Optimal Starting Conditions for the Rendezvous Maneuver, Part 1: Optimal Control Approach |
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Authors: | A Miele M Ciarcià |
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Institution: | (1) Aero-Astronautics Group, Rice University, Houston, TX 77005-1892, USA |
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Abstract: | We consider the three-dimensional rendezvous between two spacecraft: a target spacecraft on a circular orbit around the Earth
and a chaser spacecraft initially on some elliptical orbit yet to be determined. The chaser spacecraft has variable mass,
limited thrust, and its trajectory is governed by three controls, one determining the thrust magnitude and two determining
the thrust direction. We seek the time history of the controls in such a way that the propellant mass required to execute
the rendezvous maneuver is minimized. Two cases are considered: (i) time-to-rendezvous free and (ii) time-to-rendezvous given,
respectively equivalent to (i) free angular travel and (ii) fixed angular travel for the target spacecraft.
The above problem has been studied by several authors under the assumption that the initial separation coordinates and the
initial separation velocities are given, hence known initial conditions for the chaser spacecraft. In this paper, it is assumed
that both the initial separation coordinates and initial separation velocities are free except for the requirement that the
initial chaser-to-target distance is given so as to prevent the occurrence of trivial solutions.
Analyses performed with the multiple-subarc sequential gradient-restoration algorithm for optimal control problems show that
the fuel-optimal trajectory is zero-bang, namely it is characterized by two subarcs: a long coasting zero-thrust subarc followed
by a short powered max-thrust braking subarc. While the thrust direction of the powered subarc is continuously variable for
the optimal trajectory, its replacement with a constant (yet optimized) thrust direction produces a very efficient guidance
trajectory: Indeed, for all values of the initial distance, the fuel required by the guidance trajectory is within less than
one percent of the fuel required by the optimal trajectory.
For the guidance trajectory, because of the replacement of the variable thrust direction of the powered subarc with a constant
thrust direction, the optimal control problem degenerates into a mathematical programming problem with a relatively small
number of degrees of freedom, more precisely: three for case (i) time-to-rendezvous free and two for case (ii) time-to-rendezvous
given.
In particular, we consider the rendezvous between the Space Shuttle (chaser) and the International Space Station (target).
Once a given initial distance SS-to-ISS is preselected, the present work supplies not only the best initial conditions for
the rendezvous trajectory, but simultaneously the corresponding final conditions for the ascent trajectory. |
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Keywords: | Space trajectories Rendezvous Optimization Guidance Optimal control Calculus of variations Mayer problems Bolza problems Transformation techniques Multiple-subarc sequential gradient-restoration algorithm |
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