Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints
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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 controlPreview
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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.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 MathSciNetMATHGoogle Scholar
- 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.Arnol’d, V. Encyclopaedia of mathematical sciences: Dynamical systems III. Springer, New York (1988)Google Scholar
- 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.Becerra, V. M. Tech. Rep.: PSOPT Optimal Control Solver User Manual. Reading, United Kingdom (2010)Google Scholar
- 6.Bell, D. J., Jacobson, D. H.: Singular Optimal Control Problems. Academic Press, New York (1975)MATHGoogle Scholar
- 7.Bershanskiy, Y. M.: Conjugation of singular and nonsingular parts of optimal control. Autom. Remote. Control. 40, 325–330 (1979)Google Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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.Bloch, A. M., Marsden, J. E., Zenkov, D. V.: Quasivelocities and symmetries in non-holonomic systems. Dyn. Syst. 24(2), 187–222 (2009)MathSciNetCrossRefMATHGoogle Scholar
- 11.Bonnard, B., Chyba, M. Singular Trajectories and Their Role in Control THeory: Mathématiques et Applications. Springer, Berlin (2003)Google Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 13.Bryson, A. E.: Applied Optimal Control. Hemisphere, New York (1975)Google Scholar
- 14.Bullo, F., Lewis, A. D. Texts in applied mathematics: Geometric control of mechanical systems. Springer, New York (2004)Google Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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 MATHGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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.Gabasov, R., Kirillova, F. M.: High order necessary conditions for optimality. SIAM J. Control 10(1), 127–168 (1972). doi: 10.1137/0310012 MathSciNetCrossRefMATHGoogle Scholar
- 20.Gabasov, R., Kirillova, F. M.: Singular Optimal Controls. Nauka (1973)Google Scholar
- 21.Giaquinta, M., Hildebrandt, S.: Calculus of Variations I. Springer-Verlag, Berlin (1996)MATHGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 23.Gorokhovik, V. V.: High-order necessary optiMality conditions for control problems with terminal constraints. Opt. Control Appl. Meth. 4(2), 103–127 (1983)MathSciNetCrossRefMATHGoogle Scholar
- 24.Greenwood, T. D.: Advanced Dynamics. Cambridge University Press, New York (2003)CrossRefGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 28.Kane, T. R., Levinson, D. A.: Dynamics: Theory and Applications. McGraw Hill, New York (1985)Google Scholar
- 29.Kelley, H. J.: A second variation test for singular extremals. AIAA J. 2(8), 1380–1382 (1964). doi: 10.2514/3.2562 MathSciNetCrossRefMATHGoogle Scholar
- 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.Kopp, R. E., Moyer, H. G.: Necessary conditions for singular extremals. AIAA J. 3(8), 1439–1444 (1965). doi: 10.2514/3.3165 CrossRefMATHGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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.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.Lamnabhi-Lagarrigue, F.: Singular optimal control problems: On the order of a singular arc. Syst. Control Lett. 9(2), 173–182 (1987)MathSciNetCrossRefMATHGoogle Scholar
- 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 MathSciNetCrossRefMATHGoogle Scholar
- 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.Maruskin, J.: Introduction to Dynamical Systems and Geometric Mechanics. Solar Crest, San Jose (2012)Google Scholar
- 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.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 MathSciNetCrossRefMATHGoogle Scholar
- 41.Montgomery, R.: Abnormal minimizers. SIAM J. Control. Optim. 32(6), 1605–1620 (1994). doi: doi: 10.1137/S0363012993244945 MathSciNetCrossRefMATHGoogle Scholar
- 42.Neimark, J. I., Fufaev, N. A.: Dynamics of Nonholonimic Systems. American Mathematical Society, New York (1972)MATHGoogle Scholar
- 43.Paul, R. P.: Robot manipulators: Mathematics, Programming, and Control : the Computer Control of Robot Manipulators. MIT Press, Boston (1981)Google Scholar
- 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.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 CrossRefMATHGoogle Scholar
- 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.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 CrossRefMATHGoogle Scholar
- 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.Shuster, M. D.: Survey of attitude representations. J. Astronaut. Sci. 11, 439–517 (1993)MathSciNetGoogle Scholar
- 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.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.Speyer, J. L.: On the fuel optimality of cruise. J. Aircr. 10(12), 763–765 (1973). doi: 10.2514/3.60304 CrossRefGoogle Scholar
- 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.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 MathSciNetCrossRefMATHGoogle Scholar