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.

A family of mixed double-Goldberg 6 R linkages

A complete family of double-Goldberg 6 R linkages is reported in this article by combining a subtractive Goldberg 5 R linkage and a Goldberg 5 R linkage through the common link-pair or common Bennett-linkage method. A number of distinct types of overconstrained linkages are built, namely the mixed double-Goldberg 6 R linkages. They all have one degree of freedom and their closure equations are derived in detail. One of them degenerates into a Goldberg 5 R linkage. From the construction process and geometry conditions, the corresponding relationship between the newly found 6 R linkages and the double-Goldberg 6 R linkages, constructed from two Goldberg 5 R linkages or two subtractive Goldberg 5 R linkages, has been established.

A Line Geometric Foundation for Finite Screw Systems Associated With Spatial Linkages

This paper investigates the line varieties corresponding to finite screw systems associated with spatial linkages. This research is based on the correspondence between a screw and a linear complex, and a screw system corresponds to the intersection of the linear complexes. In finite kinematics, two screw systems associated with the finite motions of the revolute-revolute (R-R) and prismatic-revolute (P-R) open chains have been discovered. These two screw systems also led to the discovery of the screw systems associated with the finite motions of the spatial 4R, spatial RPRP, and other overconstrained linkages. By using the intersection operation of linear complexes, this paper finds the linear reguli corresponding to the finite motions of R-R and P-R chains. Then we utilize the sum operation of the linear reguli corresponding to the R-R and P-R chains to obtain the hyperbolic linear congruences corresponding to the finite motions of the spatial 4R and RPRP linkages. The result presented here serves as a line geometric foundation for finite screw systems associated with spatial linkages. In addition, with CAD drawings, this paper enables the visualization of the obtained line varieties and their operations.

Linkages That Transfer Rotations to Radially Reciprocating Motion

Stemming from study of polyhedral and spheroidal linkages and investigation of reciprocating motion of the PRRP chain, this paper presents four overconstrained linkages that are capable of transferring rotations to radially reciprocating motion. The linkages connected by revolute joints are of symmetrical arrangement and mobility one and are analysed by using the screw-loop equation method. The paper further investigates geometry and kinematics of the linkages and reveals their kinematic characteristics, leading to the constraint equation.

A New Approach for the Orders of the Bennett-Based Linkages in Mobility Analysis

Among the 3D single-loop overconstrained linkages, quite a number of them are combinations of Bennett linkages. Mobility on the overconstrained linkages including the Bennett-based linkages is known to be one of the difficult topics in kinematics. In the paper, a new approach based on the linear superposition principle for determining the orders of Bennett-based linkages is proposed, and the mobility of some typical Bennett-based linkages is calculated with the Modified Gru¨bler-Kutzbach criterion. In addition, geometric properties of some of the screw systems are employed to identify whether the mobility is global.