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.
翻译:监视机器人是机器人, 至少有两个反向运动式的机器人, 可以通过单一路径连接。 过去曾研究过如何决定通用 3R 机器人的累积性, 但将研究扩展至 6 度自由机器人可能是一个具有挑战性的问题 。 许多机器人可以与一个真正的代数组一起, 模拟成一个多级地图, 从而将亲化的概念扩展至这些数据 。 在本文中, 我们设计了一个算法, 在输入一个以美元固定路径输入的多数值地图上输入一个多数值图, 而在同样不确定的情况下, 确定通用 3R 机器人的累积性, 但是将研究扩展至 6 度自由机器人, 可能是一个具有挑战性的问题 。 许多机器人可以模拟地图限制为真实的代数地图, 以及一个真正的代数组。 此外, 如果 $D 和 $taautau 的概念, 分别是输入多数值的比位大小, 这个算法从 $\ 美元 和 数级数的高级算法, 在 AL2 的精度 的高级算值 上, Asal- licalbalbalbal log log log log 、 a ligal ligal ligal li li ligal ligal ma 。