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Mathematical Problems in Engineering
Volume 2013, Article ID 435725, 8 pages
Research Article

Solution for Nonlinear Three-Dimensional Intercept Problem with Minimum Energy

1Department of Aerospace Engineering, Chosun University, Gwangju 501-759, Republic of Korea
2Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA

Received 5 July 2013; Accepted 23 September 2013

Academic Editor: Mufid Abudiab

Copyright © 2013 Henzeh Leeghim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Classical orbit intercept applications are commonly formulated and solved as Lambert-type problems, where the time-of-flight (TOF) is prescribed. For general three-dimensional intercept problems, selecting a meaningful TOF is often a difficult and an iterative process. This work overcomes this limitation of classical Lambert’s problem by reformulating the intercept problem in terms of a minimum-energy application, which then generates both the desired initial interceptor velocity and the TOF for the minimum-energy transfer. The optimization problem is formulated by using the classical Lagrangian and coefficients, which map initial position and velocity vectors to future times, and a universal time variable . A Newton-Raphson iteration algorithm is introduced for iteratively solving the problem. A generalized problem formulation is introduced for minimizing the TOF as part of the optimization problem. Several examples are presented, and the results are compared with the Hohmann transfer solution approaches. The resulting minimum-energy intercept solution algorithm is expected to be broadly useful as a starting iterative for applications spanning: targeting, rendezvous, interplanetary trajectory design, and so on.