79 research outputs found
Fast polynomial evaluation and composition
The library \emph{fast\_polynomial} for Sage compiles multivariate
polynomials for subsequent fast evaluation. Several evaluation schemes are
handled, such as H\"orner, divide and conquer and new ones can be added easily.
Notably, a new scheme is introduced that improves the classical divide and
conquer scheme when the number of terms is not a pure power of two. Natively,
the library handles polynomials over gmp big integers, boost intervals, python
numeric types. And any type that supports addition and multiplication can
extend the library thanks to the template design. Finally, the code is
parallelized for the divide and conquer schemes, and memory allocation is
localized and optimized for the different evaluation schemes. This extended
abstract presents the concepts behind the \emph{fast\_polynomial} library. The
sage package can be downloaded at
\url{http://trac.sagemath.org/sage_trac/ticket/13358}
Uniqueness domains and non singular assembly mode changing trajectories
Parallel robots admit generally several solutions to the direct kinematics
problem. The aspects are associated with the maximal singularity free domains
without any singular configurations. Inside these regions, some trajectories
are possible between two solutions of the direct kinematic problem without
meeting any type of singularity: non-singular assembly mode trajectories. An
established condition for such trajectories is to have cusp points inside the
joint space that must be encircled. This paper presents an approach based on
the notion of uniqueness domains to explain this behaviour
New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems
We present a new data structure to approximate accurately and efficiently a
polynomial of degree given as a list of coefficients. Its properties
allow us to improve the state-of-the-art bounds on the bit complexity for the
problems of root isolation and approximate multipoint evaluation. This data
structure also leads to a new geometric criterion to detect ill-conditioned
polynomials, implying notably that the standard condition number of the zeros
of a polynomial is at least exponential in the number of roots of modulus less
than or greater than .Given a polynomial of degree with
for , isolating all its complex roots or
evaluating it at points can be done with a quasi-linear number of
arithmetic operations. However, considering the bit complexity, the
state-of-the-art algorithms require at least bit operations even for
well-conditioned polynomials and when the accuracy required is low. Given a
positive integer , we can compute our new data structure and evaluate at
points in the unit disk with an absolute error less than in
bit operations, where means
that we omit logarithmic factors. We also show that if is the absolute
condition number of the zeros of , then we can isolate all the roots of
in bit operations. Moreover, our
algorithms are simple to implement. For approximating the complex roots of a
polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an
order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for
high degree polynomials with random coefficients
Fast real and complex root-finding methods for well-conditioned polynomials
Given a polynomial of degree and a bound on a condition
number of , we present the first root-finding algorithms that return all its
real and complex roots with a number of bit operations quasi-linear in . More precisely, several condition numbers can be defined
depending on the norm chosen on the coefficients of the polynomial. Let . We call the
condition number associated with a perturbation of the the hyperbolic
condition number , and the one associated with a perturbation of the
the elliptic condition number . For each of these condition
numbers, we present algorithms that find the real and the complex roots of
in bit
operations.Our algorithms are well suited for random polynomials since
(resp. ) is bounded by a polynomial in with high
probability if the (resp. the ) are independent, centered Gaussian
variables of variance
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
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
On the determination of cusp points of 3-R\underline{P}R parallel manipulators
This paper investigates the cuspidal configurations of 3-RPR parallel
manipulators that may appear on their singular surfaces in the joint space.
Cusp points play an important role in the kinematic behavior of parallel
manipulators since they make possible a non-singular change of assembly mode.
In previous works, the cusp points were calculated in sections of the joint
space by solving a 24th-degree polynomial without any proof that this
polynomial was the only one that gives all solutions. The purpose of this study
is to propose a rigorous methodology to determine the cusp points of
3-R\underline{P}R manipulators and to certify that all cusp points are found.
This methodology uses the notion of discriminant varieties and resorts to
Gr\"obner bases for the solutions of systems of equations
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