Answering connectivity queries in semi-algebraic sets is a long-standing and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number of connected components of the complementary of the singularities of the kinematic map. Another task is to design a continuous path joining two given points lying in the same connected component of such a set. In this paper, we push forward the current capabilities of computer algebra to obtain computer-aided proofs of the analysis of the kinematic singularities of various robots used in industry. We first show how to combine mathematical reasoning with easy symbolic computations to study the kinematic singularities of an infinite family (depending on paramaters) modelled by the UR-series produced by the company ``Universal Robots''. Next, we compute roadmaps (which are curves used to answer connectivity queries) for this family of robots. We design an algorithm for ``solving'' positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter's space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. Practical experiments confirm our computer-aided proof and show that such an algorithm can already be used to analyze the kinematic singularities of the UR-series family. The number of connected components of the complementary of the kinematic singularities of generic robots in this family is $8$.

中文翻译：

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机器人、计算机代数和八个连接组件

回答半代数集中的连通性查询是机器人技术应用中一个长期存在且具有挑战性的计算问题，特别是对于运动学奇点的分析。一项任务是计算运动学映射奇点互补的连通分量的数量。另一个任务是设计一条连续路径，连接位于此类集合的同一连通分量中的两个给定点。在本文中，我们推进了计算机代数的当前能力，以获得工业中使用的各种机器人运动学奇点分析的计算机辅助证明。我们首先展示如何将数学推理与简单的符号计算相结合，以研究由“Universal Robots”公司生产的 UR 系列建模的无限族（取决于参数）的运动学奇点。接下来，我们为这个机器人系列计算路线图（用于回答连接查询的曲线）。我们设计了一种根据参数“求解”正维多项式系统的算法。此处求解的含义是将参数空间划分为半代数分量，在这些半代数分量上输入系统定义的半代数集的连通分量数是不变的。实际实验证实了我们的计算机辅助证明，并表明这种算法已经可以用于分析 UR 系列的运动学奇点。