Motion control algorithms for simple mechanical systems with symmetry

Acta Applicandae Mathematicae

Volumn 76, Issue 3, 2003, Pages 221-264

  Martínez, Sonia ^a   Cortés, Jorge ^b  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b University of Illinois at Urbana Champaign   (United States)

Author keywords

Controllability; Nonlinear control; Reduction; Series expansions; Small amplitude periodic forcing; Symmetric product

Indexed keywords

ALGORITHMS; FEEDBACK CONTROL; MOTION CONTROL; PROBLEM SOLVING;

MOTION CONTROL ALGORITHM;

CONTROL SYSTEM ANALYSIS;

EID: 0038369884     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1023275502000     Document Type: Review

References (51)

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.
3. Agračhev, A. A. and Gamkrelidze, R. V.: Local controllability and semigroups of diffeomorphisms, Acta Appl. Math. 32 (1993), 1-57.
4. Bloch, A. M. and Crouch, P. E.: Nonholonomic control systems on Riemannian manifolds, SIAM J. Control Optim. 33(8) (1995), 541-552.
5. Bloch, A. M. and Crouch, P. E.: Newton's law and integrability on nonholonomic systems, SIAM J. Control Optim. 36 (1998), 2020-2039.
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.
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.
8. Bullo, F.: Series expansions for the evolution of mechanical control systems, SIAM J. Control Optim. 40(1) (2001), 166-190.
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.
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.
12. Bullo, F. and Zefran, M.: On mechanical control systems with nonholonomic constraints and symmetries, Systems Control Lett. 45(2) (2002), 133-143.
13. Chen, K. T.: Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math. 67 (1957), 8615-8645.
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.
16. Crouch, P. E.: Geometric structures in systems theory, Proc. IEE 128D(5) (1981), 242-252.
17. Fliess, M.: Fonctionelles causales non linéares et indeterminées non commutatives, Bull. Soc. Math. 109 (1981), 3-40.
18. Hermes, H.: Control systems which generate decomposable Lie algebras, J. Differential Equations 44 (1982), 166-187.
19. Isidori, A.: Nonlinear Control Systems, 2nd edn, Springer-Verlag, Berlin, 1995.
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.
23. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Interscience, New York, 1963.
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.
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.
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.
27. Lewis, A. D.: Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys. 42(1/2) (1998), 135-164.
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.
30. Lewis, A. D. and Murray, R. M.: Configuration controllability of simple mechanical control systems, SIAM J. Control Optim. 35(3) (1997), 766-790.
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.
33. Marsden, J. E. and Ratiu, T. S.: Introduction to Mechanics and Symmetry, 2nd edn, Springer-Verlag, New York, 1999.
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.
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.
36. Murray, R. M., Li, Z. X. and Sastry, S. S.: A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Ratón, 1994.
37. Nijmeijer, H. and van der Schaft, A. J.: Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990.
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.
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.
40. Ostrowski, J. P. and Burdick, J. W.: The geometric mechanics of undulatory robotic locomotion, Int. J. Robotics Res. 17(7) (1998), 683-701.
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.
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.
44. Sussmann, H. J.: Lie brackets and local controllability: a sufficient condition for scalar-input systems, SIAM J. Control Optim. 21 (1983), 686-713.
45. Sussmann, H. J.: A general theorem on local controllability, SIAM J. Control Optim. 25(1) (1987), 158-194.
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.
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.
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.
49. Synge, J. Z.: Geodesics in nonholonomic geometry, Math. Annal. 99 (1928), 738-751.
50. Warner, F.: Foundations of Differentiable Manifolds and Lie Groups, Scott Foresman, Glenview, IL, 1973.
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.