Automation in Construction 122 (2021) 103488 0926-5805/© 2020 Elsevier B.V. All rights reserved. The convergence surface method for the design of deployable scissor structures Carlos J. García-Mora * , Jose S´ anchez-S´ anchez Department of Building Structures and Geotechnical Engineering, University of Seville, Spain A R T I C L E INFO Keywords: Geometry Deployable structure Scissor Mechanism Folding Kinematics Bistable ABSTRACT In this paper, the operability of the most recent method to design bistable and non-bistable deployable scissors structures (the method of the convergence surface) is extended and this operability will be divided into two types of formulas: The exact formula and the approximate formulas. The exact formula involves the obtaining of the convergence surface using its own equations and this paper will prove that this surface is a triaxial two-leaf hyperboloid (for translational units) and a non-standard surface (for polar units). On the other hand, the approximate formulas are designed due to the need to obtain the convergence surface when the exact formula cannot be used. Finally, this research will demonstrate the potential of these approximate strategies to compete against the exact formulas and to boost an improvement in the mathematical results in terms of precision in the scissor design and speed in the calculation process. 1. Introduction Historically, the frst author who designed a deployable structure with straight scissors was Leonardo Da Vinci [1]. This inventor devel- oped a scissor system that was controlled using a screw and it was used to lift a weight (Fig. 1). Although this structure was quite simple, this was the frst deployable structure with straight scissors and to design this mechanism Leonardo used design concepts of symmetry or geo- metric compatibility. Many centuries later, Emilio P´ erez Pi˜ nero (1935–1972) revolution- ized the realm of the deployable systems with his system of “Triaspas and Tetraspas” (Three-scissors Four-scissors) and [2,3] and with such innovative projects as the famous “Teatro Ambulante” (Mobile Theater) [4–7] or the order done by Salvador Dalí to design the “Vidriera Hipercúbica” (Hypercubic Stained Glass) (Fig. 2). The difference between the Three-scissors and the Four-scissors in comparison with the straight scissors is quite considerable. However, Emilio P´ erez Pi˜ nero designed an effective design method that allowed him to create deployable structures in a relative short time period (without using any computer tools). Some years later, in the 1990s, a new generation of authors began the extension of this method with the creation of new technologies: The straight scissors, whose main designer was F´ elix Escrig Pallar´ es (1950–2013) [8–10], and the angulated scissors, whose main designer was Chuck Hoberman (1956) [11,12] (The study of the angulated scis- sors is beyond the topic of this paper). The design methods that were later developed using the studies of F´ elix Escrig have the goal of obtaining a compactness level of 100%. This situation means that in the next scissors structure (Fig. 3): The following equations must be satisfed: k 1 + k 2 = k 3 + k 4 (1) k 5 + k 6 = k 7 + k 8 (2) k 9 + k 10 = k 11 + k 12 (3) This condition can be summarized in: k i1 + k i2 = k j1 + k j2 (4) Another important aspect of this kind of mechanisms is the degrees of freedoms. The deployable structure will have 1 degree of freedom if the joints are defned using a point. However, if joints are defned with a displacement of each scissor the number of degrees of freedoms will be infnite (this situation could be avoid by blocking some axes of rota- tions). The study of the degrees of freedoms has been developed by Alexey Fomin in [13,14]. Abbreviations: CE, convergence ellipsoid. * Corresponding author. E-mail addresses: email@carlosjosegarciamora.com (C.J. García-Mora), josess@us.es (J. S´ anchez-S´ anchez). Contents lists available at ScienceDirect Automation in Construction journal homepage: www.elsevier.com/locate/autcon https://doi.org/10.1016/j.autcon.2020.103488 Received 24 May 2020; Received in revised form 30 September 2020; Accepted 8 October 2020