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A fascinating corollary to Lambert's famous problem is developed. By applying this new property of twobody orbits, a simple reformulation of the two-point boundary-value problem is possible. This is accomplished by means of a geometrical transformation of the orbital foci which converts the original problem to one for which the initial point is an apsidal point. The elementary form of Kepler's equation then provides the analytic description of the time of flight. The elements of the original orbit are shown to be simply related to the corresponding elements of the transformed orbit. Finally, a simple iterative method of solving the transformed boundary-value problem using successive substitutions is developed. In most cases of interest, convergence is seen to be quite rapid.

The trajectory analysis and mission design for inspection of a host spacecraft by a microsatellite is motivated by the current developments in designing and building prototypes of a microsatellite inspector vehicle. Two different, mission scenarios are covered in this thesis -a host spacecraft in orbit about Earth and in deep space. Some of the key factors that affect the design of an inspection mission are presented and discussed. For the Earth orbiting case, the range of available trajectories -natural and forced -is analyzed using the solution to the Clohessy-Wiltshire (CW) differential equations. Utilizing the natural dynamics for inspection minimizes fuel costs, while still providing excellent opportunities to inspect and image the surface of the host spacecraft. The accessible natural motions are compiled to form a toolset, which may be employed in planning an inspection mission. A baseline mission concept for a microsatellite inspector is presented in this thesis. The mission is composed of four primary modes: deployment mode, global inspection mode, point inspection mode, and disposal mode. Some figures of merit that may be used to rate the success of the inspection mission are also presented. A simulation of the baseline mission concept for the Earth orbiting scenario is developed from the trajectory toolset. The hardware simulation is based on the current microinspector hardware developments at the Jet Propulsion Laboratory. Through the figures of merit, the quality of the inspection mission is shown to be excellent, when the natural dynamics are utilized for trajectory design. The baseline inspection mission is also extended to the deep space case.

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