Skip to main content
SpringerLink
Log in

Maximum Divert for Planetary Landing Using Convex Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper presents a real-time solution method of the maximum divert trajectory optimization problem for planetary landing. In mid-course, the vehicle is to abort and retarget to a landing site as far from the nominal as physically possible. The divert trajectory must satisfy velocity constraints in the range and cross range directions and a total speed constraint. The thrust magnitude is bounded above and below so that once on, the engine cannot be turned off. Because this constraint is not convex, it is relaxed to a convex constraint and lossless convexification is proved. A transformation of variables is introduced in the nonlinear dynamics and an approximation is made so that the problem becomes a second-order cone problem, which can be solved to global optimality in polynomial time whenever a feasible solution exists. A number of examples are solved to illustrate the effectiveness and efficiency of the solution method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Açıkmeşe, B., Ploen, S.R.: Convex programming approach to powered descent guidance for Mars landing. AIAA J. Guid. Control Dyn. 30(5), 1353–1366 (2007)

    Article  Google Scholar 

  2. Blackmore, L., Açıkmeşe, B., Scharf, D.P.: Minimum landing error powered descent guidance for Mars landing using convex optimization. AIAA J. Guidance Control Dyn. 33(4) (2010)

  3. Açıkmeşe, B., Aung, M., Casoliva, J., Mohan, S., Johnson, A., Scharf, D., Masten, D., Scotkin, J., Wolf, A., Regehr, M.W.: Flight testing of trajectories computed by G-FOLD: fuel optimal large divert guidance algorithm for planetary landing. In: AAS/AIAA Spaceflight Mechanics Meeting (2013)

    Google Scholar 

  4. Acikmese, B., Carson, J., Blackmore, L.: Lossless convexification of nonconvex control bound and pointing constraints of the soft landing optimal control problem. IEEE Trans. Control Syst. Technol. 21(6), 2104–2113 (2013)

    Article  Google Scholar 

  5. Hartl, R., Sethi, S., Vickson, R.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Pergamon, Elmsford (1964)

    MATH  Google Scholar 

  7. Hull, D.G.: Thrust programs for minimum propellant consumption during take-off and landing maneuvers of a rocket vehicle in a vacuum. Tech. Rep. 59. Boeing Scientific Research Laboratories, Flight Sciences Laboratory (1962)

  8. Hull, D.G.: Thrust programs for minimum propellant consumption during the vertical take-off and landing of a rocket. J. Optim. Theory Appl. 1(1), 53–69 (1967)

    Article  MathSciNet  Google Scholar 

  9. Miele, A.: Application of Green’s theorem to the extremization of linear integrals. Tech. Rep. 40. Boeing Scientific Research Laboratories, Flight Sciences Laboratory (1961)

  10. Meditch, J.S.: On the problem of optimal thrust programming for a lunar soft landing. IEEE Trans. Autom. Control AC-9(4), 477–484 (1964)

    Article  MathSciNet  Google Scholar 

  11. Hull, D.G.: Optimal guidance for quasi-planar lunar descent with throttling. Presented at the 21st AAS/AIAA Space Flight Mechanics Meeting. AAS 11–169 (2011)

  12. Hull, D.G.: Optimal guidance for quasi-planar lunar descent with throttling. In: Proceedings of the 21st AAS/AIAA Spaceflight Mechanics Meeting. Advances in the Astronautical Sciences Series, vol. 140. Univelt, San Diego (2011)

    Google Scholar 

  13. Harris, M.W., Hull, D.G.: Optimal solutions for quasi-planar ascent over a spherical Moon. J. Guid. Control Dyn. 35(4), 1218–1223 (2012)

    Article  Google Scholar 

  14. Klumpp, A.R.: Apollo lunar descent guidance. Automatica 10, 133–146 (1974)

    Article  Google Scholar 

  15. Martin, D., Sievers, R., O’Brien, R., Rice, A.: Saturn V guidance, navigation, and targeting. J. Spacecr. Rockets 4(7), 891–898 (1967)

    Article  Google Scholar 

  16. Chandler, D., Smith, I.: Development of the iterative guidance mode with its applications to various vehicles and missions. J. Spacecr. Rockets 4(7), 898–903 (1967)

    Article  Google Scholar 

  17. McHenry, R.L., Long, A.D., Cockrell, B.F., Thibodeau, J.R. III, Brand, T.J.: Space shuttle ascent guidance, navigation, and control. Journal of the Astronautical Sciences 27(1), 1–38 (1979)

    Google Scholar 

  18. Topcu, U., Casoliva, J., Mease, K.: Fuel efficient powered descent guidance for Mars landing. J. Spacecr. Rockets 44(2) (2007)

  19. Sostaric, R., Rea, J.: Powered descent guidance methods for the Moon and Mars. In: AIAA Guidance, Navigation, and Control Conference San Francisco, CA (2005)

  20. Hull, D.: Conversion of optimal control problems into parameter optimization problems. J. Guid. Control Dyn. 20(1), 57–60 (1997)

    Article  MATH  Google Scholar 

  21. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1995)

    Article  MathSciNet  Google Scholar 

  22. Açıkmeşe, B., Blackmore, L.: Lossless convexification for a class of optimal control problems with nonconvex control constraints. Automatica 47(2), 341–347 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Harris, M.W., Açıkmeşe, B.: Minimum time rendezvous of multiple spacecraft using differential drag. J. Guid. Control Dyn. (2013). To Appear

  24. Lu, P., Liu, X.: Autonomous trajectory planning for rendezvous and proximity operations by conic optimization. J. Guid. Control Dyn. 36(2), 375–389 (2013)

    Article  Google Scholar 

  25. Lee, U., Mesbahi, M.: Spacecraft reorientation in presence of attitude constraints via logarithmic barrier potentials. In: American Control Conference (2011)

    Google Scholar 

  26. Mattingley, J., Boyd, S.: Cvxgen: a code generator for embedded convex optimization. Optim. Eng. 13(1), 1–27 (2012)

    Article  MathSciNet  Google Scholar 

  27. Chu, E., Parikh, N., Domahidi, A., Boyd, S.: Code generation for embedded second-order cone programming. In: European Control Conference, Zurich, Switzerland (2013)

    Google Scholar 

  28. Domahidi, A., Chu, E., Boyd, S.: Ecos: an SOCP solver for embedded systems. In: European Control Conference, Zurich, Switzerland (2013)

    Google Scholar 

  29. Grant, M., Boyd, S.C.: Matlab software for disciplined convex programming, version 2.0 beta (2012). http://cvxr.com/cvx

  30. Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control. Lecture Notes in Control and Information Sciences, pp. 95–110. Springer, Berlin (2008)

    Chapter  Google Scholar 

  31. Nesterov, Y., Nemirovsky, A.: Interior-point Polynomial Methods in Convex Programming. SIAM, Providence (1994)

    Book  Google Scholar 

  32. Berkovitz, L.D.: On control problems with bounded state variables. J. Math. Anal. Appl. 5(3), 488–498 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  33. Bryson, A.E., Denham, W.F., Dreyfus, S.E.: Optimal programming problems with inequality constraints I: Necessary conditions for extremal solutions. AIAA J. 1(11), 2544–2550 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  34. Denham, W.F., Bryson, A.E.: Optimal programming problems with inequality constraints II: Solution by steepest-ascent. AIAA J. 2(1), 25–34 (1964)

    Article  MathSciNet  Google Scholar 

  35. Berkovitz, L.D., Dreyfus, S.E.: The equivalence of some necessary conditions for optimal control in problems with bounded state variables. J. Math. Anal. Appl. 10(2), 275–283 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  36. Speyer, J.L., Mehra, R.K., Bryson, A.E.: The separate computation of arcs for optimal flight paths with state variable inequality constraints. Tech. Rep. 256. National Aeronautics and Space Administration (1967)

  37. McIntyre, J., Paiewonsky, B.: Optimal control with bounded state variables. In: Advances in Control Systems, vol. 5, pp. 389–419. Academic Press, New York (1967)

    Google Scholar 

  38. Speyer, J.L., Bryson, A.E.: Optimal programming problems with a bounded state space. AIAA J. 6(8), 1488–1491 (1968)

    Article  MATH  Google Scholar 

  39. Jacobson, D., Lele, M.: A transformation technique for optimal control problems with a state variable inequality constraint. IEEE Trans. Autom. Control 14(5), 457–464 (1969)

    Article  Google Scholar 

  40. Bell, D.J.: A transformation of partially singular trajectories to state constrained arcs. Int. J. Control 28(1), 63–66 (1978)

    Article  MATH  Google Scholar 

  41. Jacobson, D., Lele, M., Speyer, J.: New necessary conditions of optimality for control problems with state variable inequality constraints. J. Math. Anal. Appl. 35(2), 255–284 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  42. Maurer, H.: On the minimum principle for optimal control problems with state constraints. Schriftenreihe des Rechenzentrums der Univ. Munster 41 (1979)

  43. Milyutin, A., Osmolovskii, N.: Calculus of Variations and Optimal Control. American Mathematical Socity, Providence (1998)

    MATH  Google Scholar 

  44. Rugh, W.J.: Linear System Theory, 2nd edn. Prentice Hall, New York (1996)

    MATH  Google Scholar 

  45. Kim, J.H.R., Maurer, H.: Sensitivity analysis of optimal control problems with bang-bang controls. In: Decision and Control. 42nd IEEE Conference on Proceedings, vol. 4, pp. 3281–3286 (2003).

    Google Scholar 

  46. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  47. Yildirim, A.E., Todd, M.: Sensitivity analysis in linear programming and semidefinite programming using interior-point methods. Math. Program. 90(2), 229–261 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, 1 University Station, C0600, Austin, TX, 78712, USA

    Matthew W. Harris & Behçet Açıkmeşe

Authors
  1. Matthew W. Harris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Behçet Açıkmeşe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew W. Harris.

Additional information

Communicated by David G. Hull.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harris, M.W., Açıkmeşe, B. Maximum Divert for Planetary Landing Using Convex Optimization. J Optim Theory Appl 162, 975–995 (2014). https://doi.org/10.1007/s10957-013-0501-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0501-7

Keywords

Search

Navigation