Some Mobile Overconstrained Parallel Mechanisms

Abstract

The Griffis-Duffy platform is an example of an overconstrained parallel mechanism. Although it has 6 SS legs joining its platform to its base it is still mobile. In this work similar structures are found but with different types of legs. The key to finding these structures is a pair of theorems concerning 3 degree-of-freedom mechanisms subjected to a translation or a half-turn. Although these results are not new concise statements and proofs are given. These constructions are then applied to parallel mechanisms consisting of 3 RPS legs and 3UPU legs. Some details of the rigid-body motions that the platform of these mechanisms can execute are found. This is facilitated by the observations that rigid displacements permitted by an RPS leg are the displacements which constrain a point to a fixed plane, while the displacements of a UPU leg constrain a line to be coplanar to a fixed line.

On the Plane Symmetric Bricard Mechanism

“…The fourth joint must still remain in Π but now as Π moves we can see that 4 will remain in the special line complex with axis 1 . Note that this motion can be mechanically generated by a UPU linkage, see [8]. As mentioned in the previous section, the variety of rigid-body motions that keep a line in a fixed linear line complex is given by the intersection of the Study quadric with another quadric hypersurface.…”

Synthesis and Singularity Analysis of N-UU Parallel Wrists: A Symmetric Space Approach

“…Here, we reveal the axial symmetry more elegantly using commutativity of coaxial translation and rotation. We would like to draw a connection between this new approach and some recent work of Selig [26], where the role of commutativity in synthesis of several novel mechanisms is emphasized.…”

“…Therefore, Eqn. (26) Comparison with the Collapsed N-UU PM In [15], we introduced a collapsed N-UU PM, where each leg is a spherical chain (w + i1 , w + i2 , w − 12 , w − 11 ), and moreover, w + i2 = w − 12 . When N = 3, we have shown that its constraint singularity is given by the following identity:…”