Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry

by   Damien Chablat, et al.

Cuspidal robots are robots with at least two inverse kinematic solutions that can be connected by a singularity-free path. Deciding the cuspidality of generic 3R robots has been studied in the past, but extending the study to six-degree-of-freedom robots can be a challenging problem. Many robots can be modeled as a polynomial map together with a real algebraic set so that the notion of cuspidality can be extended to these data. In this paper we design an algorithm that, on input a polynomial map in n indeterminates, and s polynomials in the same indeterminates describing a real algebraic set of dimension d, decides the cuspidality of the restriction of the map to the real algebraic set under consideration. Moreover, if D and τ are, respectively the maximum degree and the bound on the bit size of the coefficients of the input polynomials, this algorithm runs in time log-linear in τ and polynomial in ((s+d)D)^O(n^2). It relies on many high-level algorithms in computer algebra which use advanced methods on real algebraic sets and critical loci of polynomial maps. As far as we know, this is the first algorithm that tackles the cuspidality problem from a general point of view.


page 1

page 2

page 3

page 4


Robots, computer algebra and eight connected components

Answering connectivity queries in semi-algebraic sets is a long-standing...

A p-adic Descartes solver: the Strassman solver

Solving polynomials is a fundamental computational problem in mathematic...

Critical Point Computations on Smooth Varieties: Degree and Complexity bounds

Let V ⊂ C n be an equidimensional algebraic set and g be an n-variate po...

On Annihilators of Explicit Polynomial Maps

We study the algebraic complexity of annihilators of polynomials maps. I...

Computing the dimension of real algebraic sets

Let V be the set of real common solutions to F = (f_1, …, f_s) in ℝ[x_1,...

Deciding One to One property of Boolean maps: Condition and algorithm in terms of implicants

This paper addresses the computational problem of deciding invertibility...

An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting

We study the problem of deciding if a given triple of permutations can b...

Please sign up or login with your details

Forgot password? Click here to reset