45,158 research outputs found
The Cost of Bounded Curvature
We study the motion-planning problem for a car-like robot whose turning
radius is bounded from below by one and which is allowed to move in the forward
direction only (Dubins car). For two robot configurations ,
let be the shortest bounded-curvature path from
to . For , let be the supremum of
, over all pairs that are at
Euclidean distance . We study the function \dub(d) = \ell(d) - d, which
expresses the difference between the bounded-curvature path length and the
Euclidean distance of its endpoints. We show that \dub(d) decreases
monotonically from \dub(0) = 7\pi/3 to \dub(\ds) = 2\pi, and is constant
for d \geq \ds. Here \ds \approx 1.5874. We describe pairs of
configurations that exhibit the worst-case of \dub(d) for every distance
Existence and uniqueness of optimal maps on Alexandrov spaces
The purpose of this paper is to show that in a finite dimensional metric
space with Alexandrov's curvature bounded below, Monge's transport problem for
the quadratic cost admits a unique solution
Efficient Path Interpolation and Speed Profile Computation for Nonholonomic Mobile Robots
This paper studies path synthesis for nonholonomic mobile robots moving in
two-dimensional space. We first address the problem of interpolating paths
expressed as sequences of straight line segments, such as those produced by
some planning algorithms, into smooth curves that can be followed without
stopping. Our solution has the advantage of being simpler than other existing
approaches, and has a low computational cost that allows a real-time
implementation. It produces discretized paths on which curvature and variation
of curvature are bounded at all points, and preserves obstacle clearance. Then,
we consider the problem of computing a time-optimal speed profile for such
paths. We introduce an algorithm that solves this problem in linear time, and
that is able to take into account a broader class of physical constraints than
other solutions. Our contributions have been implemented and evaluated in the
framework of the Eurobot contest
The five gradients inequality on differentiable manifolds
The goal of this paper is to derive the so-called five gradients inequality
for optimal transport theory for general cost functions on two class of
differentiable manifolds: locally compact Lie groups and compact Riemannian
manifolds with Ricci curvature bounded from below.Comment: 29 page
The Cartan-Hadamard conjecture and The Little Prince
The generalized Cartan-Hadamard conjecture says that if is a domain
with fixed volume in a complete, simply connected Riemannian -manifold
with sectional curvature , then the boundary of
has the least possible boundary volume when is a round -ball with
constant curvature . The case and is an old result
of Weil. We give a unified proof of this conjecture in dimensions and
when , and a special case of the conjecture for \kappa
\textless{} 0 and a version for \kappa \textgreater{} 0. Our argument uses a
new interpretation, based on optical transport, optimal transport, and linear
programming, of Croke's proof for and . The generalization to
and is a new result. As Croke implicitly did, we relax the
curvature condition to a weaker candle condition
or .We also find counterexamples to a na\"ive
version of the Cartan-Hadamard conjecture: For every \varepsilon
\textgreater{} 0, there is a Riemannian 3-ball with
-pinched negative curvature, and with boundary volume bounded
by a function of and with arbitrarily large volume.We begin with
a pointwise isoperimetric problem called "the problem of the Little Prince."
Its proof becomes part of the more general method.Comment: v3: significant rewritting of some proofs, a mistake in the proof of
the ball counter-example has been correcte
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