# Higher Order Algorithm for Solving Lambert’s Problem

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## Abstract

This work presents a high-order perturbation expansion method for solving Lambert’s problem. The necessary condition for the problem is defined by a fourth-order Taylor expansion of the terminal error vector. The Taylor expansion partial derivative models are generated by Computational Differentiation (CD) tools. A novel derivative enhanced numerical integration algorithm is presented for computing nonlinear state transition tensors, where only the equation of motion is coded. A high-order successive approximation algorithm is presented for inverting the problems nonlinear necessary condition. Closed-form expressions are obtained for the first, second,third, and fourth order perturbation expansion coefficients. Numerical results are presented that compare the convergence rate and accuracy of first-through fourth-order expansions. The initial p-iteration starting guess is used as the Lambert’s algorithm initial condition. Numerical experiments demonstrate that accelerated convergence is achieved for the second-, third-, and fourth-order expansions, when compared to a classical first-order Newton method.

## Keywords

Lambert’s problem Computational differentiation p-iteration## References

- 1.Lancaster, E.R., Blanchard, R.C.: A unified form of Lambert’s theorem. NASA technical note TN D-5368,1969Google Scholar
- 2.Gooding, R.H.: A procedure for the solution of Lambert’s orbital boundary-value problem. Celest. Mech. Dyn. Astron.
**48**, 145–165 (1990)zbMATHGoogle Scholar - 3.Woollands, R.M., Read, J.L., Macomber, B., Probe, A., Younes, A.B., Junkins, J.L.: Method of Particular Solutions and Kustaanheimo-Stiefel Regularized Picard Iteration for Solving Two-Point Boundary Value Problems. Paper No. AAS 15-373 Presented at the 25th AAS/AIAA Space Flight Mechanics Meeting, Williamsburg (2015)Google Scholar
- 4.Woollands, R., Younes, A.B., Junkins, J.: A new Solution for the General Lambert’s Problem. 37th Annual AAS Guidance Control Conference (2014)Google Scholar
- 5.Woollands, R.M., Younes, A.B., Junkins, J.: New solutions for the perturbed lambert problem using regularization and picard iteration. JGCD
**38**, 1548–1562 (2015)Google Scholar - 6.Woollands, R.M., Read, J.L., Probe, A.B., Junkins, J.L.: Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration, JAS, 1–18. ISSN: 0021-9142. (2017). https://doi.org/10.1007/s40295-017-0116-6
- 7.Battin, R.: An introduction to the mathematics and methods of astrodynamics AIAA. Education Series (1999)Google Scholar
- 8.Schaub, H., Junkins, J.L.: Analytical Mechanics of Space Systems, 2nd edn. AIAA Education Series, Editor-in-Chief Joheph A. Schetz (2009)Google Scholar
- 9.Bani Younes, A., Turner, J.: Derivative Enhanced Optimal Feedback Control Using Computational Differentiation. Int. J. Appl. Exper. Math.
**1**, 112 (2016). https://doi.org/10.15344/ijaem/2016/112 Google Scholar - 10.Bani Younes, A., Turner, J.: Generalized algorithms for least squares optimization for nonlinear observation models and newton’s method. J. Astron. Sci.
**60**(3), 517–540 (2013)CrossRefGoogle Scholar - 11.Bani Younes, A., Turner, J.: Semi-Analytic Probability density function for system uncertainty, vol. 2 (2016)Google Scholar
- 12.Bani Younes, A., Alhulayil, M., Turner, J.: Efficient Uncertainty Propagation of Perturbed Satellite Motion. 27
^{th}AAS/AIAA Space Flight Mechanics Meeting AAS 17–266, San Antonio (2017)Google Scholar - 13.Alhulayil, M., Bani Younes, A., Turner, J.: Higher-Order Differential Correction Solver for Perturbed Lambert’s Problem. 27
^{th}AAS/AIAA Space Flight Mechanics Meeting AAS 17–266, San Antonio (2017)Google Scholar - 14.Bani Younes, A., Turner, J.: System uncertainty propagation using automatic differentiation. Proceedings of the ASME 2015 International Mechanical Engineering Technical Congress and Exposition, IMECE2015-51439, Houston (2015)Google Scholar
- 15.Bani Younes, A., Turner, J.: High-order State Transition Tensors of Perturbed Orbital Motion using Computational Differentiation. 26
^{th}AAS/AIAA Space Flight Mechanics Meeting, AAS 16-342, Napa (2016)Google Scholar - 16.Bani Younes, A., Turner, J.: Feedback control sensitivity calculations using computational differentiation. Proceedings of the ASME 2015 International Mechanical Engineering Technical Congress and Exposition, IMECE2015-51439, HoustonGoogle Scholar
- 17.Wengert, R.E.: A simple automatic derivative evaluation program. Comm. AGM
**7**(8), 463–464 (1964)zbMATHGoogle Scholar - 18.Wilkins, R.D.: Investigation of a new analytical method for numerical derivative evaluation. Comm ACM
**7,8**, 465–471 (1964)CrossRefzbMATHGoogle Scholar - 19.Griewank, A.: On Automatic Differentiation. In: Iri, M., Tanabe, K. (eds.) Mathematical Programming: Recent Developments and Applications, pp 83–108. Kluwer Academic Publishers, Amsterdam (1989)Google Scholar
- 20.Bischof, C., Carle, A., Corliss, G., Griewank, A., Hovland, P.: ADIFOR: Generating Derivative codes from fortran programs. Sci. Program.
**1**, 1–29 (1992)Google Scholar - 21.Bischof, C., Carle, A., Khademi, P., Mauer, A., Hovland, P.: ADIFOR 2.0 User’s Guide (Revision CO, Technical Report ANL/MCS-TM-192. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne (1995)Google Scholar
- 22.Eberhard, P.: C. Bischof. Automatic Differentiation of Numerical Integration Algorithms Technical Report ANL/MCS-P621-1196. Mathematics and Computer Science Division Argonne National Laboratory, Argonne (1996)Google Scholar
- 23.Turner, J.D.: Automated generation of High-Order partial derivative models. AIAA J.
**41**(8), 1590–1599 (2003)CrossRefGoogle Scholar - 24.Bani Younes, A.H., Turner, J.D., Majji, M., Junkins, J.L.: An Investigation of State Feedback Gain Sensitivity Calculations. Presented to AIAA/AAS Astrodynamics Specialist Conference of Held 2-5, Toronto (2010)Google Scholar
- 25.Turner, J.D., Bani Younes, A.H.: On the Integration of m-Dimensional Expectation Operators. Presented to AIAA Houston Annual Technical Symposium, Gilruth Center, NASA/JSC (2012)Google Scholar
- 26.Junkins, J.L.: Investigation of Finite-Element representations of the geopotential. AIAA J.
**14**(6), 803–808 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Pines, S.: Uniform Representation of the Gravitational Potential and its Derivatives. AIAA J.
**11**, 15081511 (1973)zbMATHGoogle Scholar - 28.Junkins, J.L., Bani Younes, A., Woollands, R., Bai, X.: Picard iteration, chebyshev polynomials and chebyshev picard methods: Application in astrodynamics. J. Astron. Sci.
**60**(3), 623–653 (2015)Google Scholar