24 research outputs found
Radial kinetic nonholonomic trajectories are Riemannian geodesics!
No full textNonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed point q are true geodesics for a family of Riemannian metrics on the image submanifold Mnh
q of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point q, for sufficiently small times, minimize length in Mnhq!D. Martín de Diego and A. Anahory Simoes acknowledge financial support from the Spanish Ministry of Science and Innovation, under grants PID2019-106715GB-C21 and ?Severo OchoaProgramme for Centres of Excellence in R&D? (CEX2019-000904-S). A. Anahory Simoes is supported by the FCT (Portugal) research fellowship SFRH/BD/129882/2017 partially funded by the European Union
(ESF). J.C. Marrero acknowledges the partial support by European Union (Feder) grant PGC2018-098265-B-C32.Peer reviewe
A nonholonomic Newmark method
No full textUsing the nonholonomic exponential map, we obtain a new version of Newmark-type methods for nonholonomic systems (see also Jay and Negrut(2009) for a different extension). We give numerical examples including a test problem where the structure of reversible integrability responsible for good energy behaviour as described in Modin and Verdier (2020) is lost. We observe that the composition of two Newmark methods is able to produce good energy behaviour on this test problem.Fil: Anahory Simoes, Alexandre. IE Universidad; EspañaFil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Marrero González, Juan Carlos. Universidad de La Laguna; EspañaFil: Martín de Diego, David. Universidad Nebrija; España. Instituto de Ciencias Matemáticas; Españ
Jacobi fields in nonholonomic mechanics
No full textIn this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.D Martín de Diego and A Simoes are supported by I-Link Project (Ref: linkA20079), Ministerio de Ciencia e Innovación (Spain) under Grants PID2019-106715GB-C21P and from the Spanish Ministry of Science and Innovation, through the 'Severo Ochoa Programme for Centres of Excellence in R&D' (CEX2019-000904-S). A Anahory Simoes is supported by the FCT research fellowship SFRH/BD/129882/2017, partially funded by the European Union (ESF). J C Marrero acknowledges the partial support by European Union (Feder) Grant PGC2018-098265-B-C32. The authors would like to thank the referees for the helpful comments improving the correctness and clarity of the paper
Euler–Lagrange–Herglotz equations on Lie algebroids
No full textWe introduce Euler–Lagrange–Herglotz equations on Lie algebroids. The methodology is to extend the Jacobi structure from TQ× R and T∗Q× R to A× R and A∗× R , respectively, where A is a Lie algebroid and A∗ carries the associated Poisson structure. We see that A∗× R possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems on Lie algebroids, generalizing previous models on TQ× R and introducing new ones as for instance for reduced systems on Lie algebras, semidirect products (action Lie algebroids) and Atiyah bundles.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Alexandre Anahory Simoes, Leonardo Colombo and Manuel de León acknowledge financial support from Grant PID2019-106715GB-C21 and PID2022-137909NB-C2 funded by MCIN/AEI/ 10.13039/501100011033. Modesto Salgado and Silvia Souto acknowledge financial support of the Ministerio de Ciencia, Innovación y Universidades (Spain), projects PGC2018-098265-B-C33 and D2021-125515NB-21. We acknowledge the reviewers of the paper for the fruitful comments to improve the presentation of this work.Peer reviewe
Virtual nonholonomic constraints: A geometric approach
No full textVirtual constraints are relations imposed in a control system that become invariant via feedback instead of real physical constraints acting on the system. Nonholonomic systems are mechanical systems with non-integrable constraints on the velocities. In this work, we introduce the notion of virtual nonholonomic constraints in a geometric framework. More precisely, it is a controlled invariant distribution associated with an affine connection mechanical control system. We show the existence and uniqueness of a control law defining a virtual nonholonomic constraint and we characterize the trajectories of the closed-loop system as solutions of a mechanical system associated with an induced constrained connection. Moreover, we characterize the dynamics for nonholonomic systems in terms of virtual nonholonomic constraints, i.e., we characterize when can we obtain nonholonomic dynamics from virtual nonholonomic constraints.Peer reviewe
VIRTUAL AFFINE NONHOLONOMIC CONSTRAINTS
No full textirtual constraints are relations imposed on a control system via feedback control that become invariant via feedback, as opposed to physical constraints acting on the system. Nonholonomic systems are mechanical systems with non-integrable constraints on the velocities. In this work, we introduce the notion of virtual affine nonholonomic constraints in a geometric framework. More precisely, it is a controlled invariant affine distribution associated with an affine connection mechanical control system. We show the existence and uniqueness of a control law defining a virtual affine nonholonomic constraint.Peer reviewe
Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups
No full textWe study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on S E(2) in the presence of a static obstacle.The authors acknowledge financial support from the Spanish Ministry of Science and Innovation,
under grants PID2019-106715GB-C21, MTM2016-76702-P.Peer reviewe
Exact discrete Lagrangian mechanics for nonholonomic mechanics
No full textWe construct the exponential map associated to a nonholonomic system that allows us
to define an exact discrete nonholonomic constraint submanifold. We reproduce the
continuous nonholonomic flow as a discrete flow on this discrete constraint submanifold deriving an exact discrete version of the nonholonomic equations. Finally, we
derive a general family of nonholonomic integrators that includes as a particular case
the exact discrete nonholonomic trajectoryD. Martín de Diego and A. Simoes acknowledge financial support from the Spanish Ministry of Science and Innovation, under grants PID2019-106715GB-C21, MTM2016-76702-P, and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” R&D (CEX2019-000904-S). A. Simoes is supported by the FCT research fellowship SFRH/BD/129882/2017 partially funded by the European Union (ESF). J.C. Marrero acknowledges the partial support by European Union (Feder) grant MTM 2015-64166-C2-2P and PGC2018-098265-B-C32
Análisis Geométrica y Numérica de Sistemas No-holónomos
No full textTesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 26-11-202
Hamel equations and quasivelocities for nonholonomic systems with inequality constraints
No full textIn this paper we derive Hamel equations for the motion of nonholonomic
systems subject to inequality constraints in quasivelocities. As examples, the
vertical rolling disk hitting a wall and the Chaplygin sleigh with a knife edge
constraint hitting a circular table are shown to illustrate the theoretical
results.Comment: 7 page
