In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of “vertical Darboux joints”. They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

在本文中，我们研究了将四元数环上的多项式分解为线性因子。虽然较早的结果假设范式多项式是实数（“运动多项式”），但我们只要求在本原部分不存在实多项式因子，并且要求将对偶数的范式多项式分解为一元二次因子。这个显然必要的条件对于分解的存在也足够了。我们提出了一种计算这些多项式的因式分解的算法，并将其用于机制的新构造，而现有的运动多项式因式分解算法无法获得这种机制的新构造。当它们产生带有旋转或平移关节的机构时，我们的方法产生了由“垂直达布克斯关节”组成的机构。