SINGULAB - A Graphical user Interface for the Singularity Analysis of Parallel Robots based on Grassmann-Cayley Algebra

This paper presents SinguLab, a graphical user interface for the singularity analysis of parallel robots. The algorithm is based on Grassmann-Cayley algebra. The proposed tool is interactive and introduces the designer to the singularity analysis performed by this method, showing all the stages along the procedure and eventually showing the solution algebraically and graphically, allowing as well the singularity verification of different robot poses.Comment: Advances in Robot Kinematics, Batz sur Mer : France (2008

A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization

In symbolic computing, a major bottleneck is middle expression swell. Symbolic geometric computing based on invariant algebras can alleviate this difficulty. For example, the size of projective geometric computing based on bracket algebra can often be restrained to two terms, using final polynomials, area method, Cayley expansion, etc. This is the "binomial" feature of projective geometric computing in the language of bracket algebra. In this paper we report a stunning discovery in Euclidean geometric computing: the term preservation phenomenon. Input an expression in the language of Null Bracket Algebra (NBA), by the recipe we are to propose in this paper, the computing procedure can often be controlled to within the same number of terms as the input, through to the end. In particular, the conclusions of most Euclidean geometric theorems can be expressed by monomials in NBA, and the expression size in the proving procedure can often be controlled to within one term! Euclidean geometric computing can now be announced as having a "monomial" feature in the language of NBA. The recipe is composed of three parts: use long geometric product to represent and compute multiplicatively, use "BREEFS" to control the expression size locally, and use Clifford factorization for term reduction and transition from algebra to geometry. By the time this paper is being written, the recipe has been tested by 70+ examples from \cite{chou}, among which 30+ have monomial proofs. Among those outside the scope, the famous Miquel's five-circle theorem \cite{chou2}, whose analytic proof is straightforward but very difficult symbolic computing, is discovered to have a 3-termed elegant proof with the recipe

Algorithms for detecting dependencies and rigid subsystems for CAD

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract based on v1

The Dotted straightening algorithm

AbstractIf a homogeneous bracket polynomial is antisymmetric in certain subsets of its points, then it can be represented in an abbreviated form called a dotted bracket expression. These dotted bracket expressions lead to a more compact expression in terms of tableaux than the usual representation. Consequently, we can derive a much more efficient straightening algorithm than the ordinary one for bracket polynomials already given in dotted form. This dotted straightening algorithm gives precisely the same result as the ordinary one, and preserves the dotted property at every step

A Pascal's theorem for rational normal curves

Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$, $\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in $\mathbb{P}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of $d+4$ ordered points in $\mathbb{P}^d$ that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.Comment: 16 pages, 1 figure. Comments are welcom