J2 Effect Research Articles

Traveling Salesman Problem of optimal debris removal sequence using non-population gradient search

A new approach for debris removal sequence optimization with multiple spacecraft is developed and tested. The motion planning of the multiple rendezvous involves complex Traveling Salesman Problem (TSP) combinatorial decision, in which both the continuous parameters of the orbital maneuvers and integer indices of the debris are determined. Expected value and variance of trajectory conditioned on preceding visit, not the state of target itself, are propagated and analyzed. With the metric, likelihood of expected trajectory can be given, and the mixed integer problem is reformulated into continuous parameter estimation with maximum likelihood. A non-population gradient based optimizer is then used to search the target sequence. The method of TSP optimization is compared to the continuous parameter estimator of Extended Kalman Filter. Using this method, the classic static TSP-LIB benchmarks with up to 100 nodes are tested. We then extend the method to the TSP with space dynamics. Effectiveness and efficiency of the proposed method are tested by a sequential visit of near-earth debris considering J2 effects with a single and multiple spacecrafts.

Controllability Analysis of Motion of Artificial Satellite Under the Effect of Oblateness of the Earth

In this article we have studied the controllability of artificial satellite under the effect of zonal harmonic J2 in cylindrical polar coordinates systems. Seven different cases of thrusters in various directions have been analyzed and it is found that the system is controllable if we apply thrusters in either r, θ and z or θ and z direction. The equations governing motion of satellite have been linearized and Kalman controllability test is applied to check the controllability of the system. We have also derived controller u for the linearized system. The trajectory of the system has been plotted to show the controllability of the system.

Shaping low-thrust multi-target visit trajectories via theory of functional connections

This paper proposes a shaping method via theory of functional connections (TFC) to solve the coplanar low-thrust multi-target visit problems under the J2 perturbation. Three types of targets are considered including ground targets, space targets, and hybrid ground-space targets. The visit constraints for the ground and space targets under the linear J2-perturbed model are transformed into those under the two-body model with a correction term, which avoids directly solving the transfer trajectories with the J2 effect. Then, a unified shaping method via the TFC is proposed for the three types of targets. The multi-target visit problem is transformed into a nonlinear programming problem that analytically satisfies the initial conditions and the multi-target visit constraints. Several numerical examples show that the proposed method can realize rapid optimization for the whole transfer trajectories. Furthermore, the solved trajectories are successfully applied to the initial guess estimation of the Gauss pseudospectral method for further optimization.

Continuous Low-Thrust Maneuver Planning for Space Gravitational Wave Formation Reconfiguration Based on Improved Particle Swarm Optimization Algorithm

This study proposes a three-spacecraft formation reconfiguration strategy of minimum fuel for space gravitational wave detection missions in the high Earth orbit (105 km). For solving the limitations of measurement and communication in long baseline formations, a control strategy of a virtual formation is applied. The virtual reference spacecraft provides a desired relative state between the satellites, which is then used to control the motion of the physical spacecraft to maintain the desired formation. A linear dynamics model based on relative orbit elements' parameterization is used to describe the relative motion in the virtual formation, which facilitates the inclusion of J2, SRP, and lunisolar third-body gravity effects and provides a direct insight into the relative motion geometry. Considering the actual flight scenarios of gravitational wave formations, a formation reconfiguration strategy based on continuous low thrust is investigated to achieve the desired state at a given time while minimizing interference to the satellite platform. The reconfiguration problem is considered a constrained nonlinear programming problem, and an improved particle swarm algorithm is developed to solve this problem. Finally, the simulation results demonstrate the performance of the proposed method in improving the maneuver sequence distribution and optimizing maneuver consumption.

Design and guidance of a multi-active debris removal mission

Space debris have become exceedingly dangerous over the years as the number of objects in orbit continues to increase. Active debris removal (ADR) missions have gained significant interest as effective means of mitigating the risk of collision between objects in space. This study focuses on developing a multi-ADR mission that utilizes controlled reentry and deorbiting. The mission comprises two spacecraft: a Servicer that brings debris to a low altitude and a Shepherd that rendezvous with the debris to later perform a controlled reentry. A preliminary mission design tool (PMDT) was developed to obtain time and fuel optimal trajectories for the proposed mission while considering the effect of J2, drag, eclipses, and duty cycle. The PMDT can perform such trajectory optimizations for multi-debris missions with computational time under a minute. Three guidance schemes are also studied, taking the PMDT solution as a reference to validate the design methodology and provide guidance solutions to this complex mission profile.

A planning tool for optimal three-dimensional formation flight maneuvers of satellites in VLEO using aerodynamic lift and drag via yaw angle deviations

Differential drag is a promising option to control the relative motion of distributed satellites in the Very Low Earth Orbit regime which are not equipped with dedicated thrusting devices. A major downside of the methodology, however, is that its control authority is (mainly) limited to the in-plane relative motion control. By additionally applying differential lift, however, all three translational degrees-of-freedom become controllable. In this article, we present a tool to flexibly plan optimal three-dimensional formation flight maneuvers via differential lift and drag. In the planning process, the most significant perturbing effects in this orbital regime, namely the J2 effect and atmospheric forces, are taken into account. Moreover, varying atmospheric densities as well as the co-rotation of the atmosphere are considered. Besides its flexible and high-fidelity nature, the major assets of the proposed methodology are that the in-and out-of-plane relative motion are controlled simultaneously via deviations in the yaw angles of the respective satellites and that the planned trajectory is optimal in a sense that the overall decay during the maneuver is minimized. Thereby, the remaining lifetime of the satellites is maximized and the practicability and sustainability of the methodology significantly increased. To the best of the authors knowledge, a tool with the given capabilities has not yet been presented in literature. The resulting trajectories for three fundamentally different relevant formation flight maneuvers are presented and discussed in detail in order to indicate the vast range of applicability of the tool.

A parametric constellation using relative critical inclination of satellites

This study presents a strategy for designing a swarm configuration that minimises the J2 effect through the combination of the recently proposed parametric constellations and relative critical inclinations that set to zero the J2-induced drifts of satellites. Parametric constellations are a theory of optimised multiple deputy satellites to have a single identical constellation pattern with respect to the chief satellite. However, when J2 perturbation is considered in the design of parametric constellations, the relative orbit drifts over time, and the closed periodic condition is not established. Thus, the application of parametric constellations is limited, and control is required to maintain consistent relative orbits between satellites. This study efficiently solves the complicated multiple satellite dynamics problem under J2 perturbation by presenting a set of methods to minimise the control effort and fuel budget by designing parametric constellations based on relative critical inclinations.

HSPB1 inhibitor J2 attenuates lung inflammation through direct modulation of Ym1 production and paracrine signaling

Heat shock protein beta-1 (HSPB1) is a multifaceted protein that controls cellular stress, modulates cell differentiation and development, and inhibits apoptosis of cancer cells. Increased HSPB1 expression is highly associated with poor outcomes in lung cancer by enhancing cell migration and invasion; therefore, targeting HSPB1 may be a promising therapeutic for lung cancer and fibrosis. Although the HSPB1 inhibitor J2 has been reported to exhibit potent antifibrotic effects, it remains unclear whether and how J2 directly modulates inflammatory immune responses in pulmonary fibrosis. In this study, we found that J2 potently attenuated irradiation or bleomycin-induced pulmonary fibrosis by significantly inhibiting the infiltration and activation of T cells and macrophages. J2 inhibited T-cell proliferation and subsequently suppressed T helper cell development. Although there was no significant effect of J2 on cell proliferation of M1 and M2 macrophages, J2 specifically increased the expression of Ym1 in M2 macrophages without affecting the expression of other M2 markers. Interestingly, J2 increased lysosomal degradation of HSPB1 and inhibited HSPB1-induced repression of signal transducer and activator of transcription 6 (STAT6), which simultaneously increased STAT6 and Ym1 expression. Ym1 production and secretion by J2-treated M2 macrophages substantially decreased IL-8 production by airway epithelial cells in vitro and in vivo, resulting in attenuation of airway inflammation. Taken together, we suggest that J2 has potential as a therapeutic agent for pulmonary fibrosis with increased HSPB1 expression through direct immune suppression by Ym1 production by M2 macrophages as well as T-cell suppression.

Time\u2013frequency analysis of the Galileo satellite clocks: looking for the J2 relativistic effect and other periodic variations

When observed from the ground, the frequency of the atomic clocks flying on the satellites of a Global Navigation Satellite System is referred to as apparent frequency, because it is observed through the on-board signal generation chain, the propagation path, the relativistic effects, the measurement system, and the clock estimation algorithm. As a consequence, the apparent clock frequency is affected by periodic variations of different origins such as, for example, the periodic component of the J2 relativistic effect, due to the oblateness of the earth, and the clock estimation errors induced by the orbital estimation errors. We present a detailed characterization of the periodic variations affecting the apparent frequency of the Galileo clocks, obtained by applying time–frequency analysis and other signal processing techniques on space clock data provided by the European Space Agency. In particular, we analyze one year of data from three Galileo Passive Hydrogen Masers, flying on two different orbital planes. Time–frequency analysis reveals how the spectral components of the apparent frequency change with time. For example, it confirms that the amplitude of the periodic signal due to the orbital estimation errors depends on the angle between the sun and the orbital plane. Moreover, it allows to find a more precise estimate of the amplitude of the J2 effect, in agreement with the prediction of the general theory of relativity, and it shows that such amplitude suddenly decreases when the corresponding relativistic correction is applied to the data, thus validating the analytical formula used for the correction.

Satellite formation flying control using an innovative technique subject to electromagnetic acceleration

The electrostatic charge became a possible method to control both orbital and attitude motion without fuel consumption or reduce the fuel cost. This study needs to install an ion collector on the satellite to increase the level of charging as artificial charging. In this paper, we developed a linearised satellite relative motion and a combination of Lorentz forces provided by modulating spacecraft's electrostatic charge (magnetic and electric fields) that can be used to keep the desired relative distances and orientations considering J2 perturbation. The proposed controller contains a feedback control using a real-time fuel-optimal control approach is considered for satellite formation control in eccentric orbits based on the developed linear J2 dynamic model. Comprehensive simulation numerical results confirmed the capability of Lorentz force to correct the drift in the relative position of formation flying due to the effect of J2, these corrections depend on the value of charge to mass ratio and magnitude of the relative position.

PERTURBED GROUND TRACK UNDER THE INFLUENCE OF J2 AND LUNISOLAR FORCES

A several types of forces are acting on the satellite. These forces are classified into conservative and nonconservativeforce. The main concern in the present research work is to studding the effect of conservative forces on the satellite orbital motion and represent this effect on satellite ground track. Where the ground tracks are the locus of points formed by the points on the Earth directly below a satellite as it travels in orbit. A mathematical model and a program code is designed using Matlab package to calculate the perturbed ground track under J2 and luni-solar forces. Whereas the J2 and Luni-Solar are a conservative forces, the secular variation is presented only in RAAN and  . Otherwise the remaining orbital element is varies periodically. The perturbed ground track is calculated under the effect of J2 and luni-solar forces. The perturbed position vectors for a satellite are converted to its corresponding latitudes and longitudes. The satellite’s position in one revolution is displayed to represent where the satellite at the time desired.

Study the Influence of Atmospheric Drag and J2 Effect in a Close-proximity Operation at LEO

In this paper is to assess the mission stability and the influence of J2 effect and aerodynamic forces. To maintain the relative motion of satellites by using a feedback control law for tracking error bound in the presence of J2 perturbation. A constant relative orbit under the effect of earth oblateness and conservative forces is referred as J2 and targeting the presence of atmospheric drag. Although, Schweighart and Sedwick control strategy for satellite relative motion is considering both lift and drag forces. The simulation result shows a better performance with high accuracy than an elliptical orbit under J2 perturbation and atmospheric drag along in-track formation. The algorithm and control strategies is useful tools for analysing a future space mission.

CubeSat constellation management using Ionic Liquid Electrospray Propulsion

The Space Propulsion Laboratory (SPL) of the Massachusetts Institute of Technology (MIT) is developing the Ion Electrospray Propulsion System (iEPS), designed to address a current need in CubeSat technology: miniaturized electric thrusters. These could be used for different applications, ranging from attitude control to interplanetary flights. In this work, performed together with the Space Systems Laboratory of the Costa Rica Institute of Technology (SETEC Lab), we explore a case study in which the iEPS is used for constellation management in Low Earth Orbit (LEO) when integrated in a 3U CubeSat. We analyze how a 180° separation in the Right Ascension of the Ascending Node (RAAN) between two CubeSats (SatA and SatB) starting in the same orbit can be achieved by modifying one of the spacecraft's orbital altitude, resulting in a difference in their rate of nodal precession (defined as the drift rate) due to the J2 effect, and therefore a difference in their relative RAAN. The method consists of SatB increasing its semi-major axis, drifting in a higher orbit with a lower drift rate, and returning to the original semi-major axis once the desired difference in RAAN in achieved relative to the other spacecraft. SatA will stay in its original orbit, using its thruster to compensate for orbital energy loss due to atmospheric drag, therefore demonstrating another application of iEPS for constellation management. Three different simulations were studied, defined as the minimum time trajectory, minimum propellant trajectory and a hybrid trajectory, consisting of reaching a higher altitude orbit, but actively changing the RAAN using the propulsion system instead of drifting. It was observed that the difference in this orbital element could be achieved using 85 g of propellant in as little as 164 days for the minimum time trajectory. The same difference could also be achieved using only 44 g of propellant in 245 days for the minimum propellant trajectory. Furthermore, the results of the hybrid trajectory showed that the goal could be achieved in 161 days, but using 158 g of propellant mass, demonstrating the benefit of using a drift orbit. The results proved the feasibility of implementing iEPS for constellation management using 3U CubeSats in LEO.

The solution of the generalized Kepler's equation

In the context of general perturbation theories, the main problem of the artificial satellite analyses the motion of an orbiter around an Earth-like planet, only perturbed by its equatorial bulge or J2 effect. By means of a Lie transform and the Krylov-Bogoliubov-Mitropolsky method, a first-order theory in closed form of the eccentricity is produced. During the evaluation of the theory it is necessary to solve a generalization of the classical Kepler's equation. In this work, the application of a numerical technique and three initial guesses to the Generalized Kepler's equation are discussed.

Optimization of Active Debris Removal Missions with Multiple Targets

Mission design for multiple debris removal is performed by selecting the most favorable sequences of objects to be removed. Debris items among a population with similar inclination values are considered. The chaser rendezvouses with the objects and attaches a removal kit. An approximate analysis, based on the use of the J2 effect to minimize propellant consumption, provides estimations of transfer times and between any object pair in order to evaluate the costs of any possible sequence. Estimations, which are verified by using an evolutionary optimization of four-impulse transfers that take J2 perturbation into account, are fast and accurate; and they permit evaluation of all available sequences when the number of objects to be removed is limited. The object sequence determination problem is converted to a traveling salesman problem, and an ant colony optimization algorithm is introduced to analyze longer sequences. The mass of the removal kit for any debris item is then evaluated, depending on the selected removal method. The overall mission mass budget is finally computed, and the best opportunities in terms of the mass and mission time are selected. Different removal techniques exploiting chemical and electric propulsion are compared. The results prove that removal of four to eight objects in less than a year is feasible with current technologies.

On J2 short-term orbit predictions in terms of KS elements

Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in terms of KS elements in closed form in eccentricity with Earth’s zonal harmonic term J2, is improved by using King-Hele’s expression for the radial distance ‘r’ which includes the effect of J2, and is suitable for low eccentricity orbits. Numerical experimentation with four test cases with perigee altitude of 200 km and eccentricity varying from 0.01 to 0.3 for different inclinations is carried out. It is found that the orbital elements computed with the analytical expressions in a single step during half a revolution match very well with numerically integrated values and show significant improvement over the earlier theory. The solution can be effectively used for computation of mean elements for near-Earth orbits, where the short-term orbit perturbations due to J2 play most important role. The theory will be very useful in computing the state vectors during the coast phase of rocket trajectories and flight algorithms for on-board implementation.

Applications of optimal finite thrust orbital transfers

Using the theory developed by one of the authors in a previous paper, the authors discuss and present solutions for the low thrust orbital transfer in a large number of applications of practical interest: Some emerging topics of Mission Analysis have been addressed, such as the optimization of the deployment of a constellation with orbits at different RAANs, the introduction of constraints in the trajectory like the minimum perigee altitude, and the existence of many extremal solutions for the same problem. The authors have discussed and solved these problems including the J2 and eclipse effects.

Relative Dynamics and Control for Satellite Formation: Accommodating J2 Perturbation

AbstractThe J2 effect is the dominating perturbation for satellite formation flying missions. In this paper, a linearized satellite relative motion model considering J2 perturbation and eccentricity was derived first. Then, a real-time fuel-optimal continuous low-thrust control approach was proposed for satellite formation control in eccentric orbits on the basis of the developed linear J2 dynamic model. This optimal control problem was converted into a mathematical programming problem by the application of the Legendre pseudospectral method, which makes real-time optimal control feasible. Finally, comprehensive simulation results validated the proposed linearized J2 model and formation control approach.

Coplanar ground-track adjustment using time difference

The coplanar ground track adjustment problem is studied for both impulsive and continuous low-thrust maneuvers considering the J2 perturbation. For the initial orbit, semi-analytical solutions of the flight time and the longitude difference are obtained for an assigned ground-site latitude. Based on the time difference, the flight time for the two-body model is derived. This avoids directly solving the transfer orbit with the J2 effect. Then two methods including single-impulse method for free final orbit and shape-based low-thrust method for an assigned final orbit are proposed. For the single-impulse method, the analytical approximate solution is obtained by solving a cubic equation. For the low-thrust method, the shape-based approach is used with a combined inverse polynomial and trigonometric function. The results of several numerical examples show that the final longitude errors of the proposed methods are less than 0.04 deg for the nonlinear J2 model.

Satellite rendezvous using differential aerodynamic forces under J2 perturbation

Purpose – The purpose of this paper is to present a method to conduct small satellite rendezvous mission by using the differential aerodynamic forces under J2 perturbation in low earth orbit (LEO). Design/methodology/approach – Each spacecraft is assumed to be equipped with two large flat plates, which can be controlled for generating differential accelerations in all three directions. Based on the kinetic theory, the aerodynamic lift and drag generated by a flat plate are calculated. To describe the relative dynamics under J2 perturbation, a modified model is derived from the high-fidelity linearized J2 equations proposed by Schweighart and Sedwick. Findings – Simulation results demonstrate that the proposed method is valid and efficient to solve satellite rendezvous problem, and the modified model considering J2 effect shows better accuracy than the Horsley’s Clohessy–Wiltshire-based model. Research limitations/implications – Because aerodynamic force will reduce drastically as orbital altitude rises, t...