Abstract
The orbital boundary value problem, also known as Lambert problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to express the time of flight equation. In the new variable, the time of flight curves have two oblique asymptotes and they mostly appear to be conveniently approximated by piecewise continuous lines. We use and invert such a simple approximation to provide an efficient initial guess to an Householder iterative method that is then able to converge, for the single revolution case, in only two iterations. The resulting algorithm is compared, for single and multiple revolutions, to Gooding’s procedure revealing to be numerically as accurate, while having a significantly smaller computational complexity.
Similar content being viewed by others
References
Arora, N., Russell, R.: A fast and robust multiple revolution Lambert algorithm using a cosine transformation. In: Paper AAS 13–728, AAS/AIAA Astrodynamics Specialist Conference (2013)
Arora, N., Russell, R.P.: A gpu accelerated multiple revolution Lambert solver for fast mission design. In: AAS/AIAA Space Flight Mechanics Meeting, pp. 10–198. San Diego, CA, No. AAS (2010)
Bate, R., Mueller, D., White, J.: Fundamentals of Astrodynamics. Courier Dover Publications, New York (1971)
Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, New York (1999)
Gauss, C.: Theory of the motion of the heavenly bodies moving about the Sun in conic sections: a translation of Carl Frdr. Gauss “Theoria motus”: With an appendix. By Ch. H. Davis. Little, Brown and Comp (1857)
Gooding, R.: A procedure for the solution of Lambert’s orbital boundary-value problem. Celest. Mech. Dyn. Astron. 48(2), 145–165 (1990)
Izzo, D.: Lambert’s problem for exponential sinusoids. J. Guid. Control Dyn. 29(5), 1242–1245 (2006)
Klumpp, A.: Relative Performance of Lambert Solvers 1: Zero Revolution Methods. Tech. rep, Jet Propulsion Laboratory, No. 19930013090 (1991)
Lancaster, E., Blanchard, R.: A Unified Form of Lambert’s Theorem. National Aeronautics and Space Administration, Washington (1969)
Luo, Q., Meng, Z., Han, C.: Solution algorithm of a quasi-Lambert’s problem with fixed flight-direction angle constraint. Celest. Mech. Dyn. Astron. 109(4), 409–427 (2011)
Parrish N.L. (2012) Accelerating Lambert’s problem on the gpu in Matlab. PhD thesis, California Polytechnic State University
Peterson, G., et al.: Relative performance of Lambert solvers 1: zero revolution methods. Adv. Astronaut. Sci. 136(1), 1495–1510 (1991)
Rauh, A., Parisi, J.: Near distance approximation in astrodynamical applications of Lambert’s theorem. Celest. Mech. Dyn. Astron. 118(1), 49–74 (2014)
Wagner, S., Wie, B.: GPU accelerated Lambert solution methods for the orbital targeting problem. Spacefl. Mech. 140 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Izzo, D. Revisiting Lambert’s problem. Celest Mech Dyn Astr 121, 1–15 (2015). https://doi.org/10.1007/s10569-014-9587-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-014-9587-y