Skip to main content
SpringerLink
Log in

real-time motion planning for multibody systems

Real life application examples

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

The solution of constrained motion planning is an important task in a wide number of application fields. The real-time solution of such a problem, formulated in the framework of optimal control theory, is a challenging issue. We prove that a real-time solution of the constrained motion planning problem for multibody systems is possible for practical real-life applications on standard personal computers.

The proposed method is based on an indirect approach that eliminates the inequalities via penalty formulation and solves the boundary value problem by a combination of finite differences and Newton–Broyden algorithm. Two application examples are presented to validate the method and for performance comparisons. Numerical results show that the approach is real-time capable if the correct penalty formulation and settings are chosen.

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

  1. Büskens, C., Knauer, M.: Hihger order real-time approximations in optimal control of multibody-systems for industrial robots. J. Multibody Syst. Dyn. 15, 85–106 (2006)

    Article  MATH  Google Scholar 

  2. Diehl, M., Findeisen, R., Allgöwer, F.: A stabilizing real-time implementation of nonlinear model predictive control. In: Biegler, L., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloem Wanders, B. (eds.) Proceedings of Sandia Real-Time PDE Optimization Workshop, Springer-Verlag (in press)

  3. Frazzoli, E., Dahleh, M.A., Feron, E.: Real-time motion planning for agile autonomous vehicles. In: AIAA Guidance, Navigation, and Control Conference and Exhibit. Number 4056 in AIAA (2000)

  4. Milam, M.B.: Real-Time Optimal Trajectory Generation for Constrained Dynamical Systems. Dissertation, California Institute of Technology (2003)

  5. Prasanth Kumar, R., Dasgupta, A., Kumar, C.S.: Real-time optimal motion planning for autonomous underwater vehicles. Ocean Eng. 32(11–12), 1431–1447 (2005)

    Article  Google Scholar 

  6. Bertolazzi, E., Biral, F., Da Lio, M.: Future advanced driver assistance systems based on optimal control: the influence of “risk functions” on overall system behavior and on prediction of dangerous situation. In: Proceedings of IEEE Intelligent Vehicles Symposium, pp. 386–390 (2004)

  7. Biral, F., Da Lio, M.: Modelling drivers with the optimal manoeuvre method. In: ATA 2001, 7th International Conference and Exhibition (2001)

  8. Biral, F., Da Lio, M., Bertolazzi, E.: Combining safety margins and user preferences into a driving criterion for optimal control-based computation of reference maneuvers for an ADAS of the next generation. In: Proceedings of IEEE Intelligent Vehicles Symposium, pp. 36–41 (2005)

  9. Bottasso, C., Chang, C.S., Croce, A., Leonello, D., Riviello, L.: Adaptive planning and tracking of trajectories for the simulation of maneuvers with multibody models. Comput. Methods Appl. Mech. Eng. 195(50–51), 7052–7072 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Petti, S., Fraichard, T.: Safe motion planning in dynamic environments. In: Proceedings of the IEEE-RSJ International Conference on Intelligent Robots and Systems, Edmonton, AB (2005)

  11. Bruce, J., Veloso, M.: Real-time randomized path planning for robot navigation. In: Proceedings of the IEEE-RSJ Intelligent Robots and Systems, pp. 2383–2388 (2002)

  12. LaValle, S.M., Kuffner, J.J.: Randomized kinodynamic planning. Int. J. Robot. Res. 20(5), 378–400 (2001)

    Article  Google Scholar 

  13. Cerven, W.T., Bullo, F., Coverstone, V.L.: Vehicle motion planning with time-varying constraints. J. Guid. Control Dyn. 27(3), 506–509 (2004)

    Article  Google Scholar 

  14. Cuthrell, J., Biegler, L.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1–2), 49–62 (1989)

    Article  Google Scholar 

  15. von Stryk, O., Bulirsch, R.: Direct and indirect methods for trajectory optimization. Ann. Oper. Res. 37(1–4), 357–373 (1992) Nonlinear methods in economic dynamics and optimal control (Vienna, 1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Veeraklaew, T., Agrawal, S.K.: New computational framework for trajectory optimization of high-order dynamic systems. J. Guid. Control Dyn. 24(2), 228–236 (2001)

    Google Scholar 

  17. Cervantes, A.M., Biegler, L.: Optimization strategies for dynamic systems. In: Encyclopedia of Optimization, vol. 4, pp. 216–227. Kluwer, Norwell, MA (2001)

    Google Scholar 

  18. Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193–207 (1998)

    Article  Google Scholar 

  19. Biegler, L., Cervantes, A.M., Wächter, A.: Advances in simultaneous strategies for dynamic process optimization. Chem. Eng. Sci. 57(4), 575–593 (2002)

    Article  Google Scholar 

  20. Bertolazzi, E., Biral, F., Da Lio, M.: Symbolic-numeric indirect method for solving optimal control problems for large multibody, the racing vehicle example. J. Multibody Syst. Dyn. 13, 233–252 (2005)

    Article  MATH  Google Scholar 

  21. Bertolazzi, E., Biral, F., Da Lio, M.: Symbolic-numeric efficient solution of optimal control problems for multibody systems. J. Comput. Appl. Math. 185(2), 404–421 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dennis, Jr., J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. In: Classics in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996) [Corrected reprint of the 1983 original]

    Google Scholar 

  23. Gould, N., Orban, D., Toint, P.: Numerical methods for large-scale nonlinear optimization. Acta Numer. 14, 299–361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nocedal, J., Wright, S.J.: Numerical optimization. In: Springer Series in Operations Research. Springer-Verlag, New York (1999)

    Google Scholar 

  25. Bouaricha, A., Schnabel, R.B.: Tensor methods for large sparse systems of nonlinear equations. Math. Program. A 82(3), 377–400 (1998)

    Article  MathSciNet  Google Scholar 

  26. Bader, B.W.: Tensor-Krylov methods for solving large-scale systems of nonlinear equations. SIAM J. Numer. Anal. 43(3), 1321–1347 (2005) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schnabel, R.B., Frank, P.D.: Tensor methods for nonlinear equations. SIAM J. Numer. Anal. 21(5), 815–843 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Potra, F.A., Shi, Y.: Efficient line search algorithm for unconstrained optimization. J. Optim. Theory Appl. 85(3), 677–704 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shi, Z.J., Shen, J.: Convergence of descent method with new line search. J. Appl. Math. Comput. 20(1–2), 239–254 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Deuflhard, P.: Newton methods for nonlinear problems. In: Springer Series in Computational Mathematics, vol. 35, Springer-Verlag, Berlin (2004)

    Google Scholar 

  31. Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton's method and extensions to related methods. SIAM J. Numer. Anal. 16(1), 1–10 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton's method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. Amodio, P., Cash, J.R., Roussos, G., Wright, R.W., Fairweather, G., Gladwell, I., Kraut, G.L., Paprzycki, M.: Almost block diagonal linear systems: sequential and parallel solution techniques, and applications. Numer. Linear Algebra Appl. 7(5), 275–317 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Amodio, P., Paprzycki, M.: A cyclic reduction approach to the numerical solution of boundary value ODEs. SIAM J. Sci. Comput. 18(1), 56–68 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Paprzychi, M., Gladwell, I.: Using level 3 BLAS to solve almost block diagonal systems. In: Parallel Processing for Scientific Computing, Houston, TX, 1991, pp. 52–62. SIAM, Philadelphia, PA (1992)

    Google Scholar 

  36. Díaz, J.C., Keast, G., Keast, P.: Algorithm 603: Colrow and arceco: Fortran packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination. ACM Trans. Math. Softw. (TOMS) 9(3), 376–380 (1983)

    Article  Google Scholar 

  37. Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statist. Comput. 11(3), 450–481 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bellavia, S., Morini, B.: A globally convergent Newton—GMRES subspace method for systems of nonlinear equations. SIAM J. Sci. Comput. 23(3), 940–960 (2001) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  39. Broyden, C.G., Dennis, Jr., J.E., Moré, J.J.: On the local and superlinear convergence of quasi-Newton methods. J. Inst. Math. Appl. 12, 223–245 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gay, D.M.: Some convergence properties of Broyden's method. SIAM J. Numer. Anal. 16(4), 623–630 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Structural Engineering, University of Trento, via Mesiano 77, I-38050, Trento, Italy

    Enrico Bertolazzi, Francesco Biral & Mauro Da Lio

Authors
  1. Enrico Bertolazzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Biral
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mauro Da Lio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Biral.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bertolazzi, E., Biral, F. & Lio, M.D. real-time motion planning for multibody systems. Multibody Syst Dyn 17, 119–139 (2007). https://doi.org/10.1007/s11044-007-9037-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-007-9037-7

Keywords

Search

Navigation