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Zbl 1120.37314
Koon, Wang Sang; Marsden, Jerrold E.
Poisson reduction for nonholonomic mechanical systems with symmetry. (English)
[J] Rep. Math. Phys. 42, No. 1-2, 101-134 (1998). ISSN 0034-4877

Many important problems in robotics, the dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics, which typically means mechanical systems with rolling constraints. Some of the important issues are trajectory tracking, dynamic stability and feedback stabilization, bifurcation and control. Many of these systems have symmetry, such as the group of Euclidean motions in the plane or in space and this symmetry plays an important role in the theory.\par In the present paper, the authors take the point of view of mechanics on Poisson manifolds. In particular, they further develop the theory of nonholonomic Poisson reduction, and tie this theory to other work in the area concerning nonholonomic symplectic and Lagrangian reduction. They use the Poisson reduction procedure to break the Hamiltonian nonholonomic dynamics into a reconstruction equation for elements of the symmetry group, a momentum equation (the momentum is not conserved for nonholonomic systems in general), and a set of reduced Hamiltonian equations which describe the time evolution of the shape of the system. The basic theory is illustrated with the example of the snakeboard.\par The results in this paper appear to be especially important for the future development of the stability theory and of the control theory for nonholonomic mechanical systems with symmetry.
[Jürgen Scheurle (MR1656278)]

MSC 2000:: *37J60 Nonholonomic dynamical systems
37J15 Symmetries, etc.
53D20 Momentum maps; symplectic reduction
70F25 Nonholonomic systems
70H33 Symmetries
70H45 Constrained dynamics, Dirac's theory of constraints
70Q05 Control of mechanical systems

Keywords: nonholonomic mechanics; symmetry; mechanics on Poisson manifolds; nonholonomic symplectic and Lagrangian reduction; momentum equation; reduced Hamiltonian equations; stability theory; control theory

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