Controlling nonholonomic Chaplygin systems

Antunes, A. C. B.; Sigaud, C.; Moraes, Pedro Claudio Guaranho de

doi:10.1590/s0103-97332010000200002

Cited by 7 publications

(9 citation statements)

References 12 publications

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“…Sometimes the second term alone on the right-hand side of the above equation, the one containing the λs, is interpreted as the constraint force. Then the λs themselves are regarded as free parameters that can be conveniently chosen in order to force the system to follow a prescribed path in configuration space [39]. This interpretation would make vakonomic mechanics compatible with the principle of virtual work, but it does not seem to be tenable for at least two reasons.…”

Section: Vakonomic Mechanicsmentioning

confidence: 99%

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Complete inequivalence of nonholonomic and vakonomic mechanics: rolling coin on an inclined plane

Lemos̀¹

2021

Preprint

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Vakonomic mechanics has been proposed as a possible description of the dynamics of systems subject to nonholonomic constraints. The aim of the present work is to show that for an important physical system the motion brought about by vakonomic mechanics is completely inequivalent to the one derived from nonholonomic mechanics, which relies on the standard method of Lagrange multipliers in the d'Alembert-Lagrange formulation of the classical equations of motion. For the rolling coin on an inclined plane, it is proved that no nontrivial solution to the equations of motion of nonholonomic mechanics can be obtained in the framework of vakonomic mechanics. This completes previous investigations that managed to show only that, for certain mechanical systems, some but not necessarily all nonholonomic motions are beyond the reach of vakonomic mechanics. Furthermore, it is argued that a simple qualitative experiment that anyone can perform at home supports the predictions of nonholonomic mechanics.

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Section: Vakonomic Mechanicsmentioning

confidence: 99%

“…Fig. (2) shows two typical trajectories for the coin's center of mass predicted by equations (39) and (40).…”

Section: Rolling Coin On An Inclined Planementioning

confidence: 99%

Complete inequivalence of nonholonomic and vakonomic mechanics: rolling coin on an inclined plane

Lemos̀¹

2021

Preprint

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“…One of the manners of realizing the brachistochronic motion of a system, as shown, has been achieved by the active control force F → = F → 1 + F → 2 , which is acting at point C (see Figure 1(a)). Note that in Antunes and Sigaud [16] the control of the Chaplygin sleigh motion is achieved by a single active force applied at point C and a single torque of active forces. Systems (4) or (11) are used as a basis for determining the laws of change of the control forces F 1 and F 2 , as well as the reaction of the nonholonomic constraint R as a function of defined quantities and their derivatives:…”

Section: Introductionmentioning

confidence: 99%

A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

Radulović

Šalinić

Obradović

et al. 2016

Mathematics and Mechanics of Solids

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A new approach for the determination of the global minimum time for the case of the brachistochronic motion of the Chaplygin sleigh is presented. The new approach is based on the use of the shooting method in solving the corresponding two-point boundary-value problem and defining either the crossing points of surfaces or the crossing points space of curves in a three-dimensional space of two costate variables and the time of the brachistochronic motion of the sleigh. A number of examples for multiple extremals of the Chaplygin sleigh brachistochrone problem are provided. In these examples, the global minimum is the solution to which the minimum time of motion corresponds.

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“…In [42,43], the dynamics of controlled nonholonomic systems (the Chaplygin sleigh and a dynamically asymmetric balanced ball rolling without slipping) is considered. By an appropriate choice of control one obtains vakonomic motions of such systems.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of systems with servoconstraints. II

Козлов

2015

Regul. Chaot. Dyn.

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This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servoconstraint, which implies that the projection of the body's angular velocity on some body-fixed direction is zero.MSC2010 numbers: 70E18, 34C40

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Controlling nonholonomic Chaplygin systems

Cited by 7 publications

References 12 publications

Complete inequivalence of nonholonomic and vakonomic mechanics: rolling coin on an inclined plane

Complete inequivalence of nonholonomic and vakonomic mechanics: rolling coin on an inclined plane

A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

The dynamics of systems with servoconstraints. II

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