Threefold-symmetric Bricard linkages for deployable structures

Abstract

A closed-loop overconstrained spatial mechanism composed of six hinge-jointed bars, which has three planes of symmetry in any position, is called a threefold-symmetric Bricard linkage. In this paper a kinematic analysis of these linkages is presented. It is pointed out that for particular parameter values, kinematic bifurcation of the linkages can occur. Features of the kinematic bifurcation are discussed in detail. The applicability of threefold-symmetric Bricard linkages and of their alternative forms to deployable structures is investigated. In addition, by using the theory of kinematic bifurcation, a snap-through phenomenon appearing in a deployable hexagonal ring is explained.

A plane symmetric 6R foldable ring

“…An early example appears in [2], in which an even number of bars are linked together in such a way that they can be folded into a tight bundle, and unfolded to form a regular polygon. In the six bar case, a three-fold symmetric linkage results [3], while in the four bar case, a Bennett linkage is formed [4,5]. Four bar foldable frames have been extensively examined [6,4,7].…”

Self-crossing Motion Analysis of a Novel Inpipe Parallel Robot with Two Foldable Platforms

“…[12], for the plane-symmetric Bricard linkage shown in Fig. 1 (b), the output angle φ can be represented by input angle θ and twist angle α:…”

“…[12], we know that the Bricard linkage keeps moving continuously when twist angle satisfies 60°≤ α ≤ 120°, and, the linkage with twist angle α behaves the same way as one whose twist angle set as 180°− α. Meanwhile, the six linkages of the Bricard linkage with twist angle α = 60°or 120°will coincide together as both θ and φ 1 reach to 180°, which is impossible to reach this point of a physical prototype due to interference.…”

Mobility of ‘ N -loops’: bodies cyclically connected by intersecting revolute hinges

“…Mobility counting and rigidity theory provide concepts that are used and re-used in contexts from conformational analysis of small molecules (Dunitz & Waser 1972), mechanics of protein structures (Jacobs et al 2001), modelling of pH-dependent expansion of nano-scale viral particles (Kovács et al 2004), through robotics (Porta et al 2009), to the generation of designs for deployable engineering structures (Chen et al 2005). Many systems at these different length scales can be described as cycles of bodies linked by intersecting revolute hinges.…”