154 research outputs found
Real roots counting for some robotics problems
We propose two algorithms to compute the number of real roots of zerodimensional systems, using effective algebraic methods. To compare their behaviour on practical examples, we apply these methods to systems that describe some robotics problems (e.g. direct kinematic problem of parallel manipulators
Computing Real Roots of Real Polynomials ... and now For Real!
Very recent work introduces an asymptotically fast subdivision algorithm,
denoted ANewDsc, for isolating the real roots of a univariate real polynomial.
The method combines Descartes' Rule of Signs to test intervals for the
existence of roots, Newton iteration to speed up convergence against clusters
of roots, and approximate computation to decrease the required precision. It
achieves record bounds on the worst-case complexity for the considered problem,
matching the complexity of Pan's method for computing all complex roots and
improving upon the complexity of other subdivision methods by several
magnitudes.
In the article at hand, we report on an implementation of ANewDsc on top of
the RS root isolator. RS is a highly efficient realization of the classical
Descartes method and currently serves as the default real root solver in Maple.
We describe crucial design changes within ANewDsc and RS that led to a
high-performance implementation without harming the theoretical complexity of
the underlying algorithm.
With an excerpt of our extensive collection of benchmarks, available online
at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in
performance of ANewDsc over other subdivision methods also transfers into
practice. These experiments also show that our new implementation outperforms
both RS and mature competitors by magnitudes for notoriously hard instances
with clustered roots. For all other instances, we avoid almost any overhead by
integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on
Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo,
Ontario, Canad
Computing Chebyshev knot diagrams
A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t);
y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the
Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no
double points, it defines a polynomial knot. We determine all possible knots
when a, b and c are given.Comment: 8
The first rational Chebyshev knots
A Chebyshev knot is a knot which has a parametrization
of the form where
are integers, is the Chebyshev polynomial of degree and We show that any two-bridge knot is a Chebyshev knot with and also
with . For every integers ( and , coprime), we
describe an algorithm that gives all Chebyshev knots \cC(a,b,c,\phi). We
deduce a list of minimal Chebyshev representations of two-bridge knots with
small crossing number.Comment: 22p, 27 figures, 3 table
Certified Algorithms for proving the structural stability of two dimensional systems possibly with parameters
International audienceIn [1], a new method for testing the structural stability of multidimensional systems has been presented. The key idea of this method is to reduce the problem of testing the structural stability to that of deciding if an algebraic set has real points. Following the same idea, we consider in this work the specific case of two-dimensional systems and focus on the practical efficiency aspect. For such systems, the problem of testing the stability is reduced to that of deciding if a bivariate algebraic system with finitely many solutions has real ones. Our first contribution is an algorithm that answers this question while achieving practical efficiency. Our second contribution concerns the stability of two dimensional systems with parameters. More precisely, given a two-dimensional system depending on a set of parameters, we present a new algorithm that computes regions of the parameter space in which the considered system is structurally stable
An algebraic method to check the singularity-free paths for parallel robots
Trajectory planning is a critical step while programming the parallel
manipulators in a robotic cell. The main problem arises when there exists a
singular configuration between the two poses of the end-effectors while
discretizing the path with a classical approach. This paper presents an
algebraic method to check the feasibility of any given trajectories in the
workspace. The solutions of the polynomial equations associated with the
tra-jectories are projected in the joint space using Gr{\"o}bner based
elimination methods and the remaining equations are expressed in a parametric
form where the articular variables are functions of time t unlike any numerical
or discretization method. These formal computations allow to write the Jacobian
of the manip-ulator as a function of time and to check if its determinant can
vanish between two poses. Another benefit of this approach is to use a largest
workspace with a more complex shape than a cube, cylinder or sphere. For the
Orthoglide, a three degrees of freedom parallel robot, three different
trajectories are used to illustrate this method.Comment: Appears in International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference , Aug 2015, Boston,
United States. 201
Motivations for an arbitrary precision interval arithmetic and the MPFI library
This paper justifies why an arbitrary precision interval arithmetic is needed: to provide accurate results, interval computations require small input intervals; this explains why bisection is so often employed in interval algorithms. The MPFI library has been built in order to fulfill this need: indeed, no existing library met the required specifications. The main features of this library are briefly given and a comparison with a fixed-preci- sion interval arithmetic, on a specific problem, is presented: it shows that the overhead due to the multiple precision is completely admissible. Eventually, some applications based on MPFI are given: robotics, isolation of polynomial real roots (by an algorithm combining symbolic and numerical computations) and approximation of real roots with arbitrary accuracy.Cet article justifie le besoin d’une arithmétique par intervalles en précision arbitraire : pour fournir des résultats précis, un calcul par intervalles requiert des intervalles en entrée qui soient fins ; c’est pour cette raison que la bissection est un procédé si souvent employé dans les algorithmes par intervalles. La bibliothèque MPFI a été développée pour répondre à ce besoin : en effet, aucune bibliothèque existante n’offrait de spécifications satisfaisantes. Les caractéristiques de cette bibliothèque sont rapidement données puis une comparaison avec une bibliothèque d’arithmétique par intervalles en précision fixée est menée sur un problème spécifique : elle met en évidence le fait que le surcoût lié à la gestion de la précision multiple est tout à fait acceptable. Pour terminer, quelques applications basées sur MPFI sont présentées : robotique, isolation des racines réelles de polynômes (par un algorithme combinant calcul symbolique et calcul numérique) et approximation avec une précision arbitraire de zéros réel
Non-singular assembly mode changing trajectories in the workspace for the 3-RPS parallel robot
Having non-singular assembly modes changing trajectories for the 3-RPS
parallel robot is a well-known feature. The only known solution for defining
such trajectory is to encircle a cusp point in the joint space. In this paper,
the aspects and the characteristic surfaces are computed for each operation
mode to define the uniqueness of the domains. Thus, we can easily see in the
workspace that at least three assembly modes can be reached for each operation
mode. To validate this property, the mathematical analysis of the determinant
of the Jacobian is done. The image of these trajectories in the joint space is
depicted with the curves associated with the cusp points
Solving Zero-dimensional Polynomial Systems through the Rational Univariate Representation
This paper is devoted to the {\it resolution} of zero-dimensional systems in , where is a field of characteristic zero (or strictly positive under some conditions). We give a new definition for {\rm solving zero-dimensional systems} by introducing the {\it Univariate Representation} of their roots. We show by this way that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation ({\it Rational Univariate Representation}): where are polynomials of , without loosing geometrical information (multiplicities, real roots). Moreover we propose different efficient algorithms for the computation of the {\it Rational Univariate Representation}, and we make a comparison with standard known tools
Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators
This paper investigates the conditions in the design parameter space for the
existence and distribution of the cusp locus for planar parallel manipulators.
Cusp points make possible non-singular assembly-mode changing motion, which
increases the maximum singularity-free workspace. An accurate algorithm for the
determination is proposed amending some imprecisions done by previous existing
algorithms. This is combined with methods of Cylindric Algebraic Decomposition,
Gr\"obner bases and Discriminant Varieties in order to partition the parameter
space into cells with constant number of cusp points. These algorithms will
allow us to classify a family of degenerate 3-RPR manipulators.Comment: ASME Journal of Mechanisms and Robotics (2012) 1-1
- …