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Analysis and Geometry in Control Theory and its Applications pp 35–64Cite as

A Geometric Approach to the Optimal Control of Nonholonomic Mechanical Systems

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 11))

Abstract

In this paper, we describe a constrained Lagrangian and Hamiltonian formalism for the optimal control of nonholonomic mechanical systems. In particular, we aim to minimize a cost functional, given initial and final conditions where the controlled dynamics are given by a nonholonomic mechanical system. In our paper, the controlled equations are derived using a basis of vector fields adapted to the nonholonomic distribution and the Riemannian metric determined by the kinetic energy. Given a cost function, the optimal control problem is understood as a constrained problem or equivalently, under some mild regularity conditions, as a Hamiltonian problem on the cotangent bundle of the nonholonomic distribution. A suitable Lagrangian submanifold is also shown to lead to the correct dynamics. Application of the theory is demonstrated through several examples including optimal control of the Chaplygin sleigh, a continuously variable transmission, and a problem of motion planning for obstacle avoidance.

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Acknowledgements

This work has been partially supported by grants MTM 2013-42870-P, MTM 2009-08166-E, IRSES-project “Geomech-246981” and NSF grants INSPIRE-1363720 and DMS-1207693. We wish to thank Klas Modin and Olivier Verdier for allowing us to use their description of the Continuously Variable Transmission Gearbox.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Anthony Bloch & Leonardo Colombo

  2. Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    Rohit Gupta

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

    David Martín de Diego

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  1. Anthony Bloch
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  2. Leonardo Colombo
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  4. David Martín de Diego
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Correspondence to Anthony Bloch .

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Editors and Affiliations

  1. Department of Mathematics, Université de Bretagne Occidentale, Brest, France

    Piernicola Bettiol

  2. Department of Mathematics, Università di Roma "Tor Vergata", Roma, Italy

    Piermarco Cannarsa

  3. Department of Mathematics, Università di Padova, Padova, Italy

    Giovanni Colombo

  4. Department of Mathematics, Università di Padova, Padova, Italy

    Monica Motta

  5. Department of Mathematics, Università di Padova, Padova, Italy

    Franco Rampazzo

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Bloch, A., Colombo, L., Gupta, R., de Diego, D.M. (2015). A Geometric Approach to the Optimal Control of Nonholonomic Mechanical Systems. In: Bettiol, P., Cannarsa, P., Colombo, G., Motta, M., Rampazzo, F. (eds) Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-06917-3_2

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