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The purpose of this paper is to study the effect of J₃ perturbation of the Earth’s oblateness on satellite orbit compared with J₂ perturbation.

Based on the parametric variation method in the time domain, considering more accurate Earth potential function by considering J₃-perturbation effect, the perturbation equations about satellite’s six orbital elements (including semi-major axis, orbit inclination, right ascension of the ascending node, true anomaly, eccentricity and argument of perigee) has been deduced theoretically. The disturbance effects of J₂ and J₃ perturbations on the satellite orbit with different orbit inclinations have been studied numerically.

With the inclination increasing, the maximum of the semi-major axis increases weakly. The difference of inclination disturbed by the J₂ and J₃ perturbation is relative to orbit inclinations. J₃ perturbation has weak effect on the right ascension and argument of perigee. The critical angle of the right ascension and argument of perigee which decides the precession direction is 90° and 63.43°, respectively. The disturbance effects of J₂ and J₃ perturbations on the argument of perigee, right ascension and eccentricity are weakened when the eccentricity increases, simultaneously, the difference of J₂ and J₃ perturbations on argument of perigee, right ascension and argument of perigee decreases with eccentricity increasing, respectively.

In the future, satellites need to orbit the Earth much more precisely for a long period. The J₃ perturbation effect and the weight compared to J₂ perturbation in LEO can provide a theoretical reference for researchers who want to improve the control accuracy of satellite. On the other hand, the theoretical analysis and simulation results can help people to design the satellite orbit to avoid or diminish the disturbance effect of the Earth’s oblateness.

The J₃ perturbation equations of satellite orbit elements are deduced theoretically by using parametric variation method in this paper. Additionally, the comparison studies of J₂ perturbation and J₃ perturbation of the Earth’s oblateness on the satellite orbit with different initial conditions are presented.

The purpose of this research was to analyze application effects of the stable frozen orbit conditions in the spacecraft Orbital Maintenance Maneuver (OMM) reduction.

One challenge in implementing these motions is maintaining the relations as it experiences orbital perturbations (zonal harmonics), most notably due to the non-spherical Earth. A natural phenomenon exists called a frozen orbit, for which the orbital elements: argument of perigee (ω) and eccentricity (e) remain virtually fixed over extended periods of time.

Simulation results show that, using stable frozen orbit condition results in considerable propellant saving, decreased OMM, increase of accuracy position errors and thus performance improvement of the spacecraft for orbiter mission is preferable. So, from among three proposed theories, the Brouwer–Hori theory has provided better accuracy and more stable conditions in the frozen orbit.

Simulation algorithm has been achieved to solve this problem by extracting and combining the equations that govern the frozen conditions with the tangential forces (ΔV) equations for orbit correction.

In all studies with content of harmonic perturbation effects on the spacecraft motion dynamics, main goal is to obtain a solution for optimization of the operation process, so that overshadowed mission costs. The case studies about this aim, mostly to the trajectory parameters optimization by considering the vehicle orbital conditions under various control methods are formed. While in this regards, the intrinsic properties of stable Earth orbits and using them effectively is less than to analyse the problems is considered.

The purpose of this paper is to analyze the inclusion effects of the linearized time‐varying J₂‐perturbed terms for relative accuracy increase significantly over Melton's problem.

The methodology is based on the previous studies provided by Ross's paper. He gives a set of equations based on the C‐W equations which incorporates the J₂ gravitational perturbations and states in his introduction that this method can be expanded for the elliptical reference orbits as described by Melton.

One challenge in implementing the relative motions is maintaining the relations as it experiences gravitational perturbations, most notably due to non‐spherical Earth. Simulation results show that the inclusion of time‐varying J₂ perturbation terms in the derived linear equations increased the accuracy of the solution significantly in the out‐of‐orbit‐plane direction, while the accuracy within the orbit plane remained roughly unchanged.

By reason of replacing approximate terms (e, M) in this solution, for continues accuracy increase of time‐varying parameters containing θ(t) and R_O(t), this solution could be useful in the element‐errors evaluation and analysis of orbital multiple rendezvous missions, that are involved to the limited orbit periods.

The originality of this paper is to develop Melton's researches. He provided a method for generalizing the linear equations of motion to an elliptical orbit which enabled the determination of a time‐explicit, approximate solution. In this regard, there is no paper which has evaluated the inclusion effects of the linearized time‐varying J₂ perturbation terms over Melton's equations with such an approach.

The purpose of this paper is to evaluate the safety of formation flying satellites, and propose a method for practical collision monitoring and collision avoidance manoeuvre.

A general formation description method based on relative orbital elements is proposed, and a collision probability calculation model is established. The formula for the minimum relative distance in the crosstrack plane is derived, and the influence of J2 perturbation on formation safety is analyzed. Subsequently, the optimal collision avoidance manoeuvre problem is solved using the framework of linear programming algorithms.

The relative orbital elements are illustrative of formation description and are easy to use for perturbation analysis. The relative initial phase angle between the in‐plane and cross‐track plane motions has considerable effect on the formation safety. Simulations confirm the flexibility and effectiveness of the linear programming‐based collision avoidance manoeuvre method.

The proposed collision probability method can be applied in collision monitoring for the proximity operations of spacecraft. The presented minimum distance calculation formula in the cross‐track plane can be used in safe configuration design. Additionally, the linear programming method is suitable for formation control, in which the initial and terminal states are provided.

The relative orbital elements are used to calculate collision probability and analyze formation safety. The linear programming algorithms are extended for collision avoidance, an approach that is simple, effective, and more suitable for on‐board implementation.

The purpose of this paper is to propose a new fuel-balanced formation keeping reference trajectories planning method based on selecting the virtual reference center(VRC) in a fuel-balanced sense in terms of relative eccentricity and inclination vectors (E/I vectors).

By using the geometrical intuitive relative E/I vectors theory, the fuel-balanced VRC selection problem is reformulated as the geometrical problem to find the optimal point to equalize the distances between the VRC and the points determined by the relative E/I vectors of satellites in relative E/I vectors plane, which is solved by nonlinear programming method.

Numerical simulations demonstrate that the new proposed fuel-balanced formation keeping strategy is valid, and the new method achieves better fuel-balanced performance than the traditional method, which keeps formation with respect to geometrical formation center.

The new fuel-balanced formation keeping reference trajectories planning method is valid for formation flying mission whose member satellite is in circular or near circular orbit in J2 perturbed orbit environment.

The new fuel-balanced formation keeping reference trajectories planning method can be used to solve formation flying keeping problem, which involves multiple satellites in the formation.

The fuel-balanced reference trajectories planning problem is reformulated as a geometrical problem, which can provide insightful way to understand the dynamic nature of the fuel-balanced reference trajectories planning issue.

The purpose of this paper is to propose a modified fuel-balanced formation keeping strategy based on actively rotating satellites in the formation in the J2 perturbed environment.

Based on the relative orbital elements theory, the J2 perturbed relative motions between different satellites in the formation are analyzed, and then, the method to estimate fuel required to keep the in-plane and out-of-plane relative motions is presented, based on which a modified fuel-balanced formation keeping strategy is derived by considering both in-plane and out-of-plane J2 perturbations.

Numerical simulations demonstrate that the modified fuel-balanced formation keeping strategy is valid, and the modified fuel-balanced formation keeping strategy requires less total fuel consumption than original Vadali and Alfriend’s method.

The modified fuel-balanced formation keeping strategy is valid for formation flying mission whose member satellite is in circular or near-circular orbit.

The modified fuel-balanced formation keeping strategy can be used to solve formation flying keeping problem, which involves multiple satellites in the formation.

The modified fuel-balanced formation keeping strategy is proposed by considering both in-plane and out-of-plane J2 perturbations, which further reduce the fuel consumption than the original Vadali and Alfriend’s method.

The purpose of this paper is to propose an intuitive and effective cluster flight orbit design method for fractionated spacecraft.

Based on the concept of fractionated spacecraft, orbit design requirements for cluster flight in the case of fractionated spacecraft are proposed, and categorized into three requirements: stabilization requirement, passive safety requirement, and the maximum inter‐satellite distance requirement. These design requirements are then reformulated in terms of relative eccentricity and inclination vectors (E/I vectors) using a relative motion model based on relative orbital elements (ROEs). By using ROEs theory, the cluster flight orbit design issue is modelled as the distribution of relative E/I vectors for each member satellite in the cluster, and solved by combining three different heuristic search methods and one nonlinear programming (NLP) method.

The simulation results show that the NLP method is valid and efficient in solving the cluster flight orbit design problem and that for some cluster flight scenarios, the heuristic search methods can be adopted to give feasible solutions without the NLP method.

The cluster flight scenario in this paper is limited because the cluster should be in the near‐circular low earth orbit (LEO), and the relative distance between the member satellites should be small enough to satisfy the relative motion linearization assumption.

The cluster flight orbit design method proposed in this paper can be applied by fractionated spacecraft mission designers to propose potential cluster flight orbit solutions.

In this paper, the relative E/I vectors method is adopted to propose an intuitive and effective cluster flight orbit design method for fractionated spacecraft.

The purpose of this paper is to consider relative navigation – a vital technology to satellites formation flying, and to propose a new concept for relative navigation determination along with a technical approach for its practical implementation.

The determination of relative orbit is considered with the relative distance elevation and azimuth measurements about formation flying while the primary satellite is in a circle or ellipse orbit. This measurement is obtained by laser range finder and the estimations of the intersatellite relative position and velocity are obtained by utilizing the unscented Kalman filter instead of extended Kalman filter.

The simulation results show that the error of the relative position and velocity can be estimated with the order of cm and mm/s, respectively, under the effect of J2, converge faster than EKF, and then demonstrate that the approach is feasible.

The paper proposes a new concept for relative navigation determination and describes a technical approach for its practical implementation.

In this paper, an online convex optimization method for the exoatmospheric ascent trajectory of space interceptors is proposed. The purpose of this paper is to transform the original trajectory optimization problem into a sequence of convex optimization subproblems.

For convenience in calculating accuracy and efficiency, the complex nonlinear terminal orbital elements constraints are converted into several quadratic equality constraints, which can be better computed by a two-step correction method during the iteration. First, the nonconvex thrust magnitude constraint is convexified by the lossless convexification technique. Then, discretization and successive linearization are introduced to transform the original problem into a sequence of one convex optimization subproblem, considering different flight phases. Parameters of trust-region and penalty are also applied to improve the computation performance. To correct the deviation in real time, the iterative guidance method is applied before orbit injection.

Numerical experiments show that the algorithm proposed in this paper has good convergence and accuracy. The successive progress can converge in a few steps and 3–4 s of CPU time. Even under engine failure or mission change, the algorithm can yield satisfactory results.

The convex optimization method presented in this paper is expected to generate a reliable optimal trajectory rapidly in different situations and has great potential for onboard applications of space interceptors.

The originality of this paper lies in the proposed online trajectory optimization method and guidance algorithm of the space inceptors, especially for onboard applications in emergency situations.

The purpose of this paper is to establish an orbital launch window for manned Moon‐to‐Earth trajectories to support China's manned lunar landing mission requirements of high‐latitude landing and anytime return, i.e. the capability of safely returning the crew exploration vehicle at any time from any lunar parking orbit. The launch window is a certain time interval during which the transearth injection may occur and result in a safe lunar return to the specified landing site on the surface of the Earth.

Using the patched conic technique, an analytical design method for determining the transearth trajectories is developed with a finite sphere of influence model. An orbital launch window has been established to study the mission sensitivities to transearth trip time and energy requirements. The results presented here are limited to a single impulsive maneuver.

The difference between the results of the analytical model and high‐fidelity model is compared. This difference is relatively small and can be easily eliminated by a simple differential correction procedure. The launch window duration varies with launch date, from less than one hour to greater than 20 h, and the launch window occurs every day in the sidereal month.

The solution can be used to serve as an initial estimate for future optimization procedures.

The orbital launch window can be used to provide the basis for the preparation of an orbital launch timetable compatible with lunar missions and re‐entry conditions requirements.

Previous studies were mainly concentrated on the launch windows for the departure from the Earth. This paper investigates and establishes the orbital launch window for Moon‐to‐Earth trajectories.