American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space
  • PROC Home
  • This Issue
  • Next Article
    Investigation of the long-time evolution of localized solutions of a dispersive wave system
2013, 2013(special): 149-158. doi: 10.3934/proc.2013.2013.149

Optimal control of underactuated mechanical systems with symmetries

Leonardo Colombo 1, and  David Martín de Diego 2,

1. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera 15, 28049, Madrid, Spain
2. 

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid

Received  August 2012 Revised  November 2012 Published  November 2013

The aim of this paper is to study optimal control problems for underactuated mechanical systems with symmetries using higher-order Lagrangian mechanics. We variationally derive the corresponding Lagrange -Poincaré equations for second-order Lagrangians with constraints defined on trivial principal bundles and apply them to study an optimal control problem for an underactuated vehicle.
Keywords: optimal control, Euler-Poincaré, underactuated systems..
Mathematics Subject Classification: Primary: 70Q05, 70G75, 70G65; Secondary: 70H50, 70H0.
Citation: Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149
References:
[1]

A. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series vol.24, Springer-Verlag, New-York, 2003.  Google Scholar

[2]

F. Bullo and A. Lewis, Geometric control of mechanical systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, Springer Verlang, New York, 2005.  Google Scholar

[3]

H. Cendra, J. Marsden, T. Ratiu., Lagrangian reduction by stages. Memoirs of the American Mathematical Society, 152 (722). pp. 1-108. (2001)  Google Scholar

[4]

L. Colombo and D. Martín de Diego, On the geometry of higher-order variational problems on Lie groups, arXiv:1104.3221v1 (2011). Google Scholar

[5]

A. Lewis, R. Murray, Variational principles for constrained systems: Theory and experiment. Int. J. nonlinear mech. 30 (1995), no. 6, 793-815.  Google Scholar

show all references

References:
[1]

A. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series vol.24, Springer-Verlag, New-York, 2003.  Google Scholar

[2]

F. Bullo and A. Lewis, Geometric control of mechanical systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, Springer Verlang, New York, 2005.  Google Scholar

[3]

H. Cendra, J. Marsden, T. Ratiu., Lagrangian reduction by stages. Memoirs of the American Mathematical Society, 152 (722). pp. 1-108. (2001)  Google Scholar

[4]

L. Colombo and D. Martín de Diego, On the geometry of higher-order variational problems on Lie groups, arXiv:1104.3221v1 (2011). Google Scholar

[5]

A. Lewis, R. Murray, Variational principles for constrained systems: Theory and experiment. Int. J. nonlinear mech. 30 (1995), no. 6, 793-815.  Google Scholar

[1]

Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261
[2]

Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063
[3]

Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1495-1518. doi: 10.3934/dcdss.2020377
[4]

Sonia Martínez, Jorge Cortés, Francesco Bullo. A catalog of inverse-kinematics planners for underactuated systems on matrix groups. Journal of Geometric Mechanics, 2009, 1 (4) : 445-460. doi: 10.3934/jgm.2009.1.445
[5]

David González-Sánchez, Onésimo Hernández-Lerma. On the Euler equation approach to discrete--time nonstationary optimal control problems. Journal of Dynamics & Games, 2014, 1 (1) : 57-78. doi: 10.3934/jdg.2014.1.57
[6]

William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control & Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028
[7]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[8]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial & Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311
[9]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[10]

Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367
[11]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
[12]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[13]

Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial & Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63
[14]

Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107
[15]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279
[16]

Luca Galbusera, Sara Pasquali, Gianni Gilioli. Stability and optimal control for some classes of tritrophic systems. Mathematical Biosciences & Engineering, 2014, 11 (2) : 257-283. doi: 10.3934/mbe.2014.11.257
[17]

Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187
[18]

Qiying Hu, Wuyi Yue. Optimal control for discrete event systems with arbitrary control pattern. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 535-558. doi: 10.3934/dcdsb.2006.6.535
[19]

Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541
[20]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020

 Impact Factor: 

Tools

Metrics

  • PDF downloads (105)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences