Skip to main content
SpringerLink
Log in

Generating locally optimal trajectories for an automatically driven car

  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

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.

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

Similar content being viewed by others

References

  • Betts JT (2001) Practical methods for optimal control using nonlinear programming. Advances in design and control, vol 3. SIAM, Philadelphia

    MATH  Google Scholar 

  • 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 

  • 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

  • Büskens C (1998) Optimierungsmethoden und sensitivitätsanalyse für optimale steuerprozesse mit steuer- und zustandsbeschränkungen. PhD thesis, Fachbereich Mathematik, Westfälische Wilhems-Universität Münster

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Gerdts M (2003b) A moving horizon technique for the simulation of automobile test-drives. ZAMM 83(3):147–162

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Gerdts M (2006) OC-ODE—optimal control of ordinary differential equations: User’s guide. Technical report, Department Mathematik, Universität Hamburg

  • Gill PE, Murray W, Saunders MA (2002) Snopt: an SQP algorithm for large-scale constrained optimization. SIAM J Optim 12(4):979–1006

    Article  MATH  MathSciNet  Google Scholar 

  • Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, London

    MATH  Google Scholar 

  • Goh CJ, Teo KL (1988) Control parametrization: a unified approach to optimal control problems with general constraints. Automatica 24:3–18

    Article  MATH  MathSciNet  Google Scholar 

  • Grötschel M, Krumke SO, Rambau J (2001) Online optimization of large scale systems. Springer, Berlin

    MATH  Google Scholar 

  • Hager W (1990) Multiplier methods for nonlinear optimal control. SIAM J Numer Anal 27:1061–1080

    Article  MATH  MathSciNet  Google Scholar 

  • Hager WW (2000) Runge-Kutta methods in optimal control and the transformed adjoint system. Numer Math 87(2):247–282

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    MATH  Google Scholar 

  • 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. http://www.cado.uwa.edu.au/miser

  • 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 

  • Martin R, Teo K (1994) Optimal control of drug administration in cancer chemotherapy. World Scientific, Singapore. xiii, 187 p

    MATH  Google Scholar 

  • Mayr R (1991) Verfahren zur Bahnfolgeregelung für ein automatisch geführtes Fahrzeug. PhD thesis, Fakultät für Elektrotechnik, Universität Dortmund

  • Mitschke M (1990) Dynamik der Kraftfahrzeuge, Band c: Fahrverhalten, 2nd edn. Springer, Berlin

    Google Scholar 

  • Moder T (1994) Optimale Steuerung eines KFZ im fahrdynamischen Grenzbereich. Master’s thesis, Mathematisches Institut, Technische Universität München

  • Müller-Beßler B, Stock G, Hoffmann J (2006) Reproducible driving near the stability limit. Technical report, presented at race.tech 2006 in Munich, Volkswagen AG, Wolfsburg, Germany

  • Neculau M (1992) Modellierung des Fahrverhaltens: Informationsaufnahme, Regel- und Steuerstrategien in Experiment und Simulation. PhD thesis, Fachbereich 12: Verkehrswesen, Technische Universität Berlin

  • Pacejka H, Bakker E (1993) The magic formula tyre model. Suppl Veh Syst Dyn 21:1–18

    Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Chapter  Google Scholar 

  • Risse H-J (1991) Das Fahrverhalten bei normaler Fahrzeugführung. VDI Fortschrittberichte Reihe 12: Verkehrstechnik/Fahrzeugtechnik, vol 160. Springer, Verlag

  • 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

    MATH  MathSciNet  Google Scholar 

  • 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 

  • 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 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Wu CZ, Teo KL (2006) Global impulsive optimal control computation. J Ind Manag Optim 2(4):435–450

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    Matthias Gerdts

  2. Volkswagen AG, Konzernforschung, 38436, Wolfsburg, Germany

    Simon Karrenberg, Bernhard Müller-Beßler & Gregor Stock

Authors
  1. Matthias Gerdts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon Karrenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernhard Müller-Beßler
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gregor Stock
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Gerdts.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gerdts, M., Karrenberg, S., Müller-Beßler, B. et al. Generating locally optimal trajectories for an automatically driven car. Optim Eng 10, 439–463 (2009). https://doi.org/10.1007/s11081-008-9047-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11081-008-9047-1

Keywords

Search

Navigation