The problem of optimal control of a Chaplygin ball by internal rotors
type: Item
instance of: scholarly article
title: The problem of optimal control of a Chaplygin ball by internal rotors
zbMATH Open document ID: 1264.37015
DOI: 10.1134/S156035471206007X
published in: Regular and Chaotic Dynamics
publication date: 2013-03-22T00:00:00Z
review text: The author of this interesting paper studies the problem of optimal control of a Chaplygin ball on a plane by three internal rotors. In 1983, V. V. Kozlov introduced a new model of mechanical system with nonintegrable constraints [Mosc. Univ. Mech. Bull. 37, No. 3--4, 27--34 (1982; Zbl 0558.70012); ibid. 37, No. 3--4, 74--80 (1982; Zbl 0508.70012); libid. 38, No. 3, 40--51 (1983; Zbl 0558.70013)]. In contrast to the classical nonholonomic mechanics for this model, called the vaconomic mechanics, Hamilton's variational principle holds, and trajectories of the system are extremals of the constraint variational problem. The equations of vaconomic mechanics can be used as an approximation of the motions of bodies in an ideal fluid under certain limit relations between masses and moments of inertia, which leads to nonintegrable constraints. These equations appear also in the geometric control theory. The control system on \({\mathcal{M}}=SO(3)\times \mathbb{R}^2\) has the form \[ \dot{q}=\mathbf{f}(q,\pmb{\omega }), \;\;q =( \pmb{\alpha }, \pmb{\beta }, \pmb{\gamma } , \mathbf{r} ) \in\mathcal{M}, \] where \(\pmb{\alpha , \beta , \gamma }\) are the basic vectors represented in the moving frame, \(\pmb{\omega }\in \mathbb{R}^3\) is the vector of angular velocity, the orientation of the ball is given by the orthogonal matrix \((\pmb{ \alpha , \beta , \gamma })\in \mathrm{SO}(3)\), and the position of the center of the ball is expressed by \( \mathbf{r} \in\mathbb{R}^2\); \(\mathcal{M}\) is the configuration space; \(\pmb{\omega }=\pmb{\omega }(t)\in \mathbb{R}^3\) is the control. The vector field \(\mathbf{f}\) on \(\mathcal{M}\) is given by \[ \dot{\alpha }=\pmb{\alpha }\times\pmb{\omega }, \;\;\dot{\pmb{\beta }} =\pmb{\beta }\times\pmb{\omega } \;\;\dot{\pmb{\gamma }}=\pmb{\gamma }\times\pmb{\omega } \] \[ \dot{x} = R\langle\pmb{\beta } ,\pmb{\omega } \rangle, \;\;\dot{y} = - R\langle\pmb{\alpha } ,\pmb{\omega } \rangle. \] The optimal control problem is to move the ball…
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 70E18
Mathematics Subject Classification ID: 70F25
Mathematics Subject Classification ID: 70H45
Mathematics Subject Classification ID: 93B27
Mathematics Subject Classification ID: 37N35
zbMATH DE Number: 6148384
zbMATH Keywords: nonholonomic constraint
zbMATH Keywords: vaconomic mechanics
zbMATH Keywords: optimal control
zbMATH Keywords: maximum principle
zbMATH Keywords: Hamiltonian
author: S. V. Bolotin
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
cites work: Q5690486
cites work: How to control Chaplygin's sphere using rotors
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cites work: Q3702552
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cites work: Variational principles for constrained systems: Theory and experiment
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description: scientific article; zbMATH DE number 6148384