The operational airspace of aerospace vehicles, including airplanes and unmanned aerial vehicles, is often restricted so that constraints on three-dimensional climbs, descents and other maneuvers are necessary. In this paper, the problem of determining constrained, three-dimensional, minimum-time-to-climb trajectories for an airplane in an airspace defined by a square cross section cylinder of arbitrary height is considered. Since a helical geometry appears to be a natural choice for climbing and descending trajectories subject to horizontal constraints, the optimal control problem is transformed to a parameter optimization problem by choosing helical starting trajectories and using a collocation method. A sequential quadratic programming algorithm has been applied to calculate "optimal" trajectories. Trajectories within airspace cylinders with different footprints are presented and compared with each other and with their two-dimensional (vertical plane), minimum-time-to-climb counterparts.
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