INITIAL-VALUE AND 2 POINT BOUNDARY-VALUE SOLUTIONS TO THE CLOHESSY-WILTSHIRE EQUATIONS

The nonhomogeneous Clohessy-Wiltshire (C-W) equations are formulated and solved as an initial value problem in the formal structure of linear systems theory. The nonhomogeneous C-W equations could have many conceivable forcing functions but one useful in many satellite studies, that of constant differential drag (or of constant low level thrusting) along or opposite the velocity vector, is assumed in this paper. The state transition matrix (STM) (obtained from the solution of the homogeneous C-W equations) and its inverse (required to solve the general nonhomogeneous set of equations in this theory) are obtained explicitly in both Newtonian and Hamiltonian form. In general, the analytic inverse of a 6 X 6 matrix is not easily constructed unless that matrix possesses some special symmetry property such as being orthogonal or symplectic. It is shown that the STM for the C-W equations does possess a special property making its inverse easily obtainable. To construct an effective mission design tool for orbit transfers, the C-W solutions also need to be available in the form of a two-point boundary value problem; therefore, the initial value solutions, obtained via the STM and its inverse, are recast with relative ease as closed-form solutions to a Lambert problem. The initial value solutions are free of singularities but the Lambert solutions possess isolated singularities which are identified.