The solution of the problem of impulsive minimization of a weighted sum of characteristic velocities of a spacecraft subject to linear equations of motion is presented without the use of calculus of variations or optimal control theory. The geometric structure of the set of boundary points associated with an optimal primer vector is found to be a simplex composed of convex conical sets. Eachk-dimensional open face of the simplex consists of boundary points having nondegeneratek-impulse solutions. This geometric structure leads to a simple proof that at mostn-impulses are required to solve a problem inn-dimensional space. This work is applied to the problem of planar rendezvous of a spacecraft with a satellite in Keplerian orbit using the Tschauner-Hempel equations of motion, with special emphasis on four-impulse solutions. Primer vectors representing four-impulse solutions are sought out and found for elliptical orbits, but none were found for orbits of higher eccentricity. For highly eccentric elliptical orbits, degenerate fiveimpulse solutions were found. In this situation, computer simulations reveal vastly different optimal trajectories having identical boundary conditions and cost.
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Communicated by D. G. Hull
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Carter, T.E., Brient, J. Linearized impulsive rendezvous problem. J Optim Theory Appl 86, 553–584 (1995). https://doi.org/10.1007/BF02192159
- primer vector
- linear problems
- rendezvous problems
- impulsive minimization
- conical sets