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.

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.

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.

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.

Integrated method for measuring distance and time difference between small satellites

The advancement of small satellites is promoting the development of distributed satellite systems,and for the latter,it is essential to coordinate the spatial and temporal relations between mutually visible satellites.By now,dual one-way ranging(DOWR)and two-way time transfer(TWTT)are generally integrated in the same software and hardware system to meet the limitations of small satellites in terms of size,weight and power(SWaP)consumption.However,studies show that pseudo-noise regenerative ranging(PNRR)performs better than DOWR if some advanced implementation technologies are employed.Besides,PNRR has no requirement on time synchronization.To apply PNRR to small satellites,and meanwhile,meet the demand for time difference measurement,we propose the round-way time difference measurement,which can be combined with PNRR to form a new integrated system without exceeding the limits of SWaP.The new integrated system can provide distributed small satellite systems with on-orbit high-accuracy and high-precision distance measurement and time difference measurement in real time.Experimental results show that the precision of ranging is about 1.94 cm,and that of time difference measurement is about 78.4 ps,at the signal to noise ratio of 80 dBHz.

Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

This paper considers nonlinear dynamics of teth- ered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy sur- face, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. （3,3）- and （2,1）-heteroclinic trajecto- ries from the neighborhood of one collinear equilibrium to that of another one, and （3,6）- and （2,1）-homoclinic trajecto- ries from and to the neighborhood of the same equilibrium, are obtained based on the Poincar6 mapping technique.

Adaptive repetitive learning control for trajectory-keeping of satellite formation flying

An adaptive repetitive control scheme is proposed for trajectory-keeping of satellite formation flying in the leader–follower mode which is described by Lawden equation.The system is parameterised by power series approximation and the unknown timevarying parameters are estimated by adaptive repetitive learning law.Through rigorous analysis by constructing a Lyapunov-like composite energy function(CEF),the stability of the closed-loop system is proved.Finally,a simulation example is provided to illustrate the effectiveness of the control algorithms proposed in this paper.