math.MGApr 1, 2026

Bistable Quad-Nets Composed of Four-Bar Linkages

Authors: Gudrun Szewieczek, Daniel Huczala, Martin Pfurner, Hans-Peter Schröcker

Organizations: University of Innsbruck, Department of Basic Sciences in Engineering Sciences, Innsbruck, Austria · Seoul National University, Robotics Laboratory, Seoul, Korea

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Abstract

We study a novel type of mechanical structures, composed of spatial four-bar linkages, that are bistable, that is, they allow for two distinct configurations. These structures have an interpretation as quad nets in the Study quadric which we use to prove existence of assemblies with an unbounded number of links and joints. We propose a purely geometric construction of such objects, starting from infinitesimally flexible quad nets in Euclidean space and applying Whiteley de-averaging. This point of view situates the problem within the broader framework of discrete differential geometry and enables the construction of bistable structures from well-known classes of quad nets, such as discrete minimal surfaces. In contrast to many other construction methods for bistable structures, our approach does not rely on numerical optimization and it allows for simple control of relevant geometric parameters such as axis positions and snap angles.

Figures & tables

Figure 1: Two configurations (orange and blue) of a snapping four-bar ( center ), in poses where two axes are aligned (a, c, e, g) and in intermediate poses (b, d, f, h). Coinciding axes are indicated by using both colors.
Figure 2: Schematic sketch of a planar scissor linkage. Its linkgraph consists of two quads. Note that graph vertices (black dots) correspond to links while edges correspond to revolute joints (white dots).
Figure 3: Construction of a Q\mathcal{Q} -net on Z2\mathbb{Z}^{2} . We prescribe the origin ( left ), then compute the values along the two coordinate axes ( center ) before consecutively determining the remaining values ( right ).
Figure 4: Top: A graph with Z2\mathbb{Z}^{2} -combinatorics with black and white coloring of the faces and the related dual graph. Bottom: The quad net f+f^{+} rolls on the quad net f−f^{-} . Both f−f^{-} and f+f^{+} correspond to the original graph while the discrete rolling motion corresponds to the dual graph ( right ).
Figure 5: Initial data and iterative construction for Theorem 1 .
Figure 6: By introducing a black-and-white coloring on vertices and switching to the diagonal net, we turn the vertex isometric quad net ff ( left ) into the face isometric quad net gg ( center ). A checkerboard coloring of gg gives rise to a discrete rolling motion and hence an S\mathcal{S} -net in the Study quadric. Note that we omit the superscript “ ±t\pm t ” in all labels because the drawings refer to both nets.
Figure 7: Top left : The Koenigs dual f∗f^{*} is obtained via inverse stereographic projection. Top right : The Koenigs net ff is infinitesimally flexible with IID qq . Bottom left : The discrete quad nets f+f^{+} and f−f^{-} are obtained by adding and subtracting ff and qq . A portion of ff is depicted as well. Bottom right : The diagonal nets of f+f^{+} and f−f^{-} have congruent (non-planar) faces. The checkerboard patterns define a discrete rolling motion and hence a snapping quad net of snapping four-bars.
Figure 8: Illustration for the derivation of Equation ( 8 ).
Figure 9: CAD renderings of a prototype snapping four-bar net in its two stable configurations. Numbers printed on the link parts are helpful during manual assembly.
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