The hyperoctahedral group $B_n$ is the group of symmetries of the hypercube ${[-1,1]}^n$ of ${\mathbb{R}}^n$. For instance permutations, or symmetries along each of the n canonical planes of ${\mathbb{R}}^n$ all belong to $B_n$. Now, many sets of equations contain symmetries in $B_n$. This is the case of the addition constraint: $x_1+x_2=x_3$ or the multiplication $x_1\cdot x_2=x_3$. In robotics, many specific geometrical constraints such as for instance constraints involving distances or angles used for localization also have these symmetries. This paper shows the fundamental role of the hyperoctahedral group for interval-based methods. These methods use operators, called contractors, which contract axis-aligned boxes, without removing any point of the solution set defined by a conjunction of constraints (typically equations, or inequalities). More precisely, the paper presents an algorithm which allows us to build minimal contractors associated to constraints with symmetries in $B_n$. As an application, we will consider the geometrical constraint associated to the angle between vectors. The corresponding contractor will then be used in a constraint propagation framework in order to localize a robot using several radars.

超八面体群${乙}_n$是超立方体的对称群${[-1,1]}^n$的${\mathbb{R}}^n$. 例如排列，或沿n 个规范平面中的每一个的对称性${\mathbb{R}}^n$都属于${乙}_n$. 现在，许多方程组包含对称性${乙}_n$. 这是添加约束的情况：$X_1+X_2=X_3$或乘法$X_1\cdot X_2=X_3$. 在机器人技术中，许多特定的几何约束，例如涉及用于定位的距离或角度的约束也具有这些对称性。本文展示了超八面体群在基于区间的方法中的基本作用。这些方法使用称为contractors的算子，它收缩轴对齐的盒子，而不删除由约束（通常是方程或不等式）的结合定义的解集的任何点。更准确地说，本文提出了一种算法，该算法允许我们构建与对称约束相关的最小承包商${乙}_n$. 作为一个应用程序，我们将考虑与向量之间的角度相关的几何约束。然后将在约束传播框架中使用相应的承包商，以便使用多个雷达定位机器人。