Factorization of Motions

We define motion polynomials as polynomials with coefficients in the dual quaternions and study their factorizations. The motion polynomials correspond to motions in 3D space, and factoring into linear factors means to compose the motion into translations and rotations. This allows to realize the motion by a linkage with revolute or prismatic joints. This is joint work with G. Hegedüs (Univ. Oboda), Z. Li (RICAM), and H.-P. Schröcker (Univ. Innsbruck). The results are published in [1]. This research has been supported by the Austrian Science Fund (FWF): DKW1214N15. LetH = 〈1, i, j,k〉R be the skew field of quaternions. It is well known [2] thatH is algebraically closed, in the sense that every univariate left polynomial P ∈ H[t] can be written as a product of linear polynomials. Here, the variable t is supposed to commute with the coefficients. To decompose P , one looks for right zeroes in H: if P(q) = 0, then (t −q) is a right factor of P , and the polynomial quotient has degree one less. In order to find right zeroes, we compute the norm polynomial N (t) = P(t)P(t), where P is obtained by conjugating all coefficients. It is a real polynomial that does not assume negative values when evaluated at real numbers. Generically, it has no real zeroes, so that it can be written as a product of irreducible quadratic factors. For any such factor Q, there is a unique common right zero of P and Q in H, and this common right zero can be computed by polynomial division: the polynomial remainder of P mod Q is linear. The factorization algorithm can be extended to skew ring DH = 〈1, i, j,k〉D of dual quaternions, where D = R ⊕ R is the two-dimensional R-algebra generated by R and with 2 = 0. We are especially interested in polynomials with real