Constellation Design via Projection of an Arbitrary Shape onto a Flower Constellation Surface

Abstract

This paper demonstrates several techniques for constellation and formation design via projection of an arbitrarily defined shape onto a surface constructed of an infinite set of a particular Flower Constellation (FC). Each of the set of Flower Constellations is differentiated only by its initial value of the right ascension of the ascending node (RAAN). There are three projection techniques presented in this paper. One is a projection from an arbitrary outlying point beyond the FC surface while the other two methods are projections from the surface of the Earth. It is important to note that the shape of the resultant constellation is not rigid and will deform as it approaches apogee and perigee. Also, the FC surface chosen will affect how long the formation shape will hold as well as the out-of-plane curvature of the shape. The constellation shape will reform twice on each petal of the Flower Constellation, and the reformation time depends upon the base FC surface. As examples, geometric shapes such as squares, circles, and triangles are projected onto the surface of several different base FC types. These formations can be reoriented to place the apogee of the formation in a different region of the Earth. Thus, the point where the formation starts to deform can be specified. The effect of perturbations on maintaining the constellation/formation will be briefly discussed.