use it to derive formation geometries that are resilient to the gravitational and third-body perturbations. Two invariant mean-distance conditions and one bounded mean-distance conditions are derived. Numerical simulations indicate that the newly-derived invariance conditions yield long-term bounded relative motion.

This paper develops cluster control algorithms for the Satellite Swarm Sensor Network project, for which the main aim is to enable disaggregation of space-based remote sensing, imaging, and observation satellites. A methodological development of orbit control algorithms is provided, supporting the various use cases of the mission. Emphasis is given on outlining the algorithm's structure, information flow, and implementation. The methodology presented herein enables operation of multiple satellites in coordination to facilitate disaggregation of space sensors and augmentation of data provided therefrom.

Satellite navigation constellations orbit the Earth in medium and geosynchronous orbits. Their high altitude raises the operational and launching costs, but provides wide coverage, which may be redundant if only regional coverage is needed. In this paper, a design scheme for regional navigation satellite constellations in low-Earth orbit is proposed. The design yields long coverage durations, which are characterized by a minimized Geometric Dilution of Precision (GDOP), with respect to a predefined mid-latitude receiver. The proposed constellations are defined for various combinations of the following parameters: repeat ground-track commensurability, receiver latitude, number of satellites, and internal satellite arrangement. Optimal constellations, with respect to the GDOP integral, are searched, design guidelines are set up, and a mathematical model is fitted. The proposed constellations provide GDOP-optimized coverage durations, recurring twice each day. The results include numerical solutions for prograde near-polar constellations in altitudes ranging between 550 and 870 km, with receivers positioned at latitudes from 30° to 60°, and number of satellites ranging from 13 to 36. The average coverage duration varies from 20 to 80 minutes, with a mean GDOP ranging from 2.5 to 3.

RegionAl coveRage gdop opTimized sEgMented posItioning System is a conceptual navigation satellite constellation design in Low-Earth Orbit (LEO), which produces relatively long coverage durations, while optimizing the Geometric Dilution of Precision. This study examines the constellation performance with respect to an aerial moving receiver. We determine the receiver’s position by using the trilateration method and velocity by using doppler estimation, and improve the accuracy thereof by using an Extended Kalman Filter (EKF). We suggest a solution for the trilateration initialization problem, which arises for LEO navigation satellites, by averaging the Earth projection of all the satellites within sight. This study examines two scenarios, one wherein the EKF’s dynamic model matches the reference dynamic model, and another with a model mismatch. The results include the trilateration and doppler estimation errors, whose medians are about 10 m, and 10 cm/sec, respectively. When the dynamic model is approximated, the EKF reduces the errors by half. When the dynamic model is fully known, the position and velocity errors are reduced by one order of magnitude.

Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic J₂, the sectorial harmonic C₂₂, and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include C₂₂, and hence does not affect the equilibria leading to frozen orbits, C₂₂ is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.

Finding satellite relative orbits that are resilient to differential gravitational perturbations has received much attention in the literature. In particular, detecting "invariant" relative orbits under the J₂ perturbation was considered a solved problem. These "invariance" conditions result in constraints on the differential mean semimajor axis, inclination, and eccentricity. In this paper, it is shown that alternative conditions can be used to further reduce the drift among J₂-perturbed satellites. These alternative conditions are found by investigating the secular part of the averaged intersatellite distance. Averaging is performed with respect to the mean anomaly and argument of perigee. A closed-form expression for the first-order J₂-perturbed mean distanceis obtained. It is found that the mean relative distance squared drifts as a quadratic function of time. Four conditions for mean-distance boundedness are derived. Using simulations, it isshown that the new conditions can improve previously obtainedresults, inthe senseofreducing the residual intersatellite distance drift.

In space systems consisting of a large number of satellites, coordinating orbits among satellites is necessary throughout the entire mission lifetime. Although previous works mainly focused on the boundedness of relative motion between satellites in the group, in this paper, an extra degree of freedom is also addressed in order to manipulate an arbitrary number of orbital elements, which is represented as coordinating a general orbital transfer and an in-space assembly. The underlying concept is using consensus theory to characterize the properties of the control objective as in a multiagent system. To that end, this paper assumes that the communication in the networked satellite system is represented as an undirected graph, and then implements the governing system dynamics in a control-affine form as described by the Gauss's variational equations. For the general orbital transfer problem, an edge-error-based controller is developed and proven asymptotically stable. Definitions of error functions are also investigated to understand the behavior of developed controllers. Several strategies for assembly control are discussed, namely, via changing of variables or in a two-phase control process based on the dynamical structure. Numerical simulations are performed to validate the analysis and demonstrate the results.

Orbit propagation algorithms for satellite relative motion relying on Runge–Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior and degraded accuracy. Thus, attempts have been made to apply symplectic methods to integrate satellite relative motion. However, so far all these symplectic propagation schemes have not taken into account the effect of atmospheric drag. In this paper, drag-generalized symplectic and variational algorithms for satellite relative orbit propagation are developed in different reference frames, and numerical simulations with and without the effect of atmospheric drag are presented. It is also shown that high-order versions of the newly-developed variational and symplectic propagators are more accurate and are significantly faster than Runge–Kutta-based integrators, even in the presence of atmospheric drag.

Global and regional satellite navigation systems are constellations orbiting the Earth and transmitting radio signals for determining position and velocity of users around the globe. The state-of-the-art navigation satellite systems are located in medium Earth orbits and geosynchronous Earth orbits and are characterized by high launching, building and maintenance costs. For applications that require only regional coverage, the continuous and global coverage that existing systems provide may be unnecessary. Thus, a nano-satellites-based regional navigation satellite system in Low Earth Orbit (LEO), with significantly reduced launching, building and maintenance costs, can be considered. Thus, this paper is aimed at developing a LEO constellation optimization and design method, using genetic algorithms and gradient-based optimization. The preliminary results of this study include 268 LEO constellations, aimed at regional navigation in an approximately 1000 km × 1000 km area centered at the geographic coordinates [30, 30] degrees. The constellations performance is examined using simulations, and the figures of merit include total coverage time, revisit time, and geometric dilution of precision (GDOP) percentiles. The GDOP is a quantity that determines the positioning solution accuracy and solely depends on the spatial geometry of the satellites. Whereas the optimization method takes into account only the Earth’s second zonal harmonic coefficient, the simulations include the Earth’s gravitational field with zonal and tesseral harmonics up to degree 10 and order 10, Solar radiation pressure, drag, and the lunisolar gravitational perturbation.

This paper investigates the long-term orbital evolution of smart dust in space. Previous studies modeled the dynamics of smart dust under solar radiation pressure and drag only, and they found conditions for long-lived orbits but without considering the gravitational perturbations. In this work, the orbital dynamics under the additional effect of gravitational perturbations are investigated, and two groups of initial conditions yielding long-lived orbits are characterized.Amongthese conditions, five representative points are selected to study the long-term orbital evolution. Based on these considerations, it is found that there are long-term stable orbits, not only with small eccentricities but also with large eccentricities, due to the influence of gravitational perturbations, solar radiation pressure, and drag.

DriveSat is a new CubeSat mission, initiated and led by the Asher Space Research Institute at the Technion. DriveSat aims to test low cost Cubesat orbit and attitude control techniques. The mission will demon- strate two new technologies: a miniaturized thruster and a cost-effective reaction wheel, both developed at the Technion. The DriveSat project will share resources with the Space Autonomous Mission for Swarming and Geo-locating Nanosatellites project. The DriveSat project will also serve as a platform for academic research and hands-on engineering edu- cation. Technion students are involved in the development of the new technologies, and in the future will operate the satellite.

Satellite navigation constellations orbit the Earth in medium and geosynchronous orbits. Their high altitude raises the operational and launching costs, but provides wide coverage, which may be redundant if only regional coverage is needed. In this paper, a design scheme for regional navigation satellite constellations in low-Earth orbit is proposed. The design yields long coverage durations, which are characterized by a minimized Geometric Dilution of Precision (GDOP), with respect to a predefined mid-latitude receiver. The proposed constellations are defined for various combinations of the following parameters: repeat ground-track commensurability, receiver latitude, number of satellites, and internal satellite arrangement. Optimal constellations, with respect to the GDOP integral, are searched, design guidelines are set up, and a mathematical model is fitted. The proposed constellations provide GDOP-optimized coverage durations, recurring twice each day. The results include numerical solutions for prograde near-polar constellations in altitudes ranging between 550 and 870 km, with receivers positioned at latitudes from 30^â—¦ to 60^â—¦, and number of satellites ranging from 13 to 36. The average coverage duration varies from 20 to 80 minutes, with a mean GDOP ranging from 2.5 to 3.

Finding relative orbits for satellite formations flying around the Earth, which are long-term bounded under various perturbations, has been a vibrant field of study. However, much less attention has been given to detecting such orbits for circumlunar formation flying missions. As opposed to low-Earth orbits, in circumlunar missions the third-body effect is large, and the magnitude of the C₂₂ sectorial harmonic has the same order as the J₂ zonal harmonic. This renders the analysis of bounded relative orbits more challenging. In this paper, we detect a new family of long-term bounded circumlunar relative orbits, which can be used for circumlunar formation flying missions. The main idea is to utilize the benefits of circumlunar frozen orbits, and operate the satellite formation in the vicinity of such orbits. This enables to find an analytical solution for the mean relative distance among the formation satellites, and use it to derive formation geometries that are resilient to the gravitational and third-body perturbations. Two invariant mean-distance conditions and one bounded mean-distance conditions are derived. Numerical simulations indicate that the newly-derived invariance conditions yield long-term bounded relative motion.

High area-to-mass ratio space objects at geosynchronous orbits pose a threat to operational satellites because of the difficulty in tracking them from Earth. This paper develops an in-orbit onboard algorithm for tracking high area-tomass ratio space objects. The design uses relative motion dynamics and a simplified stereocamera measurement model to estimate the tracked object's position and velocity, as well as its solar radiation pressure coefficient. An underlying assumption is the dominance of the solar radiation pressure perturbation, which is confirmed using an analysis of the relative motion dynamics. Two scenarios are examined: similar orbits and crossing orbits. The simulation results show that the tracking algorithm estimates the tracked object position, velocity, and solar radiation pressure coefficient in both scenarios with high accuracy. The simulations show that, although shape information is lost in the measurement model, the solar radiation pressure coefficient can still be estimated.

Spacecraft formation flying and satellite cluster flight have seen growing interest in the last decade. However, the problem of finding the optimal debris collision avoidance maneuver for a satellite in a cluster has received little attention. This paper develops a method for choosing the timing for conducting minimum-fuel avoidance maneuvers without violating the cluster intersatellite maximal distance limits. The mean semimajor axis difference between the maneuvering satellite and the other satellites is monitored for the assessment of a maneuver possibility. In addition, three techniques for finding optimal maneuvers under the constraints of cluster keeping are developed. The first is an execution of an additional cluster-keeping maneuver at the debris time of closest approach, the second is a global all-cluster maneuver, and the third is a fuel-optimal maneuver, which incorporates the cluster-keeping constraints into the calculation of the evasive maneuver. The methods are demonstrated and compared. The first methodology proves to be the most fuel-efficient. The global maneuver guarantees boundedness of the intersatellite distances as well as fuel and mass balance. However, it is rather fuel-expensive. The last method proves to be useful at certain timings and is a compromise between fuel consumption and the number of maneuvers.

Gravitational and third-body perturbations can be modeled with sufficient precision for most applications in low Earth orbit. However, owing to severe uncertainty sources and modeling limitations, computational models of satellite aerodynamics and solar radiation pressure are bound to be biased. Aiming at orbital propagation consistent with observed satellite orbital dynamics, real-time estimation of these perturbations is desired. In this paper, a particle filter for the recursive inference and prediction of nongravitational forces is developed. Specifically, after assuming a parametric model for the desired perturbations, the joint probability distribution of the parameters is inferred by using a prescribed number of weighted particles, each consisting of one set of orbital elements and one set of parameters. The particle evolution is carried out by means of an underlying orbital propagator, and the Bayes rule is used to recursively update weights by comparing propagated orbital elements with satellite observations. The proposed formulation uses mean orbital elements as the only available measurements. This feature makes the algorithm a potentially valuable resource for space situational awareness applications, such as space debris trajectories prediction from two-line elements, or for onboard force estimation from Global Positioning System data. High-fidelity simulations show that nongravitational perturbations can be estimated with 20% accuracy.

Subjected to multiple perturbations and their complex interplay, the dynamical evolution of geostationary transfer orbits (GTOs) is sensitive to initial conditions and model parameters. As one of the most remarkable outcomes of multiple perturbations, the solar apsidal resonance, i.e., the 1:1 resonance between the solar orbital motion and the rotation of the orbital apsidal line caused by Earth's oblateness, is an important feature of the GTO evolution. It occurs when the semi-major axis is reduced by the atmospheric drag to the critical value, with which the rotation of the orbital apsidal line is commensurate with the solar orbital motion. In the present paper, we show that the solar apsidal resonance plays an important role in the evolution and decay of GTOs. To do so, we first explain the underlying dynamical mechanism of the solar apsidal resonance, which is the U-turn of the solar azimuth with respect to the orbital apsidal line and the resulting monotonic increase or decrease of the eccentricity. The resonance is then classified into three kinds, and their causes and effects are analyzed. Previous studies have regarded the solar apsidal resonance as a mechanism extending the orbital lifetime. However, we find that in most cases the GTO will re-enter Earth's atmosphere soon or only several years after the resonance, and so the solar apsidal resonance can be regarded as the prelude to the GTO final re-entry. Finally, the sensitivity of orbital dynamics is studied through numerical simulations. It is shown that the high sensitivity of the dynamics can be attributed to the resonance, which is difficult to predict or manage. With the initial state, it is possible to predict the orbit evolution of GTO only before the solar apsidal resonance. To predict the lifetime of GTO, new measurements on the orbit after the resonance are required.

Tracking a non-cooperative target is a challenge, because in unfamiliar environments most targets are unknown and unspecified. Stereovision is suited to deal with this issue, because it allows to passively scan large areas and estimate the relative position, velocity and shape of objects. This research is an experimental effort aimed at developing, implementing and evaluating a real-time non-cooperative target tracking methods using stereovision measurements only. A computer-vision feature detection and matching algorithm was developed in order to identify and locate the target in the captured images. Three different filters were designed for estimating the relative position and velocity, and their performance was compared. A line-of-sight control algorithm was used for the purpose of keeping the target within the field-of-view. Extensive analytical and numerical investigations were conducted on the multi-view stereo projection equations and their solutions, which were used to initialize the different filters. This research shows, using an experimental and numerical evaluation, the benefits of using the unscented Kalman filter and the total least squares technique in the stereovision-based tracking problem. These findings offer a general and more accurate method for solving the static and dynamic stereovision triangulation problems and the concomitant line-of-sight control.

R-bar refers to the local vertical axis pointing radially upward in a satellite-fixed reference frame. Approaching a satellite along the R-bar, especially for rendezvous and docking to geostationary satellites, is advantageous in terms of safety considerations and flight time compared to other options. In this paper, a specialized study on autonomous R-bar proximity operations with respect to a geostationary target from a separation of several kilometers to a few hundreds of meters, commonly referred to as the closing phase, is carried out and a comprehensive solution for both attitude and orbit control in this scenario is proposed. An integrative design of the guidance, navigation, and control for R-bar proximity operations is presented. Impulsive R-bar hopping maneuvers are developed for the trajectory guidance. This method is shown to be passively safe and time efficient. The onboard sensors provide measurements of the line-of-sight, range to the target, attitude and angular velocity in the inertial frame. Due to the sensitivity of the sensor's pointing in the far-range phase, a sliding mode attitude control law is introduced to align the optical axis with the line-of-sight to the target. Sensor measurements are fused and processed by an extended Kalman filter. Simulation results indicate that the proposed integrative guidance, navigation, and control algorithms are robust to uncertainties and noise, and can be used as a comprehensive solution for R-bar rendezvous and docking mission design during the closing phase.

To control the rotational state of nanosatellites, miniaturized reaction wheels have been widely used. Miniaturized reaction wheels are manufactured by several companies, and can be found in multiple CubeSat missions. These reaction wheels have space heritage and good performance, but they are extremely costly. A typical price of a reaction wheel ranges from 12,000 to 20,000 USD. This poses a major problem to most CubeSat developers. We developed a costeffective alternative to existing reaction wheels, which reduces the cost of a unit to about 300 USD. The idea is to use hard disk drive (HDD) components as reaction wheels for CubeSats. HDD-based reaction wheels are much more affordable and provide similar performance to commercial systems. Analysis and experiments have been performed in order to validate the compatibility of the HDD-based wheel to the space environment.

Spacecraft formation flying and satellite cluster flight have seen growing interest in the last decade. However, the problem of finding the optimal debris collision avoidance maneuver for a satellite in a cluster has received little attention. This paper develops a method for choosing the timing for conducting minimum-fuel avoidance maneuvers without violating the cluster inter-satellite maximal distance limits. The mean semimajor axis difference between the maneuvering satellite and the other satellites is monitored for the assessment of a maneuver possibility. In addition, three techniques for finding optimal maneuvers under the constraints of cluster-keeping are developed. The first is an execution of an additional cluster- keeping maneuver at the debris time of closest approach, the second is a global all-cluster maneuver, and the third is a fuel-optimal maneuver, which incorporates the cluster-keeping constraints into the calculation of the evasive maneuver. The methods are demonstrated and compared. The first methodology proves to be the most fuel efficient. The global maneuver guarantees boundedness of the inter-satellite distances, as well as fuel and mass balance. However, it is rather fuel-expensive. The last method proves to be useful at certain timings, and is a compromise between fuel consumption, and the number of maneuvers.

Miniaturized reaction wheels are manufactured by several companies, and can be found in multiple CubeSat missions. These reaction wheels have space heritage and good performance, but they are extremely costly. This paper examines the feasibility of using hard disk drive (HDD) components as reaction wheels for CubeSats. HDD-based reaction wheels are much more affordable and provide similar performance to commercial systems. Analysis and experiments are performed in order to validate the compatibility of the HDD-based wheel to the space environment. Numerical simulations using real satellite parameters show that the suggested concept is feasible. Experimental studies performed at the Distributed Space Systems Lab demonstrate that the proposed reaction wheel is capable of controlling the CubeSat rotational state to reasonable accuracy. Consequently, the HDD-based reaction wheel can significantly reduce the overall cost of CubeSats and thereby increase the accessibility of satellite missions.

High area-to-mass ratio space objects at geosynchronous orbits pose a threat to operational satellites because of the difficulty to track them from Earth. This paper develops an in-orbit on-board algorithm for tracking high area-to-mass ratio space objects. The design utilizes relative motion dynamics and a simplified stereo-camera measurement model to estimate the tracked object's position and velocity, as well as its solar radiation pressure coefficient. An underlying assumption is the dominance of the solar radiation pressure perturbation. Two scenarios are examined: similar orbits and crossing orbits. The simulation results show that the tracking algorithm estimates the tracked object position, velocity, and solar radiation pressure coefficient in both scenarios with high accuracy. The simulations show that although shape information is lost in the measurement model, the solar radiation pressure coefficient can still be estimated.

A minimum-time station change of geostationary satellites is formulated using the indirect method. An analytical method is presented for the evaluation of the final time, as well as the initial costates of semimajor axis, longitude, and mass. The analytical method does not require the properties of the initial costates, which were obtained in previous studies by establishing a relationship between the initial costates and the transfer time. The indirect optimization method demonstrates rapid convergence when employing the initial costates evaluated by the presented method. revolutions be close to an even integer. A low-thrust minimum time station change with perturbations included requires no significant change in flight time relative to that without perturbations. Under the effect of the most significant perturbations acting on geostationary Earth orbits, the low-thrust station change maneuver achieves similar accuracy as an impulsive maneuver.

The early medium-term evolution of a debris cloud refers to the phase when the debris cloud has evolved into a toroid after several revolutions since the breakup event and when effects of natural perturbations are negligible. A debris cloud is described by two essential factors: geometry of the cloud, and spatial debris density within the cloud. In this study, a new approach, the reachable domain method, is introduced for modeling the toroid-shaped medium-term debris cloud. The conventional reachable domain algorithm is first modified to simplify the computations and, more importantly, enable a further volume calculation. Then, the geometry of the medium-term debris cloud, a toroid, is defined by the envelope of the reachable domain, and the volume of the debris cloud is obtained based on the reachable domain volume computation. Thus, a mean uniform spatial debris density is evaluated by dividing the total number of fragments by the volume. Moreover, a nonuniform spatial density is determined by dividing the debris cloud into several subclouds corresponding to different ejecting velocity bins. The utility of the proposed method is demonstrated by simulating a typical breakup event on a medium Earth orbit.

The dynamics and lifetime reduction of geostationary transfer orbits (GTOs) are of great importance to space debris mitigation. The orbital dynamics, subjected to a complex interplay of multiple perturbations, are complicated and sensitive to the initial conditions and model parameters. In this paper, a simple but effective non-singular orbital dynamics model in terms of Milankovitch elements is derived. The orbital dynamics, which include the Earth oblateness, luni-solar perturbations, and atmospheric drag, are averaged over the orbital motion of the GTO object, or, as needed, also over the orbital motions of the Moon and Sun, to eliminate the short-period terms. After the averaging process, the effect of the atmospheric drag assumes a simple analytical form. The averaged orbital model is verified through a numerical simulation compared with commercial orbit propagators. GTO lifetime reduction by using the luni-solar perturbations is studied. It is shown that the long-period luni-solar perturbation is induced by the precession of the GTO orbital plane and apsidal line, whereas the short-period perturbation is induced by the periodic luni-solar orbital motions. The long- and short-period perturbations are isolated and studied separately, and their global distribution with respect to the orbital geometry is given. The desired initial orbital geometry with a short orbital lifetime is found and verified by a numerical simulation.

The current work develops a distributed controller for steering a satellite cluster to the same orbit as well as for assembly under limited low thrust. The underlying concept is using consensus theory to characterize the control objective in a multi-agent system. This concept is leveraged by implementing the governing dynamics in a control-affine form as described by Gauss's variational equations. For general orbital transfer, the controller is asymptotically stable. The assembly is accomplished by changing of variables, as well as in a two-phase control process by separately controlling subsets of orbital elements based on the dynamics structure. Numerical simulations validate the analysis and demonstrate the results.

One possibility for satellite cluster flight is to control relative distances using differential drag. The idea is to increase or decrease the drag acceleration on each satellite by changing its attitude, and use the resulting small differential acceleration as a controller. The most significant advantage of the differential drag concept is that it enables cluster flight without consuming fuel. However, any drag-based control algorithm must cope with significant aerodynamical and mechanical uncertainties. The goal of the current paper is to develop a method for examination of the differential drag-based cluster flight performance in the presence of noise and uncertainties. In particular, the differential drag control law is examined under measurement noise, drag uncertainties, and initial condition-related uncertainties. The method used for uncertainty quantification is the Linear Covariance Analysis, which enables us to propagate the augmented state and filter covariance without propagating the state itself. Validation using a Monte-Carlo simulation is provided. The results show that all uncertainties have relatively small effect on the inter-satellite distance, even in the long term, which validates the robustness of the used differential drag controller.

Estimating the pose, motion, and structure of noncooperative dynamic targets using on board sensors is a challenging problem. This work suggests using multiple-baseline stereovision for noncooperative dynamic target relative pose, motion, and structure estimation, with a particular emphasis on space applications. A computer-vision feature-matching algorithm is designed, which produces input data for a recursive filtering algorithm. A newly-developed initialization scheme is proposed, which decreases the ambiguity in the target center of mass location. An extensive numerical study is provided, comparing between two estimators and different relative motion models. It is shown that a simple kinematic model can work well when combined with initial structure estimation. The effect of varying the number of cameras and stereo-rig geometry is investigated. The proposed vision-based relative motion estimation method was validated at the Technion's Distributed Space Systems Laboratory. A scalability analysis indicates that the proposed method may be potentially useful for space applications.

PN is the method most commonly used for guidance of homing missiles. A vast amount of literature exists on the subject (see, e.g., the works of Shneydor1 and Zarchan2 and the references therein). Modern guidance laws have also been thoroughly analyzed.3-5 PN is known to yield reasonable miss distance when applied against nonmaneuvering or moderately maneuvering.

Cluster flight algorithms enable the operation of multiple satellites within given distance bounds for long periods of time. Synchronizing the states of the satellites in the cluster is important for both cluster establishment and cluster keeping. This paper offers two distributed orbit control laws with fixed-magnitude thrust for satellite cluster flight based on mean-orbital elements. These controllers are capable of synchronizing the convergence of orbital elements among all satellites. Intersatellite communication is represented by weighted digraphs, and the related stability properties of the closed-loop control system are examined. Global asymptotic stabilityis proven for the first controller using nonsmooth analysis, whereas the second controller is shown to be only locally asymptotically stable. Furthermore, it is proven that chatteringin the thrust directionis unavoidable when using fixed-magnitude thrusters and continuous weights. The occurrence of chattering is analyzed with respect to constant weights and time-varying weights. Two approaches are suggested to mitigate chattering: fast-varying weights and thrust modulation. Numerical simulations are performed to validate the analysis.

Space debris constitutes a major and growing threat to space missions. To assess the probability of collision, the time of closest approach between the spacecraft and the debris must first be calculated. This paper presents three new methods for the calculation of the time of closest approach. The first is a surrogate-based optimization algorithm, using the Alfano-Negron Close Approach Software as the model, allowing a compromise between calculation speed and accuracy. The second is a generalization of this software, treating a cloud of debris as a continuous object and searching for the time of closest approach over initial conditions as well as time. The third uses Alfano-Negron Close Approach Software generalization as a model for surrogate-based optimization. Using the third method, a strategy for safe operation of a spacecraft cluster in a debris-rich environment is proposed. All methods are demonstrated and compared. It is shown that Alfano-Negron Close Approach Software and the generalization thereof are very fast in estimating the time of closest approach, and that the surrogate-based optimization algorithms are somewhat slower, but are highly accurate.

The objective of this paper is to find the minimum-fuel transfer needed for servicing client geostationary satellites while considering perturbations that have been neglected in previous studies. The effect of Earth's triaxiality on the semimajor axis and longitude is derived by the method of averaging and used for designing a two-impulse planar phasing maneuver. The phasing maneuver is then extended to a three-impulse planar maneuver for matching the eccentricity. The inclination variation due to lunisolar perturbations is modeled to determine the maneuver needed for eliminating the normal direction excursion. Then, an optimal servicing model is built with Earth's triaxiality and lunisolar perturbations included and applied for servicing a sparsely distributed geostationary constellation. The results show that the duration of each rendezvous of the optimized solution is mainly determined by third-body effects. The proposed maneuver strategy with a tesseral-term correction can achieve relatively high accuracy in longduration orbital transfers under the effects of Earth's actual gravitational field, lunisolar attraction, and solar radiation pressure.

The reachable domain is a geometric description of the maneuverability of a spacecraft under control and/or measurement uncertainties. In this paper, the reachable domain conceptis introduced into the field of satellite relative motion and termed the relative reachable domain. The relative reachable domain geometrically represents the effects of the initial control and/or measurement uncertainties on relative motion. A general method is developed to find the envelope of the coplanar relative reachable domain for arbitrary reference orbits, by evaluating the extremal values of the potentially accessible radius with respect to a given reference position. The relative reachable domain formulations are analytically expressed for circular reference orbits. It is shown that the relative reachable domain is a convenient tool for analyzing a variety of cluster flight mission problems. Several practical applications are suggested, including determining the upper bound on the separation distances among satellites. It is also shown that the relative reachable domain can be used for assessing the collision risk and that, in some situations, the relative reachable domain is more reliable than existing collision assessment methods. Numerical examples demonstrate the usefulness of the proposed method for finding the relative reachable domain and show that the analytical calculation of the relative reachable domain is accurate even in the case of noncoplanar perturbed motion.

This volume is designed as an introductory text and reference book for graduate students, researchers and practitioners in the fields of astronomy, astrodynamics, satellite systems, space sciences and astrophysics. The purpose of the book is to emphasize the similarities between celestial mechanics and astrodynamics, and to present recent advances in these two fields so that the reader can understand the inter-relations and mutual influences. The juxtaposition of celestial mechanics and astrodynamics is a unique approach that is expected to be a refreshing attempt to discuss both the mechanics of space flight and the dynamics of celestial objects. “Celestial Mechanics and Astrodynamics: Theory and Practice” also presents the main challenges and future prospects for the two fields in an elaborate, comprehensive and rigorous manner. The book presents homogenous and fluent discussions of the key problems, rendering a portrayal of recent advances in the field together with some basic concepts and essential infrastructure in orbital mechanics. The text contains introductory material followed by a gradual development of ideas interweaved to yield a coherent presentation of advanced topics.

Calculating the projected cross-sectional area (PCSA) of a satellite along a given direction is essential for implementing attitude control modes such as Sun pointing or minimum-drag. The PCSA may also be required for estimating the forces and torques induced by atmospheric drag and solar radiation pressure. This paper develops a new analytical method for calculating the PCSA, the concomitant torques and the satellite exposed surface area, based on the theory of convex polygons. A scheme for approximating the outer surface of any satellite by polygons is developed. Then, a methodology for calculating the projections of the polygons along a given vector is employed. The methodology also accounts for overlaps among projections, and is capable of providing the true PCSA in a computationally-efficient manner. Using the Space Autonomous Mission for Swarming and Geo-locating Nanosatellites mechanical model, it is shown that the new analytical method yields accurate results, which are similar to results obtained from alternative numerical tools.

One of the methods for maintaining a cluster of satellites in long-term bounded relative distances is keeping the satellites on near-circular orbits having the same mean semimajor axes and mean inclinations. This approach allows some freedom in determining the reference mean semimajor axis and reference mean inclination for the cluster. In this paper, this freedom is used to find the optimal target values of the mean semimajor axis and mean inclination, with the optimization criteria being either the total propellant consumption of the cluster or the fuel consumption differences among satellites. The optimization problems are solved analytically, assuming that a fixed-magnitude thruster is used for closed-loop orbit control, and new results are presented, providing simple closed-form expressions for the optimal target states. Simulations are used for validating the results, showing that much propellant can be saved by properly setting the cluster reference orbit.

Cluster flight is defined as the operation of multiple cooperative satellites under minimum and maximum distance constraints. Unlike traditional orbit injection of multiple noncooperative satellites, judicious constraints should be imposed on the orbit injection of a cluster. Moreover, the orbital injection should enable a fuel-efficient cluster establishment maneuver. This study develops constraints for the orbit injection of a satellite cluster. Evaluation of feasible orbit injection scenarios is performed. Particular launch vehicles and a concomitant set of orbital injection scenarios are found, which are the most suitable for long-term operation of a cluster. In addition, an efficient cluster establishment control method is developed, which prevents the cluster from drifting apart too rapidly. Several orbit injection scenarios are simulated and their feasibility for a planned cluster flight mission is evaluated. It is shown that both the Polar Satellite Launch Vehicle and the Dnepr launcher can be used. In addition, a new customized release proced ureisdeveloped, which guarantees cross-track separation among the satellites, adesirable featureina number of practical missions.

Nano-satellite clusters and disaggregated satellites are new concepts in the realm of distributed satellite systems, which require complex cluster management - mainly regulating the maximal and minimal inter-satellite distances on time scales of years - while utilizing simple on-off propulsion systems. The simple actuators and long time scales require judicious astrodynamical modeling coupled with specialized orbit control. This paper offers a satellite cluster orbit control law which works for long time scales in a perturbed environment while utilizing fixed-magnitude thrusters. The main idea is to design a distributed controller which balances the fuel consumption among the satellites, thus mitigating the effect of differential drag perturbations. The underlying methodology utilizes a cyclic control algorithm based on a mean orbital elements feedback. Stability properties of the closed-loop cyclic control system do not adhere to the classical Lyapunov stability theory, so an effort is made to define and implement a suitable stability theory of noncompact equilibria sets. A state selection scheme is proposed for efficiently establishing a low Earth orbit cluster. Several simulations, including a real mission study, and several comparative investigations, are performed to show the strengths of the proposed control law.

Calculating the projected cross-sectional area (PCSA) of a satellite along a given direction is essential for evaluating the forces and torques induced by atmospheric drag and solar radiation pressure. The PCSA is also required for implementing attitude control modes such as Sun pointing or minimum-drag. This paper develops a new analytical method for calculating the PCSAs, based on the theory of convex polygons. A scheme for approximating the outer surface of any satellite by polygons is developed. Then, a methodology for calculating the projections of the polygons along a given vector is employed. The methodology also accounts for overlaps among projections, and is capable of provid-ing the true PCSA in a computationally-efficient manner. Using the Space Autonomous Mission for Swarming and Geo-locating Nanosatellites mechanical model, it is shown that the new analytical method yields accurate results, which are similar to results obtained from alternative numerical tools.

Satellite clusters constitute one of the future trends in space systems. However, thus far, there has not been an efficient method for guaranteeing a safe operation of clusters in debris-rich environments. This paper suggests a method for a fast and accurate calculation of the Time of Critical Approach (TC A) between each member of a cluster of satellites and cataloged space debris. A cluster is a group of satellites flying under minimum and maximum distance constraints. All members of a cluster are in danger of collisions among themselves, as well as with other orbiting objects, such as other spacecraft and debris. The TCA is crucial in situational awareness; it is at that point in time in which the maximum probability of collision and the evasive maneuver are calculated. There are numerous methods today for finding the TCA between two orbiting objects; these methods vary in accuracy and speed. However, the problem of quickly calculating the TCA's of N satellites belonging to a cluster with M exterior objects has not been addressed. In this paper, we first suggest a method which is an effective compromise between speed and accuracy for finding the TCA between two space objects. The proposed method is a Surrogate Based Optimization algorithm (SB0), using the Alfano/Negron Close Approach Software (ANCAS) as the surrogate function. ANCAS fits a cubic polynomial to the relative speed, searching for the minimum distance in the critical points where the speed is null. As in SBO, the true position and speed are calculated at the TCA estimated by ANCAS; if the error is too large, the information of the new calculated point is used to fit a cubic polynomial and repeat the process. The described method is compared with exiting methods and the advantages are shown. Then, a generalization of the search to large groups of objects is suggested based on the known characteristics of the clusters members. The method is compared with an all-on-all search that is currently in common use.

One of the most widely considered topics in satellite cluster flight is the control of relative positions among satellites, referred to as cluster keeping. In general, satellite cluster keeping is aimed at keeping the satellites in a specific range of distances. In order to maintain relative distances within given bounds, maneuvers are required. Fuel consumption should be minimized in order to decrease mass. However, in practical situations, thrust parameters are often uncertain. The main purpose of this research is to robustify cluster control methods under thrusting errors and uncertainties, and minimize the effect of thrusting errors and noise on the evolution of relative distances and on the consumption of fuel. In particular, this research develops algorithms that minimize the fuel consumption during the maneuver and the time of the maneuver in spite of thrusting errors and noise. An inverse dynamics controller was implemented, using mean orbital elements feedback, while an adaptive control scheme was used for the thrust pointing error reduction in order to obtain optimality in terms of fuel consumption and operation time. In order to verify the performance of the cluster keeping controller under these conditions, a Kalman Filter was designed for estimating the mean elements.

The idea of differential drag (DD) as a means for fuelless satellite cluster keeping emerged in the mid-Eighties, when the feasibility of DD-based control was proven assuming linearized relative dynamics for two satellites. The present work develops a new DD-based cluster keeping method suitable for implementation in long-term cluster flight missions consisting of multiple satellites. This algorithm has been implemented on the nanosatellites planned to be launch as part of the Space Autonomous Mission for Swarming and Geo-locating Nanosatellites (SAMSON). Obviously, any drag-based algorithm must cope with aerodynamical and mechanical uncertainties. The overall error related to drag calculation is inevitable and could be as high as one or two orders of magnitude, which can be crucial for any drag-based control. Hence, a covariance analysis of the closed-loop system was developed, in the presence of drag uncertainties, initial condition-related uncertainties and measurement noise. A Kalman filter is designed in order to generate inputs to the differential drag controller. The variance of the differential mean semimajor axis is propagated analytically using the Linear Covariance Analysis (LCA) technique, which enables to propagate the augmented state and filter covariance without propagating the state itself. The results show that all these uncertainties have relatively small affect on the inter-satellite distance, even for long term, which prove the robustness of the differential drag controller.

The features of CAMILA Hall thruster operation at the strong longitudinal component of the magnetic field in the anode cavity, which is comparable with a typical radial component in the acceleration area, are experimentally studied. A special attention is paid to operation of the anode cavity in a self-maintaining mode. It was found that at typical values of the discharge currents, the potential drop in the anode cavity is ∼ 25 V. The conductivity of plasma transverse to the magnetic field in the cavity is Bohm's one. The reason of such high conductivity of the plasma is Raleigh-Taylor (convective) instability. It is shown that increasing the longitudinal magnetic field in the anode cavity leads to growth of the effectiveness of propellant ionization in the thruster and to increasing the fraction of ion current in the discharge current.

Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit's radial intermediary, obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on Deprit's radial intermediary outperforms a Dormand-Prince fixed-step Runge-Kutta integrator as the inclination grows.

Satellite cluster consisting of large number of satellites is becoming a promising topic in space research, due to its low cost and robustness. The current work offers a distributed control framework for the convergence of an autonomous satellite cluster to the same orbit under limited low thrust capability. The target orbit is a priori unknown, however, it is merely a result of cooperative protocol on the basis of locally exchanged information. The communication between satellites is represented as an undirected graph. The control is developed by introducing and analyzing an appropriate Lyapunov function. Under dynamics modeled by Gauss's variational equations, a closed-form expression is obtained for the controller. A rigorous analysis showed that the controller is globally asymptotic stable, if the underlying communication graph is connected and the information function is well-defined.

Differential drag (DD) as a means for passive satellite cluster keeping is an old idea, but so far using DD-based cluster keeping while relying on mean orbital elements feedback has not been proposed. This paper develops a DD-based maximum distance keeping method that uses Brouwer-Lyddane differential mean elements feedback for long-term control of the secular drift among satellites. The stability of the maximum distance keeping controller is proven using finite-time stability theory, and high-precision simulation results confirm that the new controller is able to arrest satellite relative drift for mission lifetimes exceeding a year. The maximum distance controller is automatically activated, and does not require a pre-determined activation time. Moreover, as a part of a complete DD-based solution for cluster keeping, a collision-avoidance method based on the same controller structure, albeit with differential osculating elements feedback, is developed and validated. Finally, the possibility to regulate cross-track drift with DD is examined, but it is shown that DD can only provide weak controllability in this case.

Disaggregated satellites distribute the functionalities of a single monolithic satellite among multiple wirelessly linked heterogenous modules. One of the challenges of designing a disaggregated satellite cluster is the development of a guidance system that is suited to the diverse cluster operational requirements. This paper presents the development of a guidance approach for cluster deployment and scatter-regather maneuvering in the case of a space debris collision threat. Intermodule collision avoidance is included through the use of artificial potential functions to ensure safe operational distances. As opposed to previous studies, the artificial potential formulation directly addresses the issue of moving obstacles and includes the complete J₂-perturbed nonlinear dynamical model through a backstepping approach. Numerical simulations that include the effects of the zonal harmonics J₂ to J₅ are used to demonstrate the effectiveness of the proposed guidance law.

Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit’s radial intermediary (DRI), obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. The only remaining equation to be solved by iterations is the Kepler equation, which in most of cases does not impact the total computation time. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on DRI outperforms a Dormand–Prince fixed-step Runge–Kutta integrator as the inclination grows.

One of the emerging topics in the realm of distributed space systems is cluster flight of nanosatellites. As opposed to formation flight, cluster flight does not dictate strict limits on the geometry of the cluster, and is hence more suitable for implementation in nanosatellites, which usually do not carry highly accurate sensors and actuators. The actuators are usually simple fixed-magnitude thrusters, which are prone to many sources of errors, such as attitude determination and control errors. In this context, the purpose of this paper is to develop a cluster-keeping control law that is capable of long-term operation under thrust uncertainties, assuming fixed-magnitude thrust provided by a simple cold-gas thruster. To that end, mean orbital elements are used for designing an inverse-dynamics controller. It is shown that, in the differential mean elements space, this controller is time-optimal. An adaptive enhancement is developed to mitigate the thrust pointing errors and restore the original optimal performance, thus saving much fuel. Several simulations and comparative studies are performed to validate the analytical results.

Differential drag (DD) as a means for fuelless satellite cluster keeping is an old idea, but so far using DD-based cluster keeping while relying on mean orbital elements feedback has not been proposed. This paper develops a DD-based maximum distance keeping method that uses Brouwer-Lyddane differential mean elements feedback for long-term control of the secular drift among satellites. The stability of the maximum distance keeping controller is proven using finite-time stability theory, and high-precision simulation results confirm that the new controller is able to arrest satellite relative drift for mission lifetimes exceeding a year. The maximum distance controller is automatically activated, and does not require a pre-determined activation time. Moreover, as a part of a complete DD-based solution for cluster keeping, a collision-avoidance method based on the same controller structure, albeit with differential osculating elements feedback, is developed and validated. Finally, the possibility to regulate cross-track drift with DD is examined, but it is shown that DD can only provide weak controllability in this case.

Optimal spacecraft orbit control has been the subject of extensive research, which resulted in solutions for optimal orbit transfers. A common orbital maneuver problem is the fuel-optimal impulsive transfer between coplanar circular orbits. Three such well-known transfers are the Hohmann transfer, which is an optimal biimpulsive transfer, the bi-elliptic tri-impulsive transfer, and the bi-parabolic transfer. These solutions were developed based on the Keplerian restricted two-body problem. However, the omission of perturbations results in deviated target orbits and leads to maneuvers that are not actually fuel-optimal. In this paper, the well-known Hohmann, bi-elliptic, and bi-parabolic transfers are modified to accommodate the J₂ zonal harmonic, and new closed-form solutions for the optimal maneuvers are presented. An improvement in maneuver precision is obtained by using an analytical model based on closed-form solutions of motion in the equatorial plane under the effect of J₂. The performance improvement is validated using high-fidelity simulations, which include a myriad of orbital perturbations.

Estimating the relative pose and motion of cooperative satellites using on-board sensors is a challenging problem. When the satellites are noncooperative, the problem becomes even more complicated, as there might be poor a priori information about the motion and structure of the target satellite. In this paper, the mentioned problem is solved by using only visual sensors, which measurements are processed through robust filtering algorithms. Using two cameras mounted on a chaser satellite, the relative state with respect to a target satellite, including the position, attitude, and rotational and translational velocities, is estimated. The new approach employs a stereoscopic vision system for tracking a set of feature points on the target spacecraft. The perspective projection of these points on the two cameras constitutes the observation model of an iterated extended Kalman filter (IEKF) estimation scheme. Using new theoretical results, the information contained in the visual data is quantified using the Fisher information matrix. It is shown that, even in the noncooperative case, there is information that can be extracted pertaining to the relative attitude and target structure. Finally, a method is proposed for rendering the relative motion filtering algorithm robust to uncertainties in the target's inertia tensor. This is accomplished by endowing the IEKF with a maximum a posteriori identification scheme for determining the most probable inertia tensor from several available hypotheses. The performance of the new filtering algorithm is validated by Monte-Carlo simulations. Also a preliminary experimental evaluation is provided.

This paper develops new methods for vision-based satellite attitude control aimed at space-based optical tracking of resident space objects (RSOs). An Earth-orbiting chaser satellite equipped with either one or two body-fixed cameras can successfully track an RSO provided that the target is kept within the camera field of view. Because the cameras are body fixed, the attitude of the satellite needs to be controlled to maintain target lock. Novel vision-based control algorithms are developed to align the chaser camera's optical axis with the chaser-target line of sight. Two control architectures are presented for the cases of monocular and stereoscopic vision. In the case of the monocular architecture, relative chaser-target acceleration information is not available. Moreover, in both cases unknown perturbations can impair the tracking performance. To increase the tracking algorithm's robustness to these effects, a variable structure attitude control technique is employed. The stability of the developed control laws are substantiated based on Lyapunov's direct method and demonstrated using Monte Carlo simulations. The results clearly show that using stereoscopic vision yields faster target tracking, increased robustness to noise and field-of-view limits, and reduced fuel consumption compared with monocular vision-based attitude tracking.

Keeping a cluster of satellites within bounded relative distances requires active control, because natural perturbations (the most significant of which is the J₂ term in the geopotential) tend to move the satellites apart. Whereas there is abundant literature on controlling the relative drift using multiple impulsive maneuvers and approximate astrodynamical models involving mean orbital elements, the works that attempt to minimize the number of impulses while using the inertial position and velocity vectors of the satellites are scarce. In this paper, singleimpulse distance-keeping maneuvers are derived, without approximating the J₂-perturbed dynamics. An analytical derivation of minimum-fuel impulsive maneuvers between equatorial orbits is provided, while relying on radial period and orbital angle matching conditions. Then, a continuation procedure is used to obtain single-impulse relative distance control between inclined orbits. The development of the impulsive maneuvers relies on the inertial position and velocity vectors of the satellites and does not involve mean elements. In each step, necessary and sufficient conditions for the existence of a single-impulse maneuver are provided. The results are illustrated using a number of realistic scenarios, such as a simulation that includes a 21 × 21 gravitational model, drag, and lunisolar attraction.

Cluster flight is a term used for describing multiple satellites that are being held within pre-defined minimum and maximum distances for long time intervals, possibly the entire mission. This technology is required for a myriad of space architectures and missions, including disaggregated space architectures. Whereas the literature is abundant with works on control laws for satellite formation flying, there are only a handful of works on control of cluster flight. The purpose of the current work is to develop a cluster flight control algorithm, which is able to keep the satellites of the cluster within pre-specified minimum and maximum distances, while utilizing small amounts of propellant. The newly developed algorithm relies on the natural inter-satellite distance dynamics. The algorithm incorporates realistic mission constraints, such as constant-magnitude thrust, and is implemented in feedback form, steering the mean elements to judiciously selected reference values. Simulations indicate that a few tens of grams of propellent are sufficient for operating a cluster flight mission in excess of 1 year, using low specific-impulse thrusters.

Close encounters of Near Earth Objects (NEOs) with large asteroids are a possible source of systematic errors in trajectory propagations and asteroid mitigation. It is, thus, necessary to identify those large asteroids that have to be considered as perturbers in NEO orbit modeling. Using the Standard Dynamical Model we searched for encounters between the 1649 numbered Near Earth Asteroids (NEAs) and 2191 large asteroids having sizes greater than 20 km. Investigating the 21^st century A. D. we have found 791 close encounters with 195 different large asteroids that lead to a substantial scattering of NEOs.

Differential drag (DD) as a means for passive satellite cluster keeping is an old idea, but so far using DD-based cluster keeping while relying on mean orbital elements feedback has not been proposed. This paper develops a DD-based maximum distance keeping method that uses Brouwer-Lyddane differential mean elements feedback for long-term control of the secular drift among satellites. The stability of the maximum distance keeping controller is proven using finite-time stability theory, and high-precision simulation results confirm that the new controller is able to arrest satellite relative drift for mission lifetimes exceeding a year. The maximum distance controller is automatically activated, and does not require a pre-determined activation time.

Estimating the pose, motion and structure of non-cooperative dynamic targets using on board sensors is a challenging problem. This work suggests using multiple-baseline stereovision for non-cooperative dynamic target relative pose, motion and structure estimation, with a particular emphasis on space applications. A computer-vision feature-matching algorithm is designed, which produces input data for a recursive filtering algorithm. A newly-developed initialization scheme is proposed, which decreases the ambiguity in the target center of mass location. The effect of varying the number of cameras is investigated. The proposed vision-based relative motion estimation method was validated at the Technion's Distributed Space Systems Laboratory. A scalability analysis indicates that the proposed method may be potentially useful for space applications.

When a group of satellites is equipped with a particularly simple propulsion system (e.g., cold-gas thrusters), constraints on the thrust level and total propellant mass renders cluster keeping extremely challenging. This is even more pronounced in disaggregated space architectures, in which a satellite is formed by clustering a number of heterogonous free-flying modules. The research described in this paper develops guidance laws aimed at keeping the relative distances between the cluster modules bounded for long mission lifetimes, typically more than a year, while using constant-magnitude low thrust, with a characteristic on-off profile. A cooperative guidance law capable of cluster establishment and maintenance under realistic environmental perturbationsis developed. The guidance law is optimized for fuel consumption, subject to relative distance constraints. Some of the solutions found to the optimal guidance problem require only a single maneuver arc to keep the cluster within relatively close distances for an entire year.

Satellite cluster flight is an enabling technology for disaggregated space architecture. A nonlinear distributed control law is developed considering fixed-magnitude thrust, in order to establish satellite cluster flight under perturbations. Mean orbital elements are used as feedback. Notation of partial stability is adopted to describe the stability. Uniform stability and asymptotic stability are proven for the relative motion control. State selection for establishing a low Earth orbit cluster is also discussed. Several numerical studies are performed to assess the performance of the control law. Comparisons are provided to show the fuel-balancing merits of the current control law. The effects of drag on the long-term performance are also investigated. The current control law is shown to be feasible and effective for satellite cluster flight.

Sensor selection is an NP-hard problem involving the selection of S out of N sensors such that optimal (in some predefined sense) filtering performance is attained. We present a novel approach for sensor selection that utilizes a measure quantifying the incoherence of the vector space spanned by the sensors with respect to the system's principal directions. This approach provides a formulation of a convex relaxation problem that can be efficiently modeled and solved using compressed sensing (CS) algorithms. One of the key requirements in the theory of CS is that of a sufficiently incoherent sensing matrix. Such matrices are normally neither encountered in the sensor selection problem nor in many other engineering and scientific problems, which, to some extent, limits the applicability of the theory. In this work, this requirement is alleviated via the concept of semi-random sensing, where the standard sensing matrix pertaining to the problem at hand is deliberately contaminated by noise. We provide relations between the noise intensity and the incoherence properties of the contaminated sensing matrix. The viability of this concept is demonstrated and analyzed in the context of the sensor selection problem.

Generally, any initially-close satellites-chief and deputy-moving on orbits with slightly different orbital elements, will depart each other on locally unbounded relative trajectories. Thus, constraints on the initial conditions must be imposed to mitigate the chief-deputy mutual departure. In this paper, it is analytically proven that choosing the chief's orbit to be a frozen orbit can mitigate the natural relative drift of the satellites. Using mean orbital element variations, it is proven that if the chief's orbit is frozen, then the mean differential eccentricity is periodic, leading to a periodic variation of the differential mean argument of latitude. On the other hand, if the chief's orbit is non-frozen, a secular growth in the differential mean argument of latitude leads to a concomitant along-track separation of the deputy from the chief, thereby considerably increasing the relative distance evolution over time. Long-term orbital simulation results indicate that the effect of choosing a frozen orbit vis-à-vis a non-frozen orbit can reduce the relative distance drift by hundreds of meters per day.

While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J₂ harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J₂ zonal harmonic. Numerical simulations are used to quantify these differences.

In disaggregated satellites, the functional capabilities of a single monolithic satellite are distributed among multiple free-flying, wirelessly communicating modules. One of the main challenges associated with disaggregated satellites is cluster flight, i.e., keeping the modules within a bounded distance, typically less than 100 km, for the entire mission lifetime. This paper presents a methodological development of cluster flight algorithms for disaggregated satellite systems in low Earth orbits. To obtain distance-bounded relative motion a new constraint on the initial conditions of the modules is developed. A concomitant analytical bound on the relative distance between the modules is proven based on a design model assuming time invariance of the environmental perturbations. It is then shown that if the actual astrodynamical model includes other possible time-varying effects, mild drifts between the modules are obtained. Furthermore, this paper presents a detailed impulsive cluster establishment and cluster-keeping algorithm for tracking a given nominal orbit, whose characteristics satisfy the previously developed no-drift constraint. This algorithm provides fuel balancing among the maneuvering modules, as well as the minimization of the total fuel consumption, while guaranteeing a collision-free operation. Numerical simulations using representative astrodynamical models are used to validate the analysis.

The idea to use differential drag (DD) for satellite formationkeeping emerged in the mid-Eighties, when the feasibility of DD-based control was proven assuming linearized relative dynamics for two satellites. Although almost three decades have passed, the most prevalent approach for investigating DD-based formationkeeping still utilizes linear models written for only a pair of satellites. However, such models are not adequate for long-term cluster flight of multiple satellites or multiple modules forming, e.g., a disaggregated satellite, with typical mission lifetimes exceeding a year. In the current work, an alternative, nonlinear method for DD-based cluster-keeping is developed. The method relies on orbital elements instead of Cartesian coordinates. The results are verified using simulations based on the forthcoming Space Autonomous Mission for Swarming and Geolocation with Nanosatellites.

Space Autonomous Mission for Swarming and Geolocation with Nano-satellites (SAMSON) is a new satellite mission, led by the Distributed Space Systems Lab at the Technion - Israel Institute of Technology. SAMSON will include three nanosatellites, built based on the CubeSat standard. The mission is planned for at least one year, and has two main goals: (i) Demonstrate long-term autonomous cluster flight of multiple satellites, and (ii) Determine the position of a radiating electromagnetic terrestrial source based on time difference of arrival and/or frequency difference of arrival. In this paper, the cluster flight control strategy for SAMSON is discussed. This strategy performs cooperative maneuvers upon necessity, when any inter-satellite distance reaches either the upper or lower bound. The maneuvers are conceived to avoid that the secular component of the inter-satellite distances exceed the prescribed distance bounds. To compute the maneuvers, a logic scheme is first applied, which establishes constraints on the differential mean semimajor axes to provide desired post-maneuver behavior. Then, a Lyapunov based control law steers the mean semimajor axis, eccentricity and inclination, to hold the aforementioned constraints. The considered actuators are constant-thrust-magnitude thrusters. Simulations for 1 year are shown, validating the potential implementability of the proposed algorithm on-board the SAMSON satellites.

Deploying a cluster of satellites could provide accurate tracking of a Mars rover, a redundant navigation system in a jammed GNNS environment, or a cost-effective system for autonomously locating distress signals. The SAMSON mission conducted by the Distributed Space Systems Laboratory is planed to demonstrate such capabilities by deploying a cluster of three Low Earth Orbit (LEO) nano-satellites. One of the mission goals is to accurately determine the position of a terrestrial source emitting electromagnetic pulses. This study is aimed at providing a theoretical basis for achieving optimal positioning performance based on sequential time difference of arrival measurements with a satellite cluster.

The idea to use differential drag for satellite formation keeping emerged in the mid-1980s, when the feasibility of differential-drag-based control was proven, assuming linearized relative dynamics for two satellites. Unlike previous work in differential-drag-based formation keeping, the present work develops a nonlinear method suitable for missions in excess of a year. It is shown that the differential mean eccentricity is uncontrollable for near-circular orbits, and hence a nonlinear differential-drag-based controller for matching the drag-related secular component of the semimajor axis is developed. An asymptotic stability proof for the controller is provided. Moreover, two new methods for differential-drag-based cluster keeping of multiple modules are developed, thus expanding existing literature, which usually deals with two satellites only. The results are validated using simulations based on the forthcoming Space Autonomous Mission for Swarming and Geolocation with Nanosatellites, showing that differential-drag-based cluster keeping can be effective for altitudes reaching about 600 km.

Satellite guidance and orbit control often use mean elements as inputs. Whereas traditional missions can use ground-station-based calculation of mean elements, this is not possible in autonomous satellites, which are required to perform onboard estimation of the mean elements. This problem is not trivial, because analytical satellite theories are not robust to modeling errors and cannot easily accommodate thrust. The purpose of this paper is to develop an effective filtering algorithm for onboard estimation of the mean orbital elements in small-eccentricity low Earth orbits. To that end, a semianalytical astrodynamical model that includes zonal/tesseral/sectorial harmonics and drag is formulated to capture the daily, long-periodic, and secular evolution of the mean orbital elements. The mapping from mean to osculating elements is used as a measurement equation by adding the short-periodic terms. This unique formulation is then fed into a spherical-simplex square-root unscented Kalman filter, which serves as the meanelements estimator. A comprehensive performance evaluation for both controlled and uncontrolled orbits shows the potential applicability of the method and its advantages compared with Brouwer-based approaches.

We present algorithms for tracking and reasoning of local traits in the subsystem level based on the observed emergent behavior of multiple coordinated groups in potentially cluttered environments. Our proposed Bayesian inference schemes, which are primarily based on (Markov chain) Monte Carlo sequential methods, include: 1) an evolving network-based multiple object tracking algorithm that is capable of categorizing objects into groups, 2) a multiple cluster tracking algorithm for dealing with prohibitively large number of objects, and 3) a causality inference framework for identifying dominant agents based exclusively on their observed trajectories.We use these as building blocks for developing a unified tracking and behavioral reasoning paradigm. Both synthetic and realistic examples are provided for demonstrating the derived concepts.

A computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead of solving the CS problem as an optimization problem, it is suggested to transform the optimization problem into a convex feasibility problem (CFP), and solve it using feasibility-seeking sequential and simultaneous subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the commonly-used CS algorithms, such as Bayesian CS and Gradient Projections for sparse reconstruction, which become inefficient as the problem dimension and sparseness degree increase, the proposed methods exhibit robustness with respect to these parameters. Moreover, it is shown that the CFP-based projection methods are superior to some of the state-of-the-art methods in recovering the signal's support. Numerical experiments show that the CFP-based projection methods are viable for solving large-scale CS problems with compressible signals.

Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J₂ perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J₂ Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J₂ problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.

Cooperative navigation (CN) enables a group of cooperative robots to reduce their individual navigation errors. For a general multi-robot (MR) measurement model that involves both inertial navigation data and other onboard sensor readings, taken at different time instances, the various sources of information become correlated. Thus, this correlation should be solved for in the process of information fusion to obtain consistent state estimation. The common approach for obtaining the correlation terms is to maintain an augmented covariance matrix. This method would work for relative pose measurements, but is impractical for a general MR measurement model, because the identities of the robots involved in generating the measurements, as well as the measurement time instances, are unknown a priori. In the current work, a new consistent information fusion method for a general MR measurement model is developed. The proposed approach relies on graph theory. It enables explicit on-demand calculation of the required correlation terms. The graph is locally maintained by every robot in the group, representing all of the MR measurement updates. The developed method calculates the correlation terms in the most general scenarios of MR measurements while properly handling the involved process and measurement noise. A theoretical example and a statistical study are provided, demonstrating the performance of the method for vision-aided navigation based on a three-view measurement model. The method is compared, in a simulated environment, with a fixed-lag centralized smoothing approach. The method is also validated in an experiment that involved real imagery and navigation data. Computational complexity estimates show that the newly developed method is computationally efficient.

Much effort has been invested during the past decades in design of parafoils for a wide range of payloads and in development of means for their guidance. Existing parafoils are capable of autonomous navigation using the global positioning system and other onboard sensors. The purpose of this study is to explore the advantages of coordination among multiple autonomous parafoils. Each parafoil is able to navigate to the target on its own by following a real-time-generated reference trajectory. A new method for trajectory generation is presented and behaviour-based rules are developed that control the relative motion of the descending parafoils. The set of simple rules results in an emergent behaviour known as flocking. The coupling between trajectory following and flocking is studied in a multiagent simulation. The simulation uses a realistic six-degrees-of-freedom model of a heavy cargo parafoil. The obtained results demonstrate the possibility of flocking behaviour for guided parafoils. The flocking rules ensure safe separation between the vehicles headed for the same target and allow the parafoils to follow a reference trajectory as a group.

This paper presents a new method for distributed vision-aided cooperative localization and navigation for multiple inter-communicating autonomous vehicles based on three-view geometry constraints. Each vehicle is equipped with a standard inertial navigation system and an on-board camera only. In contrast to the traditional approach for cooperative localization, which is based on relative pose measurements, the proposed method formulates a measurement whenever the same scene is observed by different vehicles. Each such measurement is comprising of three images, which are not necessarily captured at the same time. The captured images, to which some navigation parameters are attached, are stored in repositories by some of the vehicles in the group. A graph-based approach is applied for calculating the correlation terms between the navigation parameters associated to images participating in the same measurement. The proposed method is examined using a statistical simulation and is further validated in an experiment that involved two vehicles in a holding pattern scenario. The experiments show that the cooperative three-view-based vision-aided navigation may considerably improve the performance of an inferior INS.

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun-Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.

The paper develops the closed-form solution for the motion around an oblate planet in the situation when the orbit is unbounded. It is proven that when the effect of the J₂ zonal harmonic is taken into account, the orbit is different from its Keplerian counterpart, having a marked influence on the deflection angle, thereby changing the Keplerian flyby geometry. Numerical simulations quantify this difference, which is closely related to the minimum flyby altitude of the spacecraft. The analytic developments can be applied to the preliminary design of gravity-assisted maneuvers.

Space Autonomous Mission for Swarming and Geolocation with Nanosateliites (SAMSON) is a new satellite mission, led by the Distributed Space Systems Lab at the Technion - Israel Institute of Technology. SAMSON will include three nanosateliites, built based on the CubeSat standard. The mission is planned for at least one year, and has two main goals: (i) Demonstrate long-term autonomous cluster flight of multiple satellites, and (ii) Determine the position of a radiating electromagnetic terrestrial source based on time difference of arrival (TDOA) and/or frequency difference of arrival (FDOA). In this paper, the cluster flight orbit control law for SAMSON is discussed. The control law is aimed at regulating the mean semi- major axis, eccentricity and inclination so as to provide long-term bounded relative motion while utilizing small amount of fuel. The considered actuators are constant-thrust-magnitude thrusters. Simulations for 1 year are shown, validating the potential implementability of the proposed algorithm on-board the SAMSON satellites. Copyright

The basic idea behind disaggregated satellites is to distribute the functionality of a single monolithic satellite among multiple physically-separated inter-communicating modules. The research described in this paper develops optimal guidance laws aimed at keeping the relative distances between the modules bounded for extended mission lifetimes while using constant-magnitude chemical low-thrust. A cooperative guidance law capable of cluster establishment and maintenance under environmental perturbations is proposed. The optimal guidance law is constructed using a bang-off-bang thrust profile. The cost functional is the total fuel consumption, and the constraints are maximum and minimum distances, balanced fuel utilization - so as to minimize differential drag - and stationkeeping about a nominal reference orbit.

The main idea of disaggregated satellites is to distribute the functionality of a single monolithic satellite among multiple physically-independent wireless-communicating modules. These new architectures raises the emerging concept of cluster flight, in which the distance between any two modules of the cluster must be kept between prescribed upper and lower thresholds. This work is concerned with strategies to efficiently perform cluster-establishment. The main goal of the establishment is to deploy the modules in such a manner that the necessity of future corrective maneuvers, after the establishment, is reduced as much as possible. To that end, the establishment maneuver steers the cluster to relative initial conditions, which are favorable to avoid distance drifts, as well as enable the payload module, to track a prescribed reference trajectory. A cooperative guidance law, for cluster establishment and re-establishment, is proposed. As the modules are assumed to be equipped with low-thrust chemical (cold-gas) propulsion systems, thrust of type bang-off-bang thrust is considered, as well as perturbations due to the Earth oblateness and drag. Copyright

A solution to the satellite relative motion problem is constructed based on a radial intermediary of the J₂-problem. The new solution accounts for nonlinearities of the model to a high extent, and hence is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations without limiting to the case of tightly-controlled formations. The integrability of the radial intermediary is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and to determine an initialization scheme that yields long-term boundedness of the relative distance.

The paper introduces a unified methodology for the study of bounded motion in central force fields, which serves both for qualitative insights, as well as for the derivation of closed-form vectorial solutions for the equations of motion. The paper offers a full regularization of the equations of motion in a central force field, starting from the polar equations in the plane of motion. A time transformation, as well as a coordinate transformation, are performed together, and the motion is (i) reduced to a Kepler motion in a rotating frame, with respect to a new time variable (ii) regularized further to a harmonic oscillator. In addition, some new results pertaining to the existence of a Laplace-Runge- Lenz vector in a central-force motion are presented.

Interferometry in space has marked advantages: long integration times and observation in spectral bands where the atmosphere is opaque. When installed on separate spacecraft, it also has extended and flexible baselines for better filling of the uv plane. Intensity interferometry has an additional advantage, being insensitive to telescope and path errors, but is unfortunately much less light-sensitive. In planning towards such a mission, we are experimenting with some fundamental research issues. Towards this end, we constructed a system of three vehicles floating on an air table in formation flight, with an autonomous orbit control. Each such device holds its own light collector, detector, and transmitter, to broadcast its intensity signal towards a central receiving station. At this station we implement parallel radio receivers, analogue to digital converters, and a digital three-way correlator. Current technology limits us to ∼1GHz transmission frequency, which corresponds to a comfortable 0.3m accuracy in light-bucket shape and in its relative position. Naïve calculations place our limiting magnitude at ∼7 in the blue and ultraviolet, where amplitude interferometers are limited. The correlation signal rides on top of this huge signal with its own Poisson noise, requiring a very large dynamic range, which needs to be transmitted in full. We are looking at open questions such as deployable optical collectors and radio antennae of similar size of a few meters, and how they might influence our data transmission and thus set our flux limit.

Disaggregated satellites distribute the functionalities of a single monolithic satellite among multiple wirelessly-linked heterogenous modules. One of the challenges of designing a disaggregated satellite cluster is the development of a control system that is suited to the diverse cluster operational requirements. This paper presents a development of a backstepping approach for controlling the relative motion of the cluster modules in order to facilitate cluster initialization, maintenance, and scatter/re-gather maneuvering in case of a space debris collision threat. Inter-module collision avoidance is included through the use of artificial potential functions to ensure safe operational distances. As opposed to previous studies, the artificial potential formulation directly addresses the issue of moving targets and obstacles, as well as proper inclusion of the system dynamics within the control scheme. Numerical simulations that include the effects of J₂ to J₅ are used to demonstrate the effectiveness of the proposed top-level control for disaggregated satellites.

One of the hard issues that arises in distributed navigation is keeping an up-to-date and consistent estimation of the dependency between the solutions computed by each one of the involved agents. This issue is critical for the consistent information fusion in distributed cooperative navigation and was recently tackled using a graph-based approach for the on-demand calculation of cross-covariance terms. In particular, the approach was applied to a method for visual aided, distributed cooperative navigation based on three-view geometry constraints, in which a measurement is formulated whenever the same scene is observed by several robots, not necessarily at the same time. The purpose of this paper is twofold. First, the claim that on-demand calculation of cross-covariance terms in three-view-based cooperative navigation is further substantiated, and the difficulties with other existing techniques are emphasized. Second, the efficiency of using the on-demand calculations is validated by comparing the results to those obtained by assuming the three-view multi-robot measurements schedule is known a priori. In this latter method, the required cross-covariance terms are calculated using a fixed-lag centralized smoother. The comparison clearly shows the advantages of using the on-demand scheme.

A new method for vision-aided navigation based on three-view geometry is presented. The main goal of the proposed method is to provide position estimation in GPS-denied environments for vehicles equipped with a standard inertial navigation system (INS) and a single camera only, without using any a priori information. Images taken along the trajectory are stored and associated with partial navigation data. By using sets of three overlapping images and the concomitant navigation data, constraints relating the motion between the time instances of the three images are developed. These constraints include, in addition to the well-known epipolar constraints, a new constraint related to the three-view geometry of a general scene. The scale ambiguity, inherent to pure computer vision-based motion estimation techniques, is resolved by utilizing the navigation data attached to each image. The developed constraints are fused with an INS using an implicit extended Kalman filter. The new method reduces position errors in all axes to the levels present while the first two images were captured. Navigation errors in other parameters are also reduced, including velocity errors in all axes. Reduced computational resources are required compared with bundle adjustment and simultaneous localization and mapping (SLAM). The proposed method was experimentally validated using real navigation and imagery data. A statistical study based on simulated navigation and synthetic images is presented as well.

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J₂ is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J₂-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J_2n, n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.

Article presents a closed-form solution of the optimal biimpulsive transfer between coplanar initial and terminal orbits for satellites exerting thrust by using three orthogonal thrusters aligned with the local-vertical local-horizontal frame. Coplanar maneuvers usually consume less propellant than more general transfers wherein the orbital planes are modified. An implicit assumption in most of the aforementioned works is that the propellant consumption is measured by the Euclidean norm of the velocity change vectors. Therefore, for satellites using orthogonal thrusters the optimization approach should be modified. With this consideration in mind, this work formulates the cost function for a general coplanar transfer between two noncollinear position vectors with prescribed terminal velocities, and analytically determines the set of points that satisfy the necessary conditions for minima.

A different approach is proposed for the study of satellite relative motion in an axially-symmetric gravitational field. Instead of using the Keplerian motion as the generating nominal orbit for the absolute motion, another "unperturbed" orbit is proposed instead: An equatorial orbit about an oblate planet. Based on the superintegrability of such motion, closed-form solutions for the equatorial relative motion are obtained. Analytic conditions for the periodicity of the relative motion in a generic central force field are presented, and utilized to design long-term bounded relative motion under high-order even zonal perturbations.

Fractionated spacecraft constitutes a satellite design methodology wherein the functional capabilities of a single monolithic satellite are distributed among multiple free-flying, wirelessly-communicating modules. One of the main challenges of a fractionated spacecraft system is cluster flight, i.e. keeping the various modules within a bounded distance, typically less than 100 km, for the entire mission lifetime. This paper presents a methodological development of cluster flight algorithms for fractionated spacecraft systems. To obtain distance- bounded relative motion, a new constraint on the initial conditions of the modules is developed. A concomitant analytical bound on the relative distance between the modules is proven based on a design model assuming timeinvariance of the environmental perturbations. It is then shown that if the actual astrodynamical model includes other, possibly time-varying effects, mild drifts between the modules are obtained. Furthermore, this paper presents a cluster establishment algorithm for tracking a given nominal orbit, whose characteristics satisfy the previously-develop no-drift constraint. This algorithm provides fuel balancing among the maneuvering modules as well as minimization of the total fuel consumption. Numerical simulations using realistic astrodynamical models are used to validate the analysis.

This work presents a three degrees-of-freedom laboratory experiment in which both the states and the moment of inertia of a non-cooperative target satellite were successfully estimated by a chaser satellite using stereovision measurements only. In the experiments conducted at the Technion's Dis- tributed Space Systems Laboratory, two satellite models, a target and a chaser, were floated on an air-bearing table. A stereovision system com- posed of two cameras mounted on the chaser satellite model was used to generate sequences of image frames of the target satellite model. The rela- tive range, body angle and angular velocity between the chaser and the tar- get were estimated using the left and right camera movies only. Moreover, the estimation algorithm also successfully identified the target moment of inertia using a multiple hypotheses-based filter.

The main objective of this paper is to develop methods for carrier-phase differential GPS sensing of the relative attitude between satellites using a single-frequency receiver. Two methods for relative attitude determination were developed, implemented, tested and compared using the Gravity Recovery and Climate Experiment satellite data. The first method estimates the absolute satellite attitude and then derives the relative attitude between the satellites. The relative attitude accuracy achieved by using this method was about 0.4 degrees. The second method uses a differential solution between all the GPS antennae. The main idea here is to find minimal differences between the entire GPS baseline measurements and the baseline vectors determined in the satellite's body-fixed frame. The relative attitude determined by using the improved method was about 0.1 degrees.

We present a novel framework for identifying and tracking dominant agents in groups. Our proposed approach relies on a causality detection scheme that is capable of ranking agents with respect to their contribution in shaping the system's collective behaviour based exclusively on the agents' observed trajectories. Further, the reasoning paradigm is made robust to multiple emissions and clutter by employing a class of recently introduced Markov chain Monte Carlo-based group tracking methods. Examples are provided that demonstrate the strong potential of the proposed scheme in identifying actual leaders in swarms of interacting agents and moving crowds.

This paper presents a new approach for designing orbits for infrared (IR) space-borne observatories using multiple gravity assists. A large displacement normal to the ecliptic plane mitigates the noise generated by the local zodiacal dust, thereby reducing the size, weight, and complexity of the telescope. Although previous works focused on long-term missions, allowing very long transfer times to out-of-ecliptic orbits, this paper considers short-duration missions, for which the transfer times are tightly constrained. To reduce the transfer time, the flyby sequence includes the inner planets only: Venus, Earth, and Mars. Moreover, in order to reduce energy requirements, a fuel-optimal multiple gravity-assisted trajectory is designed. The problem is modeled using the patched-conic approximation and solved using a hybrid genetic algorithm coupled to a pattern search. Efficient trajectories requiring a minimum velocity addition, although providing a maximum observation time, are found and validated using an N-body simulation. The TPF-I mission is used as a benchmark in order to quantify the benefits of an out-of-ecliptic orbit. It is shown that the newly found orbits allow a reduction in the collector area compared to halo orbits.

This paper suggests a new approach for generating bounded motion between two space- craft orbiting an oblate planet. As opposed to most works, the suggested approach does not rely on averaging the osculating orbital elements. Instead, the problem is formulated in local-vertical local-horizontal coordinates, without approximating the gravitational po- tential. The underlying methodology relies on the concept of 1:1 orbit commensurability, which, unlike the Kepler unperturbed case, splits into two separate necessary and sufficient conditions: An equality between the radial periods of the two spacecraft and two appro- priately defined angles. For equatorial orbits, closed-form expressions for the required impulsive maneuvers are found.

This paper presents an approach for designing out-of-ecliptic orbits for infrared (IR) space-borne observatories. A large displacement normal to the ecliptic plane mitigates the noise generated by the local zodiacal dust, thereby reducing the size, weight and complexity of the telescope. While previous works focused on long-term missions, for which transfers to out-of-ecliptic orbits are allowed to be very long, this paper considers relatively short-duration missions, for which the programmatic constraints are more pronounced. In order to reduce energy requirements, an optimal multiple gravity-assisted trajectory is designed. To reduce the transfer time, the flyby sequence includes the inner planets only: Venus, Earth and Mars. The problem is modeled using the patched-conic approximation and solved using a hybrid genetic-algorithm coupled to a pattern search. Efficient trajectories requiring a minimum velocity addition while providing a maximum observation time are found and validated using an N-body simulation. The TPF-I mission is used as benchmark in order to quantify the systematic benefits of an out-of-ecliptic orbit. It is shown that the newly-found orbits are very promising for IR missions, as they allow a considerable reduction in the collector area and the concomitant cost per image.

Sensor selection is an NP-hard problem involving the selection of S out of N sensors such that optimal filtering performance is attained. We present a novel approach for sensor selection that utilizes a heuristic measure quantifying the incoherence of the vector space spanned by the sensors with respect to the system's principal directions. This approach facilitates the formulation of a convex relaxation problem that can be efficiently modeled and solved using compressed sensing (CS) algorithms. We subsequently develop a new CS algorithm based on subgradient projections. The new CS algorithm for sensor selection is shown to outperform existing methods in a number of applications.

Estimating the relative pose and motion of cooperative satellites using on-board sensors is a challenging problem. When the satellites are non-cooperative, the problem becomes far more complicated, as there might be poor or no a priori information about the motion or structure of the target satellite. In this work we develop robust algorithms for solving the said problem by assuming that only visual sensory information is available. Using two cameras mounted on a chaser satellite, the relative state of a target satellite, including the position, attitude, and rotational and translational velocities is estimated. Our approach employs a stereoscopic vision system for tracking a set of feature points on the target spacecraft. The perspective projection of these points on the two cameras constitutes the observation model of an EKF-based filtering scheme. In the final part of this work, the relative motion filtering algorithm is made robust to uncertainties in the inertia tensor. This is accomplished by endowing the plain EKF with a maximum a posteriori identification scheme for determining the most probable inertia tensor from several available hypotheses.

This paper presents a new method for distributed vision-aided cooperative localization and navigation for multiple autonomous platforms based on constraints stemming from the three-view geometry of a general scene. Each platform is assumed to be equipped with a standard inertial navigation system and an on-board, possibly gimbaled, camera. The platforms are also assumed to be capable of intercommunicating. No other sensors, or any a priori information is required. In contrast to the traditional approach for cooperative localization that is based on relative pose measurements, the proposed method formulates a measurement whenever the same scene is observed by different platforms. Each such measurement is constituted upon three images, which are not necessarily captured at the same time. The captured images, attached with some navigation parameters, are stored in repositories by each, or some, of the platforms in the group. A graph-based approach is applied for calculating the correlation terms between the navigation parameters associated to images participating in the same measurement. The proposed method is examined using a statistical simulation in a leader-follower scenario, and is demonstrated in an experiment that involved two vehicles in a holding pattern scenario.

This paper addresses the problem of distributed cooperative navigation. A new graph-based method is developed for on-demand calculation of the required correlation terms, considering a general multi-robot measurement model. These correlation terms are necessary for the consistent EKF-based data fusion when several statistically-dependent sources of information are used. The measurement model relates between the navigation information transmitted by any number of robots and the actual readings taken by the available onboard sensors. The transmitted information is not necessarily of the current time instant, but may actually belong to some time instant from the past. Experiment results and a theoretical example of the developed method are presented considering a three-view measurement, formulated upon receiving three images of the same scene, captured by different robots at different a priori unknown time instances.

This paper presents a new method for navigation aiding based on three-view geometry. An airborne platform is assumed to be equipped with a standard inertial navigation system and a gimbaled camera, which captures images along the flight. The images are stored and associated with a partial data taken from the navigation system. No additional sensors or a-priori information are assumed. Given a set of three overlapping images, and the associated navigation data, we develop constraints that relate the platform motion between the time instants of the three images. The scale ambiguity, embedded with pure vision-based motion estimation techniques, is resolved by incorporating navigation data attached to each image. The developed constraints are fused with an inertial navigation system using an Implicit Extended Kaiman Filter, allowing to reduce the position error, and errors in some other parameters, to the levels present when the first image was captured. Reduced computational resources are required, in comparison with other existing methods such as bundle adjustment and SLAM. The proposed method was examined for handling loop scenarios in indoor and outdoor navigation based on real images, allowing to maintain constant levels of navigation errors over time.

In the case of a wide-scale disaster, an accurate airdrop of emergency supplies is crucial. We present a novel, top-down approach for designing and executing airdrop missions using guided parafoils. We develop a guidance algorithm and a cooperative task management method for the autonomous handling of faults and exceptional events by the parafoil group. The autonomous operation is based on inter-parafoil ad-hoc communication. A parafoil or a number of parafoils can dynamically react to events that prevent one or more parafoils from successfully completing their mission. Two recovery methods are presented: Swap, which enables parafoils to dynamically exchange their targets, and Replace, which gives precedence to prioritized targets over low-priority targets. The parafoil guidance method, combined with the task management algorithm, significantly increases the probability of successful airdrop. The small overhead of the communication layer and of the low complexity of the swap and replace recovery algorithms enable these procedures to run in a distributed environment under real-time limitations.

We present two simple methods for recovering sparse signals from a series of noisy observations. The theory of compressed sensing (CS) requires solving a convex constrained minimization problem. We propose solving this optimization problem by two algorithms that rely on a Kalman filter (KF) endowed with a pseudo-measurement (PM) equation. Compared to a recently-introduced KF-CS method, which involves the implementation of an auxiliary CS optimization algorithm (e.g., the Dantzig selector), our method can be straightforwardly implemented in a stand-alone manner, as it is exclusively based on the well-known KF formulation. In our first algorithm, the PM equation constrains the l₁ norm of the estimated state. In this case, the augmented measurement equation becomes linear, so a regular KF can be used. In our second algorithm, we replace the l₁ norm by a quasi-norm l_p, 0 ≤p ≤. This modification considerably improves the accuracy of the resulting KF algorithm; however, these improved results require an extended KF (EKF) for properly computing the state statistics. A numerical study demonstrates the viability of the new methods.

When studying scalability in ad-hoc networks, most works present experimental results for a limited number of nodes (100-200). Various "explicit" clustering techniques have been proposed to improve scalability, obtaining successful sessions for 400-800 nodes. However, explicit clustering may damage the performance, e.g., cause session breaks due to fast movements of cluster heads. An alternative to explicit clustering is the use of algorithms that are "naturally clustered", i.e., arrange the nodes in dynamic hierarchical structures. In this work, we study the effect of explicit clustering by comparing an advanced version of the Ad Hoc Distance Vector Algorithm (AODV) with the Metrical Routing Algorithm (MRA) that possesses the natural clustering property. We cover fundamental aspects of scalability and experimentally prove the superiority of implicit clustering over explicit clustering. In particular, we consider heterogeneous theaters with several types of transmitters including personal, car-mounted, helicopters and a Geostationary (GEO) satellite. Natural clustering is more effective in heterogeneous theaters as the more powerful transmitters can serve as cluster heads. A formal bound based on general probabilistic assumptions shows that all existing ad-hoc algorithms cannot scale infinitely, thus rendering scalability as an experimental issue.

Compressed sensing is a new emerging field dealing with the reconstruction of a sparse or, more precisely, a compressed representation of a signal from a relatively small number of observations, typically less than the signal dimension. In our previous work we have shown how the Kalman filter can be naturally applied for obtaining an approximate Bayesian solution for the compressed sensing problem. The resulting algorithm, which was termed CSKF, relies on a pseudomeasurement technique for enforcing the sparseness constraint. Our approach raises two concerns which are addressed in this paper. The first one refers to the validity of our approximation technique. In this regard, we provide a rigorous treatment of the CSKF algorithm which is concluded with an upper bound on the discrepancy between the exact (in the Bayesian sense) and the approximate solutions. The second concern refers to the computational overhead associated with the CSKF in large scale settings. This problem is alleviated here using an efficient measurement update scheme based on Krylov subspace method.

This work pertains to the exploitation of the intrinsic dynamics arising from n-body models together with the optimal control techniques related to continuous low-thrust propulsion. The results show that it is possible to obtain accurate high-performance interplanetary trajectories from both the point of view of mass consumption and flight time. In addition, special distant periodic orbits around Mars - previously defined in the Hill problem and then numerically evaluated in the restricted three-body problem - have been reached by means of coast, low-thrust and stable manifold transfer stages. Finally, the transfers are optimized within wider dynamical n-body models using a direct method approach and multiple shooting technique.

Transfers from Earth to Mars have been extensively studied in the literature, with the prevalent approaches being two-body patched-conics-based trajectory optimization or three-body halo-to-halo transfers augmented by lowthrust coasting arcs. In this work, a new transfer method is suggested. The main idea is to combine manifold theory with low-thrust propulsion in order to reach distant prograde orbits about Mars. The most prominent feature of the said distant prograde orbits is the associated hyperbolic structure, permitting a free capture by coasting on the stable manifold without any additional injection maneuvers. The preliminary design process includes identification of Martian distant prograde orbits via a numerical continuation of known orbits in Hill's three-body problem and derivation of a three-phase coupled restricted three-body transfer comprising an Earth-escape stage, a heliocentric orbit, and a Martian rendezvous.Adirect nonlinear programming-based low-thrust optimization in a full-ephemeris model including the Earth, the moon, and Mars is used to improve the preliminary design and evaluate the performance of the suggested transfer method.

Compressive sensing (CS) is a popular technique used to reconstruct a signal from few training examples, a problem which arises in many machine learning applications. In this paper, we introduce a technique to guarantee that our data obeys certain isometric properties. In addition, we introduce a bayesian approach to compressive sensing, which we call ABCS, allowing us to obtain complete statistics for estimated parameters. We apply these ideas to fMRI classification and find that by isometrically transforming our data, significant improvements in classification accuracy can be achieved using the LASSO and Dantzig selector methods, two standard techniques used in CS. In addition, applying the ABCS method offers improvements in classification accuracy over both LASSO and Dantzig. Finally, we find that applying both the ABCS method together with isometric transformations, we are able to achieve an error rate of 0.0%.

The on-line construction of an image mosaic or panorama can be exploited to aid the navigation system of an airborne platform. The purpose of the paper is threefold. First, the paper presents some of the tools required for computing a mosaic and using the information collected as a side product within a navigation filter. These tools include a special variation of the Kalman filter and a new formulation of the tri-focal tensor from multi-frame vision. Second, the paper summarizes a general method for fusing the motion information obtained during the mosaicking process and also shows how "loop-closure" can be used to preserve navigation errors at their initial levels. Third, the paper discusses a number of illustrative examples to show how mosaic aiding can indeed result in a substantial improvement of the navigation solution.

This paper presents a new method for vision-aided navigation of airborne platforms. The method is based on online mosaicking using images acquired by an onboard gimballed camera, which scans ground regions in the vicinity of the flight trajectory. The coupling of the scanning and mosaicking processes improves image-based motion estimation when operating in challenging scenarios such as narrow field-of-view cameras observing low-texture scenes. These improved motion estimations are fused with an inertial navigation system. The mosaic used for navigation is constructed in two levels. A small mosaic based on recently captured images is computed in real-time, and a larger mosaic including all the images is computed in a background process. The low-level mosaic is used for immediate motion estimation, while the higher-level mosaic is used for global navigation. The correlation terms between the navigation system and the mosaic construction process are not maintained in the proposed approach. The advantage of this architecture is the low computational load required for navigation aiding. However, the accuracy of the proposed method could be compromised compared with bearing-only simultaneous localization and mapping. The new method was examined using statistical simulation runs and experiments based on oogle Earth imagery, showing its superior performance compared with traditional methods for two-view navigation aiding.

Space agencies are now realizing that much of what has previously been achieved using hugely complex and costly single platform projects-large unmanned and manned satellites (including the present International Space Station)-can be replaced by a number of smaller satellites networked together. The key challenge of this approach, namely ensuring the proper formation flying of multiple craft, is the topic of this second volume in Elsevier's Astrodynamics Series, Spacecraft Formation Flying: Dynamics, control and navigation. In this unique text, authors Alfriend et al. provide a coherent discussion of spacecraft relative motion, both in the unperturbed and perturbed settings, explain the main control approaches for regulating relative satellite dynamics, using both impulsive and continuous maneuvers, and present the main constituents required for relative navigation. The early chapters provide a foundation upon which later discussions are built, making this a complete, standalone offering.

A large normal displacement relative to the ecliptic is necessary for mitigating the effect of the zodiacal dust cloud on space telescopes. In this paper, a memetic algorithm is used to optimize flyby paths using multiple gravity-assisted maneuvers near Earth (E) and Venus (V), which are used for reaching Jupiter (J), where the inclination is increased. The global search for optimal trajectories with minimal energy requirements and short transfer times to the highly-inclined destination orbit is performed using a niching genetic algorithm improved by a gradient-based local optimization. The optimization yields three candidate paths: EVEJ, EVEEJ and EVVEJ.

This paper presents an approach for the design of out-of-ecliptic orbits for space-borne telescopes using multiple gravity-assist maneuvers with respect to Venus, Earth and Jupiter. Gravity-assist is a proven technique for energy cost reduction, implemented by a velocity change relatively to the sun due to the gravity of a passed-by massive planet. Trajectories having minimal maneuver energy requirements and maximal time in which the telescope is distant from the ecliptic plane were obtained. Such trajectories results in considerable reduced noise from the interplanetary dust, and therefore allows significantly reduction in payload mass and complex. The problem was modeled using the patched-conic approximation and solved using a hybrid method fusing genetic algorithms and a local search. Some of the feasible trajectories are simulated and discussed.

A kinematically coupled relative rotational and translational motion model, describing the 6 degrees of freedom relative dynamics between two rigid-body spacecraft has been demonstrated. The quantification of the kinematics coupling effect which is key for high-precision modeling of tight SFF, rendezvous, and docking along with kinematics coupling is also important. This effect is important in vision-based relative attitude and position control where arbitrary feature points on a target vehicle can be tracked. The new approximation is aimed at alleviating an apparent contradiction in linearized relative motion theories to obtain linear equations of motion and the spacecraft are assumed to operate in close proximity. The kinematics coupling effect does not depend on environmental perturbations and is an inherent part of the nominal relative motion equations.

The increasing satellite congestion at the geostationary altitude requires positioning a few satellites in the same geostationary slot, a technique known as collocation. In this paper, we develop a geostationary orbit satellite collocation algorithm using relative orbital element corrections, which represent the differences between the impulsive orbital element corrections of any two spacecraft in a geosynchronous slot. The main idea is that formulating the problem of collocation in terms of relative orbital element corrections leaves some of the final values of the orbital elements unconstrained. The freedom rendered by this modeling can be used to find impulsive maneuvers minimizing a given performance index. The minimum distance between satellites that guarantees collision-free motion is incorporated into the design process to find necessary and sufficient conditions for the relative eccentricity and inclination vectors, guaranteeing safe collocation. The proposed collocation algorithm is illustrated in a simulation.

This paper investigates trajectories for spaceborne observatories with long times of flight high above the ecliptic plane. A large normal displacement is necessary for mitigating the effect of noise generated by the interplanetary (zodiacal) dust cloud, thus increasing resolution and reducing the sizes of the telescopes. Flyby paths using multiple-gravity-assist maneuvers near Earth and Venus are used to reach Jupiter, where the inclination is increased and the final orbit is produced. The trajectory design is performed using a memetic algorithm, which is a combination of global and local optimizers. The global search for optimal trajectories with minimal energy requirements and short transfer times to the highly inclined destination orbit is performed using a niching genetic algorithm improved by a gradient-based local optimization. The optimization yielded three candidate paths, each accompanied by, at most, one deep-space maneuver. The resulting energy-efficient orbits are characterized by large displacements normal to the ecliptic plane, thereby providing prolonged observation times in the interplanetary-dust-noise-free celestial sphere.

The ranging accuracy provided by pseudorange-only techniques is usually no better than a few metres when no differential corrections are applied. Carrier-phase algorithms, on the other hand, yield higher-precision estimates - down to a few millimetres - but are prone to ambiguities difficult to resolve. An easier-to-implement method, using single-frequency pseudorange measurements only, is presented. It allows for a decimetre-level relative positioning accuracy. Results, derived from the GPS Relative Positioning Equations, are validated with actual satellite data from the Gravity Recovery and Climate Experiment (GRACE) mission.

Previous studies of geosynchronous orbits indicated that the equinoctial precession (EP) of the Earth affects the long-term behavior of geosynchronous satellites for missions exceeding ten years. However, these studies did not include the lunisolar gravitation and tesseral resonance. In the present study, a model that includes the latter effects is developed. In particular, it is shown that the EP affects motion in the vicinity of the stable and unstable geostationary points. This effect is pronounced in the vicinity of the unstable points, shifting the satellite away from the geosynchronous altitude. Moreover, it is shown that secular inclination growth on time scales of 10-20 years is induced by the EP. This requires additional stationkeeping maneuvers that may increase the overall fuel usage by about 1%. An additional contribution of the present study is an analysis of EP-perturbed orbits with free inclination drift. An optimal initial node location, minimizing the inclination drift, is calculated while taking into account the effect of the EP. It is shown that the classical optimal initial node locations are changed due to the effect of the EP. A maneuvering program in the presence of EP is developed. It is shown that the timing and number of stationkeeping maneuvers is affected by the EP. The models developed herein utilize non-singular orbital elements.

This paper presents a kinematically-coupled spacecraft relative motion model. In contrast to the traditional non-linear point-mass and linear Clohessy-Wiltshire models, which describe the translational motion between the center-of-masses of the spacecraft, in this model equations of motion are derived to model the relative motion between any two arbitrary points on the spacecraft. The model is constructed by including the effect of relative rotation on relative translation. Numerical simulations illustrate that this effect is important for tight formation flying and rendezvous. Thus, although the Clohessy-Wiltshire model is written for relatively close spacecraft distances, it becomes invalid when the spacecraft shapes is taken into account.

This paper develops a method for relative spacecraft pose and motion estimation using stereoscopic vision. Two cameras mounted on a leader spacecraft acquire images of feature points on a follower spacecraft. The rotational and translational dynamics of the leader relative to the follower are estimated using camera measurements only. An estimation algorithm is developed based on a relative spacecraft kinematically-coupled dynamical model fused with the image velocities and the measurements of the projected feature points. The estimation process is performed in two stages: First, an algebraic solution based on stereo-vision tracking of feature points provides an approximation of all the relative states. This procedure results in a set of linear measurements, which are fed into an extended Kalman filter. The newly-developed estimation algorithm reduces the sensitivity to initial uncertainties without requiring linearization of the original measurement equation. The developed methods were evaluated using numerical simulations, showing convergence of the relative pose and motion estimation errors for a relatively small baseline distance between the cameras.

This paper presents a method for improving the performance of an inertial navigation airborne system utilizing information from an on-line mosaicking construction process. The system setup consists of an airborne platform equipped with an inertial navigation system and a gimbaled camera, which scans the area in the flight vicinity. The images acquired by the camera are used to update the mosaic images and also to reduce the developing navigation errors. The main contribution of this paper is an updated measurements model and the performance evaluation of the proposed method based on an extensive simulation study and experiments conduction using real images acquired from Google Earth. Through these experiments we have shown that a significant improvement may be obtained in some of the navigation states under realistic scenarios. As the relative motion measurements, upon which the proposed method is constituted, are a by-product of the on-line mosaicking process, the navigation aiding is achieved without investing much additional effort.

This work presents a method for real-time mosaic-aided aircraft navigation. The method utilizes an on-line mosaic image construction process based on images acquired by a gim- balled camera attached to an airborne platform, which scans ground regions in the vicinity of the flight trajectory. The images captured by the camera are used to update the mosaic image while simultaneously estimating the platform's motion. The work is divided into two parts: The current paper addresses mosaic-based motion estimation using a scanning camera and a concomitant mosaic construction procedure. Part II focuses on fusion of the motion estimation with a standard navigation system. An extensive performance evaluation of the proposed method was carried out, involving real imagery and an implementation of the camera scanning and mosaic image construction processes. The current paper, concerned with the motion estimation procedure, shows a significant improvement of motion estimation in challenging scenarios, such as using a narrow field-of-view camera and tracking low-texture scenes. The approach proposed in this work is an alternative to Simultaneous Localization and Mapping (SLAM) in the following sense: We assume that the mosaic image construction is an independent process, to be utilized for improv- ing the navigation system. Therefore, the proposed architecture alleviates some of the computational load associated with SLAM.

The Gyldén-Meshcherskii Problem (GMP) extends the classical two-body problem of Newton and Kepler by considering time-varying gravitating masses, an important extension of the two-body problem for modeling cometary motion and cosmological phenomena. In this paper, we consider the GMP in a specialized setup, the setup of gauge theory. We show that the variational equations, modeling the effect of the mass time variation on the orbital elements, can be derived by fixing a gauge of a particular form that is different from the Lagrange gauge. Thus, the orbital elements modeling the effect of a time-varying secondary mass are non-osculating. This implies that the trajectory of a celestial body whose mass is continuously changing cannot be approximated by a series of Keplerian orbits. Finally, we provide the first-order averaged equations written in terms of non-osculating elements.

This work develops distributed hierarchical task assignments and communication algorithms for flocks of unmanned aerial vehicles dispersed in an unknown theater while engaging multiple moving targets. The dynamical changes require distributed algorithms, without relying on a central agent, which may constitute a single point-of-failure and is exposed to communication breaks. Our methodology overcomes the typical deficiencies of a centralized solution by organizing the agents in spanning trees. Whenever possible, the spanning trees merge into a single tree that clusters the maximum number of agents. Using relaxation methods and inputs from its parent and children, every agent attempts to optimize its own solution for task assignment while communicating using an ad-hoc protocol. Simulation experiments show clear-cut advantages of using the proposed task assignment and communication algorithms.

We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user flexibility but gives a mathematical guarantee for the algorithm's behavior in the inconsistent case. We present the numerical results of computational experiments that illustrate the computational advantage of the new method.

This study develops a novel distributed algorithm for task assignment (TA), coordination, and communication of multiple unmanned aerial vehicles (UAVs) engaging multiple targets and conceives an ad hoc routing algorithm for synchronization of target lists utilizing a distributed computing topology. Assuming limited communication bandwidth and range, coordination of UAV motion is achieved by implementing a simple behavioral flocking algorithm utilizing a tree topology for distributed flight coordination. Distributed TA is implemented by a relaxation process, wherein each node computes a temporary TA based on the union of the TAs of its neighbors in the tree. The computation of the temporary TAs at each node is based on weighted matching in the UAV-target distances graph. A randomized sampling mechanism is used to propagate TAs among different parts of the tree. Thus, changes in the location of the UAVs and targets do not pass through the root of the tree. Simulation experiments show that the combination of the flocking and the TA algorithms yields the best performance.

In case of a wide-scale disaster, an accurate airdrop of emergency supplies is crucial. In this paper, we present a novel top-down approach for designing and executing airdrop missions using guided parafoils. We develop a guidance algorithm and a cooperative task management method for autonomous handling of faults and exceptional events by the parafoil group. The autonomous operation is based on inter-parafoil ad-hoc communication. A parafoil or a number of parafoils can dynamically react to events that prevent one or more parafoils from successfully completing their mission. Two recovery methods are presented - Swap, which enables parafoils to dynamically exchange their targets, and Replace, which gives precedence to prioritized targets over low-priority targets. The parafoil guidance method, combined with the task management algorithm, significantly increases the probability of successful airdrop. The small overhead of the communication layer and the low complexity of the swap and replace recovery algorithms enable these procedures to run in a distributed environment under real-time limitations.

In this study, we investigate the effect of Earth's precession on the orbital dynamics of geostationary satellites. Our astrodynamical model includes second-order zonal and tesseral harmonics, and lunisolar gravitation. We show that the equinoctial precession induces secular inclination growth and thus bares a non-negligible effect on north-south stationkeeping for long mission lifetimes.

A spacecraft collocation control algorithm using relative orbital element corrections is developed. The freedom rendered by this modeling is used to find optimal impulsive maneuvers minimizing the squared l ²-norm of the velocity corrections vector, which can be used for collocation initialization. The optimization is solved using the method of least squares. Synchronous elements are used for linear approximation of the relative motion in order to detect optimal impulsive velocity corrections for the collocation problem. The minimum distance between satellites that guarantees collision-free motion was taken into account in order to find necessary and sufficient conditions for the relative eccentricity and inclination vectors guaranteeing safe collocation.

This paper develops an impulsive spacecraft formation-flying control algorithm using relative-orbital-element corrections. This formalism introduces an inherent freedom that is used for deriving an optimal formationkeeping law, balancing the fuel consumption among the spacecraft based on the impulsive Gauss variational equations. The main idea is that formulating the problem of formationkeeping in terms of relative-orbital-element corrections leaves the flnal values of the orbital elements unconstrained, thus allowing the spacecraft to create a natural energy-balanced formation. The freedom rendered by this modeling is used to find optimal impulsive maneuvers, minimizing the squared l²-norm of the velocity-correction vector, which can be used for formation initialization and control. The optimization is solved using the least-squares method. The optimal formationkeeping method is designed to accommodate the effects of oblateness and drag. Based on graph theory, it is shown that the spacecraft will naturally form a stable energy-balanced formation and that the optimal formationkeeping strategy is invariant to the spanning tree. The algorithm is illustrated by simulating the motion of a formation of spacecraft possessing different ballistic coefficients subject to oblateness and drag.

The increasing satellite congestion at the geostationary altitude requires positioning a few satellites in the same geostationary slot, a technique known as collocation. In this paper, we develop a generic geostationary orbit (GEO) satellite collocation al gorithm using relative orbital element corrections, which represent the differences between the impulsive orbital element corrections of any two spacecraft in a geosyn chronous slot. The main idea is that formulating the problem of collocation in terms of relative orbital element corrections leaves some of the final values of the orbital elements unconstrained. The freedom rendered by this modeling can be used to find optimal impulsive maneuvers minimizing the squared Z2-norm of the velocity cor rections vector. If some of the final orbital elements are to be specified, such as the geosynchronous altitude, a virtual leader can be added to the collocated group, steering the entire GEO formation into the GEO deadband. Synchronous elements are used for modeling the relative motion. The minimum distance between satellites that guarantees collision-free motion is taken into account in order to find necessary and sufficient conditions for the relative eccentricity and inclination vectors guar anteeing safe collocation. The proposed collocation algorithm was simulated while incorporating J2 perturbations.

In this study, we investigate the effect of Earth's precession on the orbital dynamics of geostationary satellites. Our astrodynamical model includes second-order zonal and tesseral harmonics, and lunisolar gravitation. We show that the equinoctial precession induces secular inclination growth and thus bares a non-negligible effect on north-south stationkeeping for long mission lifetimes.

Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2004, 2005, 2006a, b). Here we continue this line of research by combining that analytical machinery with numerical tools. Our model includes three factors: the J ₂ of the planet, its nonuniform equinoctial precession described by the Colombo formalism, and the gravitational pull of the Sun. This semianalytical and seminumerical theory, based on the Lagrange planetary equations for the Keplerian elements, is then applied to Deimos on very long time scales (up to 1 billion years). In parallel with the said semianalytical theory for the Keplerian elements defined in the co-precessing equatorial frame, we have also carried out a completely independent, purely numerical, integration in a quasi-inertial Cartesian frame. The results agree to within fractions of a percent, thus demonstrating the applicability of our semianalytical model over long timescales. Another goal of this work was to make an independent check of whether the equinoctial-precession variations predicted for a rigid Mars by the Colombo model could have been sufficient to repel its moons away from the equator. An answer to this question, in combination with our knowledge of the current position of Phobos and Deimos, will help us to understand whether the Martian obliquity could have undergone the large changes ensuing from the said model (Ward 1973; Touma and Wisdom 1993, 1994; Laskar and Robutel 1993), or whether the changes ought to have been less intensive (Bills 2006; Paige et al. 2007). It has turned out that, for low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations. This is an extension, to non-uniform equinoctial precession given by the Colombo model, of an old result obtained by Goldreich (1965) for the case of uniform precession and a low initial inclination. However, near-polar initial inclinations may exhibit considerable variations for up to ±10 deg in magnitude. This result is accentuated when the obliquity is large. Nevertheless, the analysis confirms that an oblate planet can, indeed, afford large variations of the equinoctial precession over hundreds of millions of years, without repelling its near-equatorial satellites away from the equator of date: the satellite inclination oscillates but does not show a secular increase. Nor does it show secular decrease, a fact that is relevant to the discussion of the possibility of high-inclination capture of Phobos and Deimos.

Ad hoc networks are characterized by fast, dynamic changes in the topology of the network. Introduction of new routing algorithms requires a deep and reliable evaluation process. In this paper, we describe an interactive flexible ad hoc simulator (IFAS) that presents a modern and novel approach to the family of ad hoc simulators. This simulator supports unique viewing, debugging, tuning and interactive capabilities that shorten significantly the design and debug process of new algorithms. The development of the IFAS is ongoing; however, we consider this new simulator as an innovative tool that will supplement traditional simulators.

This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics, and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, that stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret-Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into action-angle ones, using the method of Sadov.

This paper continues the review of the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics, commenced by [1], and presents some new result. We discuss the applications of the SA formalism to control theory. Considerable attention is devoted to the geometry of the Andoyer variables and to the modeling of control torques. We develop a new approach to Stabilization of rigid-body dynamics, an approach wherein the state-space model is formulated through sets of canonical elements that partially or completely reduce the unperturbed Euler-Poinsot problem. The controllability of the system model is examined using the notion of accessibility, and is shown to be accessible. Based on the accessibility proof, a Hamiltonian controller is derived by using the Hamiltonian as a natural Lyapunov function for the closed-loop dynamics. It is shown that the Hamiltonian controller is both passive and inverse optimal with respect to a meaningful performance-index. Finally, we point out the possibility to apply methods of structure-preserving control using the canonical Andoyer variables, and we illustrate this approach on rigid bodies containing internal rotors.

This paper investigates the problem of continuous-thrust orbital transfer using orbital elements feedback from a nonlinear control standpoint, utilizing concepts of controllability, feedback stabilizability and their interaction. Gauss's variational equations (GVEs) are used to model the state-space dynamics of motion under a central gravitational field. First, the notion of accessibility is reviewed. It is then shown that the GVEs are globally accessible. Based on the accessibility result, a nonlinear feedback controller is derived which asymptotically steers a spacecraft form an initial elliptic orbit to any given elliptic orbit. The performance of the new controller is illustrated by simulating an orbital transfer between two geosynchronous Earth orbits. It is shown that the low-thrust controller requires less fuel than an impulsive maneuver for the same transfer time. Closed-form, analytic expressions for the new orbital transfer controller are given. Finally, it is proven, based on a topological nonlinear stabilizability test, that there does not exist a continuous closed-loop controller that can transfer a spacecraft onto a parabolic escape trajectory.

This paper develops generalized analytic solutions of relative spacecraft dynamics in the presence of arbitrary perturbations and presents new perturbation-invariant relative orbits. The extension of existing results is achieved by using non-osculating orbital elements, i.e. removing the constraint that the instantaneous velocity vector be tangential to an instantaneous Keplerian ellipse. Generalized planetary equations are developed and averaged to yield a non-osculating description of the long-term effects of first-order small perturbations. The generalized planetary equations are linearized, and expressions for the relative dynamics are presented in terms of non-osculating classical orbital element differences. The developed methodology is utilized to find previously undetected J₂-invariant relative orbits.

This paper develops a nonsingular model for the effect of equinoctial precession on natural and artificial satellite orbits based on the Euler parameters instead of the Euler angles. The use of Euler parameters removes the zero-inclination singularity in the variational equations, thus facilitating numerical integration of low-inclination orbits. Euler-parameter-based planetary and variational equations are developed. These equations are subsequently used for modeling the long-periodic effect of a uniformly precessing reference frame on a given orbit. The Euler parameter-based model is used for simulating the orbit of Deimos, taking into account the Martian oblateness and precession of the spin axis. It is shown that the new model yields an order-of-magnitude faster simulation than the classical element-based model.

This paper studies the dynamics and stability of a rigid spacecraft subjected to gravity gradient torques exerted by the Sun and the Earth in the circular restricted three-body problem. We focus on the dynamics in a close vicinity to the Lagrangian collinear equilibrium points, and show that the linear stability domain predicted by the Beletskii-DeBra-Delp method in the two-body problem is modified due to the presence of an additional gravitating primary. The nonlinear differential equations are derived using a Hamiltonian formalism and are subsequently investigated using Poincaré maps. The effect of the gravity gradient torque is accentuated using difference Poincaré maps. The Melnikov integral method is utilized for studying the chaotic behavior of the gravity-gradient-perturbed system.

This paper studies the dynamics of a rigid spacecraft subjected to gravity gradient torques exerted by the Sun and the Earth in the circular restricted three-body problem. We focus on the dynamics in a close vicinity to the Lagrangian collinear equilibrium points, and show that the linear stability domains predicted by the Delp theory in the two-body problem are modified due to the presence of an additional gravitating primary. The nonlinear differential equations are investigated using Poincaré maps. The Melnikov integral method is utilized for studying the chaotic behavior of the gravity-gradient disturbed system. The main conclusion is that the perturbation due to the combined gravity gradient torques is a dominant factor in the rotational dynamics of a spacecraft flying on a libration-point orbit. We further conclude that the rotational motion is chaotic, although stable regions may be found, and hence passive stabilization on libration points is possible.

In this paper, we develop a new nonlinear relative spacecraft motion control law based on line-of-sight measurements only. Each spacecraft tracks its neighboring spacecraft to produce a line-of-sight vector measurement, and the last spacecraft tracks again the first spacecraft to create a cyclic formation. We show that cyclic leaderless formations controlled by line-of-sight measurements only are stable in the sense of energy matching. The initial semimajor axis errors relative to a reference semimajor axis all tend to the same value, which is the centroid of the initial semimajor axis errors. The stable behavior is achieved by using both full line-of-sight feedback (range and bearing) and partial line-of-sight feedback (bearing only). We also investigate the case of leader formations, where a designated leader does not track any other spacecraft. In this case, the semimajor axes of all follower spacecraft will converge to the initial semimajor axis of the leader. We also discuss an intermediate case,' in which only part of the spacecraft forms a cycle. In this case, the cyclic formation members will determine the semimajor axes of all the other, noncyclic, spacecraft. We use elementary graph theory to describe the resulting formation topologies and conclude the development with an illustrating example.

The long-periodic effects of the equinoctial precession on geosynchronous Earth orbit satellites are investigated. The equations of motion in a reference frame that coprecesses with the Earth are developed, and the resulting variational equations are derived using mean classical orbital elements. The Earth gravitational model includes the J₂ and J₃ zonal harmonics, which induce the equinoctial precession due to the lunisolar gravitational torque. It is shown that the ever-growing lifetime and mass of geosynchronous Earth orbit satellites render the equinoctial precession a significant factor, which should be taken into account during mission design, as it affects north-south stationkeeping maneuvers. The equilibria of the variational equations including the zonal harmonics and the equinoctial precession are investigated and a class of stable frozen orbits which are equinoctial precession invariant is derived.

In this article, a hierarchical algorithm was developed for autonomous formation flying, communication, and task assignment for cooperating unmanned aerial vehicles (UAVs) and Dudek's taxonomy was used to evaluate the performance of the autonomous UAV group cooperating in mission execution against a group of enemy agents in an unknown environment. The authors showed that increasing the number of UAVs in the group proves to be beneficial as it allows the group to react to more enemy events, even if the overall search and detect capabilities remain invariant with the number of UAVs. It is also showed that, in general, using communication improves the cooperation among the flock members; however, using infinite communication range or infinite communication bandwidth results in considerable computational complexity. In conclusion, it is sufficient to use finite-bandwidth communication, keeping the computational complexity constant with the number of UAVs in the group, thus allowing the group to be scalable to large numbers of UAVs. The authors use a behavioural flocking algorithm to control the movement of the UAVs when searching, and show that flocking assists the UAVs in dispersing in the environment and improves the overall detection probability of the flock. Finally, it is showed that using flocking improves the group performance only if the group is capable of mission task collaboration.

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This concept has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems. We focus on the concept of rescriptive gauge symmetry, which is, in essence, rescaling of an independent variable. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a strong connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse fields, including quantum mechanics, chemistry, rigid-body dynamics, and information theory.

We develop a spacecraft formation flying control algorithm using relative orbital element corrections, which represent the differences between the orbital element corrections of any two spacecraft in a formation. This formalism introduces an inherent freedom that is used for deriving a formationkeeping algorithm optimally balancing the fuel consumption among the formation members based on the impulsive Gauss variational equations. The main idea is that formulating the problem of formationkeeping in terms of relative orbital element corrections leaves the final values of the orbital elements unconstrained, thus allowing the spacecraft to create a natural energy-balanced formation. The freedom rendered by this modeling is used to find optimal impulsive maneuvers minimizing the squared l²-norm of the velocity corrections vector, which can be used for formation initialization and control. The optimization is solved using the method of least squares. The optimal formationkeeping method is designed to accommodate the effects of oblateness and drag. Based on graph theory, it is shown that the spacecraft will naturally form a stable energy-balanced formation, and that the optimal formationkeeping strategy is invariant to the spanning tree. The algorithm is illustrated by simulating the motion of a formation of spacecraft possessing different ballistic coefficients subject to oblateness and drag.

The ranging accuracy provided by pseudorange-only techniques is usually no better than a few meters when no differential corrections are applied. Carrier-phase algorithms, on the other hand, yield higher-precision estimates - down to a few millimeters - but are prone to ambiguities difficult to resolve. An unambiguous method, using single-frequency pseudorange measurements only, is presented. It allows for a decimeter-level relative positioning accuracy. Results, derived from the GPS Relative Positioning Equations, are validated with actual satellite data from the Gravity Recovery and Climate Experiment (GRACE) mission.

A new fully numerical method is presented which employs multiple Poincaré sections to find quasi-periodic orbits. The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasi-periodic orbits and the minimal memory required to store these orbits. This method reduces the calculation of the search for the two-dimensional invariant torus to a search for the closed orbits, which are the intersection of the invariant torus with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant torus is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme makes use of the invariancy of the circles of the maps on these Poincaré sections in order to find the Fourier coefficient that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasi-periodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-Halo and Lissajous families of the Sun-Earth Restricted Three-Body Problem (RTBP) around the L1 and L2 libration points are obtained via this method. Results are compared with the existing literature.

Gauge theories in physics constitute a fundamental tool for modeling interactions among electromagnetic, weak and strong forces. They have been used in a myriad of fields, ranging from sub-atomic physics to cosmology. The basic mathematical tool generating the gauge theories is that of symmetry, i.e. a redundancy in the description of the system. Although symmetries have long been recognized as a fundamental tool for solving ordinary differential equations, they have not been formally categorized as gauge theories. In this paper, we show how simple systems described by ordinary differential equations are prone to exhibit gauge symmetry, and discuss a few practical applications of this approach. In particular, we utilize the notion of gauge symmetry to question some common engineering misconceptions of chaotic and stochastic phenomena, and show that seemingly "disordered" (deterministic) or "random" (stochastic) behaviors can be "ordered". This brings into play the notion of observation; we show that temporal observations may be misleading when used for chaos detection. From a practical standpoint, we use gauge symmetry to considerably mitigate the numerical truncation error of numerical integrations.

A new computationally efficient semianalytical method is developed for the calculation of the transition and monodromy matrices for the linearized Elliptic restricted three-body problem (ER3BP). The state transition matrix is expanded into orthogonal Chebyshev polynomials of the first and second kind, shifted to fit the time interval in use. The Chebyshev approximation transforms the nonautonomous differential equations required to calculate the state-transition matrix into a set of algebraic equations. The eccentricity of the motion of the Earth around the Sun is incorporated into the equations of motion to renders the more general model, known as the elliptic restricted three body problem. The most popular coordinate system used to model the ER3BP dynamics has its origin set at the barycenter of the large primary M₁ and the small primary M₂. The proposed method is essential for trajectory design of future liberation-point missions.

Euler's integration methods are frequently used for numerical integration as well as real-time implementation of linear systems. However, when the integrated system is either undamped or stiff, Euler's explicit integration becomes unstable regardless of the forcing input. In this work, it is shown that this instability can be avoided by a judicious selection of the state variables. Instead of the generalized coordinates and velocities, it is proposed to define state variables based on the method of variation-of-parameters (VOP). It is proven that the VOP-based state variables yield a bounded numerical error for undamped as well as stiff systems provided that the forcing inputs are bounded. The analysis is performed for both deterministic and stochastic inputs. In the stochastic case, the numerically-calculated covariance matrix entries diverge when using the generalized coordinates and velocities but remain bounded when implementing the VOP-based approach. The newly developed formalism is illustrated by a number of examples of practical interest, showing that the VOP-based approach is also more computationally efficient than the standard approach.

Euler's integration methods are frequently used for numerical integration as well as for real-time implementation of linear systems. However, when the integrated system is either undamped or stiff, Euler's explicit integration becomes unstable, regardless of the forcing input In this work, it is shown that this instability can be avoided by a judicious selection of the state variables. Instead of the generalized coordinates and velocities, it is proposed to define state variables based on the method of variation of parameters. It is proven that the variation-of-parameters-based state variables yield a bounded numerical error for undamped and stiff systems, provided that the forcing inputs are bounded. The analysis is performed for both deterministic and stochastic inputs. In the stochastic case, the numerically calculated covariance matrix entries diverge when using the generalized coordinates and velocities, but remain bounded when implementing the variation-of-parameters-based approach. The newly developed formalism is illustrated by a number of examples of practical interest, showing that the variation-of-parameters-based approach is also more computationally efficient than the standard approach.

The purpose of this work is to show that using the hyperbolic network around L₁, the diversity of transfer options to the Distant Retrograde Orbits (DROs) in the Sun-Earth Planar Circular Restricted Three Body Problem can be increased. In contrast to previous works, the current paper will concentrate on the range of smallest available DROs. The attractiveness of the transfer options to these orbits will be revealed by including the stable manifolds of the Horizontal Lyapunov Orbits (HLOs) as a part of the transfer path. It will become clear that for a large range of DROs, lower ΔV budgets and/or far lower travel times are achievable.

The objective of this work is to improve the navigation system performance by using the information obtained from an online mosaic construction process. The system setup consists of an airborne platform with a pan-tilt onboard camera, which scans areas in the vicinity of the platform's flight path. The mosaic image is constructed from the captured images during the scan process. A measurement development is presented that utilizes the mosaic construction process to achieve reduced navigational errors. The measurement uses camera rotation and translation motion information extracted from the registration phase of each new scanned image with the mosaic image. Simulation performance analysis of the proposed measurement is presented.

The purpose of this work is to develop transfer trajectories from Earth to prespecified distant retrograde orbits in the sun-Earth planar circular restricted three-body problem by using orbits about the collinear equilibrium point L₁. More specifically, we examine whether it is possible to use the hyperbolic network associated with the horizontal Lyapunov orbits around L₁ to find transfer trajectories to a wide range of distant retrograde orbits that are more energy-efficient and/or time-efficient compared with standard impulsive maneuvers. We point out how to apply manifold theory in the transfer-trajectory design process and show that for a certain class of distant retrograde orbits, the dynamic systems approach reveals the availability of transfer trajectories having reduced energy requirements or considerably reduced transfer times.

We introduce a method for mitigating the numerical integration errors of linear, second-order initial value problems. We propose a methodology for constructing an optimal state-space representation that gives minimum numerical truncation error, and in this sense, is the optimal state-space representation for modelling given phase-space dynamics. To that end, we utilize a simple transformation of the state-space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time-step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one degree-of-freedom ODE. For the n degrees-of-freedom case we describe MATLAB® code capable of finding the optimal gauge functions and integrating the given system using the gauge-optimized integration algorithm. In all of our illustrating examples, the gauge-optimized integration outperforms the integration using standard state variables by a few orders of magnitude.

In this paper, we develop a new nonlinear relative spacecraft motion control law based on line-of-sight (LOS) measurements only. Each spacecraft tracks its neighboring spacecraft to produce a LOS vector measurement, and the last spacecraft tracks again the first spacecraft to create a cyclic formation. We show that cyclic leaderless formations controlled by LOS measurements only are stable in the sense of energy matching. The initial semi-major axis (SMA) errors relative to a reference SMA all tend to the same value, which is the centroid of the initial SMA errors. The stable behavior is achieved by using both full LOS feedback (range and bearing) and partial LOS feedback (bearing only). We also investigate the case of leader formations, where a designated leader does not track any other spacecraft. In this case, the SMAs of all follower spacecraft will converge to the initial SMA of the leader. We also discuss an intermediate case, where only part of the spacecraft form a cycle. In this case, the cyclic formation members will determine the SMAs of all the other, non-cyclic, spacecraft. We use elementary graph theory to describe the resulting formation topologies and conclude the development with an illustrating example.

The Advanced Space Systems Technology Laboratory (ASSTL) is a new facility to be designed and built at the Technion, aimed at high-end research in the field of space systems technology. Primarily, the laboratory will conduct research in the innovative area of distributed spacecraft systems, specifically in dynamics and control of multiple spacecraft formation flying (SFF). This document presents the research areas for which the laboratory will serve as an experimental testbed, including nonlinear cooperative control, relative position and attitude sensing and space borne sparse-aperture imaging. A considerable effort is devoted to describing the laboratory components, including an air bearing table and model spacecraft to be developed and designed according to strict performance specifications.

The successful launch of SOHO, ISEE-3 and Genesis demonstrated that the circular restricted three-body problem is a reliable framework for trajectory planning. In this work, we propose a scheme for deriving periodic orbits about the collinear libration points based on a more general formalism, the elliptic restricted three-body problem (ER3BP), which accounts for the eccentricity of the primaries. Calculation of periodic halo-like orbits was accomplished by formulating the restricted three-body problem as a control problem. Although the position of the libration points is moving in a rotating frame, it is constant in pulsating coordinates. By performing linearization about the librating points in pulsating coordinates, we obtain an unstable linear time-varying system with periodic coefficients. We introduce a continuous acceleration control term into the state-space system and use a generalized version of the pole-assignment technique to find a linear periodic reference trajectory. To that end, we develop a semi-analytical method for calculating the monodromy matrix for the linearized ER3BP system based on approximating the monodromy matrix entries using Chebyshev polynomials. The resulting periodic orbits are used as reference trajectories for the nonlinear ER3BP by applying disturbance-accommodating control.

We present an application of an evolutionary programming method, niching genetic algorithms, to the search for orbits in the spatial elliptic restricted three-body problem (ER3BP) that is suitable for deep-space science missions. The niching method used is deterministic crowding, which renders a global optimization while permitting for several optimal and suboptimal solutions to coexist. This novel approach yields diverse probing of the state space of the ER3BP. From the practical standpoint, the orbits found remain within a bounded distance from Earth, thus allowing high data-rate communication while ensuring safe operational environment, far from thermal perturbations and visual occultation as well as Earth's magnetic and radiation fields.

This works develops a novel hierarchical algorithm for task assignment (TA), coordination and communication of multiple UAVs engaging multiple targets and conceives an ad-hoc routing algorithm for synchronization of target lists utilizing a distributed computing topology. Assuming limited communication bandwidth and range, coordination of UAV motion is achieved by implementing a simple behavioral flocking algorithm utilizing a tree topology for target list routing. The TA algorithm is based on a graph-theoretic approach, in which a node locates all the detectable targets, identifies them and computes its distance to each target. The node then produces an attack plan that minimizes the sum of distances of the UAVs in the subtree of a given node to the targets. Simulation experiments show that the combination of flocking and TA algorithms gives the best performance. Clear-cut advantages of the TA algorithm are shown to exist in cases where the hit probability tends to one. An improvement of efficiency, representing the ratio between killed targets and the number of missile launches, is obtained for larger numbers of UAVs only if the engagement times are long enough, utilizing the improved coverage achieved by more UAVs.

This works develops a novel hierarchical algorithm for task assignment (TA), coordination and communication of multiple UAVs engaging multiple targets and conceives an ad-hoc routing algorithm for synchronization of target lists utilizing a distributed computing topology. Assuming limited communication bandwidth and range, coordination of UAV motion is achieved by implementing a simple behavioral flocking algorithm utilizing a tree topology for target list routing. The TA algorithm is based on a graph-theoretic approach, in which a node locates all the detectable targets, identifies them and computes its distance to each target. The node then produces an attack plan that minimizes the sum of distances of the UAVs in the subtree of a given node to the targets. Simulation experiments show that the combination of flocking and TA algorithms gives the best performance. Clear-cut advantages of the TA algorithm are shown to exist in cases where the hit probability tends to one. An improvement of efficiency, representing the ratio between killed targets and the number of missile launches, is obtained for larger numbers of UAVs only if the engagement times are long enough, utilizing the improved coverage achieved by more UAVs.

We introduce a method for mitigating the numerical integration errors of initial value problems. We propose a methodology for constructing an optimal state-space representation that gives minimum numerical truncation error, and in this sense, is the optimal state-space representation for modeling given phase-space dynamics. To that end, we utilize a simple transformation of the state-space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time-step. We illustrate the method using a few examples taken from the space systems and aeroelasticity fields. In all of our illustrating examples, the gauge-optimized integration outperforms the conventional integration.

We introduce a method for eliminating the truncation error produced when numerically integrating an initial value problem using Runge-Kutta-based algorithms. We propose a methodology for constructing an optimal state-space representation that gives zero local numerical truncation error, and in this sense, is the optimal state-space representation for modeling given phase-space dynamics. To that end, we utilize a simple transformation of the state-space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in physics. We then utilize the gauge function to eliminate the numerical truncation error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time-step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one degree-of-freedom ODE. For the n degrees-of-freedom case we describe a MATLAB® code capable of finding the optimal gauge functions and integrating the given system using the gauge-optimized integration algorithm. In all of our illustrating examples, the gauge-optimized integration outperforms the integration using standard state variables by a few orders of magnitude.

The problem of low-thrust transfer between two low-Earth sun-synchronous circular orbits is discussed. Since the Gauss variational equations (GVEs) exhibit singularities when using the classical orbital elements for modeling circular orbits, equinoctial elements are used instead to formulate the combined effects of low-thrust control inputs and the J ₂ perturbation. The problem of LEO to LEO orbit transfer for a mini-satellite using an ion thruster actuator and a Jurdjevic-Quinn closed-loop orbital transfer control law is developed. Altitude and inclination corrections were carried out while considering J ₂ perturbations, assuming that the engine is active only in a third of the orbit due to power limitations. Simulation results show that the total ΔV needed for the transfer is less than the ΔV required in an Hohman transfer with an inclination correction. The main conclusion is that ion electric propulsion utilized as a closed-loop orbital elements-based feedback control facilitates mini-satellite orbital transfers in realistic scenarios and reduces fuel usage.

This paper establishes a methodology for obtaining the general solution to the spacecraft relative motion problem by utilizing the Cartesian configuration space in conjunction with classical orbital elements. The geometry of the relative motion configuration space is analyzed, and the relative motion invariant manifold is determined. Most importantly, the geometric structure of the relative motion problem is used to derive useful metrics for quantification of the minimum, maximum, and mean distance between spacecraft for commensurable and noncommensurable mean motions. A number of analytic solutions as well as useful examples are provided, illustrating the calculated bounds. A few particular cases that yield simple solutions are given.

The need to rapidly deploy military forces in unknown areas without the ability to use existing ground-based communication infrastructure requires the use of ad-hoc communication networks. Transmission limitations motivates the use of a hierarchal routing mechanism that enables each soldier to communicate with distant soldiers using intermediate powerful nodes that hover within the transmission radius. Our research combines various types of transmitters, including short-range personal transmitters, vehicle-mounted transmitters, helicopters and a geostationary Earth-orbit (GEO) satellite. Each of these entities possesses a different communication range, velocity, and altitude. We consider the various tradeoffs rising from such a heterogeneous theater, and compare two ad-hoc protocols - AODV, which is based on dynamic updates of local cached routing tables, and MRA, which is based on dynamic updates of virtual coordinates. Our results show that for the MRA algorithm one "flat" network is sufficient, and there is no need to use clustering methods. We further show that the GEO satellite constitutes a ubiquitous mediator that contributes to the connectivity and stability of the network.

In this paper, we develop a new nonlinear relative spacecraft motion control law based on line-of-sight (LOS) measurements only. Each spacecraft tracks its neighboring spacecraft to produce a LOS vector measurement, and the last spacecraft tracks again the first spacecraft to create a cyclic formation. We show that cyclic leaderless formations controlled by LOS measurements only are stable in the sense of energy matching. The initial semi-major axis (SMA) errors relative to a reference SMA all tend to the same value, which is the centroid of the initial SMA errors. The stable behavior is achieved by using both full LOS feedback (range and bearing) and partial LOS feedback (bearing only). We also investigate the case of leader formations, where a designated leader does not track any other spacecraft. In this case, the SMAs of all follower spacecraft will converge to the initial SMA of the leader. We also discuss an intermediate case, where only part of the spacecraft form a cycle. In this case, the cyclic formation members will determine the SMAs of all the other, non-cyclic, spacecraft. We use elementary graph theory to describe the resulting formation topologies and conclude the development with an illustrating example.

We develop new methods for generating periodic orbits about the collinear libration points and for stabilizing motion on libration orbits using the general formalism of the elliptic restricted three-body problem (ER3BP). Calculation of periodic orbits was accomplished by formulating the ER3BP as a control problem. Linearization about the libration points in pulsating coordinates yields an unstable linear parameter-varying (LPV) system with periodic coefficients. We introduce a continuous acceleration control term into the state-space dynamics and use an LPV-generalized version of the poleassignment technique to find linear periodic reference trajectories. The nonlinear terms of the equations of motion are then treated as periodic disturbances. A disturbance accommodating control is used to track the libration-point reference trajectory under the nonlinear periodic disturbances. Simulation experiments show that small amounts of propellant are required.

We develop new methods for generating periodic orbits about the collinear libration points and for stabilizing motion on libration orbits using the general formalism of the elliptic restricted three-body problem (ER3BP). Calculation of periodic orbits is accomplished by formulating the ER3BP as a control problem. This approach yields halo-like orbits that do not exist without applying active control, having arbitrarily small amplitudes. Linearization about the libration points in pulsating coordinates yields an unstable linear parameter-varying (LPV) system with periodic coefficients. We introduce a continuous acceleration control term into the state-space dynamics and use an LPV-generalized version of the pole-assignment technique to find linear periodic reference trajectories. The nonlinear terms of the equations of motion are then treated as periodic disturbances. A disturbance-accommodating control is used to track the libration-point reference orbit in the presence of nonlinear periodic disturbances. Simulation experiments show that stationkeeping is robust to propulsive dispersions.

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy-Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth's oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J ₂-invariant orbits are obtained.

This study presents an adaptation of a novel estimation methodology to the general nonlinear three-dimensional problem of tracking a maneuvering target. The two-step optimal estimator (TSE) suggests an attractive alternative to the standard extended Kalman filter (EKF). A superior performance is accomplished by dividing the estimation problem into two steps: a linear first step and a nonlinear second step. The target tracking performance of the TSE is shown to be better than an EKF implemented in either inertial or modified spherical coordinates. In the passive case, where bearing/elevation angles only are measured, the TSE yields excellent range and target acceleration estimates. In the active case, where range measurement is available as well, a homing missile employing closed-loop optimal guidance based on the TSE state estimates obtains smaller miss distances than with either versions of the EKF.

This paper derives a new non-linear guidance law aimed at interception of highly manoeuvring targets. The guidance law is developed based on the theory of control Lyapunov functions (CLFs), a methodology for universal stabilization of non-linear systems which is also inverse optimal with respect to some performance measure. The three-dimensional guidance dynamics are formulated in a fixed-line-of-sight coordinate system, yielding matching between the target and missile accelerations. Closed-form expressions for the CLF guidance commands are given. Simulation shows that the new guidance scheme significantly out-performs augmented proportional navigation in short-range engagements.

This paper develops a new paradigm for stabilization of rigid body dynamics. The state-space model is formulated using canonical elements, known as the Serret-Andoyer (SA) variables, thus far scarcely used for engineering applications. The main feature of the SA formalism is the reduction of the dynamics via the underlying symmetry stemming from conservation of angular momentum and rotational kinetic energy. We use the Hamiltonian as a natural Lyapunov function for the closed-loop dynamics. It is shown that the Hamiltonian controller is both passive and inverse optimal with respect to a meaningful performance index.

This paper develops a new paradigm for stabilization of rigid-body dynamics. The state-space model is formulated using canonical elements, known as the Serret-Andoyer (SA) variables, thus far scarcely used for engineering applications. The main feature of the SA formalism is the reduction of the dynamics via the underlying symmetry stemming from conservation of angular momentum and rotational kinetic energy. The controllability of the system model is examined using the notion of accessibility, and is shown to be accessible from all points. Based on the accessibility proof, two nonlinear asymptotic feedback stabilizers are developed: a damping feedback is designed based on the Jurdjevic-Quinn method, and a Hamiltonian controller is derived by using the Hamiltonian as a natural Lyapunov function for the closed-loop dynamics. It is shown that the Hamiltonian control is both passive and inverse optimal with respect to a meaningful performance index. The performance of the new controllers is examined and compared using simulations of realistic scenarios from the satellite attitude dynamics field.

This paper establishes a methodology for obtaining the general solution to the spacecraft relative motion problem by utilizing the Cartesian configuration space in conjunction with classical orbital elements. The geometry of the relative motion configuration space is analyzed, and the relative motion manifold is determined. Most importantly, the geometric structure of the relative motion problem is used to derive useful metrics for quantification of the minimum, maximum, and mean distance between spacecraft for commensurable and non-commensurable mean motions. A number of analytic solutions as well as useful examples are provided, illustrating the calculated bounds. A few particular cases are given which yield simple solutions.

The use of Euler parameters as a benign set of orbital elements, that replace the classical elements is discussed. The singularity of Euler angles requires a regularization procedure, which results in a modified set of orbital elements. Euler parameters have a clear physical and geometrical interpretation, that gives the variational equations which are less complex than the equinoctial equations. Euler parameters are also suitable for astrodynamic and astronomical applications and are a reliable alternative to the classical elements-based nonsingular elements.

We use Dudek's taxonomy in order to investigate the performance of a group of autonomous UAVs cooperating in mission execution against a group of enemy agents acting in an unknown environment. We show that increasing the number of UAVs in the group proves to be beneficial as it allows the group to react to more enemy events. We also show that using communication helps creating better cooperation between the flock members; however, using infinite communication range or infinite communication bandwidth results in considerable computational complexity. We conclude that it may be sufficient to use finite-bandwidth communication, keeping the computational complexity constant with the number of UAVs in the group, thus allowing the group to be scalable to large numbers of UAVs. We use flocking behavior to control the movement of the UAVs when searching, as the flocking helps the UAVs to disperse in the environment and improve the detection probability of the flock. We show that using flocking improves the group performance only if the group is capable of mission task collaboration.

A precise analytic model for the relative motion of a group of satellites in slightly elliptic orbits is introduced. With this aim, we describe the relative motion of an object relative to a circular or slightly elliptic reference orbit in the rotating Hill frame via a low-order Hamiltonian, and solve the Hamilton-Jacobi equation. This results in a first-order solution to the relative motion identical to the Clohessy-Wiltshire approach; here, however, rather than using initial conditions as our constants of the motion, we utilize the canonical momenta and coordinates. This allows us to treat perturbations in an identical manner, as in the classical Delaunay formulation of the two-body problem. A precise analytical model for the base orbit is chosen with the included effect of zonal harmonics (J₂, J₃, J₄). A Hamiltonian describing the real relative motion is formed and by differing this from the nominal Hamiltonian, the perturbing Hamiltonian is obtained. Using the Hamilton equations, the variational equations for the new constants are found. In a manner analogous to the center manifold reduction procedure, the non-periodic part of the motion is canceled through a magnitude analysis leading to simple boundedness conditions that cancel the drift terms due to the higher order perturbations. Using this condition, the variational equations are integrated to give periodic solutions that closely approximate the results from numerical integration (1mm/per orbit for higher order and eccentricity perturbations and 30cm/per orbit for zonal perturbations). This procedure provides a compact and insightful analytic description of the resulting relative motion.

One of the physical settings emerging in the galaxy and stellar dynamics is motion of a single star and a stellar cluster about a galaxy center. The potential availability of analytical treatment of this problem stems from the smallness of mass of the star and cluster relative to the galactic mass, giving rise to Hill's restricted three-body problem in the galaxy-cluster-star context. Based on this observation, this paper presents a Hamiltonian approach to modelling stellar motion by the derivation of canonical coordinates for the dynamics of a star relative to a star cluster. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic orbital elements. The effect of an arbitrary cluster potential is incorporated into the analysis by a variation of parameters procedure. A numerical optimization technique is developed based on the new orbital elements, and quasiperiodic stellar orbits are found.

This paper establishes a methodology for obtaining the general solution to the spacecraft relative motion problem by utilizing Cartesian configuration space in conjunction with classical orbital elements. The geometry of the relative motion configuration space is analyzed, and the relative motion invariant manifold is determined. Most importantly, the geometric structure of the relative motion problem is used to derive useful metrics for quantification of the minimum, maximum, and mean distance between spacecraft for commensurable and non-commensurable mean motions. A number of analytic solutions, as well as useful examples, are provided, illustrating the calculated bounds. A few particular cases are given that yield simple solutions.

Based on the concept of orbital commensurability, this paper presents necessary and suffcient conditions for bounded relative motion between any two spacecraft flying on elliptic Keplerian orbits. The proposed approach does not involve any simplifying assumptions regarding the relative dynamics but rather treats the general, nonlinear, eccentric relative motion problem. The methodology presented in the paper alleviates the diffculty in computing corrections to the linear equations of motion to account for nonlinearities and eccentricities. Instead of dealing with the local relative motion problem, the global relative motion problem is addressed by transforming the orbital resonance requirement into an energy matching condition. The newly developed setup is then utilized to derive an optimal single-impulse formationkeeping maneuver based on relative state variables. The orbital elements interpretation of the optimal formationkeeping maneuver is also discussed.

Based on the concept of orbital commensurability, this paper presents necessary and sufficient conditions for bounded relative motion between any two spacecraft flying on elliptic Keplerian orbits. The proposed approach does not involve any simplifying assumptions regarding the relative dynamics but rather treats the general, nonlinear, eccentric relative motion problem. The methodology presented in the paper alleviates the difficulty in computing corrections to the linear equations of motion to account for nonlinearities and eccentricities. Instead of dealing with the local relative motion problem, the global relative motion problem is addressed by transforming the orbital resonance requirement into an energy matching condition. The newly developed setup is then utilized to derive an optimal single-impulse formationkeeping maneuver based on relative state variables. The orbital elements interpretation of the optimal formationkeeping maneuver is also discussed.

Based on the concept of orbital commensurability, necessary and sufficient conditions are presented for bounded relative motion between any two spacecraft flying on elliptic Keplerian orbits. The proposed approach does not involve any simplifying assumptions regarding the relative dynamics but rather treats the general, nonlinear, eccentric relative motion problem. The methodology presented alleviates the difficulty in computing corrections to the linear equations of motion to account for nonlinearities and eccentricities. Instead of dealing with the local relative motion problem, the global relative motion problem is addressed by transforming the orbital resonance requirement into an energy-matching condition. The newly developed setup is then utilized to derive an optimal single-impulse formationkeeping maneuver based on relative state variables. The orbital elements interpretation of the optimal formationkeeping maneuver is also discussed.

This paper develops a new controller for stabilization of rigid body dynamics. The state-space model is formulated using canonical elements, known as the Serret-Andoyer variables, thus far unused for engineering applications. The controllability of the problem is examined and a damping feedback is derived using the Jurdjevic-Quinn method. It is shown that the new feedback controller is an asymptotic smooth feedback stabilizer. The performance of the new controller is examined in a simulation, showing excellent dynamic closed-loop behavior.

Utilizing collinear libration points of restricted three body problem for mission trajectory planning has been intensively investigated over the past few decades. Around these libration points quasiperiodic and periodic orbits also known as Lissajous and Halo orbits exist. Calculation of periodic halo-like orbit was accomplished by stating the restricted three-body problem as a control problem. Quasiperiodic solution of linearized system is calculated thru selection of initial conditions that map the stable modes around libration point of Earth-Moon system. Next, a continuous acceleration control term was introduced into the state-space system and by using pole-assignment technique a linear periodic trajectory found. Then, linear periodic trajectory introduced into the non-linear part of the exact equation of motion and addressed as a periodic disturbance. The frequencies of these disturbances were found numerically. A disturbance accommodating LQR control was then found to follow the path of this periodic disturbance. The work incorporates effect of oblateness of the primaries for each one of the three collinear libration points. Such continuous control acceleration renders reasonable control effort for missions targeted to stay closely to libration point.

Groups of unmanned aerial vehicles (UAVs) are believed to be useful in wide-area search and destroy (WASD) missions. While many researches focus on developing better algorithms that govern the mission capabilities of a group of UAVs (flock), we take a novel approach and examine the effect of flock properties on mission execution performance. Using Dudek's taxonomy we investigate the performance of a group of autonomous Unmanned Aerial Vehicles (UAVs) cooperating in mission execution against a. group of enemy agents acting in an unknown environment. We show that increasing the number of UAVs in the group proves to be beneficial as it allows the group to react to more enemy events. We also show that using communication helps creating better cooperation between the flock members: however, using infinite communication range or infinite communication bandwidth results in considerable computational complexity. We conclude that it may be sufficient to use finite-bandwidth communication, keeping the computational complexity constant with the number of UAVs in the group, thus allowing the group to be scalable to large numbers of UAVs. We conclude that using flocking behavior improves the group performance only if the group is capable of mission task collaboration. Finally, we show that using heterogeneous flocks, comprised of identifying UAVs and shooter UAVs, provide better mission performance, especially in civilian-rich environment.

This paper investigates the long-period and secular dynamics of a satellite about an oblate primary while relieving the assumption that the perturbed orbit is instantaneously parameterized by osculating Keplerian orbits. The inherent freedom obtained by transforming the orbital dynamics from the Cartesian inertial space to the orbital elements space, termed gauge freedom, is utilized to nullify four planetary equations. It is shown that there exists an orbit representation in which the mean non-osculating perigee is stable under the oblateness perturbation and that nodal precession, apsidal rotation and epoch drift may be simultaneously nullified on the expense of secular eccentricity and inclination variations. These observations considerably expand the standard description of J₂-perturbed motion using mean osculating orbital elements, which predicts secular variation nullification of semi-major axis, eccentricity and inclination only.