Study on relative orbital configuration in satellite formation flying

In this paper, the relative orbital configurations of satellites in formation flying with non-perturbation and J2 perturbation are studied, and an orbital elements method is proposed to obtain the relative orbital configurations of satellites in formation. Firstly, under the condition of non-perturbation, we obtain many shapes of relative orbital configurations when the semi-major axes of satellites are equal. These shapes can be lines, ellipses or distorted closed curves. Secondly, on the basis of the analysis of J2 effect on relative orbital configurations, we find out that J2 effect can induce two kinds of changes of relative orbital configurations. They are distortion and drifting, respectively. In addition, when J2 perturbation is concerned, we also find that the semi-major axes of the leading and following satellites should not be the same exactly in order to decrease the J2 effect. The relationship of relative orbital elements and J2 effect is obtained through simulations. Finally, the minimum relation perturbation conditions are established in order to reduce the influence of the J2 effect. The results show that the minimum relation perturbation conditions can reduce the J2 effect significantly when the orbital element differences are small enough, and they can become rules for the design of satellite formation flying.

Robust Adaptive Gain Higher Order Sliding Mode Observer Based Control-constrained Nonlinear Model Predictive Control for Spacecraft Formation Flying

This work deals with the development of a decentralized optimal control algorithm, along with a robust observer,for the relative motion control of spacecraft in leader-follower based formation. An adaptive gain higher order sliding mode observer has been proposed to estimate the velocity as well as unmeasured disturbances from the noisy position measurements.A differentiator structure containing the Lipschitz constant and Lebesgue measurable control input, is utilized for obtaining the estimates. Adaptive tuning algorithms are derived based on Lyapunov stability theory, for updating the observer gains,which will give enough flexibility in the choice of initial estimates.Moreover, it may help to cope with unexpected state jerks. The trajectory tracking problem is formulated as a finite horizon optimal control problem, which is solved online. The control constraints are incorporated by using a nonquadratic performance functional. An adaptive update law has been derived for tuning the step size in the optimization algorithm, which may help to improve the convergence speed. Moreover, it is an attractive alternative to the heuristic choice of step size for diverse operating conditions. The disturbance as well as state estimates from the higher order sliding mode observer are utilized by the plant output prediction model, which will improve the overall performance of the controller. The nonlinear dynamics defined in leader fixed Euler-Hill frame has been considered for the present work and the reference trajectories are generated using Hill-Clohessy-Wiltshire equations of unperturbed motion. The simulation results based on rigorous perturbation analysis are presented to confirm the robustness of the proposed approach.

GPS/VISNAV integrated relative navigation and attitude determination system for ultra-close spacecraft formation flying

For the improvement of accuracy and better fault-tolerant performance, a global position system （GPS）/vision navigation （VISNAV） integrated relative navigation and attitude determination approach is presented for ultra-close spacecraft formation flying. Onboard GPS and VISNAV system are adopted and a federal Kalman filter architecture is used for the total navigation system design. Simulation results indicate that the integrated system can provide a total improvement of relative navigation and attitude estimation performance in accuracy and fault-tolerance.

Nonlinear control of spacecraft formation flying with disturbance rejection and collision avoidance

A nonlinear controller for disturbances rejection and collision avoidance is proposed for spacecraft formation flying.The formation flying is described by a nonlinear model with the J2 perturbation and atmospheric drag. Based on the theory of the state-dependent Riccati equation（SDRE）, a finite time nonlinear control law is developed for the nonlinear dynamics involved in formation flying. Then, a compensative internal mode（IM） control law is added to eliminate disturbances.These two control laws compose a finite time nonlinear tracking controller with disturbances rejection. Moreover, taking safety requirements into account, the repulsive control law is incorporated in the composite controller to perform collision avoidance manoeuvres. A numerical simulation is presented to demonstrate the effectiveness of the proposed method.Compared to the conventional control method, the proposed method provides better performance in the presence of the obstacles and external disturbances.

Finite-time coordination control for formation flying spacecraft

This paper investigates a distributed coordination control scheme using an adaptive terminal sliding mode for formation flying spacecraft with coupled attitude and translational dynamics. In order to overcome the singularity of the traditional fast terminal sliding manifold, a novel fast terminal sliding manifold is given. And then, based on the adaptive control method, a continuous robust coordinated controller is designed to compensate external disturbances and to alleviate the chattering phenomenon. The theoretical analysis shows that the coordinated controller can guarantee the finite-time stability of the overall closed-loop system through local information exchange, and numerical simulations also demonstrate its effectiveness.

Solar radiation pressure used for formation flying control around the Sun-Earth libration point

Solar radiation pressure is used to control the formation flying around the L2 libration point in the Sun-Earth system. Formation flying control around a halo orbit requires a very small thrust that cannot be satisfied by the latest thrusters. The key contribution of this paper is that the continuous low thrust is produced by solar radiation pressure to achieve the tight formation flying around the libration point. However, only certain families of formation types can be controlled by solar radiation pressure since the direction of solar radiation pressure is restricted to a certain range. Two types of feasible formations using solar radiation pressure control are designed. The conditions of feasible formations are given analytically. Simulations are presented for each case, and the results show that the formations are well controlled by solar radiation pressure.

COMPARISON OF TWO METHODS IN SATELLITE FORMATION FLYING

Recently, the research of dynamics and control of the satellite formation flying has been attracting a great deal of attentions of the researchers. The theory of the research was mainly based on Clohessy-Wiltshire' s (C-W's) equations, which describe the relative motion between two satellites. But according to some special examples and qualitative analysis , neither the initial parameters nor the period of the solution of C-W' s equations accord with the actual situation, and the conservation of energy is no longer held. A new method developed from orbital element description of single satellite , named relative orbital element method ( ROEM) , was introduced. This new method, with clear physics conception and wide application range, overcomes the limitation of C-W s equation , and the periodic solution is a natural conclusion. The simplified equation of the relative motion is obtained when the eccentricity of the main satellite is small. Finally, the results of the two methods (C-W' s equation and ROEM) are compared and the limitations of C-W s equations are pointed out and explained.

Orbit and attitude control of spacecraft formation flying

Formation flying is a novel concept of distributing the functionality of large spacecraft among several smaller, less expensive, cooperative satellites. Some applications require that a controllable satellite keeps relative position and attitude to observe a specific surface of another satellite among the cluster. Specially, the target space vehicle is malfunctioning. The present paper focuses on the problem that how to control a chaser satellite to fly around an out-of-work target satellite closely in earth orbit and to track a specific surface. Relative attitude and first approximate relative orbital dynamics equations are presented. Control strategy is derived based on feedback linearization and Lyapunov theory of stability. Further, considering the uncertainty of inertia, an adaptive control method is developed to obtain the correct inertial ratio. The numerical simulation is given to verify the validity of proposed control scheme.

An improved nonlinear control strategy for deep space formation flying spacecraft

This paper aims to provide further study on the nonlinear modeling and controller design of formation flying spacecraft in deep space missions. First, in the Sun-Earth system, the nonlinear formation dynamics for the circular restricted three-body problem （CRTBP） and elliptic restricted three-body problem （ERTBP） are presented. Then, with the Floquet mode method, an impulsive controller is developed to keep the Chief on the desired Halo orbit. Finally, a nonlinear adaptive control scheme based on Nonzero set- point LQR and neural network is proposed to achieve high precision formation maneuver and keeping. The simulation results indicate that the proposed nonlinear control strategy is reasonable as it considers not only the orbit keeping of the Chief, but also the formation modeling inaccuracy. Moreover, the nonlinear adaptive control scheme is effective to improve the control accuracy of the formation keeping.

USEFUL RELATIVE MOTION DESCRIPTION METHOD FOR PERTURBATIONS ANALYSIS IN SATELLITEFORMATION FLYING

A set of parameters called relative orbital elements were defined to describe the relative motion of the satellites in the formation flying. With the help of these parameters, the effect of the perturbations on the relative orbit trajectory and geometric properties of satellite formation can be easily analyzed. First, the relative orbital elements are derived, and pointed out： if the eccentricity of the leading satellite is a small value, the relative orbit trajectory is determined by the intersection between an elliptic cylinder and a plane in the leading satellite orbit frame reference; and the parameters that describe the elliptic cylinder and the plane can be used to obtain the relative orbit trajectory and the relative orbital elements. Second, by analyzing the effects of gravitational perturbations on the relative orbit using the relative orbital elements,it is found that the propagation of a relative orbit consists of two parts ： one is the drift of the elliptic cylinder; and the other is the rotation of the plane resulted from the rotation of the normal of the plane. Meanwhile, the analytic formulae for the drift and rotation rates of a relative trajectory under gravitational perturbations are presented. Finally, the relative orbit trajectory and the corresponding changes were analyzed with respect to the J2 perturbation.

Robust attitude control for rapid multi-target tracking in spacecraft formation flying

A robust attitude tracking control scheme for spacecraft formation flying is presented. The leader spacecraft with a rapid mobile antenna and a camera is modeled. While the camera is tracking the ground target, the antenna is tracking the follower spacecraft. By an angular velocity constraint and an angular constraint, two methods are proposed to compute the reference attitude profiles of the camera and antenna, respectively. To simplify the control design problem, this paper first derives the desired inverse system （DIS）, which can convert the attitude tracking problem of 3D space into the regulator problem. Based on DIS and sliding mode control （SMC）, a robust attitude tracking controller is developed in the presence of mass parameter uncertainties and external disturbance. By Lyapunov stability theory, the closed loop system stability can be achieved. The numerical simulations show that the proposed robust control scheme exhibits significant advantages for the multi-target attitude tracking of a two-spacecraft formation.

Design and control of multiple spacecraft formation flying in elliptical orbits

Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.

A stealthy method to determine the relative position and velocity for satellite formation flying

To achieve the satellite formation control and the succeed formation missions, we present a new stealthy method to determine the relative states between formation satellites. In this method, the combination of a CCD camera and laser radar is used as the relative measure sensors. To reduce electromagnetic radiation, the laser radar works intermittently to minimize the probability of being discovered. And an unscented Kalman filter (UKF) is applied to estimate the relative states. The observability of this method is analyzed. The validity and effectiveness of the method is demonstrated in a typical application of formation relative navigation.

Integrating BDS and GPS for precise relative orbit determination of LEO formation flying

Low-Earth-Orbit（LEO） formation-flying satellites have been widely applied in many kinds of space geodesy. Precise Relative Orbit Determination（PROD） is an essential prerequisite for the LEO formation-flying satellites to complete their mission in space. The contribution of the BeiDou Navigation Satellite System（BDS） to the accuracy and reliability of PROD of LEO formation-flying satellites based on a Global Positioning System（GPS） is studied using a simulation method. Firstly, when BDS is added to GPS, the mean number of visible satellites increases from9.71 to 21.58. Secondly, the results show that the 3-Dimensional（3 D） accuracy of PROD, based on BDS-only, GPS-only and BDS ＋ GPS, is 0.74 mm, 0.66 mm and 0.52 mm, respectively. When BDS co-works with GPS, the accuracy increases by 29.73%. Geostationary-Earth-Orbit（GEO） satellites and Inclined Geosynchronous-Orbit（IGSO） satellites are only distributed over the Asia-Pacific region; however, they could provide a global improvement to PROD. The difference in PROD results between the Asia-Pacific region and the non-Asia-Pacific region is not apparent. Furthermore, the value of the Ambiguity Dilution Of Precision（ADOP）, based on BDS ＋ GPS, decreases by 7.50% and 8.26%, respectively, compared with BDS-only and GPS-only. Finally, if the relative position between satellites is only a few kilometres, the effect of ephemeris errors on PROD could be ignored. However, for a several-hundred-kilometre separation of the LEO satellites, the SingleDifference（SD） ephemeris errors of GEO satellites would be on the order of centimetres. The experimental results show that when IGSO satellites and Medium-Earth-Orbit（MEO） satellites co-work with GEO satellites, the accuracy decreases by 17.02%.

Fixed-time position coordinated tracking control for spacecraft formation flying with collision avoidance

In this paper,the fixed-time stability of spacecraft formation reconfiguration(position tracking)is studied.Firstly,a novel nonsingular terminal sliding mode surface is designed and based on which a fixed-time coordinated controller is designed to keep the closed-loop system states have a finite settling time bounded by some predefined constants.Secondly,another nonsingular terminal sliding mode surface is designed by combining the artificial potential function and the aforementioned sliding surface,which meets the mutual distance constraint during transition process among spacecraft when it is bounded.Then another coordinated controller with fixed-time observer considering mutual distance constraint is presented,which guarantees the closed-loop system states stable also in bounded settling time.Finally,simulation results are shown to validate the correctness of the proposed theorems.It is worth mentioning that the control schemes also work even though there is a properly limit on the control input.

Discrete time multiple spacecraft formation flying attitude optimal control in presence of relative state constraints

This paper addresses the challenge of synchronized multiple spacecraft attitude reorientation in presence of pointing and boundary constraints with limited inter-spacecraft communication link.Relative attitude pointing constraint among the fleet of spacecraft has also been modeled and considered during the attitude maneuvers toward the desired states.Formation fling control structure that consists of decentralized path planners based on virtual structure approach joint with discrete time optimal local controller is designed to achieve the mission's goals.Due to digital computing of spacecraft's onboard computer,local optimal controller based on discrete time prediction and correction algorithm has been utilized.The time step of local optimal algorithm execution is designed so that the spacecraft track their desired attitudes with appropriate error bound.The convergence of the proposed architecture and stability of local controller's tracking error within appropriate upper bound are proved.Finally,a numerical simulation of a stereo imaging scenario is presented to verify the performance of the proposed architecture and the effectiveness of the algorithm.

Robust nonlinear control of spacecraft formation flying using constraint forces

A robust nonlinear control method is presented for spacecraft precise formation flying.With the constraint forces and consid-ering nonlinearity and perturbations,the problem of the formation keeping is changed to the Lagrange systems with the holonomic constraints and the differential algebraic equations (DAE).The nonlinear control laws are developed by solving the DAE.Because the traditional numerical solving methods of DAE are very sensitive to the various errors and the resulting con-trol laws are not robust in engineering application,the robust control law designed method is further developed by designing the correct coefficients to correct the errors of the formation array constraints.A numeral study simulated the robustness of this method for the various errors in the formation flying mission,including the initial errors of spacecraft formation,the reference satellite orbit determination errors,the relative perturbation forces model errors,and so on.

Floquet-based design and control approach to spacecraft formation flying in libration point orbits

A method for spacecraft formation flying (SFF) design and control near libration point orbits was developed by making use of the Floquet theory for periodic orbits. Firstly, the Floquet theory used in libration point orbits was introduced and the coefficients of four Floquet periodic modes were proved to be nearly constant when the amplitude in z direction of earth-moon L1 halo orbits is less than 20000 km. On this basis, a configuration design approach to SFF in L1 halo orbits was proposed, and several types of special configurations were obtained with periodic mode 3 and mode 5 or mode 4 and mode 6. Then, in order to control the SFF configuration concisely, those coefficients of the 5 modes (except the stable one) must be kept constant. A stationkeeping method for SFF was developed, which controls 5 Floquet modes simultaneously. Finally, simulations showed that the Floquet-based approach of configuration design and control for SFF is effective, simple and convenient. The research may be of value for deep space explorations.

Robust image-based coordinated control for spacecraft formation flying

This paper addresses a coordinated control problem for Spacecraft Formation Flying(SFF). The distributed followers are required to track and synchronize with the leader spacecraft.By using the feature points in the two-dimensional image space, an integrated 6-degree-of-freedom dynamic model is formulated for spacecraft relative motion. Without sophisticated threedimensional reconstruction, image features are directly utilized for the controller design. The proposed image-based controller can drive the follower spacecraft in the desired configuration with respect to the leader when the real-time captured images match their reference counterparts. To improve the precision of the formation configuration, the proposed controller employs a coordinated term to reduce the relative distance errors between followers. The uncertainties in the system dynamics are handled by integrating the adaptive technique into the controller, which increases the robustness of the SFF system. The closed-loop system stability is analyzed using the Lyapunov method and algebraic graph theory. A numerical simulation for a given SFF scenario is performed to evaluate the performance of the controller.

Nonlinear programming control using differential aerodynamic drag for CubeSat formation flying

Because of their volume and power limitation, it is difficult for CubeSats to configure a traditional propulsion system. Atmospheric drag is one of the space environmental forces that low-orbit satellites can use to realize orbit adjustment. This paper presents an integrated control strategy to achieve the desired in-track formation through the atmospheric drag difference, which will be used on ZJUCubeSat, the next pico-satellite of Zhejiang University and one of the participants of the international QB50 project. The primary mission of the QB50 project is to explore the near-Earth thermosphere and ionosphere at the orbital height of 90-300 km. Atmospheric drag cannot be ignored and has a major impact on both attitude and orbit of the satellite at this low orbital height. We conduct aerodynamics analysis and design a multidimensional nonlinear constraint programming （MNLP） strategy to calculate different desired area-mass ratios and corresponding hold times for orbit adjustment, taking both the semimajor axis and eccentricity into account. In addition, area-mass ratio adjustment is achieved by pitch attitude maneuver without any deployable mechanism or corresponding control. Numerical simulation based on ZJUCubeSat verifies the feasibility and advantage of this design.