The general objective is the development of efficient techniques for preliminary design of trajectory arcs in nonlinear autonomous dynamic systems in which the desired solution is subject to algebraic interior and/or exterior constraints. For application to the n-body problem, trajectories must satisfy specific requirements, e.g., periodicity in terms of the states, interior or boundary constraints, and specified coverage. Thus, a strategy is formulated in a sequence of increasingly complex steps: 1) a trajectory is first modeled as a series of arcs (analytical or numerical) and general trajectory characteristics and timing requirements are established; 2) the specific constraints and associated partials are formulated; 3) a corrections process ensures position and velocity continuity while satisfying the constraints; and finally, 4) the solution is transitioned to a full model employing ephemerides. Though the examples pertain to spacecraft mission design, the methodology is generally applicable to autonomous systems subject to algebraic constraints. For spacecraft mission design applications, an immediate advantage of this approach, particularly for the identification of periodic orbits, is that the startup solution need not exhibit any symmetry to achieve the objectives.

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The Micro-Arcsecond X-ray Imaging Mission (MAXIM), a proposed concept for the Structure and Evolution of the Universe (SEU) Black Hole Imager mission, is designed to make a ten million-fold improvement in X-ray image clarity of celestial objects by providmg better than 0.1 micro-arcsecond imaging. Currently the mission architecture comprises 25 spacecraft, 24 as optics modules and one as the detector, which will form sparse sub-apertures of a grazing incidence X-ray interferometer covering the 0.3-10 keV bandpass. This formation must allow for long duration continuous science observations and also for reconfiguration that permits re-pointing of the formation. To achieve these mission goals, the formation is required to cooperatively point at desired targets. Once pointed, the individual elements of the MAXIM formation must remain stable, maintaining their relative positions and attitudes below a critical threshold. These pointing and formation stability requirements impact the control and design of the formation.In this paper, we provide analysis of control efforts that are dependent upon the stability and the configuration and dimensions of the MAXIM formation. We emphasize the utilization of natural motions in the Lagrangian regions to minimize the control efforts and we address continuous control via input feedback linearization (IFL). Results provide control cost, configuration options, and capabilities as guidelines for the development of this complex mission.

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