46,143 research outputs found
Provably Safe Robot Navigation with Obstacle Uncertainty
As drones and autonomous cars become more widespread it is becoming
increasingly important that robots can operate safely under realistic
conditions. The noisy information fed into real systems means that robots must
use estimates of the environment to plan navigation. Efficiently guaranteeing
that the resulting motion plans are safe under these circumstances has proved
difficult. We examine how to guarantee that a trajectory or policy is safe with
only imperfect observations of the environment. We examine the implications of
various mathematical formalisms of safety and arrive at a mathematical notion
of safety of a long-term execution, even when conditioned on observational
information. We present efficient algorithms that can prove that trajectories
or policies are safe with much tighter bounds than in previous work. Notably,
the complexity of the environment does not affect our methods ability to
evaluate if a trajectory or policy is safe. We then use these safety checking
methods to design a safe variant of the RRT planning algorithm.Comment: RSS 201
Algorithms for Colourful Simplicial Depth and Medians in the Plane
The colourful simplicial depth of a point x in the plane relative to a
configuration of n points in k colour classes is exactly the number of closed
simplices (triangles) with vertices from 3 different colour classes that
contain x in their convex hull. We consider the problems of efficiently
computing the colourful simplicial depth of a point x, and of finding a point,
called a median, that maximizes colourful simplicial depth.
For computing the colourful simplicial depth of x, our algorithm runs in time
O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For
finding the colourful median, we get a time of O(n^4). For comparison, the
running times of the best known algorithm for the monochrome version of these
problems are O(n log(n)) in general, improving to O(n) if the points are sorted
around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
Finding an ordinary conic and an ordinary hyperplane
Given a finite set of non-collinear points in the plane, there exists a line
that passes through exactly two points. Such a line is called an ordinary line.
An efficient algorithm for computing such a line was proposed by Mukhopadhyay
et al. In this note we extend this result in two directions. We first show how
to use this algorithm to compute an ordinary conic, that is, a conic passing
through exactly five points, assuming that all the points do not lie on the
same conic. Both our proofs of existence and the consequent algorithms are
simpler than previous ones. We next show how to compute an ordinary hyperplane
in three and higher dimensions.Comment: 7 pages, 2 figure
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