計算機代数と実代数幾何アルゴリズムによるマニピュレータの尖点性判定
Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry
多項式写像と実代数集合で表されるロボットの尖点性（特異点なしで結べる逆運動学解の存在）を判定する一般アルゴリズムを提案し、その計算量を解析した。
著者: Damien Chablat, Rémi Prébet, Mohab Safey El Din, Durgesh Salunkhe, Philippe Wenger
分類: cs.SC, cs.RO, math.AG
原文アブストラクト
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 $\tau$ 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 $\tau$ 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.
