The convergence surface method for the design of deployable scissor structures

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is an accepted manuscript of a paper published in: Automation in Construction (vol. 122): February 2021 DOI: https://doi.org/10.1016/j.autcon.2020.103488 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This is an Accepted Manuscript of an article published by Elsevier in Automation in Construction on February 2021, available at: https://doi.org/10.1016/j.autcon.2020.103488” The convergence surface method for the design of 1 deployable scissor structures 2 3 4 Abstract 5 6 In this paper, the operability of the most recent method to design bistable and non-bistable 7 deployable scissors structures (the method of the convergence surface) is extended and this 8 operability will be divided into two types of formulas: The exact formula and the approximate 9 formulas. The exact formula involves the obtaining of the convergence surface using its own 10 equations and this paper will prove that this surface is a triaxial two-leaf hyperboloid (for 11 translational units) and a non-standard surface (for polar units). On the other hand, the 12 approximate formulas are designed due to the need to obtain the convergence surface when 13 the exact formula cannot be used. Finally, this research will demonstrate the potential of these 14 approximate strategies to compete against the exact formulas and to boost an improvement in 15 the mathematical results in terms of precision in the scissor design and speed in the 16 calculation process. 17 18 19 20 Keywords Geometry; Deployable structure; Scissor; Mechanism; Folding; Kinematics; Bistable 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Abbreviations 41 In this paper, the notation "CE" will sometimes be used due to the abbreviation (CE = 42 Convergence ellipsoid). 43 1. Introduction 44 45 Historically, the first author who designed a deployable structure with straight scissors was 46 Leonardo Da Vinci [1]. This inventor developed a scissor system that was controlled using a 47 screw and it was used to lift a weight (Fig. 1). Although this structure was quite simple, this 48 was the first deployable structure with straight scissors and to design this mechanism 49 Leonardo used design concepts of symmetry or geometric compatibility. 50 51 Fig. 1. Original drawings of the Leonardo Da Vinci design and three dimensional perspective of the mechanism [1]. 52 Many centuries later, Emilio Pérez Piñero (1935 – 1972) revolutionized the realm of the 53 deployable systems with his system of “Triaspas and Tetraspas” (Three-scissors Four-scissors) 54 and [2] [3] and with such innovative projects as the famous “Teatro Ambulante” (Mobile 55 Theater) [4] [5] [6] [7] or the order done by Salvador Dalí to design the “Vidriera Hipercúbica” 56 (Hypercubic Stained Glass) (Fig. 2). 57 58 Fig. 2. (Left) Deployable structure with a doublé curvature for a Mobile Theater (Right) Emilio Pérez Piñero 59 presenting the Hypercubic Stained Glass to Salvador Dalí. 60 The difference between the Three-scissors and the Four-scissors in comparison with the 61 straight scissors is quite considerable. However, Emilio Pérez Piñero designed an effective 62 design method that allowed him to create deployable structures in a relative short time period 63 (without using any computer tools). 64 65 Some years later, in the 1990s, a new generation of authors began the extension of this 66 method with the creation of new technologies: The straight scissors, whose main designer was 67 Félix Escrig Pallarés (1950 - 2013) [8] [9] [10], and the angulated scissors, whose main designer 68 was Chuck Hoberman (1956) [11] [12] (The study of the angulated scissors is beyond the topic 69 of this paper). 70 71 The design methods that were later developed using the studies of Félix Escrig have the goal of 72 obtaining a compactness level of 100%. This situation means that in the next scissors structure 73 (Fig. 3): 74 75 76 77 Fig. 3. Deployable structure composed with some straight scissors. 78 The following equations must be satisfied: 79 80 𝑘1+𝑘2=𝑘3+𝑘4 (1) 81 𝑘5+𝑘6=𝑘7+𝑘8 (2) 82 𝑘9+𝑘10=𝑘11+𝑘12 (3) 83 This condition can be summarized in: 84 85 𝑘𝑖1+𝑘𝑖2=𝑘𝑗1+𝑘𝑗2 (4) 86 Another important aspect of this kind of mechanisms is the degrees of freedoms. The 87 deployable structure will have 1 degree of freedom if the joints are defined using a point. 88 However, if joints are defined with a displacement of each scissor the number of degrees of 89 freedoms will be infinite (this situation could be avoid by blocking some axes of rotations). The 90 study of the degrees of freedoms has been developed by Alexey Fomin in [13] and [14]. 91 92 It is important to highlight that the previous information is only focused on the fulfilment of 93 geometric conditions to guarantee a full deployment process. However, a different way to 94 study this type of structures is from the energy point of view and, in consequence, two groups 95 can be identified: bistable and non-bistable deployable structures. Basically, the main 96 difference between both is the existence of geometric incompatibilities during the deployment 97 process: 98 99 - Bistable structure: the rods are going to have elongations and reductions of the length due to 100 a geometric incompatibility and, in consequence, the structure will accumulate energy of 101 deformation during a part of the deployment process. This energy of deformation will be null 102 in two positions of the deployment process (the folded position and the unfolded position) 103 because the rods will not have variations of the length in these two cases. An example of a 104 bistable deployable structure with the evolution of the energy of deformation has been 105 represented in Figure 4 and more information about the optimisation of these structures can 106 be found in [15] [16]. 107 108 109 110 111 Fig. 4. (Left) Evolution of the energy of deformation in a spherical deployable structure (Right) Deployment process. 112 - Non-bistable structure: the rods will not have a deformation of the length during the 113 deployment process and, in consequence, the energy of deformation will be null if the weight 114 of the structure is not considered. 115 116 Thereby, a design method to create these deployable structures (bistable and non bidtables) is 117 required. Firstly, this research will develop a brief description of the most important methods 118 to obtain deployable structures with straight scissors and the advantages and disadvantages of 119 each one will be underlined. This section will be called as “Previous methods to design 120 deployable scissor structures”. 121 122 The last method that will be introduced in this section is the Method of the convergence 123 surface, which is not only quite useful but also complex. The research works that have been 124 proposed previously only develop basic equations and general ideas about the practical 125 applications but they don’t consider important factors such as the automation of the method 126 using an algorithm or the obtaining of approximate solutions where the loss of resolution is 127 not important. 128 129 To improve and to enhance this situation, this research will develop the whole formulas that 130 guarantee the operability and the practical use of the method of the convergence surface. 131 These formulas can be divided into two groups: 132 133 - Exact formula: This process is based on the obtaining of the convergence surface equation. 134 The paper will demonstrate that this equation is a two-leaf hyperboloid in case of translational 135 units and there is not a standard geometric shape in case of polar units. The exact formula is 136 useful when the designer wants a high level in the resolution of the scissor length. 137 138 - Approximate formulas: This section is the most important contribution of this research. The 139 exact formula is important from a curious point of view but a calculation program must do a 140 huge work, in terms of calculation, to obtain the convergence surface. The goal of the 141 approximate formulas is to reduce and to simplify the obtaining of the convergence surface in 142 exchange of a little loss of resolution. This research will develop two approximate formulas, 143 the first one is focused on the resolution of a determinant and the second one is based on the 144 creation of the convergence surface using level curves. 145 146 The use of these formulas allows the immediate application of external programs to obtain the 147 convergence surface in a short time. Once this surface is created, all geometric solutions in the 148 space that satisfy Eq. (4) are available. 149 2. Previous methods to design deployable scissor structures 150 151 2.1. Method of the spheres (1990) (Félix Escrig Pallarés, Jose Sánchez Sánchez and Juan Pérez 152 Valcárcel): 153 154 This method [17] [18] is based on the assumption that the end of each scissor has an 155 associated sphere (with radius r) with a center at the midpoint of its focal length (focal length = 156 c). The cut point of the scissors will be in the tangency between two spheres. In the case of a 157 cylindrical surface, it is possible to work with circles instead of spheres due to the parallelism 158 between planes (Fig. 5). 159 160 161 162 Fig. 5. Deployable structure of a circumference arc with the spheres method (Magenta curve = Curve to design as 163 deployable; Orange curve = Convergence curve; Black lines = Scissors). 164 a) Advantages of the method: 165 166 - The value of the focal distance is independent of the value of the radius of the sphere. 167 - It is a very simple method because the designer only need to copy and paste the same 168 spherical module and the focal distance of each scissor will be in the intersection point 169 between the spherical modules and the original surface (surface that is going to be converted 170 to a scissor mechanism). 171 - There is little previous knowledge required to apply this method because the designer only 172 needs the commands “sphere”, “intersect”, “copy-paste” and “line” from a graphic program. 173 Therefore, people who have never designed a deployable structure should begin with this 174 method. 175 176 b) Disadvantages of the method: 177 178 - The sphere and the focal distance associated with the end of each rod must be the same in 179 the complete structure. Otherwise, the structure will not fold entirely. 180 - It can only be applied in the following cases (the use of this method on any other surface not 181 mentioned below will cause a non-complete deployment process): 182 * Flat surfaces. 183 * Straight extrusion surfaces with a circular guideline (cylinders with a circular base 184 and with the focal distance oriented to the center of each scissor plane). 185 * Spherical surfaces (With the focal distance oriented to the center of the sphere). 186 - The control of the relative position between the structure and the surface that is going to be 187 designed as deployable is not allowed because the middle point of the focal distance will 188 always be in this surface. 189 - The position of a scissor in a singular point (boundaries of the surface, supports of the 190 surface, etc.) is not allowed and, in consequence, the tessellation cannot be regulated because 191 for each pair of blades there is a unique mathematical solution. 192 193 2.2. Method of constant ellipsoids (1996) (Luis Sánchez Cuenca): 194 195 The constant ellipsoid method [19] [20] has been created as a consequence of the limited 196 cases where the sphere method can be used. This new method considers that the end of each 197 scissor has an associated ellipsoid with a circular revolution (a = major axis, b = minor axis, c = 198 focal distance), where the focal distance of the ellipsoid coincides with the focal distance of 199 the end of the scissor. To get a complete deployable process of the structure, the cross point 200 between the two rods of each scissor must belong to the surface of the ellipsoid. This method 201 uses the property of the ellipse (Eq. (5)) (Fig. 6) to satisfy Eq. (4). 202 203 204 Fig. 6. Fundamental property of the ellipse. 205 𝑓1+𝑓2=𝑓3+𝑓4=2∙𝑎 (5) 206 The ellipsoids must be constant because Luis Sánchez Cuenca only used the same ellipsoid 207 which was repeated on the entire surface. An example of an application to a curve can be seen 208 in Fig. 7 (in the case of a surface, the method is used with the same strategy but using two 209 ellipsoids simultaneously). 210 211 212 Fig. 7. Deployable structure of a random curve with the constant ellipsoids method (Magenta curve = Curve to 213 design as deployable; Orange curve = Convergence curve; Black lines = Scissors). 214 215 a) Advantages of the method: 216 217 - It can be applied to any surface because the non-constant curvature of the ellipsoid allows 218 the adaptation on any shape. 219 - As in the previous case, it is a very simple method to use: just copy and paste the same 220 ellipsoid and the middle point of each scissor will be in the intersection points. 221 - Little previous knowledge is required to apply this method. The disadvantage that can be 222 found here in comparison with the previous method is the use of ellipsoids instead of spheres 223 (There are design programs where the command ellipsoid is not available). 224 225 b) Disadvantages of the method: 226 227 - The ellipsoid associated with the end of each rod must be the same in the complete 228 structure. Otherwise, the structure will not fold entirely. 229 - The position of a scissor in a singular point (boundaries of the surface, supports of the 230 surface, etc.) is not allowed and, in consequence, the tessellation cannot be regulated because 231 for each pair of blades there is a unique mathematical solution. 232 - The control of the relative position between the structure and the surface that is going to be 233 designed as deployable is not allowed because the middle point of the focal distance will 234 always be in this surface. 235 - Only translational units can be used (the focal distance of all ellipsoids is parallel) and, in 236 consequence, the size of the structure in the folded position will be bigger in comparison with 237 the use of polar units. 238 239 2.3. Method of proportional ellipsoids (2017) (Niels De Temmerman and Kelvin Roovers): 240 241 An extension of the constant ellipsoid method is the proportional ellipsoid method [21] [22] 242 [23]. With this new design system, the tessellation of the structure can be regulated by the 243 designer (different mathematical solutions are obtained when changing the proportionality 244 constant). Like its predecessor, this method can only be used for translational units. The 245 reason for this situation is, as is going to be demonstrated later in the convergence surface 246 method, if the units are translational it is mandatory that all the ellipsoids have to be 247 proportional. An example of an application to a curve can be seen in Fig. 8 (in the case of a 248 surface, the method is used with the same strategy but using two ellipsoids simultaneously). 249 250 251 252 Fig. 8. Deployable structure of a random curve with the proportional ellipsoids method (Magenta curve = Curve to 253 design as deployable; Orange curve = Convergence curve; Black lines = Scissors). 254 a) Advantages of the method: 255 256 - It can be applied to any surface because the non-constant curvature of the ellipsoid allows 257 the adaptation on any shape. 258 - It is a very simple method to use. As in the previous cases, the designer has to copy and paste 259 an ellipsoid but having in mind that he/she can change the size of this ellipsoid using a 260 constant of proportionality. 261 - Little previous knowledge is required to apply this method. As in the previous case, the 262 graphic program must have the command “ellipsoid” to use this method. 263 - The tessellation can be controlled: the variation of the constant of proportionality allows the 264 change of the size of the ellipsoid and, in consequence, more than one mathematical solution 265 can be obtained for one input. 266 267 b) Disadvantages of the method: 268 269 - The control of the relative position between the structure and the surface that is going to be 270 designed as deployable is not allowed because the middle point of the focal distance will 271 always be in this surface. 272 - Only translational units can be used (the focal distance of all ellipsoids is parallel) and, in 273 consequence, the size of the structure in the folded position will be bigger in comparison with 274 the use of polar units. 275 - The position of a scissor in a singular point (boundaries of the surface, supports of the 276 surface, etc.) must be done using an iterative process and the final solution will never belong 277 to the boundary of the surface, the support of the surface, etc. The reason for this situation is 278 that this method only gives a unique mathematical solution for each proportionality constant. 279 280 281 282 516 517 Fig. 10. Minimum geometric convergence situation for two ellipsoids in the space with u = 0. 518 519 After this change has been done, Eq. (28) is: 520 521 [𝑧11−𝑙−𝑧𝑜 𝑎12∙(1+𝑐2 𝑐12)]2+[ 𝑥11−𝑥𝑜 𝑏12∙(1+𝑐2 𝑐12)]2+[ 𝑦11 𝑏12∙(1+𝑐2 𝑐12)]2=1 (28) 522 Consequently, and in function of the “l” value, the most important situations that can be found 523 are: 524 525 3.1.1. l = Constant with l ≠ -c2 and l ≠ c2 526 527 If c2 is isolated in Eq. (25) and in Eq. (28): 528 529 𝑐2=𝑐11∙[√(𝑧11−𝑙 𝑎11 )2+(𝑥11 𝑏11)2+(𝑦11 𝑏11)2−1] (29) 530 𝑐2=𝑐12∙[√(𝑧11−𝑙−𝑧𝑜 𝑎12 )2+(𝑥11−𝑥𝑜 𝑏12 )2+(𝑦11 𝑏12)2−1] (30) 531 In addition, ellipsoids 11 and 12 must be proportional. In consequence: 532 533 𝑐12 𝑐11=𝑎12 𝑎11=𝑏12 𝑏11=𝑘 (31) 534 Then, if Eq. (31) is replaced in Eq. (30) and after that Eq. (29) is equaled with Eq. (30): 535 536 𝑘1∙𝑥11 2+𝑘2∙𝑦11 2+𝑘3∙𝑧11 2+𝑘4∙𝑥11∙𝑧11+𝑘5∙𝑥11+𝑘6∙𝑧11=𝑘7 (32) Where: 537 538 𝑘1=[ 𝑥0 𝑏11 2∙(1−𝑘)]2−1 𝑏11 2 (33) 539 𝑘2=−1 𝑏11 2 (34) 540 𝑘3=[ 𝑧0 𝑎11 2∙(1−𝑘)]2−1 𝑎11 2 (35) 541 𝑘4=2∙𝑥𝑜∙𝑧𝑜 𝑎11 2∙𝑏11 2∙(1−𝑘)2 (36) 542 𝑘5=2∙𝑥𝑜 𝑏11 2∙[1− 𝛿 1−𝑘] (37) 543 𝑘6=2 𝑎11 2∙[𝑙+𝑧𝑜−𝛿∙𝑧𝑜 1−𝑘] (38) 544 𝑘7=(𝑥𝑜 𝑏11)2+(𝑙+𝑧𝑜 𝑎11 )2−𝛿2 (39) 545 Where: 546 547 𝛿= 1 2∙(1−𝑘)∙[(𝑥𝑜 𝑏11)2+(𝑧𝑜 𝑎11)2+2∙𝑙∙𝑧𝑜 𝑎11 2+(1−𝑘)2] (40) 548 As can be seen, Eq. (32) is the equation of a revolutionized conic whose axis is not parallel with 549 respect to the axes of the reference system due to the term x11 · z11 550 To eliminate this rotation, a change of reference system must be done. The goal is to obtain 551 the value of the angle "ƞ" that guarantees the para parallelism between X’11 and the axis of the 552 hyperboloid. Consequently: 553 554 𝑥11=𝑥11 ′∙𝑐𝑜𝑠(𝜂)−𝑧11 ′∙𝑠𝑖𝑛(𝜂) (41) 555 𝑧11=𝑥11 ′∙𝑠𝑖𝑛(𝜂)+𝑧11 ′∙𝑐𝑜𝑠(𝜂) (42) 556 𝑦11=𝑦11 ′ (43) 557 The parallelism between X’11 and the axis of the hyperboloid is going to be obtained for the 558 value of “ƞ” that makes: X'11 · Y’11 = 0. Then: 559 560 −2∙𝑘1∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)−𝑘4∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠2(𝜂)+2∙𝑘3∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)=0 (44) 561 Using the equations of the double angle for the sine and the cosine: 562 563 𝑘3−𝑘1 𝑘4=𝑠𝑖𝑛2(𝜂)−𝑐𝑜𝑠2(𝜂) 2∙𝑠𝑖𝑛(𝜂)∙𝑐𝑜𝑠(𝜂)=−𝑐𝑜𝑠(2∙𝜂) 𝑠𝑖𝑛(2∙𝜂)=− 1 𝑡𝑎𝑛(2∙𝜂) (45) 564 Consequently: 565 566 𝜂=12∙𝑡𝑎𝑛−1(𝑘4 𝑘1−𝑘3) (46) 567 If Eq. (41), Eq. (42), Eq. (43) and Eq. (46) are replaced in Eq. (32): 568 569 [𝑘1∙𝑐𝑜𝑠2(𝜂)+𝑘3∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]∙𝑥11 ′2+𝑘2∙𝑦11 ′2+ +[𝑘1∙𝑠𝑖𝑛2(𝜂)+𝑘3∙𝑐𝑜𝑠2(𝜂)−𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]∙𝑧11 ′2+ +[𝑘5∙𝑐𝑜𝑠(𝜂)+𝑘6∙𝑠𝑖𝑛(𝜂)]∙𝑥11 ′+[𝑘6∙𝑐𝑜𝑠(𝜂)−𝑘5∙𝑠𝑖𝑛(𝜂)]∙𝑧11 ′=𝑘7 (47) 570 The next step is to balance Eq. (47). If this process is done using the isolation of two auxiliary 571 variables and these variables are added to Eq. (47) in a square form, the final equation is 572 obtained: Eq. 51 573 574 [𝑥11 ′−𝑑1 𝑑2]2−[𝑦11 ′ 𝑑3]2−[𝑧11 ′−𝑑4 𝑑5]2=1 (48) 575 Where: 576 577 𝑑1=− 𝑘5∙𝑐𝑜𝑠(𝜂)+𝑘6∙𝑠𝑖𝑛(𝜂) 2∙[𝑘1∙𝑐𝑜𝑠2(𝜂)+𝑘3∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)] (49) 578 𝑑2= √ [1 𝑘1∙𝑐𝑜𝑠2(𝜂)+𝑘3∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]∙ ∙[𝑘7+[𝑘5∙𝑐𝑜𝑠(𝜂)+𝑘6∙𝑠𝑖𝑛(𝜂)]2 4∙[𝑘1∙𝑐𝑜𝑠2(𝜂)+𝑘3∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]+ +[𝑘6∙𝑐𝑜𝑠(𝜂)−𝑘5∙𝑠𝑖𝑛(𝜂)]2 4∙[𝑘1∙𝑠𝑖𝑛2(𝜂)+𝑘3∙𝑐𝑜𝑠2(𝜂)−𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]] (50) 579 𝑑3= √ −1 𝑘2∙[𝑘7+[𝑘5∙𝑐𝑜𝑠(𝜂)+𝑘6∙𝑠𝑖𝑛(𝜂)]2 4∙[𝑘1∙𝑐𝑜𝑠2(𝜂)+𝑘3∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]+ +[𝑘6∙𝑐𝑜𝑠(𝜂)−𝑘5∙𝑠𝑖𝑛(𝜂)]2 4∙[𝑘1∙𝑠𝑖𝑛2(𝜂)+𝑘3∙𝑐𝑜𝑠2(𝜂)−𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]] (51) 580 𝑑4=− 𝑘6∙𝑐𝑜𝑠(𝜂)−𝑘5∙𝑠𝑖𝑛(𝜂) 2∙[𝑘1∙𝑠𝑖𝑛2(𝜂)+𝑘3∙𝑐𝑜𝑠2(𝜂)−𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)] (52) 581 𝑑5= √ −[ 1 𝑘1∙𝑠𝑖𝑛2(𝜂)+𝑘3∙𝑐𝑜𝑠2(𝜂)−𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]∙ ∙[𝑘7+[𝑘5∙𝑐𝑜𝑠(𝜂)+𝑘6∙𝑠𝑖𝑛(𝜂)]2 4∙[𝑘1∙𝑐𝑜𝑠2(𝜂)+𝑘3∙𝑠𝑖𝑛2(𝜂)+𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]+ +[𝑘6∙𝑐𝑜𝑠(𝜂)−𝑘5∙𝑠𝑖𝑛(𝜂)]2 4∙[𝑘1∙𝑠𝑖𝑛2(𝜂)+𝑘3∙𝑐𝑜𝑠2(𝜂)−𝑘4∙𝑐𝑜𝑠(𝜂)∙𝑠𝑖𝑛(𝜂)]] (53) 582 Eq. (48) is the equation of a triaxial two-leaf hyperboloid. Likewise, if "l" were 0, the axis of the 583 convergence surface would pass through the midpoint of the line that joins the centers of the 584 two ellipsoids and the vertex of this hyperboloid would be in this point. 585 586 Another important aspect is that the “ƞ” angle is an angle with orientation. Two situations can 587 be defined: 588 - If the “ƞ” value is positive, the orientation of this angle is counter-clockwise, taking the X11 589 axis as the origin of the angle and taking the X’11 axis as the end of the angle. 590 - If the “ƞ” value is negative, the orientation of this angle is clockwise, taking the X11 axis as the 591 origin of the angle and taking the X’11 axis as the end of the angle. 592 593 3.1.2. l = -c2 594 595 Eq. (32) will be obtained if the previous process is repeated with this value of “l” parameter, 596 but, in this case, the values of ki are going to have the following equations: 597 598 𝑘1=[ 𝑎11∙𝑥𝑜 𝑎11 2∙(𝑘−1)+𝑐11∙𝑧𝑜]2−1 (54) 599 𝑘2=−1 (55) 600 𝑘3=[𝑎11∙[𝑐11∙(𝑘−1)+𝑧𝑜] 𝑎11 2∙(𝑘−1)+𝑐11∙𝑧𝑜]2−1 (56) 601 𝑘4=2∙𝑥𝑜∙𝑎11 2∙[𝑐11∙(𝑘−1)+𝑧𝑜] [𝑎11 2∙(𝑘−1)+𝑐11∙𝑧𝑜]2 (57) 602 𝑘5=2∙𝑥𝑜∙[1− 𝑎11∙𝛿 𝑎11 2∙(𝑘−1)+𝑐11∙𝑧𝑜] (58) 603 𝑘6=2∙[𝑧𝑜+𝑘∙𝑐11−𝑎11∙𝛿∙[𝑐11∙(𝑘−1)+𝑧𝑜] 𝑎11 2∙(𝑘−1)+𝑐11∙𝑧𝑜] (59) 604 𝑘7=𝑥𝑜2+(𝑧𝑜+𝑘∙𝑐11)2−𝛿2 (60) 605 Where: 606 607 𝛿=𝑎11∙[𝑥𝑜2−𝑐11 2+(𝑧𝑜+𝑘∙𝑐11)2] 2∙[𝑎11 2∙(𝑘−1)+𝑐11∙𝑧𝑜]+12∙[𝑎11∙(𝑘−1)+𝑐11∙𝑧𝑜 𝑎11 ] (61) 608 3.1.3. l = c2 609 610 Doing the same steps that have been developed in the previous section, the values of ki will 611 be: 612 613 𝑘1=[ 𝑎11∙𝑥𝑜 𝑎11 2∙(𝑘−1)−𝑐11∙𝑧𝑜]2−1 (62) 614 𝑘2=−1 (63) 615 𝑘3=[𝑎11∙[𝑐11∙(𝑘−1)−𝑧𝑜] 𝑎11 2∙(𝑘−1)−𝑐11∙𝑧𝑜]2−1 (64) 616 𝑘4=−2∙𝑥𝑜∙𝑎11 2∙[𝑐11∙(𝑘−1)−𝑧𝑜] [𝑎11 2∙(𝑘−1)−𝑐11∙𝑧𝑜]2 (65) 617 𝑘5=2∙𝑥𝑜∙[1+ 𝑎11∙𝛿 𝑎11 2∙(𝑘−1)−𝑐11∙𝑧𝑜] (66) 618 𝑘6=2∙[𝑧𝑜−𝑘∙𝑐11−𝑎11∙𝛿∙[𝑐11∙(𝑘−1)−𝑧𝑜] 𝑎11 2∙(𝑘−1)−𝑐11∙𝑧𝑜] (67) 619 𝑘7=𝑥𝑜2+(𝑧𝑜−𝑘∙𝑐11)2−𝛿2 (68) 620 Where: 621 622 𝛿=𝑎11∙[𝑥𝑜2−𝑐11 2+(𝑧𝑜−𝑘∙𝑐11)2] 2∙[𝑐11∙𝑧𝑜−𝑎11 2∙(𝑘−1)]+12∙[𝑐11∙𝑧𝑜 𝑎11 −𝑎11∙(𝑘−1)] (69) 623 3.2. For u ≠ 0 (The axes of all the ellipsoids are not parallel) 624 625 If in Eq. (6) the “u” variable is isolated, a full 4-degree equation is obtained. Only one of these 4 626 values is going to be the correct one. To get this correct value the Descartes Method is going to 627 be used: 628 629 a) The 4o degree term is modified with a multiplication by 1. 630 b) The 3o degree term is eliminated from the equation with a change of a variable. 631 c) The previous expression is factored to obtain a 6o degree equation with only even powers. 632 d) A variable change is made to convert the 6o degree equation from the previous section to a 633 3o degree grade equation with the full terms. 634 e) The 3o degree equation is solved using the Cardano Method. 635 f) With the results of this 3o degree equation, the calculation process must go back to find the 636 solutions of the full 4-degree equation. 637 638 Once the value of “u” is isolated, this process must also be done with Eq. (7) in combination 639 with Eq. (27) and, after that, the equation of the convergence surface will be obtained if the 640 previous two equations are equalized. The final result will be: 641 642 - A huge equation with an extension of more than 200 pages (this equation is not operative). 643 644 - The shape of the convergence surface is similar in comparison with the shape of a triaxial 645 two-leaf hyperboloid: the lower the angle between the ellipsoids (close to translational units), 646 the better this approximation. This is the basis of the approximate formula 1. 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 4. Approximate formula 1 of the convergence surface: Numerical approximation 665 666 The approximate formula 1 can be used in the following cases: 667 668 - When the designer is working with translational units but he/she does not want to use the 669 analytic equation. 670 - When the designer is working with polar units and he/she wants to approximate the 671 convergence surface to a triaxial two-leaf hyperboloid (this is only recommended if the angle 672 between the ellipsoids is small = less than 10o). 673 674 The goal of this formula is to obtain 6 random points that belong to the convergence surface 675 using the intersection of convergence ellipsoids with the same “u” value (The relationship 676 between these points cannot be a lineal combination). The next step is the calculation of a 677 determinant to obtain the ki parameters. Once these parameters are calculated, the values of 678 the di parameters are automatic using Eq. (48), Eq. (49), Eq. (50), Eq. (51), Eq. (52), Eq. (53). 679 680 To obtain the value of ki parameters, a system with 7 equations and 7 variables must be 681 solved: Eq. (70) 682 ( 𝑥111 2 𝑦111 2 𝑧111 2 𝑥111∙𝑧111 𝑥111 𝑧111 −1 𝑥112 2 𝑦112 2 𝑧112 2 𝑥112∙𝑧112 𝑥112 𝑧112 −1 𝑥113 2 𝑦113 2 𝑧113 2 𝑥113∙𝑧113 𝑥113 𝑧113 −1 𝑥114 2 𝑦114 2 𝑧114 2 𝑥114∙𝑧114 𝑥114 𝑧114 −1 𝑥115 2 𝑦115 2 𝑧115 2 𝑥115∙𝑧115 𝑥115 𝑧115 −1 𝑥116 2 𝑦116 2 𝑧116 2 𝑥116∙𝑧116 𝑥116 𝑧116 −1 𝑥117 2 𝑦117 2 𝑧117 2 𝑥117∙𝑧117 𝑥117 𝑧117 −1 ) ∙ ( 𝑘1 𝑘2 𝑘3 𝑘4 𝑘5 𝑘6 𝑘7 ) =0 (70) 683 However, the previous system just gives the trivial solution. To avoid this situation, the next 684 determinant must be solved: Eq. (71) 685 | | | 𝑥11 2 𝑦11 2 𝑧11 2 𝑥11∙𝑧11 𝑥11 𝑧11 1 𝑥111 2 𝑦111 2 𝑧111 2 𝑥111∙𝑧111 𝑥111 𝑧111 1 𝑥112 2 𝑦112 2 𝑧112 2 𝑥112∙𝑧112 𝑥112 𝑧112 1 𝑥113 2 𝑦113 2 𝑧113 2 𝑥113∙𝑧113 𝑥113 𝑧113 1 𝑥114 2 𝑦114 2 𝑧114 2 𝑥114∙𝑧114 𝑥114 𝑧114 1 𝑥115 2 𝑦115 2 𝑧115 2 𝑥115∙𝑧115 𝑥115 𝑧115 1 𝑥116 2 𝑦116 2 𝑧116 2 𝑥116∙𝑧116 𝑥116 𝑧116 1 | | | =0 (71) 686 If the previous determinant is calculated, a polynomial equation is going to be obtained and ki 687 parameters can be solved using the comparison with Eq. (32). 688 The last step is to calculate di parameters to get the equation of the hyperboloid. With all the 689 terms of the hyperboloid, its geometry can be represented using a graphic program. 690 691 692 693 694 5. Approximate formula 2 of the convergence surface: Geometrical approximation 695 696 The goal of the approximate formula 2 is to make the proposed method in [24] operative: to 697 find the interval of the “u” parameter where the length of the rods has a positive value. 698 699 To achieve this goal, the equation of the convergence ellipsoid 11 is going to be used (the 700 results are the same if the equation of the convergence ellipsoid 12 is used instead of that of 701 ellipsoid 11). 702 703 Likewise, if translational units are used, the limits will be those of the c2 parameter and if polar 704 units are used, the limits will be those of the “u” parameter. 705 706 5.1. Maximum value of the iteration interval: Maximum growth of the convergence surface 707 708 5.1.1. For u = 0 (the axes of the ellipsoids are parallel or translational units): 709 The maximum value of c2 will cause the semi-axes of the convergence ellipsoid 11 to grow to 710 ∞. If this situation is graphed, the following figure is obtained (Fig. 11) where the value of c2 711 has been plotted against the semi-major axis of the convergence ellipsoid 11, keeping the 712 values of a11, c11 constant. 713 714 Fig. 11. Evolution of the semi-major axis of the convergence ellipsoid 11 as a function of c2 value. 715 716 Consequently, using this curve the value of c2max can be obtained with Eq. (72): 717 718 𝑐2=𝑐2𝑚𝑎𝑥 𝑖𝑓 𝑙𝑖𝑚 𝑎11∙(1+𝑐2 𝑐11)→∞[𝑎11∙(1+𝑐2 𝑐11)] 𝑎𝑛𝑑 𝑐2=𝑐2𝑚𝑎𝑥 𝑖𝑓 𝑙𝑖𝑚 𝑏11∙(1+𝑐2 𝑐11)→∞[𝑏11∙(1+𝑐2 𝑐11)] (72) 719 This equation implies that: 720 𝑎11∙(1+𝑐2𝑚𝑎𝑥 𝑐11 )=𝑏11∙(1+𝑐2𝑚𝑎𝑥 𝑐11 )=∞ → 𝑐2𝑚𝑎𝑥=∞ (73) 721 The reader can notice that this result is valid for -ꚙ < l < ꚙ because the expression of the 722 semi-major axis and the semi-minor axis do not depend on the parameter “l”. 723 724 Furthermore, the displacement of the convergence ellipsoid 11 for c2 = c2max will be the value 725 of "l" because this function does not depend on the value of c2. 726 727 5.1.2. For u ≠ 0 (the axes of the ellipsoids are not parallel or polar units) 728 a) If - ∞ ≤ v ≤ ∞ with v ≠ u and with v ≠ -u 729 The maximum value of “u” will cause the semi-axes of the convergence ellipsoid 11 to grow to 730 ∞. This can be seen in the following graph where the value of “u” has been plotted against the 731 semi-major axis of the convergence ellipsoid 11, keeping the values of a11, c11 and h1 constant 732 and varying the value of the parameter “v” (Fig. 12). 733 734 Fig. 12. Evolution of the semi-major axis of the convergence ellipsoid 11 as a function of the value of “u” for - ∞ ≤ v 735 ≤ ∞ with v ≠ u and with v ≠ -u. 736 737 Three curves can be differentiated from the previous graph: 738 739 - The curve on the left is negligible because “u” can never have negative values. 740 - The center curve is the only one that can give coherent values of "u" and of the semi-major 741 axis of the ellipsoid. 742 - The curve on the right is also negligible because its interval is composed of negative values of 743 the semi-major axis of the ellipsoid. 744 745 Consequently, the analysis is going to be focused on the center curve. In this curve there is a 746 value of “u” where the axes of the convergence ellipsoid 11 grow to ∞ (This curve is 747 asymptotic). Mathematically this can be represented with the following equation Eq. (74). 748 𝑢=𝑢𝑚𝑎𝑥 𝑠𝑖 𝑙𝑖𝑚 𝑓21→∞[𝑓21]=𝑙𝑖𝑚 𝑓31→∞[𝑓31] (74) Consequently: 749 [𝑎11 (1+𝑣)2−𝑢𝑚𝑎𝑥 2]∙[1+𝑣+𝑢𝑚𝑎𝑥∙ℎ1 𝑐11 ]=∞ 𝑎𝑛𝑑 [ 𝑏11 (1+𝑣)2−𝑢𝑚𝑎𝑥 2]∙[1+𝑣+𝑢𝑚𝑎𝑥∙ℎ1 𝑐11 ]=∞ (75) 750 If the value of umax is isolated: 751 𝑢𝑚𝑎𝑥=|1+𝑣| (76) 752 The previous expression must have an absolute value because umax can only have positive 753 values. 754 755 The displacement of the convergence ellipsoid 11 for u = umax (f11max) is: 756 757 𝑓11𝑚𝑎𝑥= 𝑙𝑖𝑚 − ∞ ≤ 𝑣 ≤ ∞ 𝑢→|1+𝑣|[𝑓11]=𝑙𝑖𝑚 − ∞ ≤ 𝑣 ≤ ∞ 𝑢→|1+𝑣|[−ℎ1∙𝑢2+𝑐11∙𝑢−ℎ1∙𝑣−ℎ1∙𝑣2 (1+𝑣)2−𝑢2]=∞ (77) 758 b) If v = -u 759 760 As in the previous case, the maximum value of “u” is the value that causes a growth of the 761 semi-axes of the convergence ellipsoids to ∞. The reason of this situation can be seen in the 762 next graph where the value of “u” has been represented against the semi-major axis (Fig. 13). 763 764 Fig. 13. Evolution of the semi-major axis of the convergence ellipsoid 11 as a function of the value of “u” for v = -u. 765 Two curves can be differentiated from the previous graph: 766 767 - The curve on the left can be divided into two parts: The first is composed of the negative 768 values of “u” and, in consequence, this part of the curve is not acceptable (“u” cannot be 769 negative). The second is composed of the positive values of “u” and, in consequence, this part 770 of the curve is the right one. 771 - The curve on the right is not acceptable because in its interval the value of the semi-major 772 axis is negative for any value of “u” and this situation is not possible. 773 Consequently, the analysis will focus on the part of the left curve where there are positive 774 values of "u". Mathematically, the condition of this curve can be represented with the 775 following equation Eq. (78). 776 𝑢𝑚𝑎𝑥=𝑙𝑖𝑚 𝑓21(𝑢=𝑢𝑚𝑎𝑥)→∞[𝑓21(𝑢=𝑢𝑚𝑎𝑥)]=𝑙𝑖𝑚 𝑓31(𝑢=𝑢𝑚𝑎𝑥)→∞[𝑓31(𝑢=𝑢𝑚𝑎𝑥)] (78) 777 Eq. (79) implies that: 778 [𝑎11 (1−𝑢𝑚𝑎𝑥)2−𝑢𝑚𝑎𝑥 2]∙[1−𝑢𝑚𝑎𝑥+𝑢𝑚𝑎𝑥∙ℎ1 𝑐11 ]=∞ (79) 779 [𝑏11 (1−𝑢𝑚𝑎𝑥)2−𝑢𝑚𝑎𝑥 2]∙[1−𝑢𝑚𝑎𝑥+𝑢𝑚𝑎𝑥∙ℎ1 𝑐11 ]=∞ (80) 780 If umax is isolated: 781 𝑢𝑚𝑎𝑥=0.5 (81) 782 The displacement of the convergence ellipsoid 11 for u = umax (f11max) is: 783 784 𝑓11𝑚𝑎𝑥=𝑙𝑖𝑚 𝑢→0.5(𝑓11)=𝑙𝑖𝑚 𝑣→−𝑢 𝑢→0.5[−ℎ1∙𝑢2+𝑐11∙𝑢−ℎ1∙𝑣−ℎ1∙𝑣2 (1+𝑣)2−𝑢2]=∞ (82) 785 c) If v = u 786 In this case, the parameter that is used is the maximum value of the semi-axes that causes a 787 growth of “u” parameter to ꚙ. This situation can be observed in the next graph where the 788 value of “u” has been represented against the semi-major axis (Fig. 14). 789 790 Fig. 14. Evolution of the semi-major axis of the convergence ellipsoid 11 as a function of the value of "u" for v = u. 791 Two curves can be differentiated from the previous graph: 792 793 - The curve on the left is not acceptable because all the values of “u” in this curve are negative 794 and this parameter cannot be negative. 795 - The curve on the right can be divided into two parts: The first is composed of the negative 796 values of the semi-major axis and, in consequence, this part of the curve is not acceptable (a 797 semi-major axis cannot be negative). The second is composed of the positive values of this 798 parameter and, in consequence, this part of the curve is the right one. 799 Consequently, the analysis will focus on the part of the right curve where there are positive 800 values of the semi-major axis. Mathematically, the condition of this curve can be represented 801 with the following equations: Eq. (83) 802 𝑎𝑚𝑎𝑥𝑑𝑒𝑙 𝐶𝐸11=𝑙𝑖𝑚 𝑢→∞[𝑓21(𝑣=𝑢)] 𝑎𝑛𝑑 𝑏𝑚𝑎𝑥𝑑𝑒𝑙 𝐶𝐸11=𝑙𝑖𝑚 𝑢→∞[𝑓31(𝑣=𝑢)] (83) b) If two initial ellipsoids (11 and 12) are tangent: 936 937 The convergence ellipsoids 11 and 12 for umin have been represented in Fig. 20: 938 939 940 Fig. 20. Minimum geometric convergence situation for two ellipsoids in the space that are tangent with v = 0.5 and 941 with u ≠ 0. 942 To find out the minimum value of “u”, the orientation between the ellipsoids with umin = 0 is 943 going to be developed. If this value of umin is replaced in Eq. (100): 944 945 𝑥𝑐=𝑐11∙(ℎ22−𝑐12 2)∙𝑠𝑖𝑛(∝) [𝑐11∙ℎ2+𝑐12∙ℎ1]∙(1+𝑣) (106) 946 Also, the next condition is satisfied in Fig. 9: 947 𝑠𝑖𝑛(∝) 𝑑=𝑠𝑖𝑛(𝛾) ℎ2−𝑙=𝑠𝑖𝑛(180−∝−𝛾) ℎ1 (107) 948 If Eq. (107) is replaced in Eq. (106) and the variables h1 and h2 are converted to other known 949 variables, Eq. (89) is obtained. If this process is done with the equation of zc, the equation of yc 950 of [24] is obtained. In [24], the equations of xc and yc were developed to satisfy the tangency 951 between the ellipsoids and, in consequence, if the ellipsoids are tangent → umin = 0. 952 953 c) If two initial ellipsoids (11 and 12) are secants: 954 955 The convergence ellipsoids 11 and 12 for umin in this situation have been represented in Fig. 21: 956 957 958 Fig. 21. Minimum geometric convergence situation for two ellipsoids in the space that are secants with v = 0.5 and 959 with u ≠ 0. 960 To find out the value of umin, a geometric argumentation is going to be used due to the 961 complexity of the equations of this case. 962 If the value of “u” is lower, the geometry of the convergence ellipsoid is going to be smaller 963 according to Eq. (6), Eq. (7), Eq. (8), Eq. (9), Eq. (10) and Eq. (11). When “u” has the value of 0, 964 the convergence ellipsoid will only depend on the “v” parameter. In consequence, this 965 problem can be solved with the analysis of the cases of “v”: 966 The first is v = 0. In this case, the convergence ellipsoid would have the same geometry and 967 position in comparison with the initial ellipsoid. Consequently, umin = 0. 968 The second is v ≠ 0 (as the situation that is represented in Fig. 21). In this case, the 969 convergence ellipsoid would have a different geometry and a different position in comparison 970 with the initial ellipsoid. However, v = l/n, where “n” is the distance between the intersection 971 of the two convergence ellipsoids and the “l” parameter”. Consequently, if “u” is going to be 0, 972 the values of “l” are going to be in the intersection points between the initial ellipsoids. 973 This situation means that the focal distance with a value of 0 will belong to the intersection 974 between the initial ellipsoids. The “u” parameter cannot be negative, so if the initial ellipsoids 975 are secants → umin = 0. 976 977 978 979 980 6. Conclusions 981 982 Before the onset of the convergence surface Method, the obtaining of two scissors from two 983 ellipsoids in the space had to be done using the intersection of 2 proportional ellipsoids and 984 the result was a unique mathematical solution. If the designer wanted other mathematical 985 solution, a different peer of ellipsoids had to be intersected using other proportional constant. 986 In comparison with this situation, the convergence surface Method not only gives all possible 987 solutions with just one mathematical operation but also this Method allows the use of polar 988 units and, in consequence, the size of the deployable structure in the folded position will be 989 smaller. Likewise, the consideration of all solutions of the space enables the design of bistable 990 deployable structures. 991 992 However, the main drawback of the convergence surface Method is the obtaining of this 993 surface. To balance this situation, the research of this article has completed the equations that 994 control the Method of [24] and the results can be summarised into 2 types of formulas: the 995 exact formula (triaxial two-leaf hyperboloid) and the approximate formulas (determinant and 996 level curves). 997 998 Before the outcomes of this article, the convergence surface had to be created with the 999 manual intersection of ellipsoids without knowing the start point of the intersection process 1000 and the solution that limits the superior interval of valid results. Moreover, the standard shape 1001 of the convergence surface was a mystery and, consequently, the use of mathematical 1002 programs (MatLab, Maple, etc.) to create this surface was quite tedious. 1003 1004 The results of this paper have solved these disadvantages so the Method of [24] is now more 1005 operative and friendly. Once the convergence surface has been obtained, the rest of the 1006 design process is based on the intersection of this surface with the design surface and the final 1007 curve will be all possible points where there is a mathematical solution. The aim of the 1008 designer is the selection of a point of this curve in function of boundary conditions, structural 1009 behaviour, etc. 1010 1011 Finally, a further extension intended for future work is the incorporation of the formulas 1012 developed in this research into a design program where the designer will have the 1013 convergence surface just clicking a button. 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 7. Acknowledgements 1033 1034 This research did not receive any specific grant from funding agencies in the public, 1035 commercial, or not-for-profit sectors. 1036 1037 8. 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