2,663 research outputs found
Lagrange-Poincar\'e reduction for optimal control of underactuated mechanical systems
We deal with regular Lagrangian constrained systems which are invariant under
the action of a symmetry group. Fixing a connection on the higher-order
principal bundle where the Lagrangian and the (independent) constraints are
defined, the higher-order Lagrange-Poincar\'e equations of classical mechanical
systems with higher-order constraints are obtained from classical Lagrangian
reduction. Higher-order Lagrange-Poincar\'e operator is introduced to
characterize higher-order Lagrange-Poincar\'e equations. Interesting
applications are derived as, for instance, the optimal control of an
underactuated Elroy's Beanie and a snakeboard seens as an optimization problem
with higher-order constraints.Comment: This paper has been withdrawn by the author due to a crucial erro
On the Existence and Uniqueness of Poincar\'e Maps for Systems with Impulse Effects
The Poincar\'e map is widely used to study the qualitative behavior of
dynamical systems. For instance, it can be used to describe the existence of
periodic solutions. The Poincar\'e map for dynamical systems with impulse
effects was introduced in the last decade and mainly employed to study the
existence of limit cycles (periodic gaits) for the locomotion of bipedal
robots. We investigate sufficient conditions for the existence and uniqueness
of Poincar\'e maps for dynamical systems with impulse effects evolving on a
differentiable manifold. We apply the results to show the existence and
uniqueness of Poincar\'e maps for systems with multiple domains
Second-order variational problems on Lie groupoids and optimal control applications
In this paper we study, from a variational and geometrical point of view,
second-order variational problems on Lie groupoids and the construction of
variational integrators for optimal control problems. First, we develop
variational techniques for second-order variational problems on Lie groupoids
and their applications to the construction of variational integrators for
optimal control problems of mechanical systems. Next, we show how Lagrangian
submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics
for second-order systems, both unconstrained and constrained, and we study the
geometric properties of the implicit flow which defines the dynamics in the
Lagrangian submanifold. We also study the theory of reduction by symmetries and
the corresponding Noether theorem.Comment: 41 pages, 1 figure, first version. Comments welcom
Dynamic interpolation for obstacle avoidance on Riemannian manifolds
This work is devoted to studying dynamic interpolation for obstacle
avoidance. This is a problem that consists of minimizing a suitable energy
functional among a set of admissible curves subject to some interpolation
conditions. The given energy functional depends on velocity, covariant
acceleration and on artificial potential functions used for avoiding obstacles.
We derive first-order necessary conditions for optimality in the proposed
problem; that is, given interpolation and boundary conditions we find the set
of differential equations describing the evolution of a curve that satisfies
the prescribed boundary values, interpolates the given points and is an
extremal for the energy functional.
We study the problem in different settings including a general one on a
Riemannian manifold and a more specific one on a Lie group endowed with a
left-invariant metric. We also consider a sub-Riemannian problem. We illustrate
the results with examples of rigid bodies, both planar and spatial, and
underactuated vehicles including a unicycle and an underactuated unmanned
vehicle.Comment: Comments welcom
Motion Feasibility Conditions for Multi-Agent Control Systems on Lie Groups
We study the problem of motion feasibility for multiagent control systems on
Lie groups with collision avoidance constraints. We first consider the problem
for kinematic left invariant control systems and next, for dynamical control
systems given by a left-trivialized Lagrangian function. Solutions of the
kinematic problem give rise to linear combinations of the control inputs in a
linear subspace annihilating the collision avoidance constraints. In the
dynamical problem, motion feasibility conditions are obtained by using
techniques from variational calculus on manifolds, given by a set of equations
in a vector space, and Lagrange multipliers annihilating the constraint force
that prevents deviation of solutions from a constraint submanifold.Comment: L. J. Colombo was partially supported by ACCESS Linnaeus Center
(KTH), Ministerio de Econom{\i}a, Industria y Competitividad grant
MTM2016-76702-P, i-Link project and La Caixa Foundation
(LCF/BQ/PI19/11690016). D. V. Dimarogonas is supported by the Swedish
Research Council (VR), Knut och Alice Wallenberg foundation (KAW), the H2020
Project Co4Robots and the H2020 ERC Starting Grant BUCOPHSY
Variational obstacle avoidance problem on Riemannian manifolds
We introduce variational obstacle avoidance problems on Riemannian manifolds
and derive necessary conditions for the existence of their normal extremals.
The problem consists of minimizing an energy functional depending on the
velocity and covariant acceleration, among a set of admissible curves, and also
depending on a navigation function used to avoid an obstacle on the workspace,
a Riemannian manifold.
We study two different scenarios, a general one on a Riemannian manifold and,
a sub-Riemannian problem. By introducing a left-invariant metric on a Lie
group, we also study the variational obstacle avoidance problem on a Lie group.
We apply the results to the obstacle avoidance problem of a planar rigid body
and an unicycle.Comment: Paper submitted to IEEE CDC 2017 - Melbourne, Australia. This version
contain a slightly modification in the computations for the application given
in section 4, part
On the geometry of higher-order variational problems on Lie groups
In this paper, we describe a geometric setting for higher-order lagrangian
problems on Lie groups. Using left-trivialization of the higher-order tangent
bundle of a Lie group and an adaptation of the classical Skinner-Rusk
formalism, we deduce an intrinsic framework for this type of dynamical systems.
Interesting applications as, for instance, a geometric derivation of the
higher-order Euler-Poincar\'e equations, optimal control of underactuated
control systems whose configuration space is a Lie group are shown, among
others, along the paper.Comment: 20 pages, 4 figure
Unified formalism for higher-order variational problems and its applications in optimal control
In this paper we consider an intrinsic point of view to describe the
equations of motion for higher-order variational problems with constraints on
higher-order trivial principal bundles. Our techniques are an adaptation of the
classical Skinner-Rusk approach for the case of Lagrangian dynamics with
higher-order constraints. We study a regular case where it is possible to
establish a symplectic framework and, as a consequence, to obtain a unique
vector field determining the dynamics. As an interesting application we deduce
the equations of motion for optimal control of underactuated mechanical systems
defined on principal bundles.Comment: 28 pp. Revised version: Minor corrections don
Optimal Control of Quantum Purity for Systems
The objective of this work is to study time-minimum and energy-minimum global
optimal control for dissipative open quantum systems whose dynamics is governed
by the Lindblad equation. The controls appear only in the Hamiltonian.
Using recent results regarding the decoupling of such dissipative dynamics
into intra- and inter-unitary orbits, we transform the control system into a
bi-linear control system on the Bloch ball (the unitary sphere together with
its interior). We then design a numerical algorithm to construct an optimal
path to achieve a desired point given initial states close to the origin (the
singular point) of the Bloch ball. This is done both for the minimum-time and
minimum -energy control problems.Comment: Comments welcome! Paper submitted to IEEE CDC 2017 - Melbourne,
Australi
Higher-order discrete variational problems with constraints
An interesting family of geometric integrators for Lagrangian systems can be
defined using discretizations of the Hamilton's principle of critical action.
This family of geometric integrators is called variational integrators.
In this paper, we derive new variational integrators for higher-order
lagrangian mechanical system subjected to higher-order constraints. From the
discretization of the variational principles, we show that our methods are
automatically symplectic and, in consequence, with a very good energy behavior.
Additionally, the symmetries of the discrete Lagrangian imply that momenta is
conserved by the integrator. Moreover, we extend our construction to
variational integrators where the lagrangian is explicitly time-dependent.
Finally, some motivating applications of higher-order problems are considered;
in particular, optimal control problems for explicitly time-dependent
underactuated systems and an interpolation problem on Riemannian manifolds.Comment: Comments Welcome
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