Analytic Solutions for Equal Mass Four-Craft Static Coulomb Formation

Abstract

Analytic constant charge solutions are investigated for square planar and 3-D tetrahedron four-craft static Coulomb formations. The solutions are formulated in terms of the formation geometry and attitude. In contrast to the two- and three-spacecraft Coulomb formations, a four-spacecraft formation has new equality and multiple inequality constraints that need to be satisfied for the individual spacecraft charges to be both unique and real. Unique charge relative equilibriums are important to reduce the overall power requirement of the spacecraft charge emision. A spacecraft must not only satisfy three inequality three-craft constraints to yield a real charge solution, but it must also satisfy two additional equality constraints to ensure that the spacecraft charges are unique. Further, a method is presented to reduce the number of equality constraints arising due to the dynamics of a four-spacecraft formation. The unique and real spacecraft charges are determined as a function of the orientation of the square formation in a given principal orbit plane. For the 3-D tetrahedron formation scenario there is only a unique set of charged products. The implementability constraints are numerically evaluated to show that only trivial equal-mass tetrahedron formations are possible where one craft is on the along-track axis with zero charge.