126 research outputs found
On the complexity of curve fitting algorithms
We study a popular algorithm for fitting polynomial curves to scattered data
based on the least squares with gradient weights. We show that sometimes this
algorithm admits a substantial reduction of complexity, and, furthermore, find
precise conditions under which this is possible. It turns out that this is,
indeed, possible when one fits circles but not ellipses or hyperbolas.Comment: 8 pages, no figure
Investigations into the shape-preserving interpolants using symbolic computation
Shape representation is a central issue in computer graphics and computer-aided geometric design. Many physical phenomena involve curves and surfaces that are monotone (in some directions) or are convex. The corresponding representation problem is given some monotone or convex data, and a monotone or convex interpolant is found. Standard interpolants need not be monotone or convex even though they may match monotone or convex data. Most of the methods of investigation of this problem involve the utilization of quadratic splines or Hermite polynomials. In this investigation, a similar approach is adopted. These methods require derivative information at the given data points. The key to the problem is the selection of the derivative values to be assigned to the given data points. Schemes for choosing derivatives were examined. Along the way, fitting given data points by a conic section has also been investigated as part of the effort to study shape-preserving quadratic splines
Approximate Fitting of a Circular Arc When Two Points Are Known
The task of approximating points with circular arcs is performed in many
applications, such as polyline compression, noise filtering, and feature
recognition. However, the development of algorithms that perform a significant
amount of circular arcs fitting requires an efficient way of fitting circular
arcs with complexity O(1). The elegant solution to this task based on an
eigenvector problem for a square nonsymmetrical matrix is described in [1]. For
the compression algorithm described in [2], it is necessary to solve this task
when two points on the arc are known. This paper describes a different approach
to efficiently fitting the arcs and solves the task when one or two points are
known.Comment: 15 pages, 4 figures, extended abstract published at the conferenc
Robust polynomial regression up to the information theoretic limit
We consider the problem of robust polynomial regression, where one receives
samples that are usually within of a polynomial , but have a chance of being arbitrary adversarial outliers.
Previously, it was known how to efficiently estimate only when . We give an algorithm that works for the entire feasible
range of , while simultaneously improving other parameters of the
problem. We complement our algorithm, which gives a factor 2 approximation,
with impossibility results that show, for example, that a approximation
is impossible even with infinitely many samples.Comment: 19 Pages. To appear in FOCS 201
Stable Real-Time Feedback Control of a Pneumatic Soft Robot
Soft actuators offer compliant and safe interaction with an unstructured
environment compared to their rigid counterparts. However, control of these
systems is often challenging because they are inherently under-actuated, have
infinite degrees of freedom (DoF), and their mechanical properties can change
by unknown external loads. Existing works mainly relied on discretization and
reduction, suffering from either low accuracy or high computational cost for
real-time control purposes. Recently, we presented an infinite-dimensional
feedback controller for soft manipulators modeled by partial differential
equations (PDEs) based on the Cosserat rod theory. In this study, we examine
how to implement this controller in real-time using only a limited number of
actuators. To do so, we formulate a convex quadratic programming problem that
tunes the feedback gains of the controller in real time such that it becomes
realizable by the actuators. We evaluated the controller's performance through
experiments on a physical soft robot capable of planar motions and show that
the actual controller implemented by the finite-dimensional actuators still
preserves the stabilizing property of the desired infinite-dimensional
controller. This research fills the gap between the infinite-dimensional
control design and finite-dimensional actuation in practice and suggests a
promising direction for exploring PDE-based control design for soft robots
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