Overconstrained 6R Linkage Research Articles

A Network of Type III Bricard Linkages

Among Bricard's overconstrained 6R linkages, the third type has two collapsed configurations, where all joint axes are coplanar. This paper presents a one-degree-of-freedom network of such linkages. Using the two coplanar states of the constituent Bricard units, the network is able to cover a large surface with a specific outline when deployed and can be folded compactly into a stack of much smaller planar shapes. Five geometric parameters describing each type III Bricard mechanism are introduced. Their influence on the outline of one collapsed configuration is discussed and inverse calculation to obtain the parameter values yielding a desired planar shape is performed. The network is built by linking the units, either using scissor linkage elements, if the thickness of the panels can be ignored, or with hinged parallelograms, for a thicker material. Two case studies, in which the Bricard network deploys as a rectangle and a regular hexagon, respectively, are presented, validating the analysis and design methods.

Novel spherical-planar and Bennett-spherical 6R metamorphic linkages with reconfigurable motion branches

A metamorphic linkage is capable of changing its motion branches and can be used as mechanisms for reconfigurable robots for various tasks. This paper presents two novel metamorphic linkages as the spherical-planar 6R metamorphic linkage and the Bennett-spherical 6R metamorphic linkage both of which have three various distinguished motion branches. Having established the close-loop equation of the spherical-planar 6R metamorphic linkage, the paper reveals the conditions of various motion branches and a set of transformations for switching motion branches. The paper further uses to reveal the inherent properties of this over-constrained metamorphic 6R linkage that is able to perform both spherical and planar motion with mobility one. Because of geometrical constraints at bifurcation points, the linkage is able to reconfigure to the deployed spherical motion branch, the planar motion branch and the folded spherical motion branch. The two spherical motion branches could be seen on both a large sphere that presents the deployed spherical motion and a small sphere that presents the folded spherical motion. This leads to the revelation of the novel Bennett-spherical 6R metamorphic linkage that has the transition from one deployed Bennett configuration branch to a spherical configuration branch and then to another folded Bennett configuration branch. Given the geometric parameters of both metamorphic linkages, it reveals that these linkages are special cases of Bricard line-symmetric 6R linkage.

Rigid Foldability of Generalized Triangle Twist Origami Pattern and Its Derived 6R Linkages

Rigid origami is a restrictive form of origami that permits continuous motion between folded and unfolded states along the predetermined creases without stretching or bending of the facets. It has great potential in engineering applications, such as foldable structures that consist of rigid materials. The rigid foldability is an important characteristic of an origami pattern, which is determined by both the geometrical parameters and the mountain-valley crease (M-V) assignments. In this paper, we present a systematic method to analyze the rigid foldability and motion of the generalized triangle twist origami pattern using the kinematic equivalence between the rigid origami and the spherical linkages. All schemes of M-V assignment are derived based on the flat-foldable conditions among which rigidly foldable ones are identified. Moreover, a new type of overconstrained 6R linkage and a variation of doubly collapsible octahedral Bricard are developed by applying kirigami technique to the rigidly foldable pattern without changing its degree-of-freedom. The proposed method opens up a new way to generate spatial overconstrained linkages from the network of spherical linkages. It can be readily extended to other types of origami patterns.

Origami-Inspired Integrated Planar-Spherical Overconstrained Mechanisms

This paper presents two integrated planar-spherical overconstrained mechanisms that are inspired and evolved from origami cartons with a crash-lock base. Investigating the crash-lock base of the origami cartons, the first overconstrained mechanism is evolved by integrating a planar four-bar linkage with two spherical linkages in the diagonal corners. The mechanism has mobility one and the overconstraint was exerted by the two spherical linkages. This mechanism is then evolved into another integrated planar-spherical overconstrained mechanism with two double-spherical linkages at the diagonal corners. The evolved mechanism has mobility one. It is interesting to find that the double-spherical linkage at the corner of this new mechanism is an overconstrained 6R linkage. The geometry evolution is presented and the constraint matrices of the mechanisms are formulated using screw-loop equations verifying mobility of the mechanisms. The paper further reveals the assembly conditions and geometric constraint of the two overconstrained mechanisms. Further, with mechanism decomposition, geometry and kinematics of the mechanisms are investigated with closed-form equations, leading to comparison of these two mechanisms with numerical simulation. The paper further proposes that the evolved overconstrained mechanism can in reverse lead to new origami folds and crease patterns. The paper hence not only lays the groundwork for kinematic investigation of origami-inspired mechanisms but also sheds light on the investigation of integrated overconstrained mechanisms.

A Kirigami-Inspired 8R Linkage and Its Evolved Overconstrained 6R Linkages With the Rotational Symmetry of Order Two

This paper presents a novel metamorphic 8R linkage extracted from a kirigami-fold with pregrooved creases and two overconstrained 6R linkages evolved from the 8R linkage by taking the concept of metamorphosis in the sense of structural evolution. The geometric characteristics and the parametric constraints of the evolved 6R linkages are identified following the structural evolution of the 8R linkage. The paper reveals that the evolved 6R linkages are special line-symmetric Bricard 6R loops characterized by the rotational symmetry of order two. The joint space of the overconstrained 6R linkages is analyzed and the relationship between motion parameters of the evolved 6R linkages and reconfiguration parameters of the metamorphic 8R linkage are derived. The motion characteristics of the overconstrained 6R linkages are further verified in terms of screw theory. The bifurcation and trifurcation associated with various transitory positions of the evolved 6R linkages having distinct parametric constraints are consequently identified based on constraint analysis.