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The purpose of this paper is to propose a fuel‐optimal virtual centre selection method for formation flying maintenance in the J2 perturbed environment.

Based on the relative orbital elements (ROE) theory, the J2 perturbed relative motions between different satellites in the formation are analyzed, and then the fuel‐optimal virtual centre selection issue for formation flying maintenance are parameterized in terms of ROE. In order to determine the optimal virtual centre, two theories are proposed in terms of ROE.

Numerical simulations demonstrate that the fuel‐optimal virtual centre selection method is valid, and the control of the ROE of each satellite with respect to a virtual optimal centre of the formation is more efficient regarding the fuel consumption than the control of all satellites with respect to a satellite belonging to the formation.

The fuel‐optimal virtual centre selection method is valid for formation flying mission whose member satellite in circular or near circular orbit.

The fuel‐optimal virtual centre selection approach can be used to solve formation flying maintenance problem which involves multiple satellites in the formation.

The paper proposes a fuel‐optimal virtual centre selection method in terms of ROE, and shows that keeping the formation with respect the optimal virtual centre is more fuel efficient.

A decentralized dynamic neural network (DNN)-based fault detection (FD) system for the reaction wheels of satellites in a formation flying mission is proposed. The paper aims to discuss the above issue.

The highly nonlinear dynamics of each spacecraft in the formation is modeled by using DNNs. The DNNs are trained based on the extended back-propagation algorithm by using the set of input/output data that are collected from the 3-axis of the attitude control subsystem of each satellite. The parameters of the DNNs are adjusted to meet certain performance requirements and minimize the output estimation error.

The capability of the proposed methodology has been investigated under different faulty scenarios. The proposed approach is a decentralized FD strategy, implying that a fault occurrence in one of the spacecraft in the formation is detected by using both a local fault detector and fault detectors constructed specifically based on the neighboring spacecraft. It is shown that this method has the capability of detecting low severity actuator faults in the formation that could not have been detected by only a local fault detector.

The nonlinear dynamics of the formation flying of spacecraft are represented by multilayer DNNs, in which conventional static neurons are replaced by dynamic neurons. In our proposed methodology, a DNN is utilized in each axis of every satellite that is trained based on the absolute attitude measurements in the formation that may nevertheless be incapable of detecting low severity faults. The DNNs that are utilized for the formation level are trained based on the relative attitude measurements of a spacecraft and its neighboring spacecraft that are then shown to be capable of detecting even low severity faults, thereby demonstrating the advantages and benefits of our proposed solution.

The purpose of this paper is to present the obtained analytic solutions of maximal and minimal inter-satellite distances for flying-around satellite formation.

The relative motion equation is used to express the inter-satellite distance as the function of the orbital elements of two participating satellites for the flying-around satellite formation. Then by taking the derivative of the distance function with respect to the true anomaly, some possible extreme value points are obtained. According to the detailed analysis, the maximal and minimal distance solutions are found. By a reverse process, the expected initial differential orbital elements that generate the required extreme inter-satellite distances are also obtained.

The maximal and minimal distances of the flying-around formation can be analytically written as the functions of three initial orbital elements differences, i.e. the differential orbital inclination, the differential eccentricity and the differential right ascension. For the given maximal and minimal distances, there are lots of solutions of the initial differential orbital elements, which can produce the expected relative motions.

The solutions of the maximal and minimal inter-satellite distances are only accurate for the circular or near circular reference orbit. For the elliptic reference orbit, there is a need to develop new methods to find the analytic solutions.

The results here can be applied to design the factual flying-around formation with dimension requirements in mission analysis stage.

By using the solutions presented in this paper, the engineers can design the expected flying-around formation with required maximal and minimal inter-satellite distances in a very easy way.

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 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 present a novel high-precision relative navigation method for tight formation-keeping based on thrust on-line identification.

Considering that thrust acceleration cannot be measured directly, an on-line identification method of thrust acceleration is explored via the estimated acceleration of major space perturbation and the inter-satellite relative states obtained from space-borne acceleration sensors; then, an effective identification model is designed to reconstruct thrust acceleration. Based on the identified thrust acceleration, relative orbit dynamics for tight formation-keeping is established. Further, using global positioning system (GPS) measurement information, a modified extended Kalman filter (EKF) is suggested to obtain the inter-satellite relative position and relative velocity.

Compared with the normal EKF and the adaptive robust EKF, the proposed modified EKF has better estimation accuracy in radial and along-track directions because of accurate compensation of thrust acceleration. Meanwhile, high-precision relative navigation results depend on high-precision acceleration sensors. Finally, simulation studies on a chief-deputy formation flying control system are performed to verify the effectiveness and superiority of the proposed relative navigation algorithm.

This paper provides a reference in solving the problem of high-precision relative navigation in tight formation-keeping application.

This paper proposes a novel on-line identification method for thrust acceleration and shows that thrust identification-based modified EKF is more efficient in relative navigation for tight formation-keeping.

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.

Hill equations have definite limitation in the design of multiple spacecraft formation flying in eccentric orbits. To solve the problem, the design method of spacecraft formation flying in a circular reference orbit based on Hill equation can be generalized and applied to spacecraft formation flying in eccentric orbits.

In this paper, T‐H equation is expressed as the explicit function form of reference orbit true anomaly, and the state transition matrix of relative motion of spacecraft formation flying in eccentric orbits is derived. According to the requirement that relative dynamics equation of spacecraft formation flying in eccentric orbits has periodicity solution, the paper theoretically gives the initial condition needed by the long‐term close‐distance spacecraft formation flying including the relationship between relative position and relative velocity. Without perturbation the spacecraft formation, which satisfies the initial periodicity restriction, can keep long‐term close‐distance flying without the need of active control.

Based on the theoretical analysis, some numerical simulations are carried out. The results demonstrate that each spacecraft in eccentric orbits can run in a periodic motion surrounding the center spacecraft under some conditions. And spacecraft formation reconfiguration is implementing according to missions.

Combined with the periodicity restriction primary condition a new method about spacecraft formation reconfiguration is put forward. The method given by this paper can be applied to eccentric orbits of arbitrary eccentricity, and provides theoretical reference for orbit design of spacecraft formation flying in eccentric orbits.

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.

The purpose of this paper is to develop a theoretical design for the attitude control of electromagnetic formation flying (EMFF) satellites, present a nonlinear controller for the relative translational control of EMFF satellites and propose a novel method for the allocation of electromagnetic dipoles.

The feedback attitude control law, magnetic unloading algorithm and large angle manoeuvre algorithm are presented. Then, a terminal sliding mode controller for the relative translation control is put forward and the convergence is proved. Finally, the control allocation problem of electromagnetic dipoles is formulated as an optimization issue, and a hybrid particle swarm optimization (PSO) – sequential quadratic programming (SQP) algorithm to optimize the free dipoles. Three numerical simulations are carried out and results are compared.

The proposed attitude controller is effective for the sun-tracking process of EMFF satellites, and the magnetic unloading algorithm is valid. The formation-keeping scenario simulation demonstrates the effectiveness of the terminal sliding model controller and electromagnetic dipole calculation method.

The proposed method can be applied to solve the attitude and relative translation control problem of EMFF satellites in low earth orbits.

The paper analyses the attitude control problem of EMFF satellites systematically and proposes an innovative way for relative translational control and electromagnetic dipole allocation.