January  2015, 2(1): 25-50. doi: 10.3934/jcd.2015.2.25

Symmetry exploiting control of hybrid mechanical systems

1. 

Neuroscience and Robotics Laboratory, Northwestern University, Evanston, IL, United States

2. 

Chair of Applied Mathematics, University of Paderborn, Paderborn, Germany, Germany

Received  May 2014 Revised  April 2015 Published  August 2015

Symmetry properties such as invariances of mechanical systems can be beneficially exploited in solution methods for control problems. A recently developed approach is based on quantization by so called motion primitives. A library of these motion primitives forms an artificial hybrid system. In this contribution, we study the symmetry properties of motion primitive libraries of mechanical systems in the context of hybrid symmetries. Furthermore, the classical concept of symmetry in mechanics is extended to hybrid mechanical systems and an extended motion planning approach is presented.
Citation: Kathrin Flasskamp, Sebastian Hage-Packhäuser, Sina Ober-Blöbaum. Symmetry exploiting control of hybrid mechanical systems. Journal of Computational Dynamics, 2015, 2 (1) : 25-50. doi: 10.3934/jcd.2015.2.25
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley, 1987.

[2]

A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem, IEEE Transactions on Automatic Control, 45 (2000), 2253-2270. doi: 10.1109/9.895562.

[3]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, vol. 49 of Texts in Applied Mathematics, Springer, 2005. doi: 10.1007/978-1-4899-7276-7.

[4]

M. Buss, M. Glocker, M. Hardt, O. von Stryk, R. Bulirsch and G. Schmidt, Nonlinear hybrid dynamical systems: Modeling, optimal control, and applications, in Modelling, Analysis, and Design of Hybrid Systems (eds. S. Engell, G. Frehse and E. Schnieder), vol. 279 of Lecture Notes in Control and Information Sciences, Springer, 2002, 311-335. doi: 10.1007/3-540-45426-8_18.

[5]

H. Choset, K. M. Lynch, S. Hutchinson, G. A. Kantor, W. Burgard, L. E. Kavraki and S. Thrun, Principles of Robot Motion: Theory, Algorithms, and Implementations, MIT Press, 2005.

[6]

K. Flaßkamp, S. Ober-Blöbaum and M. Kobilarov, Solving optimal control problems by exploiting inherent dynamical systems structures, Journal of Nonlinear Science, 22 (2012), 599-629. doi: 10.1007/s00332-012-9140-7.

[7]

K. Flaßkamp, On the Optimal Control of Mechanical Systems - Hybrid Control Strategies and Hybrid Dynamics, PhD thesis, University of Paderborn, 2013.

[8]

E. Frazzoli, Robust Hybrid Control for Autonomous Vehicle Motion Planning, PhD thesis, Massachusetts Institute of Technology, 2001.

[9]

E. Frazzoli and F. Bullo, On quantization and optimal control of dynamical systems with symmetries, in Proceedings of the 41st IEEE Conference on Decision and Control, 1 (2002), 817-823. doi: 10.1109/CDC.2002.1184606.

[10]

E. Frazzoli, M. A. Dahleh and E. Feron, Robust hybrid control for autonomous vehicle motion planning, in Proceedings of the 39th IEEE Conference on Decision and Control, 1 (2000), 821-826. doi: 10.1109/CDC.2000.912871.

[11]

E. Frazzoli, M. A. Dahleh and E. Feron, Maneuver-based motion planning for nonlinear systems with symmetries, IEEE Transactions on Robotics, 21 (2005), 1077-1091.

[12]

M. Golubitsky and I. Stewart, The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space, vol. 200 of Progress in Mathematics, 2002. doi: 10.1007/978-3-0348-8167-8.

[13]

S. Hage-Packhäuser, Structural Treatment of Time-Varying Dynamical System Networks in the Light of Hybrid Symmetries, PhD thesis, University of Paderborn, 2012.

[14]

M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles, PhD thesis, University of Southern California, USA, 2008.

[15]

J. Lygeros, K. H. Johansson, S. N. Simić, J. Zhang and S. Sastry, Dynamical Properties of Hybrid Automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650.

[16]

J. E. Marsden, Lectures on Mechanics, no. 174 in London Mathematical Society Lecture Note Series, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.

[17]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Communications in Mathematical Physics, 199 (1998), 351-395. doi: 10.1007/s002200050505.

[18]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer, 1999. doi: 10.1007/978-0-387-21792-5.

[19]

J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, Journal of Mathematical Physics, 41 (2000), 3379-3429. doi: 10.1063/1.533317.

[20]

J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, Zeitschrift für angewandte Mathematik und Physik (ZAMP), 44 (1993), 17-43. doi: 10.1007/BF00914351.

[21]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[22]

S. Ober-Blöbaum, O. Junge and J. E. Marsden, Discrete mechanics and optimal control: an analysis, Control, Optimisation and Calculus of Variations, 17 (2011), 322-352. doi: 10.1051/cocv/2010012.

[23]

A. J. v. d. Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems, vol. 251 of Lecture Notes in Control and Information Sciences, Springer, 2000.

[24]

S. N. Simić, K. H. Johansson, S. Sastry and J. Lygeros, Towards a Geometric Theory of Hybrid Systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 12 (2005), 649-687.

[25]

J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method, Archive for Rational Mechanics and Analysis, 115 (1991), 15-59. doi: 10.1007/BF01881678.

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley, 1987.

[2]

A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem, IEEE Transactions on Automatic Control, 45 (2000), 2253-2270. doi: 10.1109/9.895562.

[3]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, vol. 49 of Texts in Applied Mathematics, Springer, 2005. doi: 10.1007/978-1-4899-7276-7.

[4]

M. Buss, M. Glocker, M. Hardt, O. von Stryk, R. Bulirsch and G. Schmidt, Nonlinear hybrid dynamical systems: Modeling, optimal control, and applications, in Modelling, Analysis, and Design of Hybrid Systems (eds. S. Engell, G. Frehse and E. Schnieder), vol. 279 of Lecture Notes in Control and Information Sciences, Springer, 2002, 311-335. doi: 10.1007/3-540-45426-8_18.

[5]

H. Choset, K. M. Lynch, S. Hutchinson, G. A. Kantor, W. Burgard, L. E. Kavraki and S. Thrun, Principles of Robot Motion: Theory, Algorithms, and Implementations, MIT Press, 2005.

[6]

K. Flaßkamp, S. Ober-Blöbaum and M. Kobilarov, Solving optimal control problems by exploiting inherent dynamical systems structures, Journal of Nonlinear Science, 22 (2012), 599-629. doi: 10.1007/s00332-012-9140-7.

[7]

K. Flaßkamp, On the Optimal Control of Mechanical Systems - Hybrid Control Strategies and Hybrid Dynamics, PhD thesis, University of Paderborn, 2013.

[8]

E. Frazzoli, Robust Hybrid Control for Autonomous Vehicle Motion Planning, PhD thesis, Massachusetts Institute of Technology, 2001.

[9]

E. Frazzoli and F. Bullo, On quantization and optimal control of dynamical systems with symmetries, in Proceedings of the 41st IEEE Conference on Decision and Control, 1 (2002), 817-823. doi: 10.1109/CDC.2002.1184606.

[10]

E. Frazzoli, M. A. Dahleh and E. Feron, Robust hybrid control for autonomous vehicle motion planning, in Proceedings of the 39th IEEE Conference on Decision and Control, 1 (2000), 821-826. doi: 10.1109/CDC.2000.912871.

[11]

E. Frazzoli, M. A. Dahleh and E. Feron, Maneuver-based motion planning for nonlinear systems with symmetries, IEEE Transactions on Robotics, 21 (2005), 1077-1091.

[12]

M. Golubitsky and I. Stewart, The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space, vol. 200 of Progress in Mathematics, 2002. doi: 10.1007/978-3-0348-8167-8.

[13]

S. Hage-Packhäuser, Structural Treatment of Time-Varying Dynamical System Networks in the Light of Hybrid Symmetries, PhD thesis, University of Paderborn, 2012.

[14]

M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles, PhD thesis, University of Southern California, USA, 2008.

[15]

J. Lygeros, K. H. Johansson, S. N. Simić, J. Zhang and S. Sastry, Dynamical Properties of Hybrid Automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650.

[16]

J. E. Marsden, Lectures on Mechanics, no. 174 in London Mathematical Society Lecture Note Series, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.

[17]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Communications in Mathematical Physics, 199 (1998), 351-395. doi: 10.1007/s002200050505.

[18]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer, 1999. doi: 10.1007/978-0-387-21792-5.

[19]

J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, Journal of Mathematical Physics, 41 (2000), 3379-3429. doi: 10.1063/1.533317.

[20]

J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, Zeitschrift für angewandte Mathematik und Physik (ZAMP), 44 (1993), 17-43. doi: 10.1007/BF00914351.

[21]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[22]

S. Ober-Blöbaum, O. Junge and J. E. Marsden, Discrete mechanics and optimal control: an analysis, Control, Optimisation and Calculus of Variations, 17 (2011), 322-352. doi: 10.1051/cocv/2010012.

[23]

A. J. v. d. Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems, vol. 251 of Lecture Notes in Control and Information Sciences, Springer, 2000.

[24]

S. N. Simić, K. H. Johansson, S. Sastry and J. Lygeros, Towards a Geometric Theory of Hybrid Systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 12 (2005), 649-687.

[25]

J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method, Archive for Rational Mechanics and Analysis, 115 (1991), 15-59. doi: 10.1007/BF01881678.

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