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Amethod for e nding fuel-optimal trajectoriesfor spacecraft subjected to avoidancerequirements is introduced. These include avoidance of collisions with obstacles or other vehicles and prevention of thruster plumes from one spacecraft impinging on another spacecraft. The necessary logical constraints for avoidance are appended to a fuel-optimizinglinearprogramby includingbinaryvariablesin theoptimization.Theresulting problem isa mixedintegerlinearprogram (MILP)thatcan besolved using availablesoftware. Thelogical constraints can also beused to express the cone guration requirements for maneuvers where only the e nal relative alignment of the vehicles is important and the assignment of spacecraft within the e eet is not specie ed. The collision avoidance, trajectory optimization, and e eet assignment problems can be combined into a single MILP to obtain the optimal solution for these maneuvers. The MILP problem formulation, including these various avoidance constraints, is presented, and then several examples of their application to spacecraft maneuvers, including recone guration of a satellite formation and close inspection of the International Space Station by a microsatellite, are shown. These examples clearly showthat the trajectory design methods presented areparticularly wellsuited to proposed formation e ying missions that involve multiple vehicles operating in close proximity. Nomenclature G = number of global cone gurations available for end states M = large number for logical constraints N = number of dimensions P = plume length T = number of time steps u = control input V = number of vehicles W = plume width x = vehicle state Subscripts g = global cone guration for e nal states i = time step l = obstacle n, m = axes in some orthogonal coordinate frame p, q = vehicles r = position within e nal cone guration
Richards et al. (Mon,) studied this question.