Zijia Li

Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on the invertibility of a certain coefficient occurring in the algorithm. If this coefficient is not invertible, factorizations may or may not exist. In the case of existence we call this an irregular factorization. We characterize quadratic motion polynomials with irregular factorizations in terms of algebraic equations and present examples whose number of unique factorizations range from one to infinitely many. For two special sub-cases we show the unique existence of such polynomials. In case of commuting factors we obtain the conformal Villarceau motion, in case of rigid body motions the circular translation.

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear factors, and we give an algorithm to compute them. We can use those linear factors to construct mechanisms because the factorization corresponds to the decomposition of the rational motion into simple rotations or translations. Bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial. Our criterion for factorizability allows us to improve on earlier algorithms to compute a suitable real or quaternion polynomial co-factor.

Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn’s work, we refine the stratification of iterated singular loci to a Whitney (a) stratification, which gives a list of candidates of varieties whose dual is an irreducible component of the algebraic boundary of the dual convex body. We also present an algorithm based on Teissier’s criterion to compute Whitney (a) stratifications, which employs conormal spaces and prime decomposition.

The configuration space (c-space) of a mechanism is the real-solution variety of a set of loop closure constraints, which is therefore the chief object in kinematic analysis. Singularities of this variety (referred to as c-space singularities) are singular configurations of the mechanism. In addition, a mechanism may exhibit other kinematic singularities that are not visible from the differential geometry of the c-space (referred to as hidden singularities). Such situations were analyzed by investigating the local geometry of the c-space and its corank stratification. It has been shown recently that hidden singularities and shakiness can be attributed to the fact that complex solution branches intersect with the c-space, i.e., with real-solution branches. This paper employs the kinematic tangent cone to identify local solution branches. While the kinematic tangent cone is an established generally applicable concept, which gives rise to a computational (numeric and symbolic) algorithm, it has yet only been applied to analyzing the real-solution set. Application of the method is shown for several examples. Further, the algebraic aspects are briefly elaborated.

Spinor polynomials are polynomials with coefficients in the even sub-algebra of conformal geometric algebra whose norm polynomial is real. They describe rational conformal motions. Factorizations of spinor polynomial corresponds to the decomposition of the rational motion into elementary motions. Generic spinor polynomials allow for a finite number of factorizations. We present two examples of quadratic spinor polynomials that admit infinitely many factorizations. One of them, the circular translation, is well-known. The other one has only been introduced recently but in a different context. We not only compute all factorizations of these conformal motions but also interpret them geometrically.

We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions. The factorization algorithm is based on the "kinematics at infinity" of the underlying rational motion. Factorizations exist generically but not generally and are typically not unique. We prove that generic multiples of non-factorizable spinor polynomials admit factorizations and we demonstrate at hand of an example how our ideas can be used to tackle the hitherto unsolved problem of "factorizing" algebraic motions.