An Explanation of Never-Special Assembly Changing Motions for 3–3 Parallel Manipulators

Abstract

When the leg rods of a fully in-parallel manipulator are fixed in their lengths, it is usual that the device can be assembled in several distinct ways. Sometimes it happens that motion between such assemblies can take place such that the linkage is never at a special configuration; that is, a configuration where the moving-platform body acquires uncontrollable freedom relative to the base. The possibility of such motion has implications for control. Focusing on 3-3 devices, we present a geometric explanation of how these motions arise, and give a sufficient condition for their existence. For the 3-3 planar-motion device, we show that never-special assembly changing motions can be excluded by making platform and base triangles similar, and we conjecture that appropriate, perhaps identical, specialization for the octahedral manipulator has the same effect.

Asymptotic Singularities of Planar Parallel 3-RPR Manipulators

“…This curve is generically an irreducible curve of degree 8, not rational. However, it is possible to obtain parameterizations by analytic functions of θ of each of the branches by substituting in Equations (2) the expressions for the trigonometric fonction of ϕ extracted from Equation (5). One has to be careful in the determination of the angle ϕ from its tangent, in order to ensure the continuity.…”

Nonsingular Change of Assembly Mode Without any Cusp

“…The possibility to execute such a motion is most frequently associated with the presence of cusps, and the nonsingular change of assembly mode is realized by turning around a cusp point, or a cuspidal edge of the singularity surface (see for instance [2,3,4,5]). It has also been reported [6,7] that nonsingular change of assembly modes can be realized by following an "alpha curve" (i.e.…”

Perturbation of Symmetric 3-RPR Manipulators and Asymptotic Singularities

“…Planar parallel manipulators have been extensively studied in the past, see [1,4,5,6,7,8,9,11,12] for example. Symmetric 3-RPR manipulators (those for which the platform triangle is congruent to the base triangle by an orientation-reversing isometry) have special kinematic properties (see [3]).…”