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Alexandre Anahory Simoes

Publications and source records attributed to Alexandre Anahory Simoes.

At least 37 records · Page 2Linked to original sources
arXiv2023

Reduction by symmetries of contact mechanical systems on Lie groups

We study the dynamics of contact mechanical systems on Lie groups that are invariant under a Lie group action. Analogously to standard mechanical systems on Lie groups, existing symmetries allow for reducing the number of equations. Thus, we obtain Euler-Poincaré-Herglotz equations on the extended reduced phase space $\mathfrak{g}\times \R$ associated with the extended phase space $TG\times \R$, where the configuration manifold $G$ is a Lie group and $\mathfrak{g}$ its Lie algebra. Furthermore, we obtain the Hamiltonian counterpart of these equations by studying the underlying Jacobi structure. Finally, we extend the reduction process to the case of symmetry-breaking systems which are invariant under a Lie subgroup of symmetries.

math-ph↗
arXiv2023

Jacobi structure for dissipative mechanical systems on Lie Algebroids

We extend the Jacobi structure from $TQ\times \mathbb{R}$ and $T^{*}Q \times \mathbb{R}$ to $A\times \mathbb{R}$ and $A^{*}\times \mathbb{R}$, respectively, where $A$ is a Lie algebroid and $A^{*}$ carries the associated Poisson structure. We see that $A^*\times \mathbb{R}$ possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems, generalizing previous models on $TQ\times \mathbb{R}$ and $\mathfrak{g}\times \mathbb{R}$.

math-ph↗
arXiv2023

Higher-order retraction maps and construction of numerical methods for optimal control of mechanical systems

Retractions maps are used to define a discretization of the tangent bundle of the configuration manifold as two copies of the configuration manifold where the dynamics take place. Such discretization maps can be conveniently lifted to a higher-order tangent bundle to construct geometric integrators for the higher-order Euler-Lagrange equations. Given a cost function, an optimal control problem for fully actuated mechanical systems can be understood as a higher-order variational problem. In this paper we introduce the notion of a higher-order discretization map associated with a retraction map to construct geometric integrators for the optimal control of mechanical systems. In particular, we study applications to path planning for obstacle avoidance of a planar rigid body.

math.OC↗
arXiv2023

Nonholonomic systems with inequality constraints

In this paper we derive the equations of motion for nonholonomic systems subject to inequality constraints, both, in continuous-time and discrete-time. The last is done by discretizing the continuous time-variational principle which defined the equations of motion for a nonholonomic system subject to inequality constraints. An example is show to illustrate the theoretical results.

math.OC↗
arXiv2023

Virtual Affine Nonholonomic Constraints

Virtual 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 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.

math.OC↗
arXiv2022

Contracting Forced Lagrangian and Contact Lagrangian Systems: application to nonholonomic systems with symmetries

In this paper we address the problem of identifying contracting systems among dynamical systems appearing in mechanics. First, we introduce a sufficient condition to identify contracting systems in a general Riemannian manifold. Then, we apply this technique to establish that a particular type of dissipative forced mechanical system is contracting, while stating immediate consequences of this fact for the stability of these systems. Finally, we use the previous results to study the stability of particular types of Contact and Nonholonomic systems.

math.OC↗
arXiv2022

Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage

Forced variational integrators are given by the discretization of the Lagrange-d'Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem.

eess.SY↗
arXiv2022

Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups

We 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 $SE(2)$ in the presence of a static obstacle.

math.OC↗
arXiv2022

Virtual Nonholonomic Constraints: A Geometric Approach

Virtual constraints are invariant relations imposed on a control system via feedback as opposed to 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 demonstrate 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.

eess.SY↗
arXiv2022

Variational problems on Riemannian manifolds with constrained accelerations

We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy functional, among a set of admissible curves defined by a constraint on the covariant acceleration. In addition, we use this framework to address the elastic splines problem with obstacle avoidance in the presence of this type of contraints.

math.OC↗
arXiv2021

Contact bundle formulation of nonholonomic Maupertuis-Jacobi principle and a length minimizing property of nonholonomic dynamics

We prove a nonholonomic version of the classical Mauper\-tuis-Jacobi principle which transforms an autonomous mechanical nonholonomic problem, determined by a kinetic minus potential energy and a distribution, in a kinetic nonholonomic problem over a fixed level set of the Lagrangian energy. To prove this result we introduce an appropriate contact bundle structure clarifying the geometric equivalence between both problems. By using the nonholonomic Maupertuis-Jacobi principle, we prove that the regular solutions of a mechanical nonholonomic problem starting from a fixed point and in the same level set of the Lagrangian energy are reparametrizations of geodesics for a family of Riemannian metrics defined on the image of the nonholonomic exponential map. In particular, these trajectories minimize Riemannian length.

math-ph↗
arXiv2020

The geometry of some thermodynamic systems

In this article, we continue the program started in our previous article of exploring an important class of thermodynamic systems from a geometric point of view. In order to model the time evolution of systems verifying the two laws of thermodynamics, we show that the notion of evolution vector field is adequate to appropriately describe such systems. Our formulation naturally arises from the introduction of a skew-symmetric bracket to which numerical methods based on discrete gradients fit nicely. Moreover, we study the corresponding Lagrangian and Hamiltonian formalism, discussing the fundamental principles from which the equations are derived. An important class of systems that is naturally covered by our formalism are composed thermodynamic systems, which are described by at least two thermal variables and exchange heat between its components.}.

math-ph↗
arXiv2020

Radial kinetic nonholonomic trajectories are Riemannian geodesics!

Nonholonomic 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 ${\mathcal M}^{nh}_q$ 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 ${\mathcal M}^{nh}_q$!

math-ph↗
arXiv2020

Jacobi fields in nonholonomic mechanics

In 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 equations 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.

math.DG↗
arXiv2020

Exact discrete Lagrangian mechanics for nonholonomic mechanics

We 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.

math-ph↗
arXiv2020

Contact geometry for simple thermodynamical systems with friction

Using contact geometry we give a new characterization of a simple but important class of thermodynamical systems which naturally satisfy the first law of thermodynamics (total energy preservation) and the second law (increase of entropy). We completely clarify its qualitative dynamics, the underlying geometrical structures and we show how to use discrete gradient methods.

math-ph↗