Formation flying and constellation station keeping

in near-circular orbits

Peter M. Bainum, Xiaodong Duan

Howard University

Washington, D.C. 20059, USA

To maintain the formation and constellation, the relative drifts due to the Earth's perturbation between the spacecraft should be carefully considered. Schaub, Alfriend et al. have presented a method to set the relative secular drift of the longitude of the ascending node, , and also the relative secular drift of the sum (mean latitude) of the argument of perigee and mean anomaly, , to be equal between two neighboring orbits. Such kinds of orbits are called J₂ invariant relative orbits. Subsequently, the types of orbits in which the primary disturbance is the differential љgravitational force are classified according to the number of constraints.

Based on an analytic solution of the perturbation motions of Earth satellites, the current authors developed a general solution, which establishes the relationship between the difference of the relative orbits and the resulting various secular drifts. Higher order harmonic terms such as J₄ in addition to the J₂ effect can also be considered under this approach. Around this general solution, different types of orbits are discussed and the corresponding solutions are formulated.

One particular interest among all the possibilities for the relative motion is: to let all the relative secular drifts (the drifts of the longitude of the ascending node, the argument of perigee and mean anomaly) vanish. One possible answer is addressed based on the zero solution for the general solution set. Two solution sets are found by the current authors as non-zero solutions for this problem. One of these two solution sets, which formation flying and constellation station keeping in near-circular orbits can satisfy, will be discussed in this paper. The significance of the solution and its difference from the zero solution will also be discussed.

In the procedure to get the non-zero solutions that result in all the relative secular drifts vanishing, the following solution sets, which have the potential to maintain the formation or constellation, and to maintain the distance between spacecraft in the neighboring orbits, are found:

(1)                                       The variation in the eccentricity ( ) can be chosen at will for the nominal eccentricity , while the variations in both the inclination ( ) and semi-major axis ( ) from their nominal values are set to zero;

or

(2)                                       The variation in the inclination ( ) can be chosen at will for the nominal inclination љ , and the variations in both the eccentricity ( ) and semi-major axis ( ) from their nominal values are set to zero.

This means that, the deployment and maintenance of the formation or constellation can be done by closely controlling two mean orbital elements. In the first solution set that can be satisfied by near-circular orbits, the differences between the inclination angle and between the semi-major axis with respect to the reference orbit should be controlled to be as small as possible, while the differences in the eccentricity љnear e=0 are not parameter sensitive (not required to be zero). In the second solution set that can be satisfied by near-equatorial orbits, the differences between the eccentricity and between the semi-major axis with respect to the reference orbit should be controlled to be as small as possible, while the differences in the inclination љnear љare not parameter sensitive (not required to be zero).

Results from high precision simulations show the accuracy and effectiveness of the method and indicate the boundary that sets the limit to the solution set. The influence of drag on this approach is also demonstrated through the simulation. Different combinations of pairs of satellites from a total of eight are included.

The solution proposed in this paper can be used for formation flying and constellation station keeping and maintaining the relative distance between spacecraft in nearby orbits. Moreover, the deployment or maintenance of the formation or constellation can be done by closely controlling two mean orbital elements, the semi-major axis and the inclination angle. Controlling the eccentricity closely is proved unnecessary for near-circular orbits. Many configurations for formations and constellations can satisfy the above requirements.

The solution is compared with the zero solution for the same kind of problem and its advantage is further demonstrated through simulations. The non-zero solution proves that the small error in the zero solution could cause serious relative drifts. However, there is a reference orbit for which the performance is not sensitive to small errors in , or . The non-zero solution provides the information about the optimal reference orbits, and it can be more easily implemented in practice.

The method still works under the influence of the atmospheric drag. Therefore, it can be effectively used on the low and mid-altitude orbits where the zonal harmonics would represent the main perturbation effect.