A classification of 3R orthogonal manipulators by the topology of their workspace

Singular curves and cusp points in the joint space of 3-RPR parallel manipulators

“…Finally, the number of cusp points stabilizes to four, defining one central foursolution region surrounded by a two-solution region (L 1 =31, …). Interestingly, this last pattern is very similar to the one often observed in a cross-section of the workspace of 3-R serial manipulators [13]. However, serial manipulators feature the same pattern in all cross-sections (the sections which passes through the first revolute joint axis), and variation in the number of cusp points arises only from a modification of the manipulator geometry.…”

An exhaustive study of the workspace topologies of all 3R orthogonal manipulators with geometric simplifications

“…The first factor defines two horizontal lines in the joint space (assuming d 3 ≤d 4 , which is the case for the manipulator in An interesting classification criterion is the topology of the singular curves in the cross section of the workspace. A way of defining the topology of these curves is to enumerate their singular points, the cusp points and the node points [14,16].…”

“…The work of [7] was completed in [8] to take into account additional features in the classification like genericity and the number of aspects. [9] established a classification of 3R orthogonal manipulators with no offset on their last joint, based on the work of [8], according to the number of cusps and node points, the parameters space was divided into 9 cells where the manipulators have the same number of cusps and nodes in their workspace. [9] classified only the family of 3R orthogonal manipulators with no offset on their last joint.…”

Computing Cusps of 3R Robots Using Distance Geometry

“…It can be shown that the elements of the third level of determinants depend quadratically on s 3,7 and s 4,7 . Then, since the two outer levels of determinants are quadratic with respect to their elements, the singularity locus can be displayed as a curve of order 2 3 in the plane defined by s 3,7 and s 4,7 .…”

“…The symbolic condition, in terms of the DH parameters of the robot, for this polynomial to have three equal roots has been considered as intractable [1]. Only the case of orthogonal 3R robots -robots whose consecutive joint axes are mutually orthogonal-has been analyzed in detail [4]. This paper puts forward a new formulation of the problem that could lead to new insights into Federico Thomas Institut de Robòtica i Informàtica Industrial (CSIC-UPC) e-mail: fthomas@iri.upc.edu the general case thanks to the simplicity and symmetry of the resulting algebraic expressions.…”