Constraint Control of Nonholonomic Mechanical Systems

Vakhtang Putkaradze; Stuart Rogers

doi:10.1007/s00332-017-9406-1

Constraint Control of Nonholonomic Mechanical Systems

Vakhtang Putkaradze, Stuart Rogers

Institute for Mathematics and its Applications

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov’s problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction ξ. We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov’s problem for the rotation group SO(3). We show that it is possible to control the system using the constraint ξ(t) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.

Original language	English (US)
Pages (from-to)	193-234
Number of pages	42
Journal	Journal of Nonlinear Science
Volume	28
Issue number	1
DOIs	https://doi.org/10.1007/s00332-017-9406-1
State	Published - Feb 1 2018

Bibliographical note

Funding Information:
This problem has been suggested to us by Prof. D. V. Zenkov during a visit to the University of Alberta. Subsequent discussions and continued interest by Prof. Zenkov to this project are gratefully acknowledged. We also acknowledge fruitful discussions with Profs. A. A. Bloch, D. D. Holm, M. Leok, F. Gay-Balmaz, T. Hikihara, A. Lewis, and H. Yoshimura. There were many helpful exchanges with Prof. H. Oberle, Prof. L. F. Shampine (bvp4c and bvp5c), Prof. F. Mazzia (TOM and bvptwp), J. Willkomm (ADiMat), M. Weinsten (ADiGator), and Prof. A. Rao (GPOPS-II) concerning ODE BVP solvers, automatic differentiation software, and direct method solvers. Both authors of this project were partially supported by the NSERC Discovery Grant and the University of Alberta Centennial Fund. In addition, Stuart Rogers was supported by the FGSR Graduate Travel Award, the IGR Travel Award, the GSA Academic Travel Award, and the AMS Fall Sectional Grad Student Travel Grant. We also thank the Alberta Innovates Technology Funding (AITF) for providing support to the authors through the Alberta Centre for Earth Observation Sciences (CEOS).

Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

Keywords

Nonholonomic mechanics
Optimal control
Suslov’s problem

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abstract = "We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov{\textquoteright}s problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction ξ. We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov{\textquoteright}s problem for the rotation group SO(3). We show that it is possible to control the system using the constraint ξ(t) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.",

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note = "Funding Information: This problem has been suggested to us by Prof. D. V. Zenkov during a visit to the University of Alberta. Subsequent discussions and continued interest by Prof. Zenkov to this project are gratefully acknowledged. We also acknowledge fruitful discussions with Profs. A. A. Bloch, D. D. Holm, M. Leok, F. Gay-Balmaz, T. Hikihara, A. Lewis, and H. Yoshimura. There were many helpful exchanges with Prof. H. Oberle, Prof. L. F. Shampine (bvp4c and bvp5c), Prof. F. Mazzia (TOM and bvptwp), J. Willkomm (ADiMat), M. Weinsten (ADiGator), and Prof. A. Rao (GPOPS-II) concerning ODE BVP solvers, automatic differentiation software, and direct method solvers. Both authors of this project were partially supported by the NSERC Discovery Grant and the University of Alberta Centennial Fund. In addition, Stuart Rogers was supported by the FGSR Graduate Travel Award, the IGR Travel Award, the GSA Academic Travel Award, and the AMS Fall Sectional Grad Student Travel Grant. We also thank the Alberta Innovates Technology Funding (AITF) for providing support to the authors through the Alberta Centre for Earth Observation Sciences (CEOS). Publisher Copyright: {\textcopyright} 2017, Springer Science+Business Media, LLC.",

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Constraint Control of Nonholonomic Mechanical Systems

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