Optimization and Engineering

, 10:439 | Cite as

Generating locally optimal trajectories for an automatically driven car

  • Matthias Gerdts
  • Simon Karrenberg
  • Bernhard Müller-Beßler
  • Gregor Stock


The test-drive of an automobile along a given test-course can be modeled by formulating a suitable optimal control problem. However, if the length of the course is very long or if it has a very complicated structure, the numerical solution of the optimal control problem becomes very difficult. Therefore a moving horizon technique is employed, which splits the optimal control problem into a sequence of local optimal control problems that are combined by suitable continuity conditions. This approach yields a reference trajectory. A controller and differential GPS are integrated in a real-world car and allows a reference trajectory to be followed in real-time. A benefit of this approach is the very high accuracy obtained in reproducing the reference trajectory. Hence, it can be used for testing different setups of cars under the same conditions while excluding the comparatively large influence of a real-world driver. In this article, we will focus on a method for generating the reference trajectory and report our experiences with this algorithm. The method allows an locally optimal solution to be computed for various handling courses in a robust way.


Optimal control Automatic test-driving Direct discretization method Moving horizon 


  1. Betts JT (2001) Practical methods for optimal control using nonlinear programming. Advances in design and control, vol 3. SIAM, Philadelphia MATHGoogle Scholar
  2. Bock HG (1981) Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert KH, Deuflhard P, Jäger W (eds) Modelling of chemical reaction systems. Springer series in chemical physics, vol 18. Springer, Berlin Google Scholar
  3. Bock HG, Plitt KJ (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC worldcongress, Budapest, Hungary Google Scholar
  4. Büskens C (1998) Optimierungsmethoden und sensitivitätsanalyse für optimale steuerprozesse mit steuer- und zustandsbeschränkungen. PhD thesis, Fachbereich Mathematik, Westfälische Wilhems-Universität Münster Google Scholar
  5. Dontchev AL, Hager WW, Malanowski K (2000a) Error bounds for Euler approximation of a state and control constrained optimal control problem. Numer Funct Anal Optim 21(5 & 6):653–682 MATHCrossRefMathSciNetGoogle Scholar
  6. Dontchev AL, Hager WW, Veliov VM (2000b) Second-order RUnge-KUtta approximations in control constrained optimal control. SIAM J Numer Anal 38(1):202–226 MATHCrossRefMathSciNetGoogle Scholar
  7. Gerdts M (2003a) Direct shooting method for the numerical solution of higher index dae optimal control problems. J Optim Theory Appl 117(2):267–294 MATHCrossRefMathSciNetGoogle Scholar
  8. Gerdts M (2003b) A moving horizon technique for the simulation of automobile test-drives. ZAMM 83(3):147–162 MATHCrossRefMathSciNetGoogle Scholar
  9. Gerdts M (2005) Solving mixed-integer optimal control problems by branch & bound: a case study from automobile test-driving with gear shift. Optim Control Appl Methods 26(1):1–18 CrossRefMathSciNetGoogle Scholar
  10. Gerdts M (2006) OC-ODE—optimal control of ordinary differential equations: User’s guide. Technical report, Department Mathematik, Universität Hamburg Google Scholar
  11. Gill PE, Murray W, Saunders MA (2002) Snopt: an SQP algorithm for large-scale constrained optimization. SIAM J Optim 12(4):979–1006 MATHCrossRefMathSciNetGoogle Scholar
  12. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, London MATHGoogle Scholar
  13. Goh CJ, Teo KL (1988) Control parametrization: a unified approach to optimal control problems with general constraints. Automatica 24:3–18 MATHCrossRefMathSciNetGoogle Scholar
  14. Grötschel M, Krumke SO, Rambau J (2001) Online optimization of large scale systems. Springer, Berlin MATHGoogle Scholar
  15. Hager W (1990) Multiplier methods for nonlinear optimal control. SIAM J Numer Anal 27:1061–1080 MATHCrossRefMathSciNetGoogle Scholar
  16. Hager WW (2000) Runge-Kutta methods in optimal control and the transformed adjoint system. Numer Math 87(2):247–282 MATHCrossRefMathSciNetGoogle Scholar
  17. Hairer E, Norsett SP, Wanner G (1993) Solving ordinary differential equations I: Nonstiff problems. Springer series in computational mathematics, vol 8, 2nd edn. Springer, Berlin MATHGoogle Scholar
  18. Jennings LS, Fisher ME, Teo KL, Goh CJ (2004) MISER3 optimal control software version 3: Theory and user manual. Department of Mathematics, The University of Western Australia, Nedlands, Australia.
  19. Malanowski K, Büskens C, Maurer H (1997) Convergence of approximations to nonlinear optimal control problems. In: Fiacco A (ed) Mathematical programming with data perturbations. Lecture notes in pure and applied mathematics, vol 195. Dekker, New York, pp 253–284 Google Scholar
  20. Martin R, Teo K (1994) Optimal control of drug administration in cancer chemotherapy. World Scientific, Singapore. xiii, 187 p MATHGoogle Scholar
  21. Mayr R (1991) Verfahren zur Bahnfolgeregelung für ein automatisch geführtes Fahrzeug. PhD thesis, Fakultät für Elektrotechnik, Universität Dortmund Google Scholar
  22. Mitschke M (1990) Dynamik der Kraftfahrzeuge, Band c: Fahrverhalten, 2nd edn. Springer, Berlin Google Scholar
  23. Moder T (1994) Optimale Steuerung eines KFZ im fahrdynamischen Grenzbereich. Master’s thesis, Mathematisches Institut, Technische Universität München Google Scholar
  24. Müller-Beßler B, Stock G, Hoffmann J (2006) Reproducible driving near the stability limit. Technical report, presented at 2006 in Munich, Volkswagen AG, Wolfsburg, Germany Google Scholar
  25. Neculau M (1992) Modellierung des Fahrverhaltens: Informationsaufnahme, Regel- und Steuerstrategien in Experiment und Simulation. PhD thesis, Fachbereich 12: Verkehrswesen, Technische Universität Berlin Google Scholar
  26. Pacejka H, Bakker E (1993) The magic formula tyre model. Suppl Veh Syst Dyn 21:1–18 Google Scholar
  27. Polak E, Yang T, Mayne D (1993) A method of centers based on barrier function methods for solving optimal control problems with continuum state and control constraints. SIAM J Control Optim 31:159–179 MATHCrossRefMathSciNetGoogle Scholar
  28. Powell MJD (1978) A fast algorithm for nonlinearily constrained optimization calculation. In: Watson G (ed) Numerical analysis. Lecture notes in mathematics, vol 630. Springer, Berlin CrossRefGoogle Scholar
  29. Risse H-J (1991) Das Fahrverhalten bei normaler Fahrzeugführung. VDI Fortschrittberichte Reihe 12: Verkehrstechnik/Fahrzeugtechnik, vol 160. Springer, Verlag Google Scholar
  30. Schittkowski K (1983) On the convergence of a sequential quadratic programming method with an augmented Lagrangean line search function. Math Operationsforsch Stat Ser Optim 14(2):197–216 MATHMathSciNetGoogle Scholar
  31. Stoer J (1985) Principles of sequential quadratic programming methods for solving nonlinear programs. In: Schittkowski K (ed) Computational mathematical programming. NATO ASI series, vol F15. Springer, Berlin, pp 165–207 Google Scholar
  32. Teo K, Goh C, Wong K (1991) In: Pitman monographs and surveys in pure and applied mathematics. A unified computational approach to optimal control problems, vol 55. Wiley, New York. ix, 329 p. Google Scholar
  33. Teo KL, Jennings LS, Lee HWJ, Rehbock V (1999) The control parametrization enhancing transform for constrained optimal control problems. J Aust Math Soc Ser B 40:314–335 MATHCrossRefMathSciNetGoogle Scholar
  34. von Stryk O (1993) Numerical solution of optimal control problems by direct collocation. In: Optimal control. International series of numerical mathematics, vol 111. Birkhäuser, Basel, pp 129–143 Google Scholar
  35. Wu CZ, Teo KL (2006) Global impulsive optimal control computation. J Ind Manag Optim 2(4):435–450 MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Matthias Gerdts
    • 1
  • Simon Karrenberg
    • 2
  • Bernhard Müller-Beßler
    • 2
  • Gregor Stock
    • 2
  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.Volkswagen AGKonzernforschungWolfsburgGermany

Personalised recommendations