Using dual quaternions, the closure equations of a kinematic loop can be expressed as a system of multiaffine equations. In this paper, this property is leveraged to introduce a branch-and-prune method specially tailored for solving such systems of equations. The new method is objectively simpler (in the sense that it is easier to understand and to implement) than previous approaches relying on general techniques such as interval Newton methods or methods based on Bernstein polynomials or linear relaxations. Moreover, it relies on two basic operations — linear interpolation and projection onto coordinate planes— that can be efficiently computed. The generality of the proposed method is evaluated on position analysis problems with 0-, 1-, and 2-dimensional solution sets, including the inverse kinematics of serial robots and the forward kinematics of parallel ones. The results obtained on these problems show that the efficiency of the method compares favorably to state-of-the-art alternatives.

使用对偶四元数，运动回路的闭合方程可以表示为多重仿射方程组。在本文中，利用这一特性引入了一种专门为求解此类方程组而量身定制的分支剪枝方法。新方法客观上比以前依赖于一般技术的方法更简单（从某种意义上说，它更容易理解和实现），例如间隔牛顿方法或基于伯恩斯坦多项式或线性松弛的方法。此外，它依赖于可以有效计算的两个基本操作——线性插值和坐标平面上的投影。在具有 0、1 和 2 维解集的位置分析问题上评估了所提出方法的通用性，包括串联机器人的逆运动学和并联机器人的正向运动学。在这些问题上获得的结果表明，该方法的效率优于最先进的替代方法。