Explicit Solution to the Full Nonlinear Problem for Satellite Formation-keeping

This paper presents simple and exact formation-keeping guidance schemes that use a new method that is rooted in some recent advances in analytical dynamics. As a result of this new approach, explicit control inputs to exactly maintain a given formation configuration are easily determined using continuous thrust propulsion systems. The complete nonlinear problem is addressed, and no linearizations and/or approximations are made. The approach provides a marked improvement over existing results in that the control forces, which cause geometric formation-keeping constraints to be exactly satisfied for arbitrary reference orbits, are found in closed form. For Keplerian reference orbits, a much simpler and explicit expression for the control needed to exactly satisfy formation-keeping constraints than hereto available is obtained. The paper also includes explicit control results when the follower is inserted into orbit with incorrect initial conditions, as usually happens in practice. The Hill reference frame, which is often more intuitive for formation-keeping, is used in the analysis. While this paper takes an example of a projected circular formation, the methodology that is developed can be applied to any desired configuration or orbital requirements. Extensive computational simulations are performed to demonstrate the ease of implementation, and the numerical accuracy provided by the approach developed herein.