Additive design of 2-dimensional scissor lattices

TL;DR

This work introduces an additive approach for the design of a class of transformable structures based on two-bar linkages joined at vertices to form a two dimensional lattice, and shows how to design karigami which unfold from a one dimensional collapsed state to two-dimensional surfaces of single and double curvature.

Abstract

We introduce an additive approach for the design of a class of transformable structures based on two-bar linkages ("scissor mechanisms") joined at vertices to form a two dimensional lattice. Our discussion traces an underlying mathematical similarity between linkage mechanisms, origami, and kirigami and inspires our name for these structures: karigami. We show how to design karigami which unfold from a one dimensional collapsed state to two-dimensional surfaces of single and double curvature. Our algorithm for growing karigami structures is provably complete in providing the ability to explore the full space of possible mechanisms, and we use it to computationally design and physically realize a series of examples of varying complexity.

Figures (4)

• Figure 1: Geometry and Kinematics (A-C) Physical examples of assemblies made from scissors arranged in 0-, 1-, and 2-D lattices. Scale bars represent 5cm. (A) The geometry of a single scissor is characterized by four leg lengths and an opening angle. (B) The scissor mount for a bathroom mirror is an example of a 1-D chain of identical scissor mechanisms. The coupling of opening angles allows for the telescopic extension of the mirror from the wall. (C) A karigami torus designed using the methods developed in this paper is characterised by a 2-D lattice of scissor elements. (D) karigami consists of a lattice of scissor-like unit cells ( $\bm{S}$$\bm{S}_{}$ ) composed of two rods (represented by black lines) joined by a scissor pivot (orange circle). Multiple cells can be joined together using vertex pivots (blue). Throughout this paper, we represent the geometry of each scissor by the combination of leg direction vectors ($\{ \IfNoValueTF{-NoValue-} {\bm{v}^{(1)}} {\bm{v}^{(1)}_{-NoValue-}} , \IfNoValueTF{-NoValue-} {\bm{v}^{(2)}} {\bm{v}^{(2)}_{-NoValue-}} \}$) and leg lengths ($\{ \IfNoValueTF{-NoValue-} {\ell^{(1)}} {\ell^{(1)}_{-NoValue-}} \ldots \IfNoValueTF{-NoValue-} {\ell^{(4)}} {\ell^{(4)}_{-NoValue-}} \}$). In addition we introduce the edge vectors, $\bm{s}$ and $\bm{t}$ which lie along the upper and left sides of the scissor facet respectively. (E) scissor and vertex pivots are distinguished by the way they connect strips. (F) For a set of cells to be intrinsically compatible, the lengths of adjacent facets must be equal (see edges highlighted in pink). (G-H) We can derive a closure condition for intrinsic compatibility by considering mapping facet edge lengths around the vertex $\bm{x}_{i,j}$ . (I) collapsibility describes whether a karigami structure can be "flattened", in the sense that there exists an intrinsically valid state where all opening angles $\{\alpha\}$ are identically 0. Note that in the lower example one cell has $\alpha=0$ while in the other $\alpha>0$. (J) collapsibility (Eq. \ref{['eq:collapsibility']}) requires that the total length of the legs on the left of each pore must equal the total length of the legs on the right.
• Figure 2: Constraints and Construction (A) An existing karigami lattice, with the growth front ($\{ \IfNoValueTF{i,1} {t} {t_{i,1}} , \ldots , \IfNoValueTF{i,n} {t} {t_{i,n}} \}$) identified using a dotted line. (B) The additive algorithm constructs a new row of scissors along the growth front. (C) Construction of the scissor pivot of a new scissor at site $j$ along the growth front. (C.1) We are free to a leg direction i,j $\bm{v}^{(2)}$$\bm{v}^{(2)}_{i,j}$ . The triangular loop highlighted in (C.1) then corresponds to the vector closure condition Eq. \ref{['eq:2D_pivot_closure']}. (C.2) Eq. \ref{['eq:1D_leg_length']} sets i,j $\ell^{(4)}$$\ell^{(4)}_{i,j}$ , and thereby fixes the location of pivot $\bm{p}_{i,j}$ (as well as i,j $\bm{v}^{(1)}$$\bm{v}^{(1)}_{i,j}$ , i,j $\ell^{(1)}$$\ell^{(1)}_{i,j}$ ). (D) Construction of an scissor pivot $\bm{p}_{i,j+1}$ adjacent to a scissor ( i,j $\bm{S}$$\bm{S}_{i,j}$ ) with an existing scissor pivot. (D.1) First, we construct the unique vertex pivot $\bm{x}_{i+1,j+1}$ making use of the vector closure condition highlighted in pink. (D.2) Second, we construct $\bm{p}_{i,j+1}$ according to the same process as shown in (C), with the direction i,j+1 $\bm{v}^{(2)}$$\bm{v}^{(2)}_{i,j+1}$ set by the location of the (now fixed) vertex $\bm{x}_{i,j+1}$ . (D.3) The new row of karigami with $\bm{p}_{i,j}$ and $\bm{p}_{1,j+1}$ constructed.
• Figure 3: Growth of Surfaces (A-C) karigami surfaces can be categorized by the topology of their constitutive growth fronts (equivalently, by the topology of the seed with which these surfaces were initialized). Thus we distinguish between helices (A) which are fully determined by their initial seed, loops (B) with one free edge, and sheet-like surfaces (C) with two free edges. (D) In the extrinsic algorithm we consider constructing a new row of scissors along the edge of an existing karigami structure---in this case, the last row of the structure shown in (C). The algorithm is initialized by first constructing a pivot at a site $j$ (D.2). Using our algorithm we can construct the scissors left and right along the front (D.3) until a full row of scissors have been built (D.4). (B-D) karigami can be classified by the topologies of the "seed" row by which they are initialized. In Fig. \ref{['fig:4_EXAMPLES']} we demonstrate structures designed with each of these seed topologies. E) The intrinsic algorithm generalizes the growth procedure presented in (D) by considering not only the growth front given by the space curve $\{t_j\}$, but a class of compatible fronts. (E.1) We begin with an existing karigami growth front and extract the sequence of edge lengths $\{|t_j|\}$ and shearing $\{k_j\}$ parameters for each site. (E.2) A compatible growth front is given by any space curve with the same sequence of edge lengths extracted from the karigami front as in (E.1). (E.3) Finally, we apply the extrinsic algorithm to the modified front to construct a strip of cells that is intrinsic compatibility and kinematic compatibility with the original karigami structure.
• Figure 4: Karigami structures designed using our extrinsic (examples A-D) and intrinisic (examples E,F) algorithms. (See §S.IV for details of surface construction.) (A) A bowl constructed from a helix-like seed with 32 steps of growth. (B) An axisymmetric torus. Here the initial seed is a loop lying along a nodal curve along the top (or bottom) of the torus. (C, D) We design a lattices which map the surface of a disk and an ellipsoid. Here we optimize over the growth parameters for each row (i.e. initial pivot locations and terminal leg lengths) to fit the target shapes. (E) Using the intrinsic algorithm we develop an eggbox-like surface with multi-stable "nodes" which can be flipped in the deployed state (and reset by closing the karigami). (F) We also use the intrinsic algorithm to design a surface of constant negative Gaussian curvature.

Theorems & Definitions (4)

• Lemma 1
• proof
• Theorem 1
• proof