Constraint and Singularity Analysis of Lower-Mobility Parallel Manipulators With Parallelogram Joints

Abstract

This paper presents a general approach to analyze the singularities of lower-mobility parallel manipulators with parallelogram joints. Using screw theory, the concept of twist graph is introduced and the twist graphs of two types of parallelogram joints are established in order to simplify the constraint analysis of the manipulators under study. Using Grassmann-Cayley Algebra, the geometric conditions associated with the dependency of six Plu¨cker vectors of finite and infinite lines in the 3-dimensional projective space are reformulated in the superbracket in order to derive the geometric conditions for parallel singularities. The methodology is applied to three lower-mobility parallel manipulators with parallelogram joints: the Delta-linear robot, the Orthoglide robot and the H4 robot. The geometric interpretations of the singularities of these robots are given.

SINGULARITY ANALYSIS OF THE 4 RUU PARALLEL MANIPULATOR USING GRASSMANN-CAYLEY ALGEBRA

“…For most manipulators, the determinant of such a matrix is highly nonlinear and unwieldy to assess even with a computer algebra system. Hence, linear algebra fails to provide satisfactory results, and therefore, the use of Grassmann Geometry (GG) [1,[13][14][15] or Grassmann-Cayley Algebra (GCA) [2,[16][17][18] can be regarded as a promising solution. The GG is a geometric approach that provides a classification for the conditions under which a set of n Plücker lines spans a variety of dimension lower than n. On the other hand, the GCA is a systematic approach to obtain a bracket representation of the determinant of J, called superbracket.…”

“…Moreover, it does not apply when some Plücker vector(s) of J correspond to the Plücker coordinate vector(s) of a line at infinity. Recently, Kanaan et al [17] and Amine et al [18] filled this gap by using some properties of projective geometry in order to formulate a superbracket with points and lines at infinity, and therefore, to extend the application of GCA to lower-mobility PMs. This paper focuses on the application of GCA to provide a compact vector expression for the singularity locus of 3T1R PMs with identical limb structures.…”

Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures

“…Moreover, it does not apply when a line at infinity is among the six Plücker lines of J m . Recently, Kanaan et al [15] and Amine et al [7,22,23] filled this gap by using some properties of projective geometry in order to formulate a superbracket with points and lines at infinity, and therefore, to extend the application of GCA to lower-mobility PMs.…”

“…Mostly, the determinant of such a matrix is highly non linear and unwieldy to assess even with a computer algebra system. In that case, linear algebra fails to give satisfactory results, and therefore, the use of Grassmann-Cayley Algebra (GCA) [7,[12][13][14][15] or Grassmann Geometry (GG) [16][17][18][19][20][21] can be regarded as promising solutions.…”

“…Nevertheless, their approach cannot provide a 6 × 6 J m for a more general lower-mobility PM and thus it does not allow the examination of all singular configurations of such a PM. Based on the theory of reciprocal screws [3][4][5][6][7], Joshi and Tsai [8] developed a general methodology to perform the Jacobian analysis of lowermobility PMs, namely, to derive a 6 × 6 J m providing information about both constraint and actuation wrench systems of such PMs. Accordingly, away from parallel singularities, the rows of J m of a (n < 6)-DOF PM are composed of n linearly independent actuation wrenches plus (6 − n) linearly independent constraint wrenches.…”