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Mathematical Problems in Engineering
Volume 2013, Article ID 502610, 7 pages
Research Article

Trajectory Estimation of Aircraft in a Double-Satellite Passive Positioning System with the Adjoint Method

1Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266003, China
2Laboratory of Coast and Island Development, Nanjing University, Nanjing 210093, China

Received 7 March 2013; Accepted 15 May 2013

Academic Editor: Antonio F. Bertachini A. Prado

Copyright © 2013 Anzhou Cao 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.


A double-satellite passive positioning system is constructed based on the theory of space geometry, where two observation coordinate systems and a fundamental coordinate system exist. In each observation coordinate system, there exists a ray from the observation satellite to the aircraft. One difficulty lies in that these two rays may not intersect due to the existence of various errors. Under this situation, this work assumes that the middle point of common perpendicular between two rays is the actual position of aircraft. Based on the theory of space geometry, the coordinates of aircraft in the fundamental coordinate system can be determined. A dynamic model with the adjoint method is developed to estimate the trajectory of aircraft during the process of rocket propulsion. By assimilating observations, the trajectory of aircraft can be calculated. Numerical experiments are designed to validate the reasonability and feasibility of this model. Simulated results indicate that even by assimilating a small number of observations, the trajectory of aircraft can be estimated. In addition, the trajectory estimation can become more accurate when more observations are assimilated to the model.