In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.

It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by considering links to be vectors in the complex plane. However, scant attention has been paid to the natural interpretation of complex vectors as isotropic coordinates. These coordinates, often considered a special trick for analyzing four-bar motion, are in fact uniquely suited to two new techniques for analyzing polynomial systems: the BKK bound and the product-decomposition bound. From this synergistic viewpoint, a fundamental formulation of planar kinematics is developed and used to prove several new results, mostly concerning the degree and circularity of the motion of planar linkages. Useful for both analysis and synthesis of mechanisms, the approach both simpli es theoretical proofs and facilitates the numerical solution of mechanism problems.

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To convexify a polygon is to reconfigure it with respect to a given set of operations until the polygon becomes convex. The problem of convexifying polygons has had a long history in a variety of fields, including mathematics, kinematics and physical chemistry. We survey its history throughout these disciplines.

Engineers have for some time known that singularities play a significant role in the design and control of robot manipulators. Singularities of the kinematic mapping, which determines the position of the end–effector in terms of the manipulator’s joint variables, may impede control algorithms, lead to large joint velocities, forces and torques and reduce instantaneous mobility. However they can also enable fine control, and the singularities exhibited by trajectories of the points in the end–effector can be used to mechanical advantage. A number of attempts have been made to understand kinematic singularities and, more specifically, singularities of robot manipulators, using aspects of the singularity theory of smooth maps. In this survey, we describe the mathematical framework for manipulator kinematics and some of the key results concerning singularities. A transversality theorem of Gibson and Hobbs asserts that, generically, kinematic mappings give rise to trajectories that display only singularity types up to a given codimension. However this result does not take into account the specific geometry of manipulator motions or, a fortiori, to a given class of manipulator. An alternative approach, using screw systems, provides more detailed information but also shows that practical manipulators may exhibit high codimension singularities in a stable way. This exemplifies the difficulties of tailoring singularity theory’s emphasis on the generic with the specialized designs that play a key role in engineering.

PRELIMINARY -- comments desirable. We survey and redo the whole mathematical area of "planar mechanisms:" machines made of rigid parts constrained to lie in a plane and linked by rotatable rivets. This area attracted great interest in England and France in the late 1800s, and later enjoyed a brief revival in Russia in the 1940s, but has largely lain dormant. Important contributions of the present work include: 1. A streamlined algebraic method is proposed to analyse simple planar mechanisms, allowing the first analysis of the 5-link anchored mechanisms. 2. The "algebraic completeness theorem" for planar mechanisms is revisited. Algebraic completeness theorems have attracted a lot of interest in the last decade, but none of the workers realized that the first such result had been shown about a century earlier! 3. We show how to enumerate all combinatorial types of planar mechanisms with N links, and provide asymptotic bounds and a table of counts for small N computed by Henry Cejtin. 4. Assuming the "downhill principle" that physical systems not at local minima of the potential energy function always go downhill, would allow planar mechanisms to solve NP-hard problems quickly. But we show that in both classical and quantum mechanics, the downhill principle cannot be relied upon. Keywords --- Downhill principle, algebraic completeness, planar mechanisms, isomorph-free generation of combinatorial objects, graphs, hypergraphs, 3-bar curve, conicographs, line drawing devices, NP-completeness, Church's thesis, quantum mechanics, classical mechanics. Contents 1

The focal points of a curve traced by a planar linkage capture essential information about the curve. In a previous paper, we showed how to determine the singular foci of planar linkages from an expression for the tracing curve derived by use of the Dixon determinant. This paper gives an alternative approach to finding the singular foci, one that lends itself to simple geometric interpretations and does not require a derivation of the tracing curve equation. In many cases, singular foci can be determined from a simple graphical construction. The method is demonstrated for one inversion each of the Stephenson-3 six-bar and the Watt-1 six-bar. A by-product of the study is a technique for illustrating the non-real points on a tracing curve. Knowledge of the singular foci will be helpful in further study of path cognates.