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On computational Poisson geometry I: Symbolic foundations
December
2021, 13(4): 629-646.
doi: 10.3934/jgm.2021022
Explicit solutions of the kinetic and potential matching conditions of the energy shaping method
| 1. | Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina |
| 2. | CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, and CONICET, Argentina |
| 3. | CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, Argentina |
In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the kinetic and potential equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the $ C^{\infty} $ category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.
Keywords:
Hamiltonian systems,
underactuated systems,
closed-loop systems,
energy shaping,
integration by quadratures.
Mathematics Subject Classification:
Primary: 93D05, 93D20; Secondary: 93C10.
Citation:
Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics,
2021, 13
(4)
: 629-646.
doi: 10.3934/jgm.2021022
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References:
| [1] |
R. Abraham and J. E. Marsden, Foundation of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. |
| [2] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.
doi: 10.1007/978-1-4757-1693-1. |
| [3] |
A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003.
doi: 10.1007/b97376. |
| [4] |
W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, 63, Academic Press, New York-London, 1975.
|
| [5] |
F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.
doi: 10.1007/978-1-4899-7276-7. |
| [6] |
D. E. Chang, Generalization of the IDA-PBC method for stabilization of mechanical systems, 18th Mediterranean Conference on Control and Automation, MED'10, Marrakech, Morocco, 2010.
doi: 10.1109/MED.2010.5547672. |
| [7] |
D. E. Chang,
The method of controlled Lagrangians: Energy plus force shaping, SIAM J. Control Optim., 48 (2010), 4821-4845.
doi: 10.1137/070691310. |
| [8] |
D. E. Chang,
On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regul. Chaotic Dyn., 19 (2014), 556-575.
doi: 10.1134/S1560354714050049. |
| [9] |
D. E. Chang,
Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation by the energy-shaping method, IEEE Trans. Automat. Control, 55 (2010), 1888-1893.
doi: 10.1109/TAC.2010.2049279. |
| [10] |
D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Wollsey,
The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM Control Optim. Calc. Var., 8 (2002), 393-422.
doi: 10.1051/cocv:2002045. |
| [11] |
B. Gharesifard,
Stabilization of systems with one degree of underactuation with energy shaping: A geometric approach, SIAM J. Control Optim., 49 (2011), 1422-1434.
doi: 10.1137/09076698X. |
| [12] |
H. Goldschmidt,
Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.
doi: 10.2307/1970689. |
| [13] |
S. D. Grillo, L. M. Salomone and M. Zuccalli,
Asymptotic stabilizability of underactuated Hamiltonian systems with two degrees of freedom, Russ. J. Nonlinear Dyn., 15 (2019), 309-326.
doi: 10.20537/nd190309. |
| [14] |
S. D. Grillo, L. M. Salomone and M. Zuccalli,
On the relationship between the energy shaping and the Lyapunov constraint based methods, J. Geom. Mech., 9 (2017), 459-486.
doi: 10.3934/jgm.2017018. |
| [15] |
J. Hamberg,
Controlled Lagrangians, symmetries and conditions for strong matching, IFAC Proceedings Volumes, 33 (2000), 57-62.
doi: 10.1016/S1474-6670(17)35547-7. |
| [16] |
J. Hamberg, General matching conditions in the theory of controlled Lagrangians, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, 1999.
doi: 10.1109/CDC.1999.831306. |
| [17] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. |
| [18] |
A. D. Lewis, Potential energy shaping after kinetic energy shaping, Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006.
doi: 10.1109/CDC.2006.376885. |
| [19] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.
doi: 10.1007/978-0-387-21792-5. |
| [20] |
J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 2001. |
| [21] |
J. G. Romero, A. Donaire and R. Ortega,
Robust energy shaping control of mechanical systems, Systems Control Lett., 62 (2013), 770-780.
doi: 10.1016/j.sysconle.2013.05.011. |
| [22] |
D. Zenkov, Matching and stabilization of linear mechanical systems, Proceedings of 2002 International Symposium of Mathematical Theory of Networks and Systems, South Bend, IN, 2002. Available from: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F5FAA6E2AB4998A2C00B8A5CDB0A461C?doi=10.1.1.12.7462&rep=rep1&type=pdf. |
show all references
References:
| [1] |
R. Abraham and J. E. Marsden, Foundation of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. |
| [2] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.
doi: 10.1007/978-1-4757-1693-1. |
| [3] |
A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003.
doi: 10.1007/b97376. |
| [4] |
W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, 63, Academic Press, New York-London, 1975.
|
| [5] |
F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.
doi: 10.1007/978-1-4899-7276-7. |
| [6] |
D. E. Chang, Generalization of the IDA-PBC method for stabilization of mechanical systems, 18th Mediterranean Conference on Control and Automation, MED'10, Marrakech, Morocco, 2010.
doi: 10.1109/MED.2010.5547672. |
| [7] |
D. E. Chang,
The method of controlled Lagrangians: Energy plus force shaping, SIAM J. Control Optim., 48 (2010), 4821-4845.
doi: 10.1137/070691310. |
| [8] |
D. E. Chang,
On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regul. Chaotic Dyn., 19 (2014), 556-575.
doi: 10.1134/S1560354714050049. |
| [9] |
D. E. Chang,
Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation by the energy-shaping method, IEEE Trans. Automat. Control, 55 (2010), 1888-1893.
doi: 10.1109/TAC.2010.2049279. |
| [10] |
D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Wollsey,
The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM Control Optim. Calc. Var., 8 (2002), 393-422.
doi: 10.1051/cocv:2002045. |
| [11] |
B. Gharesifard,
Stabilization of systems with one degree of underactuation with energy shaping: A geometric approach, SIAM J. Control Optim., 49 (2011), 1422-1434.
doi: 10.1137/09076698X. |
| [12] |
H. Goldschmidt,
Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.
doi: 10.2307/1970689. |
| [13] |
S. D. Grillo, L. M. Salomone and M. Zuccalli,
Asymptotic stabilizability of underactuated Hamiltonian systems with two degrees of freedom, Russ. J. Nonlinear Dyn., 15 (2019), 309-326.
doi: 10.20537/nd190309. |
| [14] |
S. D. Grillo, L. M. Salomone and M. Zuccalli,
On the relationship between the energy shaping and the Lyapunov constraint based methods, J. Geom. Mech., 9 (2017), 459-486.
doi: 10.3934/jgm.2017018. |
| [15] |
J. Hamberg,
Controlled Lagrangians, symmetries and conditions for strong matching, IFAC Proceedings Volumes, 33 (2000), 57-62.
doi: 10.1016/S1474-6670(17)35547-7. |
| [16] |
J. Hamberg, General matching conditions in the theory of controlled Lagrangians, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, 1999.
doi: 10.1109/CDC.1999.831306. |
| [17] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. |
| [18] |
A. D. Lewis, Potential energy shaping after kinetic energy shaping, Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006.
doi: 10.1109/CDC.2006.376885. |
| [19] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.
doi: 10.1007/978-0-387-21792-5. |
| [20] |
J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 2001. |
| [21] |
J. G. Romero, A. Donaire and R. Ortega,
Robust energy shaping control of mechanical systems, Systems Control Lett., 62 (2013), 770-780.
doi: 10.1016/j.sysconle.2013.05.011. |
| [22] |
D. Zenkov, Matching and stabilization of linear mechanical systems, Proceedings of 2002 International Symposium of Mathematical Theory of Networks and Systems, South Bend, IN, 2002. Available from: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F5FAA6E2AB4998A2C00B8A5CDB0A461C?doi=10.1.1.12.7462&rep=rep1&type=pdf. |
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