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The purpose of this paper is to present a method to achieve small satellite formation keeping operations by using the differential lift and drag to control the drift caused by J2 perturbation in circular or near-circular low earth orbits (LEOs).

Each spacecraft is equipped with five large flat plates, which can be controlled to generate differential accelerations. The aerodynamic lift and drag acting on a flat plate is calculated by the kinetic theory. To maintain the formation within tracking error bounds in the presence of J2 perturbation, a nonlinear Lyapunov-based feedback control law is designed.

Simulation results demonstrate that the proposed method is efficient for the satellite formation keeping and better accuracy advantage in comparison with classical approaches via the fixed maximum differential aerodynamic acceleration.

Because the aerodynamic force will reduce drastically as the orbital altitude increases, the formation keeping control strategy for small satellites presented in this paper should be limited to the scenarios when satellites are in LEO.

The formation keeping control method in this paper can be applied to solve satellite formation keeping problem for small satellites in LEO.

This paper proposes a Lyapunov control strategy for satellite formation keeping considering both lift and drag forces, and simulation results show better performance with high accuracy under J2 perturbation.

The purpose of this paper is to analyze the frame of a missile formation cooperative control system, and present an optimal keeping controller of a missile formation in the cooperative engagement.

A missile relative motion model is established directly based on the kinematics relationships in the relative coordinated frame, following that is the detailed process of designing an optimal formation controller, which is analyzed through the small disturbance linearized method and transforming control variables method, respectively, these two methods both have themselves properties. The equations and control variables are intuitive during the linearized analysis, but errors brought by the linearized method are unavoidable, which will reduce the control precision. As for the transforming method, the control accuracy is greatly increased although the control form is a little complex, so in this paper the transforming control variable method is mainly researched to design an optimal formation controller. Considering the states of a leader as input perturbation variables, we design an optimal formation controller based on the linear quadric theory, which has quadric optimal performances of the missile flight states and control quantity. In order to obtain a higher accurate solution, the precise integration algorithm is introduced to solve the Riccati Equation that significantly affects the accuracy of an optimal control problem.

The relative motion model established directly in the relative coordinate frame has intuitive physical significance, and the optimal controller based on this relative motion model is capable of restraining the invariable or slowly varying perturbation brought by the velocity of a leader and the input perturbations caused by the maneuver of the leader, at the same time this optimal controller can implement formation reconfiguration and keeping to an expected states rapidly, steadily and exactly; the steady errors can be greatly decreased by analyzing the relative motion model through transforming control variables method compared to the small disturbance linearized operation.

The main frame of a missile formation cooperative engagement system can be found in this paper, which shows a clear structure and relations of each part of this complex system. The relations between each subsystem including the specific input and output variables can also be used to guide and restrict how to design each subsystem. The emphasis of this paper is on designing an optimal formation keeping controller which can overcome slowly varying or invariable perturbations and implement quadric optimal keeping control rapidly, stably and accurately.

This paper provides a new method to analyze the missile relative motion model. The proposed proportional and integral (PI) optimal controller based on this model, and utilizing the Precise Integration Algorithm to solve this optimal controller are also new thoughts for formation control problems.

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 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.

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.

This paper aims to study the issue of the three-dimensional formation coordinated control for the unmanned autonomous helicopter (UAH) by using the sliding mode disturbance observer. Under the designed formation coordinated controller, the desired formation can be maintained and the closed-loop system stability is analyzed by using the Lyapunov theory.

Considering the unknown time-varying external 10; disturbance in formation flight of UAHs, a sliding mode disturbance observer has been employed to estimate them.

This work is supported in part by the National Natural Science Foundation of China under Grant 61803207, and in part by the Fundamental Research Funds for the Central Universities under Grant LGZD201806.

A sliding mode disturbance observer has been designed to estimate the unknown time-varying external disturbance in formation flight of UAHs. Aiming at the leading UAH maneuver in three-dimensional space during the formation flight progress, the formation coordinated controller has been proposed based on the output of the disturbance observer to maintain the formation.

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.

This paper aims to investigate the feasibility of using the combination of Lorentz force and aerodynamic force as a propellantless control method for spacecraft formation.

It is assumed that each spacecraft is equipped with several large flat plates, which can rotate to produce aerodynamic force. Lorentz force can be achieved by modulating spacecraft’s electrostatic charge. An adaptive output feedback controller is designed based on a sliding mode observer to account for unknown uncertainties and the absence of relative velocity measurements. Aiming at distributing the control input, an optimal control allocation method is proposed to calculate the electrostatic charge of the Lorentz spacecraft and control commands for the atmospheric-based actuators.

Numerical examples are provided to demonstrate the effectiveness of the proposed control strategy in the presence of J₂ perturbations. Simulation results show that relative motion in a formation can be precisely controlled by the proposed propellantless control method under uncertainties and unavailability of velocity measurements.

The controllability of the system is not theoretically investigated in the current work.

The proposed control method introduced in this paper can be applied for small satellites formation in low Earth orbit.

The main contribution of the paper is the proposal of the propellantless control approach for satellite formation using the combination of Lorentz force and aerodynamic force, which can eliminate the requirement of the propulsion system.

In this paper, an adaptive fuzzy sliding mode controller (AFSMC) is developed for the formation control of a team of autonomous underwater vehicles (AUVs) subjected to unknown payload mass variations during their mission.

A sliding mode controller (SMC) is designed to drive the state trajectories of the AUVs to a switching surface in the state space. The payload mass variation results in parameter variation in AUV dynamics leading to actuator failure. This further leads to loss of communication among the members of the team. Hence, an adaptive SMC based on fuzzy logic is developed to maintain the coordinated motion of AUVs with payload mass variation.

The results are obtained by employing adaptive SMC for AUVs with and without payload variations and are compared. It is observed that the proposed adaptive SMC exhibits improved performance and tracks the desired trajectory in less time even with variation in the payload. The adaptive fuzzy control algorithm is developed to handle variation in payload mass variation. Lyapunov theory is used to establish stability of AFSMC controller.

Perfect alignment is assumed between centres of gravity (OG) and buoyancy (OB), thus AUVs maintaining horizontal stability during motion. The AUVs’ body centres are aligned with centres of gravity (OG), thus the distance vector being rg = [0,0,0]T. As it is a tracking problem, sway motion cannot be neglected as the AUVs are travelling in a curved locus, hence susceptible to Coriolis and centripetal forces. The AUV is underactuated as only two thrusters at the stern plate that are employed for the surge and yaw controls and error in Y- direction are controlled by adjusting control input in surge and heave direction. Control inputs to the thruster are constants, and depth control is achieved by adjusting the rudder angle.

AUVs are employed in military mission or surveys, and they carry heavy weapons or instrument to be deployed at or picked from specific locations. Such tasks lead to variation in payload, causing overall mass variation during an AUV’s motion. A sudden change in the mass after an AUV release or pick load results in variation in depth and average velocity.

The proposed controller can be useful for military missions for carrying warfare and hydrographic surveys for deploying instruments.

A proposed non-linear SMC has been designed, and its performances have been verified in terms of tracking error in X, Y and Z directions. An adaptive fuzzy SMC has been modelled using quantized state information to compensate payload variation. The stability of AFSMC controller is established by using Lyapunov theorem, and reachability of the sliding surface is ensured.

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.