SINGULAB – A GRAPHICAL USER INTERFACE FOR THE SINGULARITY ANALYSIS OF PARALLEL ROBOTS BASED ON GRASSMANN-CAYLEY ALGEBRA Patricia Ben-Horin, Moshe Shoham Department of Mechanical Engineering Technion – Israel Institute of Technology, Haifa, Israel [patbh,shoham]@tx.technion.ac.il Stéphane Caro, Damien Chablat, Philippe Wenger Institut de Recherche en Communications et Cybernétique de Nantes 1, Rue de la Noe, 44321 Nantes, France [stephane.caro, damien.chablat, philippe.wenger]@irccyn.ec-nantes.fr Abstract 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. Keywords: Singularity, Grassmann-Cayley algebra, parallel robot, software. 1. Introduction Singularity of parallel manipulators has been thoroughly investigated using different methods, mainly including line geometry, screw theory, and Jacobian determinants analysis. Recently, Grassmann-Cayley algebra (GCA) has been used for singularity analysis too. SinguLab is the first version of a tool for singularity analysis of parallel robots. The aim of this user interface is to provide the designer an automatic tool for the analysis, geometric interpretation and visualization of singularities. It enables the user to determine the singularities of a large range of parallel robots and gives him some guidelines of GCA. SinguLab was developed within the framework of SIROPA1 – a French national project, the aim of which is to develop knowledge about the direct-kinematics singularities of parallel robots and to transmit this knowledge to the end-users – during a sojourn stay of the first author at IRCCyN. 1.1 Grassmann-Cayley algebra The algorithm used in SinguLab is based on GCA. For space limitations we only introduce the basic concepts and the readers are referred to (Ben-Horin and Shoham, 2006a) and reference therein for further details on this topic. The basic elements of this algebra are called extensors, which in fact are symbolically denoted Plücker coordinates of vectors. Two basic operations that play an essential role in GCA involving extensors are the join and meet operators. The first is associated with the union of two vector spaces, and the latter has the same geometric meaning as the intersection of two vector spaces. Further, special determinants called brackets are also defined in GCA. The brackets, of which columns are vectors, satisfy special product relations called syzygies, which are useful to manipulate and compare bracket expressions. The Grassmann-Cayley algebra functions under the projective space Pd, in which points are represented by homogeneous coordinates and lines are represented by Plücker coordinates. As mentioned above, the extensors are vectors that represent geometric entities, and are characterized by their step. Extensors of step 1, 2 and 3 stand for a point, a line and a plane, respectively. Assumming two extensors A and B, of step k and h, respectively, defined in the d- projective space, the join and meet operations are written as follows:

1 2 1 1 2 1 ∨

∨ ∨ ∨ ∨ ∨ ∨

L L L L k h k h a a a b b a a a b b A B

(1)

(1) (2) ( ) 1 ( 1) ( ) 1 2 1 1 sgn( )[ … … ] … sgn( )[ … … ] … d h h d h k d h d h k h a a a b b a a a a a b b a a σ σ σ σ σ σ σ σ σ • • • • • − −+ − −+ ∧

= ∑ ∑ A B

(2) where the sum in Eq.(2) is taken over all permutations σ of {1,2,..,k} such that σ(1)<σ(2)<…<σ(d-h) and σ(d-h+1)<σ(d-h+2)<…<σ(k). Incidences between geometric entities are obtained as extensors of step 0 (scalars). Some examples of incidences in 3D-space are the meet of four planes, the meet of two lines and the meet of a line with two planes. Three meet examples are written in GCA as follows:

1 https://wiki-sop.inria.fr/wiki/bin/view/Coprin/SIROPA Meet of four planes: • • • ⎡ ⎤⎡ ⎤⎡ ⎤ ∧ ∧ ∧ = ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦ abc def ghi jkl adef bghi c jkl
Meet of two lines: [ ] ∧

ab cd abcd
Meet of a line and two planes: • • ⎡ ⎤⎡ ⎤ ∧ ∧ = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ gh abc def gabc hdef
Let us consider a finite set of 1-extensors {a1,a2,.,ad} defined in the d- dimensional vector space over the field ϒ, V, where ai=x1,i,x2,i,..,xd,i (1≤i≤d). The bracket of these extensors is the determinant of the matrix, of which columns are vectors ai (1≤i≤d) :
[ ] 1,1 1,2 1, 1 2 ,1 ,2 , , ,…,

L M M L M L d d d d d d x x x a a a x x x , (3) For example, the bracket of points a, b, c and d defined in the 3D space is written as: [ ] 1 1 1 1

x x x x y y y y z z z z a b c d a b c d a b c d abcd

From a geometrical point of

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