Skip to main content
SpringerLink
Log in

Motion Control Algorithms for Simple Mechanical Systems with Symmetry

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

We treat underactuated mechanical control systems with symmetry, taking the viewpoint of the affine connection formalism. We first review the appropriate notions and tests of controllability associated with these systems, including that of fiber controllability. Secondly, we present a series expansion describing the evolution of the trajectories of general mechanical control systems starting from nonzero velocity. This series is then used to investigate the behavior of the system under small-amplitude periodic forcing. On this basis, motion control algorithms are designed for systems with symmetry to solve the tasks of point-to-point reconfiguration, static interpolation and stabilization problems. Several examples are given and the performance of the algorithms is illustrated in the blimp system.

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. Abraham, R. and Marsden, J. E.: Foundations of Mechanics, 2nd edn, Benjamin-Cummings, Reading, 1978.

  2. Agračhev, A. A. and Gamkrelidze, R. V.: The exponential representation of flows and the chronological calculus, Math. USSR Sb. 35(6) (1978), 727–785.

    Google Scholar 

  3. Agračhev, A. A. and Gamkrelidze, R. V.: Local controllability and semigroups of diffeomorphisms, Acta Appl. Math. 32(1993), 1–57.

    Google Scholar 

  4. Bloch, A. M. and Crouch, P. E.: Nonholonomic control systems on Riemannian manifolds, SIAM J. Control Optim. 33(8) (1995), 541–552.

    Google Scholar 

  5. Bloch, A. M. and Crouch, P. E.: Newton's law and integrability on nonholonomic systems, SIAM J. Control Optim. 36(1998), 2020–2039.

    Google Scholar 

  6. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E. and Murray, R. M.: Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. Anal. 136(1996), 21–99.

    Google Scholar 

  7. Bloch, A. M., Reyhanoglu, M. and McClamroch, N. H.: Control and stabilization of nonholonomic dynamic systems, IEEE Trans. Automatic Control 37(11) (1992), 1746–1757.

    Google Scholar 

  8. Bullo, F.: Series expansions for the evolution of mechanical control systems, SIAM J. Control Optim. 40(1) (2001), 166–190.

    Google Scholar 

  9. Bullo, F., Leonard, N. E. and Lewis, A. D.: Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups, IEEE Trans. Automatic Control 45(8) (2000), 1437–1454.

    Google Scholar 

  10. Bullo, F. and Lewis, A. D.: Configuration controllability of mechanical systems on Lie groups, In: Mathematical Theory of Networks and Systems (MTNS'96), St. Louis, Missouri, 1996.

  11. Bullo, F. and Lynch, K. M.: Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems, IEEE Trans. Robotics & Automation 17(4) (2001), 402–412.

    Google Scholar 

  12. Bullo, F. and Zefran, M.: On mechanical control systems with nonholonomic constraints and symmetries, Systems Control Lett. 45(2) (2002), 133–143.

    Google Scholar 

  13. Chen, K. T.: Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula, Ann. Math. 67(1957), 8615–8645.

    Google Scholar 

  14. Cortés, J. and Martínez, S.: Configuration controllability of mechanical systems underactuated by one control, Available at: http://motion.csl.uiuc.edu/~jorge/papers

  15. Cortés, J., Martínez, S., Ostrowski, J. P. and Zhang, H.: Simple mechanical control systems with constraints and symmetry, SIAM J. Control Optim. 41(3) (2002), 851–874.

    Google Scholar 

  16. Crouch, P. E.: Geometric structures in systems theory, Proc. IEE 128D(5) (1981), 242–252.

    Google Scholar 

  17. Fliess, M.: Fonctionelles causales non linéares et indeterminées non commutatives, Bull. Soc. Math. 109 (1981), 3–40.

    Google Scholar 

  18. Hermes, H.: Control systems which generate decomposable Lie algebras, J. Differential Equations 44 (1982), 166–187.

    Google Scholar 

  19. Isidori, A.: Nonlinear Control Systems, 2nd edn, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  20. Kawski, M.: High-order small-time local controllability, In: H. J. Sussmann (ed.), Nonlinear Controllability and Optimal Control, New York, 1990, pp. 441–477.

  21. Kawski, M. and Sussmann, H. J.: Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, In: U. Helmke, D. Pratzel-Wolters and E. Zerz (eds), Operators, Systems and Linear Algebra, Teubner, Berlin, 1997, pp. 111–128.

  22. Kelly, S. D. and Murray, R. M.: Geometric phases and robotic locomotion, J. Robotic Systems 12(6) (1995), 417–431.

    Google Scholar 

  23. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Interscience, New York, 1963.

    Google Scholar 

  24. Koon, W. S. and Marsden, J. E.: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems, Rep. Math. Phys. 40 (1997), 21–62.

    Google Scholar 

  25. Lafferriere, G. and Sussmann, H. J.: A differential geometric approach to motion planning, In: Z. X. Li and J. F. Canny (eds), Nonholonomic Motion Planning, Kluwer, Boston, 1993, pp. 235–270.

    Google Scholar 

  26. Leonard, N. E. and Krishnaprasad, P. S.: Motion control of drift-free, left-invariant systems on Lie groups, IEEE Trans. Automatic Control 40(9) (1995), 1539–1554.

    Google Scholar 

  27. Lewis, A. D.: Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys. 42(1/2) (1998), 135–164.

    Google Scholar 

  28. Lewis, A. D.: When is a mechanical control system kinematic?, In: Proc. IEEE Int. Conf. Decision & Control, Phoenix, U.S.A., 1999, pp. 1162–1167.

  29. Lewis, A. D.: Simple mechanical control systems with constraints, IEEE Trans. Automatic Control 45(8) (2000), 1420–1436.

    Google Scholar 

  30. Lewis, A. D. and Murray, R. M.: Configuration controllability of simple mechanical control systems, SIAM J. Control Optim. 35(3) (1997), 766–790.

    Google Scholar 

  31. Lewis, A. D., Ostrowski, J. P., Murray, R. M. and Burdick, J.W.: Nonholonomic mechanics and locomotion: the Snakeboard example, In: IEEE Int. Conf. Robotics & Automation, San Diego, CA, 1994, pp. 2391–2397.

  32. Magnus, W.: On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649–673.

    Google Scholar 

  33. Marsden, J. E. and Ratiu, T. S.: Introduction to Mechanics and Symmetry, 2nd edn, Springer-Verlag, New York, 1999.

    Google Scholar 

  34. Martínez, S.: Geometric methods in nonlinear control theory with applications to dynamic robotic systems, PhD dissertation, Universidad Carlos III de Madrid, Madrid, Spain, 2002.

    Google Scholar 

  35. Montgomery, R.: Nonholonomic control and gauge theory, In: Z. X. Li and J. F. Canny (eds), Nonholonomic Motion Planning, Kluwer, Boston, MA, 1993, pp. 343–378.

    Google Scholar 

  36. Murray, R. M., Li, Z. X. and Sastry, S. S.: A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Ratón, 1994.

    Google Scholar 

  37. Nijmeijer, H. and van der Schaft, A. J.: Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990.

    Google Scholar 

  38. Ostrowski, J. P.: Geometric perspectives on the mechanics and control of undulatory locomotion, PhD dissertation, California Institute of Technology, Pasadena, CA, 1995. Available at http://www.grip.cis.edu/~jpo.

    Google Scholar 

  39. Ostrowski, J. P. and Burdick, J. W.: Controllability tests for mechanical systems with symmetries and constraints, J. Appl. Math. Comput. Sci. 7(2) (1997), 101–127.

    Google Scholar 

  40. Ostrowski, J. P. and Burdick, J.W.: The geometric mechanics of undulatory robotic locomotion, Int. J. Robotics Res. 17(7) (1998), 683–701.

    Google Scholar 

  41. Reyhanoglu, M., van der Schaft, A. J., Kolmanovsky, I. V. and McClamroch, N. H.: Dynamics and control of a class of underactuated mechanical systems, IEEE Trans. Automatic Control 44 (1999), 1663–1671.

    Google Scholar 

  42. Rui, C., Kolmanovsky, I. V., McNally, P. J. and McClamroch, N. H.: Attitude control of underactuated multibody spacecraft, In: Proc. IFAC World Conference, San Francisco, CA, 1996.

  43. van der Schaft, A. J.: Linearization of Hamiltonian and gradient systems, IMA J. Math. Control & Information 1 (1984), 185–198.

    Google Scholar 

  44. Sussmann, H. J.: Lie brackets and local controllability: a sufficient condition for scalar-input systems, SIAM J. Control Optim. 21 (1983), 686–713.

    Google Scholar 

  45. Sussmann, H. J.: A general theorem on local controllability, SIAM J. Control Optim. 25(1) (1987), 158–194.

    Google Scholar 

  46. Sussmann, H. J.: A product expansion of the Chen series, In: C. I. Byrnes and A. Lindquist (eds), Theory and Applications of Nonlinear Control Theory, Elsevier, Amsterdam, 1993, pp. 343–378.

    Google Scholar 

  47. Sussmann, H. J. and Liu, W.: Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, In: Proc. IEEE Int. Conf. Decision & Control, Brighton, U.K., 1991, pp. 437–442.

    Google Scholar 

  48. Sussmann, H. J. and Liu, W.: Lie bracket extensions and averaging: the single bracket case, In: Z. Li and J. Canny (eds), Nonholonomic Motion Planning, Kluwer, Boston, 1993, pp. 109–148.

    Google Scholar 

  49. Synge, J. Z.: Geodesics in nonholonomic geometry, Math. Annal. 99 (1928), 738–751.

    Google Scholar 

  50. Warner, F.: Foundations of Differentiable Manifolds and Lie Groups, Scott Foresman, Glenview, IL, 1973.

    Google Scholar 

  51. Zhang, H. and Ostrowski, J. P.: Control algorithms using affine connections on principal fiber bundles, In: Proc. IFAC Workshop on Lagrangian and Hamiltonian methods for Nonlinear Control, Princeton, 1999.

Download references

Author information

Authors and Affiliations

  1. Escola Universitària Politècnica de Vilanova i la Geltrú, Universitat Politècnica de Catalunya, Av. V. Balaguer s/n, Vilanova i la Geltrú, 08800, Spain

    Sonia Martínez

  2. Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 W. Main St., Urbana, IL, 61801, U.S.A.

    Jorge Cortés

Authors
  1. Sonia Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Cortés
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martínez, S., Cortés, J. Motion Control Algorithms for Simple Mechanical Systems with Symmetry. Acta Applicandae Mathematicae 76, 221–264 (2003). https://doi.org/10.1023/A:1023275502000

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023275502000

Search

Navigation