# American Institute of Mathematical Sciences

June  2015, 2(2): 193-225. doi: 10.3934/jcd.2015003

## Variational integrators for mechanical control systems with symmetries

 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109 2 Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen 3 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM C/ Nicolas Cabrera, 15 - 28049 Madrid

Received  June 2013 Revised  May 2015 Published  May 2016

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems.
An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems, paying particular attention to the case of underactuated mechanical systems.
Citation: Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003
##### References:
 [1] J. Baillieul, The geometry of controlled mechanical systems,, Mathematical control theory, (1999), 322. Google Scholar [2] R. Benito, M. de León and D. Martín de Diego, Higher-order discrete lagrangian mechanics,, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421. doi: 10.1142/S0219887806001235. Google Scholar [3] A. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (2003). doi: 10.1007/b97376. Google Scholar [4] A. M. Bloch and P. E. Crouch, Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems,, Proceedings of 33rd IEEE Conference on Decision and Control, 3 (1994), 2584. doi: 10.1109/CDC.1994.411534. Google Scholar [5] A. M. Bloch and P. E. Crouch, On the equivalence of higher order variational problems and optimal control problems,, Proceedings of 35rd IEEE Conference on Decision and Control, 2 (1996), 1648. doi: 10.1109/CDC.1996.572780. Google Scholar [6] A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric structure-preserving optimal control of a rigid body,, Journal of Dynamical and Control Systems, 15 (2009), 307. doi: 10.1007/s10883-009-9071-2. Google Scholar [7] N. Borda, J. Fernandez and S. Grillo, Discrete second order constrained Lagrangian systems: First results,, Journal Geometric Mechanics, 5 (2013), 381. doi: 10.3934/jgm.2013.5.381. Google Scholar [8] N. Bou-Rabee and J. E. Marsden, Hamilton-pontryagin integrators on Lie groups,, Foundations of Computational Mathematics, 9 (2009), 197. doi: 10.1007/s10208-008-9030-4. Google Scholar [9] C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups,, Proc. R. Soc. A., 469 (2013). doi: 10.1098/rspa.2013.0249. Google Scholar [10] F. Bullo and A. Lewis, Geometric control of mechanical systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005). doi: 10.1007/978-1-4899-7276-7. Google Scholar [11] J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393. Google Scholar [12] C. M. Campos, O. Junge and S. Ober-Blöbaum, Higher order variational time discretization of optimal control problems,, In: 20th International Symposium on Mathematical Theory of Networks and Systems, (2012). Google Scholar [13] C. M. Campos, High Order Variational Integrators: A Polynomial Approach,, Advances in Differential Equations and Applications SEMA SIMAI Springer Series, 4 (2014), 249. doi: 10.1007/978-3-319-06953-1_24. Google Scholar [14] C. M. Campos, S. Ober-Blöbaum and E. Trélat, High order variational integrators in the optimal control of mechanical systems,, Discrete and Continuous Dynamical Systems - Series A, 35 (2015), 4193. doi: 10.3934/dcds.2015.35.4193. Google Scholar [15] A. Castro and J. Koiller, On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy,, Regul. Chaotic Dyn., 18 (2013), 1. doi: 10.1134/S1560354713010012. Google Scholar [16] H. Cendra, J. Marsden and T. Ratiu, Lagrangian reduction by stages,, Memoirs of the American Mathematical Society, 152 (2001). doi: 10.1090/memo/0722. Google Scholar [17] J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lec. Notes in Math., (1793). doi: 10.1007/b84020. Google Scholar [18] M. Chyba, E. Hairer and G. Vilmart, The role of symplectic integrators in optimal control,, Opt. Control Appl. Method, 30 (2009), 367. doi: 10.1002/oca.855. Google Scholar [19] L. Colombo, F. Jiménez and D. Martín de Diego, Discrete second-order euler-poincaré equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012). doi: 10.1142/S0219887812500375. Google Scholar [20] L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications,, J. Geom. Mech., 6 (2014), 451. doi: 10.3934/jgm.2014.6.451. Google Scholar [21] L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, Journal of Mathematical Physics, 51 (2010). doi: 10.1063/1.3456158. Google Scholar [22] L. Colombo, D. Martín de Diego and M. Zuccalli, On variational integrators for optimal control of mechanical systems,, RACSAM Rev. R. Acad. Cienc. Ser A. Mat, 106 (2012), 161. doi: 10.1007/s13398-011-0032-8. Google Scholar [23] L. Colombo L, D. Martín de Diego and M. Zuccalli, Higher-order variational problems with constraints,, Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4820817. Google Scholar [24] J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids,, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213. doi: 10.3934/dcds.2009.24.213. Google Scholar [25] J. Fernandez and M. Zuccalli, A geometric approach to discrete connections on principal bundles,, J. Geom. Mech., 5 (2013), 433. doi: 10.3934/jgm.2013.5.433. Google Scholar [26] F. Gay-Balmaz, D. Holm and T. Ratiu, Higher order Lagrange-Poincaré, and Hamilton-Poincaré reductions,, Bulletin of the Brazialian Mathematical Society, 42 (2011), 579. doi: 10.1007/s00574-011-0030-7. Google Scholar [27] F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Geometric dynamics of optimization,, Comm. Math. Sci., 11 (2013), 163. doi: 10.4310/CMS.2013.v11.n1.a6. Google Scholar [28] F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413. doi: 10.1007/s00220-011-1313-y. Google Scholar [29] S. Grillo, M. Zuccalli, Variational reduction of Lagrangian systems with general constraints,, J. Geom. Mech., 4 (2012), 49. doi: 10.3934/jgm.2012.4.49. Google Scholar [30] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, 31 (2002). doi: 10.1007/978-3-662-05018-7. Google Scholar [31] D. D. Holm, Geometric Mechanics. Part I and II,, Imperial College Press, (2008). Google Scholar [32] A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods,, Acta Numerica, 9 (2000), 215. doi: 10.1017/S0962492900002154. Google Scholar [33] F. Jiménez, M. de León and D. Martín de Diego, Hamiltonian dynamics and constrained variational calculus: Continuous and discrete settings,, J. Phys A, 45 (2012). doi: 10.1088/1751-8113/45/20/205204. Google Scholar [34] F. Jiménez, M. Kobilarov and D. Martín de Diego, Discrete variational optimal control,, Journal of Nonlinear Science, 23 (2013), 393. doi: 10.1007/s00332-012-9156-z. Google Scholar [35] F. Jiménez and D. Martín de Diego, A geometric approach to Discrete mechanics for optimal control theory,, Proceedings of the IEEE Conference on Decision and Control, (2010), 5426. Google Scholar [36] M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles,, Thesis, (2008). Google Scholar [37] M. Kobilarov and J. Marsden, Discrete geometric optimal control on lie groups,, IEEE Transactions on Robotics, 27 (2011), 641. doi: 10.1109/TRO.2011.2139130. Google Scholar [38] W.-S. Koon, Reduction, Reconstruction and Optimal Control for Nonholonomic Mechanical Systems with Symmetry,, PhD thesis, (1997). Google Scholar [39] T. Lee, M. Leok and N. H. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465. doi: 10.1007/s10883-008-9047-7. Google Scholar [40] M. Leok, Foundations of Computational Geometric Mechanics, Control and Dynamical Systems,, Thesis, (2004). Google Scholar [41] M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies, 12 (1985). Google Scholar [42] A. Lewis and R. Murray, Variational principles for constrained systems: Theory and experiment,, Int. J. Nonlinear Mechanics, 30 (1995), 793. doi: 10.1016/0020-7462(95)00024-0. Google Scholar [43] S. Leyendecker, S. Ober-Blöbaum, J. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained systems,, Optim. Control, 31 (2010), 505. doi: 10.1002/oca.912. Google Scholar [44] J. C. Marrero, D. Martín de Diego D and E. Martínez, Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids,, Nonlinearity, 19 (2006), 1313. doi: 10.1088/0951-7715/19/6/006. Google Scholar [45] J. C. Marrero, D. Martín de Diego and A. Stern, Symplectic groupies and discrete constrained Lagrangian mechanics,, Discrete and Continuous Mechanical Systems, 35 (2015), 367. Google Scholar [46] J. Marsden, S. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups,, J. Geom. Phys., 36 (2000), 140. doi: 10.1016/S0393-0440(00)00018-8. Google Scholar [47] J. E. Marsden and J. M. Wendlandt, Mechanical integrators derived from a discrete variational principle,, Physica D, 106 (1997), 223. doi: 10.1016/S0167-2789(97)00051-1. Google Scholar [48] J. Marsden and M. West, Discrete Mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar [49] D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar [50] S. Ober-Blöbaum, O. Junge and J. Marsden, Discrete mechanics and optimal control: An analysis,, ESAIM: COCV, 17 (2011), 322. doi: 10.1051/cocv/2010012. Google Scholar

show all references

##### References:
 [1] J. Baillieul, The geometry of controlled mechanical systems,, Mathematical control theory, (1999), 322. Google Scholar [2] R. Benito, M. de León and D. Martín de Diego, Higher-order discrete lagrangian mechanics,, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421. doi: 10.1142/S0219887806001235. Google Scholar [3] A. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (2003). doi: 10.1007/b97376. Google Scholar [4] A. M. Bloch and P. E. Crouch, Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems,, Proceedings of 33rd IEEE Conference on Decision and Control, 3 (1994), 2584. doi: 10.1109/CDC.1994.411534. Google Scholar [5] A. M. Bloch and P. E. Crouch, On the equivalence of higher order variational problems and optimal control problems,, Proceedings of 35rd IEEE Conference on Decision and Control, 2 (1996), 1648. doi: 10.1109/CDC.1996.572780. Google Scholar [6] A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric structure-preserving optimal control of a rigid body,, Journal of Dynamical and Control Systems, 15 (2009), 307. doi: 10.1007/s10883-009-9071-2. Google Scholar [7] N. Borda, J. Fernandez and S. Grillo, Discrete second order constrained Lagrangian systems: First results,, Journal Geometric Mechanics, 5 (2013), 381. doi: 10.3934/jgm.2013.5.381. Google Scholar [8] N. Bou-Rabee and J. E. Marsden, Hamilton-pontryagin integrators on Lie groups,, Foundations of Computational Mathematics, 9 (2009), 197. doi: 10.1007/s10208-008-9030-4. Google Scholar [9] C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups,, Proc. R. Soc. A., 469 (2013). doi: 10.1098/rspa.2013.0249. Google Scholar [10] F. Bullo and A. Lewis, Geometric control of mechanical systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005). doi: 10.1007/978-1-4899-7276-7. Google Scholar [11] J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393. Google Scholar [12] C. M. Campos, O. Junge and S. Ober-Blöbaum, Higher order variational time discretization of optimal control problems,, In: 20th International Symposium on Mathematical Theory of Networks and Systems, (2012). Google Scholar [13] C. M. Campos, High Order Variational Integrators: A Polynomial Approach,, Advances in Differential Equations and Applications SEMA SIMAI Springer Series, 4 (2014), 249. doi: 10.1007/978-3-319-06953-1_24. Google Scholar [14] C. M. Campos, S. Ober-Blöbaum and E. Trélat, High order variational integrators in the optimal control of mechanical systems,, Discrete and Continuous Dynamical Systems - Series A, 35 (2015), 4193. doi: 10.3934/dcds.2015.35.4193. Google Scholar [15] A. Castro and J. Koiller, On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy,, Regul. Chaotic Dyn., 18 (2013), 1. doi: 10.1134/S1560354713010012. Google Scholar [16] H. Cendra, J. Marsden and T. Ratiu, Lagrangian reduction by stages,, Memoirs of the American Mathematical Society, 152 (2001). doi: 10.1090/memo/0722. Google Scholar [17] J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lec. Notes in Math., (1793). doi: 10.1007/b84020. Google Scholar [18] M. Chyba, E. Hairer and G. Vilmart, The role of symplectic integrators in optimal control,, Opt. Control Appl. Method, 30 (2009), 367. doi: 10.1002/oca.855. Google Scholar [19] L. Colombo, F. Jiménez and D. Martín de Diego, Discrete second-order euler-poincaré equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012). doi: 10.1142/S0219887812500375. Google Scholar [20] L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications,, J. Geom. Mech., 6 (2014), 451. doi: 10.3934/jgm.2014.6.451. Google Scholar [21] L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, Journal of Mathematical Physics, 51 (2010). doi: 10.1063/1.3456158. Google Scholar [22] L. Colombo, D. Martín de Diego and M. Zuccalli, On variational integrators for optimal control of mechanical systems,, RACSAM Rev. R. Acad. Cienc. Ser A. Mat, 106 (2012), 161. doi: 10.1007/s13398-011-0032-8. Google Scholar [23] L. Colombo L, D. Martín de Diego and M. Zuccalli, Higher-order variational problems with constraints,, Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4820817. Google Scholar [24] J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids,, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213. doi: 10.3934/dcds.2009.24.213. Google Scholar [25] J. Fernandez and M. Zuccalli, A geometric approach to discrete connections on principal bundles,, J. Geom. Mech., 5 (2013), 433. doi: 10.3934/jgm.2013.5.433. Google Scholar [26] F. Gay-Balmaz, D. Holm and T. Ratiu, Higher order Lagrange-Poincaré, and Hamilton-Poincaré reductions,, Bulletin of the Brazialian Mathematical Society, 42 (2011), 579. doi: 10.1007/s00574-011-0030-7. Google Scholar [27] F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Geometric dynamics of optimization,, Comm. Math. Sci., 11 (2013), 163. doi: 10.4310/CMS.2013.v11.n1.a6. Google Scholar [28] F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413. doi: 10.1007/s00220-011-1313-y. Google Scholar [29] S. Grillo, M. Zuccalli, Variational reduction of Lagrangian systems with general constraints,, J. Geom. Mech., 4 (2012), 49. doi: 10.3934/jgm.2012.4.49. Google Scholar [30] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, 31 (2002). doi: 10.1007/978-3-662-05018-7. Google Scholar [31] D. D. Holm, Geometric Mechanics. Part I and II,, Imperial College Press, (2008). Google Scholar [32] A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods,, Acta Numerica, 9 (2000), 215. doi: 10.1017/S0962492900002154. Google Scholar [33] F. Jiménez, M. de León and D. Martín de Diego, Hamiltonian dynamics and constrained variational calculus: Continuous and discrete settings,, J. Phys A, 45 (2012). doi: 10.1088/1751-8113/45/20/205204. Google Scholar [34] F. Jiménez, M. Kobilarov and D. Martín de Diego, Discrete variational optimal control,, Journal of Nonlinear Science, 23 (2013), 393. doi: 10.1007/s00332-012-9156-z. Google Scholar [35] F. Jiménez and D. Martín de Diego, A geometric approach to Discrete mechanics for optimal control theory,, Proceedings of the IEEE Conference on Decision and Control, (2010), 5426. Google Scholar [36] M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles,, Thesis, (2008). Google Scholar [37] M. Kobilarov and J. Marsden, Discrete geometric optimal control on lie groups,, IEEE Transactions on Robotics, 27 (2011), 641. doi: 10.1109/TRO.2011.2139130. Google Scholar [38] W.-S. Koon, Reduction, Reconstruction and Optimal Control for Nonholonomic Mechanical Systems with Symmetry,, PhD thesis, (1997). Google Scholar [39] T. Lee, M. Leok and N. H. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465. doi: 10.1007/s10883-008-9047-7. Google Scholar [40] M. Leok, Foundations of Computational Geometric Mechanics, Control and Dynamical Systems,, Thesis, (2004). Google Scholar [41] M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies, 12 (1985). Google Scholar [42] A. Lewis and R. Murray, Variational principles for constrained systems: Theory and experiment,, Int. J. Nonlinear Mechanics, 30 (1995), 793. doi: 10.1016/0020-7462(95)00024-0. Google Scholar [43] S. Leyendecker, S. Ober-Blöbaum, J. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained systems,, Optim. Control, 31 (2010), 505. doi: 10.1002/oca.912. Google Scholar [44] J. C. Marrero, D. Martín de Diego D and E. Martínez, Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids,, Nonlinearity, 19 (2006), 1313. doi: 10.1088/0951-7715/19/6/006. Google Scholar [45] J. C. Marrero, D. Martín de Diego and A. Stern, Symplectic groupies and discrete constrained Lagrangian mechanics,, Discrete and Continuous Mechanical Systems, 35 (2015), 367. Google Scholar [46] J. Marsden, S. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups,, J. Geom. Phys., 36 (2000), 140. doi: 10.1016/S0393-0440(00)00018-8. Google Scholar [47] J. E. Marsden and J. M. Wendlandt, Mechanical integrators derived from a discrete variational principle,, Physica D, 106 (1997), 223. doi: 10.1016/S0167-2789(97)00051-1. Google Scholar [48] J. Marsden and M. West, Discrete Mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar [49] D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar [50] S. Ober-Blöbaum, O. Junge and J. Marsden, Discrete mechanics and optimal control: An analysis,, ESAIM: COCV, 17 (2011), 322. doi: 10.1051/cocv/2010012. Google Scholar
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