A New Mobility Formula for Spatial Mechanisms

Abstract

A

Mechanism mobility and a local dimension test

“…A different approach to generalizing the Grübler-Kutzbach formulas depends on a technique of replacing links and joints with an equivalent polyhedral model [25]. This approach overlaps with the group theory approach in the ability to deal with some exceptional mechanisms.…”

“…A collapsible cube, consisting of 12 scissors linkages each aligned with one edge of a cube, was presented in [25]. Locking the scissors, the mechanism becomes a 12-bar spherical linkage, with one link along each edge of a cube and with rotational joints at the vertices of the cube, all joint axes meeting at the center of the cube.…”

“…Locking the scissors, the mechanism becomes a 12-bar spherical linkage, with one link along each edge of a cube and with rotational joints at the vertices of the cube, all joint axes meeting at the center of the cube. Using the methodology espoused in [25], one may convert the links into triangles joined at their vertices. This comes down to a set of nine points, one for each vertex of the cube and its center as illustrated in Figure 3.…”

Generic mobility of rigid body mechanisms

“…Unlike the CKG formula, these criteria are not expressible in a single formula, but require recursive processing of screw quantities (as the first approach does). Beside these three classes other approaches exist, like the recent contribution [35] that uses polyhedral models of linkages. Remarkably, there are mechanisms (so-called overconstrained) for which none of these approaches gives the correct DOF.…”

Real Numerical Algebraic Geometry: Finding All Real Solutions of a Polynomial System

“…One mechanism where this occurs is a cubic-centered 12-bar mechanism, first presented in [11]. This mechanism is described by 17 polynomials in 18 variables so that the complex dimension is at least one.…”