Zijia Li

Classification of Angle-Symmetric 6R Linkages

Abstract In this paper, we consider a special kind of overconstrained 6R closed linkage which we call angle-symmetric 6R linkage. These are linkages with the property that the rotation angles are equal for each of the three pairs of opposite joints. We give a classification of these linkages. It turns out that there are three types. First, we have the linkages with line symmetry. The second type is new. The third type is related to cubic motion polynomials.

The theory of bonds II

In this paper, we study closed linkages with six rotational joints that allow a one-dimensional set of motions. We prove that the genus of the configuration curve of such a linkage is at most five, and give a complete classification of the linkages with a configuration curve of genus four or five. The classification contains new families.

Planar linkages following a prescribed motion

Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.

Three Types of Parallel 6R Linkages

In this paper, we consider a special kind of overconstrained 6R closed linkages which we call parallel 6R linkages. These are linkages with the property that they have three pairs of parallel joint-axes. We prove that there are three types of parallel 6R linkage. The first type is new, the other two also appear in a recent classification of linkages with angle equalities. We give constructions for each of the three types.

A technique for deriving equational conditions on the Denavit–Hartenberg parameters of 6R linkages that are necessary for movability

A closed 6R linkage is generically rigid. In this paper we give, for the first time, equational conditions on the Denavit/Hartenberg parameters that are necessary for mobility. The conditions suggest to distinguish the movable 6R linkages into various types characterized by their bond diagrams.

The rational motion of minimal dual quaternion degree with prescribed trajectory

We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the trajectorys degree minus its circularity. Hence, it is lower than the degree of a trivial curvilinear translation for circular curves.

Spatial Straight-Line Linkages by Factorization of Motion Polynomials

We use the recently introduced factorization of motion polynomials for constructing overconstrained spatial linkages with a straight line trajectory. Unlike previous examples, the end-effector motion is not translational and the link graph is a cycle. In particular, we obtain a number of linkages with four revolute and two prismatic joints and a remarkable linkage with seven revolute joints one of whose joints performs a Darboux motion.

Kempe’s Universality Theorem for Rational Space Curves

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with

Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions

\frac{9}{2} d-6c+1