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Journal of Intelligent & Robotic Systems

, Volume 89, Issue 1–2, pp 51–67 | Cite as

Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints

  • Andrea L’AfflittoEmail author
  • Wassim M. Haddad
Article
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Abstract

In optimal control problems, the Hamiltonian function is given by the weighted sum of the integrand of the cost function and the dynamic equation. The coefficient multiplying the integrand of the cost function is either zero or one; and if this coefficient is zero, then the optimal control problem is known as abnormal; otherwise it is normal. This paper provides a characterization of the abnormal optimal control problem for multi-body mechanical systems, subject to external forces and moments, and holonomic and nonholonomic constraints. This study does not only account for first-order necessary conditions, such as Pontryagin’s principle, but also for higher-order conditions, which allow the analysis of singular optimal controls.

Keywords

Optimal trajectory planning Singular controls Pontryagin’s principle Normal and abnormal optimal control 
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Notes

Acknowledgments

This work was supported in part by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement #NA16OAR4320115, the U.S. Department of Commerce, and the Air Force Office of Scientific Research under Grant FA9550-16-1-0100.

References

  1. 1.
    Agrachev, A., Sarychev, A.: On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estimation Control 8, 87–118 (1998). doi: 10.1145/1731022.1731032 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aoude, G. S.: Two-Stage Path Planning Approach for Designing Multiple Spacecraft Reconfiguration Maneuvers and Application to SPHERES Onboard ISS. Master’s Thesis, Massachusetts Institute of Technology (2007). MS Thesis, Massachusetts Institute of Technology, Cambridge, MAGoogle Scholar
  3. 3.
    Arnol’d, V. Encyclopaedia of mathematical sciences: Dynamical systems III. Springer, New York (1988)Google Scholar
  4. 4.
    Barbero-Linán, M., de León, M., de Diego, D. M., Marrero, J. C., Munoz-Lecanda, M. C.: Kinematic reduction and the Hamilton-Jacobi equation. J. Geo. Mech. 4(3), 207–237 (2012). doi: 10.3934/jgm.2012.4.207
  5. 5.
    Becerra, V. M. Tech. Rep.: PSOPT Optimal Control Solver User Manual. Reading, United Kingdom (2010)Google Scholar
  6. 6.
    Bell, D. J., Jacobson, D. H.: Singular Optimal Control Problems. Academic Press, New York (1975)zbMATHGoogle Scholar
  7. 7.
    Bershanskiy, Y. M.: Conjugation of singular and nonsingular parts of optimal control. Autom. Remote. Control. 40, 325–330 (1979)Google Scholar
  8. 8.
    Bloch, A., Krishnaprasad, P., Marsden, J., Murray, R.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136, 21–99 (1996). doi: 10.1007/BF02199365 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bloch, A. M., Crouch, P. E.: Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems Proceedings of the 33rd IEEE Conference on Decision and Control. doi: 10.1109/CDC.1994.411534, vol. 3, pp 2584–2590 (1994)
  10. 10.
    Bloch, A. M., Marsden, J. E., Zenkov, D. V.: Quasivelocities and symmetries in non-holonomic systems. Dyn. Syst. 24(2), 187–222 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bonnard, B., Chyba, M. Singular Trajectories and Their Role in Control THeory: Mathématiques et Applications. Springer, Berlin (2003)Google Scholar
  12. 12.
    Bonnard, B., Kupka, I.: Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Mathematicum 5, 111–159 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bryson, A. E.: Applied Optimal Control. Hemisphere, New York (1975)Google Scholar
  14. 14.
    Bullo, F., Lewis, A. D. Texts in applied mathematics: Geometric control of mechanical systems. Springer, New York (2004)Google Scholar
  15. 15.
    Chitour, Y., Jean, F., Trélat, E.: Singular trajectories of control-affine systems. SIAM J. Control. Optim. 47(2), 1078–1095 (2008). doi: 10.1137/060663003 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chyba, M., Haberkorn, T., Smith, R. N., Wilkens, G. R.: Controlling a Submerged Rigid Body: A Geometric Analysis, pp 375–385. Springer, Berlin (2007). doi: 10.1007/978-3-540-73890-9_30 zbMATHGoogle Scholar
  17. 17.
    Chyba, M., Leonard, N. E., Sontag, E. D.: Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems. J. Dyn. Control. Syst. 9(1), 103–129 (2003). doi: 10.1023/A:1022159318457 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Delgado-Tellez, M., Ibort, A.: On the Geometry and Topology of Singular Optimal Control Problems and Their Solutions Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations, pp 223–233 (2002)Google Scholar
  19. 19.
    Gabasov, R., Kirillova, F. M.: High order necessary conditions for optimality. SIAM J. Control 10(1), 127–168 (1972). doi: 10.1137/0310012 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gabasov, R., Kirillova, F. M.: Singular Optimal Controls. Nauka (1973)Google Scholar
  21. 21.
    Giaquinta, M., Hildebrandt, S.: Calculus of Variations I. Springer-Verlag, Berlin (1996)zbMATHGoogle Scholar
  22. 22.
    Goh, B. S.: Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4(4), 716–731 (1966). doi: 10.1137/0304052 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gorokhovik, V. V.: High-order necessary optiMality conditions for control problems with terminal constraints. Opt. Control Appl. Meth. 4(2), 103–127 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Greenwood, T. D.: Advanced Dynamics. Cambridge University Press, New York (2003)CrossRefGoogle Scholar
  25. 25.
    Jacobson, D. H.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7(4), 578–595 (1969). doi: 10.1137/0307042 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jakubczyk, B., Krynski, W., Pelletier, F.: Characteristic vector fields of generic distributions of corank 2. Annales de l’Institut Henri Poincare Non Linear Analysis 26(1), 23–38 (2009). doi: 10.1016/j.anihpc.2007.05.006 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jóźwikowski, M., Respondek, W.: A contact covariant approach to optimal control with applications to sub-riemannian geometry. Math. Control Signals Syst. 28(3), 27 (2016). doi: 10.1007/s00498-016-0176-3 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kane, T. R., Levinson, D. A.: Dynamics: Theory and Applications. McGraw Hill, New York (1985)Google Scholar
  29. 29.
    Kelley, H. J.: A second variation test for singular extremals. AIAA J. 2(8), 1380–1382 (1964). doi: 10.2514/3.2562 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kelley, H. J., Kopp, R. E., Moyer, H. G. Singular extremals. In: Leitman, G. (ed.) : Topics in Optimization, pp 63–101. Academic Press, New York (1967)CrossRefGoogle Scholar
  31. 31.
    Kopp, R. E., Moyer, H. G.: Necessary conditions for singular extremals. AIAA J. 3(8), 1439–1444 (1965). doi: 10.2514/3.3165 CrossRefzbMATHGoogle Scholar
  32. 32.
    Krener, A. J.: The high order maximal principle and its application to singular extremals. SIAM J. Control. Optim. 15(2), 256–293 (1977). doi: 10.1137/0315019 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    L’Afflitto, A., Haddad, W. M. A variational approach to the fuel optimal control. In: Agarwal, R. (ed.) : Recent Advances in Aircraft Technology, Chap. 10, pp 221–248. InTech, Croatia (2012)Google Scholar
  34. 34.
    L’Afflitto, A., Haddad, W. M.: Necessary conditions for control effort minimization of euler-lagrange systems AIAA Guidance, Navigation, and Control Conference, pp 1–18 (2015)Google Scholar
  35. 35.
    Lamnabhi-Lagarrigue, F.: Singular optimal control problems: On the order of a singular arc. Syst. Control Lett. 9(2), 173–182 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lamnabhi-Lagarrigue, F., Stefani, G.: Singular optimal control problems: On the necessary conditions of optimality. SIAM J. Control. Optim. 28(4), 823–840 (1990). doi: 10.1137/0328047 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, W.: Abnormal extremals and optimality in sub-riemannian manifolds IEEE Conference on Decision and Control. doi: 10.1109/CDC.1994.411090, vol. 2, pp 1957–1963 (1994)
  38. 38.
    Maruskin, J.: Introduction to Dynamical Systems and Geometric Mechanics. Solar Crest, San Jose (2012)Google Scholar
  39. 39.
    Maruskin, J. M., Bloch, A. M.: The Boltzmann-Hamel equations for the optimal control of mechanical systems with nonholonomic constraints. Int. J. Robust Nonlinear Control. doi: 10.1002/rnc.1598 (2010)
  40. 40.
    McDanell, J. P., Powers, W. F.: Necessary conditions joining optimal singular and nonsingular subarcs. SIAM J. Control 9(2), 161–173 (1971). doi: 10.1137/0309014 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Montgomery, R.: Abnormal minimizers. SIAM J. Control. Optim. 32(6), 1605–1620 (1994). doi: doi: 10.1137/S0363012993244945 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Neimark, J. I., Fufaev, N. A.: Dynamics of Nonholonimic Systems. American Mathematical Society, New York (1972)zbMATHGoogle Scholar
  43. 43.
    Paul, R. P.: Robot manipulators: Mathematics, Programming, and Control : the Computer Control of Robot Manipulators. MIT Press, Boston (1981)Google Scholar
  44. 44.
    Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F.: The Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962)Google Scholar
  45. 45.
    Rao, A. V., Benson, D. A., Darby, C., Patterson, M. A., Francolin, C., Sanders, I., Huntington, G. T.: Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method. ACM Trans. Math. Softw. 37, 1–39 (2010). doi: 10.1145/1731022.1731032 CrossRefzbMATHGoogle Scholar
  46. 46.
    Rao, A. V., Benson, D. A., Darby, C. L., Huntington, G. T.: User’s Manual for GPOPS Version 4.x: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp–Adaptive Pseudospectral Methods. Tech. rep. (2011)Google Scholar
  47. 47.
    Robbins, H. M.: A generalized Legendre-Clebsch condition for the singular cases of optimal control. IBM J. Res. Dev. 11(4), 361–372 (1967). doi: 10.1147/rd.114.0361 CrossRefzbMATHGoogle Scholar
  48. 48.
    Ross, I. M., Fahroo, F.: User’s Manual for DIDO 2001: A MATLAB Application for Solving Optimal Control Problems. Tech. Rep. AAS-01-03, Monterey (2001)Google Scholar
  49. 49.
    Shuster, M. D.: Survey of attitude representations. J. Astronaut. Sci. 11, 439–517 (1993)MathSciNetGoogle Scholar
  50. 50.
    Sontag, E. D.: Remarks on the time-optimal control of a class of Hamiltonian systems IEEE Conference on Decision and Control, vol. 1, pp 217–221 (1989)Google Scholar
  51. 51.
    Sontag, E. D., Sussmann, H. J.: Time-optimal control of manipulators Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, pp 1692–1697 (1986)Google Scholar
  52. 52.
    Speyer, J. L.: On the fuel optimality of cruise. J. Aircr. 10(12), 763–765 (1973). doi: 10.2514/3.60304 CrossRefGoogle Scholar
  53. 53.
    Spreyer, J. L., Jacobson, D.: Necessary and sufficient conditions for optimally for singular control problems; a transformation approach. J. Math. Anal. Appl. 33(1), 163–187 (1971). doi: 10.1016/0022-247X(71)90190-9 MathSciNetCrossRefGoogle Scholar
  54. 54.
    Zelikin, M. I., Lokutsievskiy, L. V., Hildebrand, R.: Geometry of neighborhoods of singular trajectories in problems with multidimensional control. Proc. Steklov Inst. Math. 277(1), 67–83 (2012). doi: 10.1134/S0081543812040062 MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Aerospace and Mechanical EngineeringThe University of OklahomaNormanUSA
  2. 2.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

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