Solution of the Free-Fall Guidance and Control Problem in N-Body Space

The solution of the free-fall guidance and control problem in an inverse-square central force field is treated as an introduction. Circular, elliptical, parabolic and hyperbolic trajectories are covered. Consideration of the control problem (steering and thrust cutoff signals) is applied to the case of a rocket where only the direction of the thrust vector and thrust cutoff are controllable. Consideration is given to the energy aspects, and to launch and atmospheric conditions as they affect guidance and control. These techniques are then extended to the solution of the free-fall guidance problem in N-body space. In this latter category, Bonnet´s Theorem, as extended by Egorov, is covered to show its application to the solution of the N-body problem by superimposing solutions of the inverse-square central force field problem. The theorem is applied to treat the problem of finding possible conic trajectories in two-body space. The problem of finding possible conic and nonconic trajectories in three-body space is also covered and a solution to the guidance problem for these trajectories is included. The libration points of the N-body problem are treated as special cases of the solutions discussed. The equations discussed are valid for a vehicle of negligible mass acted upon by forces due only to N inverse-square central force fields. No attempt has been made to cover the effects of noninverse-square central force fields, although Bonnet´s Theorem can be applied to constrained trajectories such as those that will be made possible with a continuous low thrust vehicle.