Skip to main content
Search
Search
Quick Search anywhere
Quick Search fdjslkfh
Advanced search
Cart
Join AIAA Institution Login
Skip main navigation
Journal of Guidance, Control, and Dynamics logo
Skip to article control options
No AccessFull-Length Papers

Exact Solution to Lambert’s Problem Using Contour Integrals

Published Online:22 Nov 2024https://doi.org/10.2514/1.G008499

Abstract

A novel method for solving Lambert’s problem based around the universal variable formulation is presented. By extending the transfer time function into the complex plane, an exact solution is derived for Lambert’s boundary value problem. This solution is a function of complex contour integrals of the transfer time function that must be approximated numerically. For integrals of this type, the composite trapezoidal rule is exponentially convergent and is used to approximate the solution to a high degree of accuracy for both zero- and n-revolution Lambert problems. Different families of contours are presented for the cases of hyperbolic, zero-revolution elliptic, and n-revolution elliptic transfers, and numerical examples are presented for all the cases. Monte Carlo analysis demonstrates a direct tradeoff between accuracy and computation speed based on the number of points used to approximate the integrals. When choosing a number of points with a balanced accuracy to speed, this method demonstrates an exceptionally fast computation speed, faster than all other methods we compare to in this paper.

References

  • [1] Lambert J. H., Insigniores Orbitae Cometarvm Proprietates, Svmtibvs Eberhardi Klett Vidvae, 1761. Google Scholar

  • [2] Albouy A. and Zhao L., “Lambert’s Theorem and Projective Dynamics,” Philosophical Transactions of the Royal Society A, Vol. 377, No. 2158, 2019, Paper 20180417. https://doi.org/10.1098/rsta.2018.0417 Google Scholar

  • [3] Battin R. H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, Reston, VA, 1999, Chap. 7. https://doi.org/10.2514/5.9781600861543.0295.0364 Google Scholar

  • [4] Gauss C. F., Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium, Vol. 7, FA Perthes, Gotha, Germany, 1877. Google Scholar

  • [5] Forbes E. G., “Gauss and the Discovery of Ceres,” Journal for the History of Astronomy, Vol. 2, No. 3, 1971, pp. 195–199. https://doi.org/10.1177/002182867100200305 Google Scholar

  • [6] Battin R. H. and Vaughan R. M., “An Elegant Lambert Algorithm,” Journal of Guidance, Control, and Dynamics, Vol. 7, No. 6, 1984, pp. 662–670. https://doi.org/10.2514/3.19910 LinkGoogle Scholar

  • [7] Lancaster E. and Blanchard R., “A Unified Form of Lambert’s Theorem,” NASA TN D-5368, 1969. Google Scholar

  • [8] Gooding R. H., “A Procedure for the Solution of Lambert’s Orbital Boundary-Value Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 48, No. 2, 1990, pp. 145–165. https://doi.org/10.1007/BF00049511 CrossrefGoogle Scholar

  • [9] Yin Y., Li Z., Liu C., Kang Z., Sun J. and Chen L., “Improved Initial Orbit Determination Based on the Gooding Method of Low Earth Orbit Space Debris Using Space-Based Observations,” Remote Sensing, Vol. 15, No. 21, 2023, p. 5217. https://doi.org/10.3390/rs15215217 CrossrefGoogle Scholar

  • [10] Izzo D., “Revisiting Lambert’s Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 121, No. 1, 2015, pp. 1–15. https://doi.org/10.1007/s10569-014-9587-y CrossrefGoogle Scholar

  • [11] Bate R. R., Mueller D. D., White J. E. and Saylor W. W., “Fundamentals of Astrodynamics,” Courier Dover Publ., Mineola, NY, 2020, Chap. 5. Google Scholar

  • [12] Arora N. and Russell R. P., “A Fast and Robust Multiple Revolution Lambert Algorithm Using a Cosine Transformation,” AAS/AIAA Astrodynamics Specialist Conference, AAS Paper 13-162, 2013. Google Scholar

  • [13] Russell R. P., “On the Solution to Every Lambert Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 131, No. 11, 2019, p. 50. https://doi.org/10.1007/s10569-019-9927-z CrossrefGoogle Scholar

  • [14] Thorne J. D., “Lambert’s Theorem – A Complete Series Solution,” Journal of the Astronautical Sciences, Vol. 52, No. 4, 2004, pp. 441–454. https://doi.org/10.1007/BF03546411 CrossrefGoogle Scholar

  • [15] Peterson G., Campbell E., Hawkins A., Balbas J., Ivy S., Merkurjev E., Hall T., Rodriguez P. and Gustafson E., “Relative Performance of Lambert Solvers 1: Zero Revolution Methods,” F. Landis Markley Astronautics Symposium, Vol. 136, No. 1, 1991, pp. 1495–1510. Google Scholar

  • [16] Torre Sangrà D.d l and Fantino E., “Review of Lambert’s Problem,” ISSFD 2015: 25th International Symposium on Space Flight Dynamics, DLR’s German Space Operations Center (GSOC) and ESA’s European Space Operations Centre (ESOC), Munich, Germany, Oct. 2015. Google Scholar

  • [17] Philcox O. H., Goodman J. and Slepian Z., “Kepler’s Goat Herd: An Exact Solution to Kepler’s Equation for Elliptical Orbits,” Monthly Notices of the Royal Astronomical Society, Vol. 506, No. 4, 2021, pp. 6111–6116. https://doi.org/10.1093/mnras/stab1296 Google Scholar

  • [18] Vallado D. A., Fundamentals of Astrodynamics and Applications, Vol. 12, Springer Science & Business Media, Berlin, 2001, Chap. 7. Google Scholar

  • [19] Jackson D., “Non-Essential Singularities of Functions of Several Complex Variables,” Annals of Mathematics, Vol. 17, No. 4, 1916, pp. 172–179. https://doi.org/10.2307/2007204 Google Scholar

  • [20] Jackson D., “Roots and Singular Points of Semi-Analytic Functions,” Annals of Mathematics, Vol. 19, No. 2, 1917, pp. 142–151. https://doi.org/10.2307/1967771 Google Scholar

  • [21] Ullisch I., “A Closed-Form Solution to the Geometric Goat Problem,” Mathematical Intelligencer, Vol. 42, No. 3, 2020, pp. 12–16. https://doi.org/10.1007/s00283-020-09966-0 Google Scholar

  • [22] Slepian Z. and Philcox O. H., “A Uniform Spherical Goat (Problem): Explicit Solution for Homologous Collapse’s Radial Evolution in Time,” Monthly Notices of the Royal Astronomical Society: Letters, Vol. 522, No. 1, 2023, pp. L42–L45. https://doi.org/10.1093/mnrasl/slac153 Google Scholar

  • [23] Trefethen L. N. and Weideman J. A., “The Exponentially Convergent Trapezoidal Rule,” SIAM Review, Vol. 56, No. 3, 2014, pp. 385–458. https://doi.org/10.1137/130932132 CrossrefGoogle Scholar

  • [24] Golub G. H. and Welsch J. H., “Calculation of Gauss Quadrature Rules,” Mathematics of Computation, Vol. 23, No. 106, 1969, pp. 221–230. https://doi.org/10.1090/S0025-5718-69-99647-1 CrossrefGoogle Scholar

  • [25] Davis P. J. and Rabinowitz P., Methods of Numerical Integration, Courier Corp., Mineola, NY, 2007, pp. 74–80. Google Scholar

  • [26] Negrete A. and Abdelkhalik O., “Elliptic Lambert’s Problem: An Exact Solution,” AAS/AIAA Astrodynamics Specialist Conference, AAS Paper 24-163, 2024. Google Scholar

  • [27] Russell R. P., “Complete Lambert Solver Including Second-Order Sensitivities,” Journal of Guidance, Control, and Dynamics, Vol. 45, No. 2, 2022, pp. 196–212. https://doi.org/10.2514/1.G006089 LinkGoogle Scholar

Previous article
Next article