335 research outputs found
Stabilization of non-admissible curves for a class of nonholonomic systems
The problem of tracking an arbitrary curve in the state space is considered
for underactuated driftless control-affine systems. This problem is formulated
as the stabilization of a time-varying family of sets associated with a
neighborhood of the reference curve. An explicit control design scheme is
proposed for the class of controllable systems whose degree of nonholonomy is
equal to 1. It is shown that the trajectories of the closed-loop system
converge exponentially to any given neighborhood of the reference curve
provided that the solutions are defined in the sense of sampling. This
convergence property is also illustrated numerically by several examples of
nonholonomic systems of degrees 1 and 2.Comment: This is the author's version of the manuscript accepted for
publication in the Proceedings of the 2019 European Control Conference
(ECC'19
On the stabilization problem for nonholonomic distributions
Let be a smooth connected and complete manifold of dimension , and
be a smooth nonholonomic distribution of rank on . We
prove that, if there exists a smooth Riemannian metric on for which no
nontrivial singular path is minimizing, then there exists a smooth repulsive
stabilizing section of on . Moreover, in dimension three, the
assumption of the absence of singular minimizing horizontal paths can be
dropped in the Martinet case. The proofs are based on the study, using specific
results of nonsmooth analysis, of an optimal control problem of Bolza type, for
which we prove that the corresponding value function is semiconcave and is a
viscosity solution of a Hamilton-Jacobi equation, and establish fine properties
of optimal trajectories.Comment: accept\'e pour publication dans J. Eur. Math. Soc. (2007), \`a
para\^itre, 29 page
Motion planning and stabilization of nonholonomic systems using gradient flow approximations
Nonlinear control-affine systems with time-varying vector fields are
considered in the paper. We propose a unified control design scheme with
oscillating inputs for solving the trajectory tracking and stabilization
problems. This methodology is based on the approximation of a gradient like
dynamics by trajectories of the designed closed-loop system. As an intermediate
outcome, we characterize the asymptotic behavior of solutions of the considered
class of nonlinear control systems with oscillating inputs under rather general
assumptions on the generating potential function. These results are applied to
examples of nonholonomic trajectory tracking and obstacle avoidance.Comment: submitte
A general framework for nonholonomic mechanics: Nonholonomic Systems on Lie affgebroids
This paper presents a geometric description of Lagrangian and Hamiltonian
systems on Lie affgebroids subject to affine nonholonomic constraints. We
define the notion of nonholonomically constrained system, and characterize
regularity conditions that guarantee that the dynamics of the system can be
obtained as a suitable projection of the unconstrained dynamics. It is shown
that one can define an almost aff-Poisson bracket on the constraint AV-bundle,
which plays a prominent role in the description of nonholonomic dynamics.
Moreover, these developments give a general description of nonholonomic systems
and the unified treatment permits to study nonholonomic systems after or before
reduction in the same framework. Also, it is not necessary to distinguish
between linear or affine constraints and the methods are valid for explicitly
time-dependent systems.Comment: 50 page
Mass Transportation on Sub-Riemannian Manifolds
We study the optimal transport problem in sub-Riemannian manifolds where the
cost function is given by the square of the sub-Riemannian distance. Under
appropriate assumptions, we generalize Brenier-McCann's Theorem proving
existence and uniqueness of the optimal transport map. We show the absolute
continuity property of Wassertein geodesics, and we address the regularity
issue of the optimal map. In particular, we are able to show its approximate
differentiability a.e. in the Heisenberg group (and under some weak assumptions
on the measures the differentiability a.e.), which allows to write a weak form
of the Monge-Amp\`ere equation
A Global Steering Method for Nonholonomic Systems
In this paper, we present an iterative steering algorithm for nonholonomic
systems (also called driftless control-affine systems) and we prove its global
convergence under the sole assumption that the Lie Algebraic Rank Condition
(LARC) holds true everywhere. That algorithm is an extension of the one
introduced in [21] for regular systems. The first novelty here consists in the
explicit algebraic construction, starting from the original control system, of
a lifted control system which is regular. The second contribution of the paper
is an exact motion planning method for nilpotent systems, which makes use of
sinusoidal control laws and which is a generalization of the algorithm
described in [29] for chained-form systems
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