Optimal low-thrust gravity perturbed orbit transfers with shadow constraints

We present a new methodology to incorporate shadow-and time-triggered constraints within the indirect optimization methods to solve low-thrust fuel-optimal orbit transfer problems. Such constraints could represent, for instance, zero thrusting during an eclipse or a time interval during which the thruster has to be shut down during a mission science phase for data collection or communication purposes. Incorporation of the constraints is achieved in a straightforward manner through a hyperbolic tangent smoothing (HTS) method, which reduces the problem to a two-point boundary-value problem (TPBVP). A unique feature of the presented construct is that non-smooth components in the dynamics (e.g., engine throttle input and shadow-or time-triggered constraints) are all approximated by smooth representations. As a consequence of smoothing, the domain of convergence of the standard single-shooting methods used for solving the ensuing TP-BVPs is drastically enlarged. The utility of the method is demonstrated through a fixed-time rendezvous-type maneuver from a geostationary transfer orbit to a geostationary equatorial orbit, where a high-fidelity spherical harmonic gravity model of the Earth is used. Moreover, the system dynamics are propagated with the Picard-Chebyshev numerical integrator and the TPBVP is solved using the method of particular solutions. The proposed construct affords several avenues for computational speedup that has appealing numerical features making them suitable for trajectory optimization using high-fidelity models.