The application of retractable roof structures has enhanced the urban 5 landscape, particularly in large -scale public buildings. Nevertheless , the utilization of 6 origami mechanism s in such structures has only been explored conceptually, with a lack 7 of practical engineering imp lementations. In this paper, three novel retractable roof 8 structure s, integrating Miura -ori (and its variations ) with scissor hinge units , are 9 proposed . We first provide a concise overview of the concepts of Miura -ori (including 10 its arch ed and fan -shaped variations), scissor joint unit, and plate thickening. 11 Subsequently, coordinated motion conditions for the three proposed retractable roof 12 structures are elaborately derived using geometric formulas , and the corresponding 13 solutions are provided when the coordinated motion is not feasible. Lastly, after 14 parameterizing the thick -plate model, the motion mechanism of the retractable roof 15 structure s is analyzed exhaustively . The results indicate that all three new structures can 16 achieve smooth opening and closing operations without motion interference. The 17 proposed novel retractable roof structures enrich the variety of architectural structures 18 and provide practical guidance for engineering applications. 19 Keywords : Retractable roof structure; Miura -ori; Scissor joint unit; K inematic analysis 20 1. Introducti on 21 2 Although the structure of retractable roofs has a long history, it was not until recent 22 decades that it gained widespread attention and became an essential category of 23 building structures [1–4]. The various forms of retractable roofs not only enrich the 24 urban landscape but also endow new functionalities to building structures . 25 The retractable roof structure is a type of deployable structure, and as such, its 26 research is closely related to that o f trus s, plate, and membrane unit systems. The 27 concept of deployable structures was first introduced by American architect 28 Buckminster Fuller in 1960 [5,6] . Inspired by his work, Spanish architect E.P. Pineo 29 proposed a scissor joint unit system that connected two bar members through a centr al 30 pin joint. This system was first applied to the design of a deployable theater, marking 31 the first use of deployable structures in such a context [7]. Hoberman , based on the 32 scissor joint unit, further developed a unique deployable unit system, which served as 33 the basis for designing a new kind of centrally -symmetric radial retractable roof 34 structure [8]. 35 British scholars You and Pellegrino , building on the research of Hoberman, further 36 expanded the research by developing a new type of fundamental scis sor joint unit and 37 constructed a more general deployable structure known as Foldable Bar structure, FBS 38 [9]. Additionally, utilizing the scissor joint unit as the fundamental element , You and 39 Pellegrino , at the same time, proposed a radial retractable structure system [10]. 40 In practice, structural performance s, particularly mechanical properties, play a 41 crucial role in the design and analysis of retractable roof structure s. Chen and his 42 3 colleagues [11] conducted a study on the structural performance of deployable planar 43 truss structure s and found that the low stiffness of the structure may lead to reduced 44 efficiency in large -span truss structure s (due to the significant bending moments to 45 internal forces ). To overcome this issue, they proposed adding diagonal bracing and 46 lower chords that are compatible with the original geometry of the scissor joint units, 47 thereby improving the mechanical performance of the structure [11]. Professor Cai [12] 48 has extensively researched various new deployable structures, including angular sci ssor 49 joint units, folding truss structure s, and tensioned cable structure s, utilizing 50 theoretical analysis, nonlinear finite element simulations, and physical model 51 experiments. His research focused on the geometric composition, motion process, and 52 mechan ical performance of these structures, resulting in a set of highly practical 53 conclusions. In short, a substantial amount of work [3,13,14] has been done on the 54 development, theoretical analy sis, and optimization of deployable structures. 55 Nevertheless , there is still a need for further integration and application of these 56 deployable structures in specific engineering areas (for instance, civil engineering) , to 57 fully realize their potential . 58 Origami has been widely applied in engineering , particularly in deployable 59 structures, due to its diverse configurations. Mechanisms based on the principles of 60 origami are an excellent choice for structures that undergo significant changes in 61 characteristic s, such as area, in different states. Origami has been used as a tool for 62 mathematical and scientific research since the end of the 19th century. In the 1990s, 63 4 research on the mathematics and theory of origami gained popularity in many countries 64 worldwide . De Focatiis et al. [15] used the folding model of the oak lea f as a basis to 65 propose various high -compaction folding methods by connecting several leaf units in 66 different ways , which provide d new avenue s for the deployable membrane structure. 67 Origami structures have found critic al applications in the aerospace field. For instance , 68 a solar panel based on Flasher origami [16], composed of a series of triangles, 69 rectangles, and trapezoids arranged according to specific rules , was developed by the 70 Jet Propulsion Laboratory of NASA, and can be readily deployed into a hexagonal 71 shape . The folding and unfoldi ng ratios of this new structure satisfy the requirement of 72 the aerospace industry for solar panel s in both the un- and folded states. Utilizing the 73 unique morphological adaptability of origami , Professor Miura Koryo [17,18] , from the 74 Institute of Space and Astronautical Science at the University of Tokyo , invented an 75 origami technique called Miura -ori, which not only saves space but also minimizes 76 losses during the un- and folding p rocesses (reduc ing the volume of an object by 25 77 times and increas ing its energy density by 14 times ). Currently, researchers are 78 exploring the potential of origami technique s in the field of architecture and civil 79 engineering [19–21]. Nevertheless , the application of origami structures in this area , 80 overall, is still relatively limited and primarily focused on temporary structures such as 81 lightweight exhibition halls, galleries, and tents. Expanding the application of origami 82 structure s in civil en gineering, particularly in large and permanent building structures 83 such as sports stadiums , is a research direction that requires further exploration and 84 5 development. 85 We, in this paper, propose a novel retractable roof structure based on origami and 86 conduc t an in -depth investigation into its kinematic mechanism , aiming to provide more 87 design options for the first party and foster innovation in building structural design by 88 enabling novel functional and structural possibilities for buildings . We first introd uce 89 the geometric principles underlying the Miura -ori, including its arch ed and fan -shaped 90 variations, and provide an overview of the concepts of rigid origami and the thick plate 91 analysis (Section 2 ). Further, we utilize scissor joint rectangular units to develop three 92 novel retractable roof structur es, derive the compatibility conditions for the coordinated 93 motion of each new retractable roof structure , and propose a corresponding 94 improvement strateg y (Section 3). After realizing the deployment of retractable roof 95 structures, a detailed analysis of the structural changes during the motion process is 96 conducted (Section 4 ). Finally, the conclusion is summarized, while the limitations of 97 the study and the prospects for future research are also described (Section 5 ). 98 2. Retractable roof based on the principle of orig ami mechanism 99 This section aim s to identify suitable origami structures that can be applied to 100 retractable roof structures . Firstly, we elaborate on the fundamental kinematic 101 characteristics of the classic Miura -ori, as well as its arched and fan -shaped variations. 102 Additionally, a detailed introduction is given to the expandable scissor joint unit and its 103 variant s. Subsequently, a compreh ensive analysis of the thick -plate transformation for 104 the retractable roof is presented. Finally, the parameterization modeling method used in 105 6 this paper is explained using the classic Miura -ori structure as an example. 106 2.1 Miura -ori and its variations 107 The Miura-ori is a classic single -degree -of-freedom origami structure with identical 108 parallelogram -shaped units [17]. The classic Miura -ori, when fully expanded, features 109 longitudinal zigzag crease s. Based on the classic Miura -ori, adjusting the longitudinal 110 angle of the zigzag creases to make adjacent longitudinal creases unparallel (with a 111 reasonable angle ) while spaced longitudinal creases remain parallel , to get the arched 112 variant of the Miura -ori [22–24]. If the creases that lie along straight lines in the 113 transverse direction are altered to become non -parallel, a different form of Miura -ori, 114 known as the fan -shaped variant , can be achieved [25]. 115 The uniformity of the basic components in the Miura -ori enables us to derive the 116 kinematic behavior of the entire mechanism by just analyzing a single unit. Utilizing 117 the principles of Babuška -Brezzi's spherical trigonometry as well as the constraints 118 governing the mechanism [26–28], more generalized kinematic patterns can be deduced. 119 This approach is further applied to derive the corresponding kinematic patterns for the 120 arched and fan -shaped variant s of the Miura -ori. 121 2.2 Expandable Scissor Joint Unit 122 We introduce expandable scissor joint units to enhance the strength of the structure 123 and to achieve more convenien t control [29,30] . Several scissor joint units, consisting 124 of two bars of equal length hinged together at the center, are connected by hinges at the 125 ends of the bars , which form a rectangular m echanism with a single degree of freedom. 126 7 Furthermore, by altering the hinge po sitions in a single scissor joint unit and satisfying 127 certain conditions, an arched scissor joint unit can also be formed [30]. A novel 128 retractable roof structure can be obtained by combining scissor joint unit s with 129 deployable grid structures. 130 2.3 Thick pla te retractable roof structures analysis 131 Rigid origami maintains its shape as a rigid plane throughout the un- and folding 132 process, making it particularly suitable for origami -derived structures with significant 133 thickness [31–33]. We draw inspiration from origami and thick plate theories, and 134 utilize the principles of origami mechanisms to design retractable roof panel s. Various 135 thick pl ate methods , including adjust ed the crease position, adjust ed the shape, and 136 expanding the gap , etc., have been proposed by different researcher s for planar 137 symmetric quadrilateral origami structures [31,32,34 –36]. The adjusted shape method 138 is applied in our study to obtain the corresponding thick pla tes for the retractable roof 139 panels. The calculation formula for the opening ratio of the retractable roof panels is 140 given by Eq. (1) (for a detailed derivation, please refer to Appendix A): 141 1 sin(arctan )h k=− (1) 142 where η denotes the opening rate, h represents the thickness of the panels being trimmed 143 on both sides o f the crease, and k is the length of one side of the symmetrical notches . 144 Furthermore, by visualizing Eq. (1), the variation spectrum of the opening rate can be 145 obtained, as depicted in Fig. 1, which vividly demonstrates the relationship between the 146 variables of plate thi ckness and notch length with max opening rate. 147 8 148 Fig. 1 The variation spectrum of the maximal opening rate . 149 As shown in Fig. 1, when the notch length is fixed and its value is relatively small, 150 the opening rate decreases with the increase in panel thickness, and the reduction 151 gradually weakens. Meanwhile, w hen the notch length is fixed and its value is relatively 152 large, the opening rate decreases in a nearly linear manner. Eq. (1), essentially, indicates 153 that the maximum opening ratio is solely determined by the trimmed thickness h and 154 the notch length k, and is irrelevant to factors such as the shape and size of the panel . 155 Therefore, the calculation of the opening rate, for the arched and fan -shaped retractable 156 roof structures, is the same as Eq. (1). 157 2.4 Parametric modeling of retractable roof structure 158 Parametric modeling steps using Grasshopper, a powerful parametric modeling 159 software widely used in building structure area s, can be briefly summarized as follows : 160 First, establish a zer o-thickness Miura -ori, without considering factors such as the 161 thicknes s of the panels and the composition of the bar s, which is used to determine the 162 shape of the roof unit at different stages of the retractable roof . Then, b ased on the zero- 163 9 thickness Miura -ori mechanism, construct a grid structure with thickness . Thereafter , a 164 practical and applicable retractable roof structure is established for specific analysis. A 165 comprehensive guide to the parameterized modeling process can be found in Appendix 166 B. 167 3. Layout of retractable roof structures 168 This section further elaborates on the different deployments and form s of the 169 retractable roofs based on the principle of Miura -ori (as well as its arched and fan- 170 shaped variants ). First, the deployment of scissor joint units is based on the co -linear 171 and coplanar poi nts that are distributed in a relatively regular manner during the un- 172 and folding process . Geometric formulas are then further employed to validate whether 173 the upper and lower parts of the structure can work synchronously and coordinately . 174 Ultimately, the corresponding optimal methods are proposed for cases where 175 coordi nated motion can not be achieved . 176 3.1 Miura -ori retractable roof 177 When th e Miura -ori mechanism is fully extended, all the mountain and valley 178 vertices of the transverse (span direction, parallel to the x -axis) creases are collinear 179 and located in the same plane （Fig. 2c）. The distances between th e mountain and valley 180 vertices in the transverse direction keep unchanged during the deploy ing process for 181 they share the same basic unit (Fig. 2a). By placing the scissor -jointed rectangular 182 mechanism on the transverse direction of the Miura -ori (the red outline section in Fig. 183 2c) where the ends of the upper sid es of the rectangular mechanism are hinged with the 184 10 valley vertices of the Miura -ori (Fig. 2e), the motion of these two parts are coordinated 185 in the transverse direct ion. 186 187 Fig. 2 Miura-ori retractable roof . (a) and (b) denote t he transverse and longitudinal 188 origami units , respectively; (c) and (d) depict t he location of the transverse and 189 longitudinal scissor joint units ; (e) and (f) are t he axonometric projection in the 190 transverse and longitudinal direction of scissor joint units . 191 Similarly, all the vertices of the longitudinal (parallel to the y -axis) creases are 192 collinear throughout the deploy ing process, and the distances between them rema in 193 11 equal ( the distance between the blue dots and the distance between the red dots in Fig. 194 2b). Additionally, the longitudinally arranged creases, which are spaced apart, remain 195 parallel to each other, and maintain in the same plane during the folding process. By 196 placing the scissor -jointed rectangular mechanism at the inflection point of the 197 longitudinal creases of the Miura -ori, as shown in Fig. 2d, where the upper ends of both 198 sides of the rec tangular mechanism are hinged with the vertices of the inflection points 199 of the creases of the Miura -ori (Fig. 2f), the motion of these two parts are coordinated 200 in the longitudinal direction . 201 The two conditions for the coordinated motion of the scissor -jointed rectangular 202 mechanisms in both x - and y -direction are as follows: 203 (1) The ratio between the distance of the upper ends of the rectangular mechanism 204 in the x - and y -direction remains constant and is equivalent to the ratio between the 205 distance of the crease vertices of the Miura -ori mechanism in bo th the x - and y -direction. 206 (2) The ratio of the relative velocity between the upper ends of the rectangular 207 mechanism in the x - and y -direction is equivalent to the ratio of the relative velocity 208 between the crease vertices of the Miura -ori mechanism in bo th the x - and y -direction, 209 and it remains constant. 210 As mentioned above , the distance between the ends of the scissor joint units in both 211 x- and y -direction is always equal to the distance between the corresponding crease 212 vertices. Therefore, the ratio of the distances of the former is equal to the ratio of the 213 distances of the latter, which satisfies condition (1). Nevertheless , for condition (2), 214 12 further discussion is needed. 215 216 Fig. 3 Geometric parameters of the Miura -ori retractabl e roof . (a) The overall layout 217 of Miura -ori; (b) Axonometric projection of a panel unit ; (c) The overall layout of 218 scissor joint units ; (d) Simplified diagram of two scissor joint units ; (e) The schematic 219 diagram of the improved scissor joint units . (W and L represent the length along the x- 220 and y -direction, respectively. n1 and n2 represent the number of units in the two 221 directions.) 222 All the critical geometric parameters of the retractable roof structure , formed by the 223 13 classic Miura -ori, are diagrammed holistically in Fig. 3. Once the side lengths (a, b) 224 and acute angle (β) of a sub-parallelogram panel unit ( Fig. 3b), with four other identical 225 sub-parallelograms , are given , the shape of the basic unit can be determined . The 226 motion of the entire unit is controlled by 𝜙, the angle between the two longitudinal 227 creases . According to [37], the le ngth (l), width (w), and height (h) of the basic unit can 228 be expressed as follows: 229 2 sin( / 2)lb= (2) 230 cos2cos( / 2)wa = (3) 231 22sin sin ( / 2) cos( / 2)ah −= (4) 232 The angles formed between the panels of the parallelogram, denoted α1 and α2, can 233 be represented as follows: 234 2 1 1 2sin ( / 2)cos 1 2sin−=− (5) 235 1 2 2 2cos 1 2cot tan ( / 2)  −=− (6) 236 Fig. 3c-d depict s the three -dimensional scissor joint units mechanism and its 237 essential geometric parameters. To form the whole structure with a single degree of 238 freedom, the scissor joint units require the length of the upper and lower portions of the 239 two scissor joint units at the red dashed line to be equal (Fig. 3d), namely Δ1 = Δ2, which 240 can be expressed as: 241 2 2 2 22 ( ) 2 ( )22xycd− = − (7) 242 where c and d are half of the length of the scissor joint unit in the x - and y -direction , x 243 14 and y denote the distance between the upper ends of the scissor joint units in the x - and 244 y-direction, respectively . We, further, formulate the values of x and y using Eqs. (8) and 245 (9): 246 2 cos cos( / 2)waxmm = =  (8) 247 2sin( / 2)lbynn = =  (9) 248 where m and n respectively denote the number of scissor joint unit s for a single Miura - 249 ori in the x - and y -direction . Substituting the abo ve two equations into Eq. (7) then, we 250 can obtain: 251 22 2 2 2 2 2 2 21( cos ) sin ( / 2)cos ( / 2)abcdmn −  = − (10) 252 The parameters β, a, b, m, and n above are constants , and the un- and fold of the 253 entire structure is controlled by the parameter φ. Since the left side of the equation is a 254 variable and the right side is a constant, it is not feasible for the three -dimensional 255 scissor joint units connected to the Miura -ori to create a structure with only one degree 256 of freedom. Therefore, an improved design is needed. Fig. 3e is a schematic diagram of 257 the improved scissor joint units . A vertical bar is added to the top of the original scissor 258 joint units to connect with the crease vertex of the Miura -ori. The sci ssor joint units in 259 both directions are connected at the top, and the bottom ends of the bars are connected 260 to the vertical bar through two sliding joints. In this way, free deploy ing in the x - and 261 y-direction can be achieved readily , and it is coordinated with the movement of the 262 Miura -ori mechanism at the top. 263 3.2 Arched variant of retractable roof 264 15 In the un- and folding process of the arched variant （Fig. 4a）, the crease vertices 265 at the longitudinal inflection points (parallel to the y -axis) are always collinear. Since 266 the basic unit s are identical , the distance between the crease vertices in th e longitudina l 267 direction (the distance between the blue dots) remains unchanged during the opening 268 and closing process. When arranging the scissor -jointed rectangular mechanism at the 269 longitudina l crease inflection point of the arched variant （the blue outl ine section in 270 Fig. 4a）, the upper ends of the scissor -jointed units on both sides of the rectangular 271 mechanism are hinged with the crease vert ices at the inflection point s of the arched 272 variant. Taking only the longitudina l direction , the motion of the upper and lower 273 mechanism is coordinated . 274 During un- and folding , likewise, the crease vertices along the x -axis are coplanar 275 and the mountain and va lley crease vertices in the same plane are located on a single 276 arc segment (red points in Fig. 4a), which creates a uniform distance between the 277 mountain and valley c rease vertices in the transverse direction due to the consistent 278 basic unit s. 279 16 280 Fig. 4 The arched variant of the retractable roof . (a) The longitudinal distribution of 281 arched variant origami and scissor joint units distribution ; (b) The layout of the 282 transverse arched scissor joint unit ; (c) The schematic diagram of the improved lower 283 mechanism; (d) Geometric of the upper origami; ( e) Geometric parameters of the 284 lower arched scissor joint mechanism. 285 On the arched variant of the Miura -ori (Fig. 4b), the arched scissor joint units are 286 17 arranged in a transverse orientation, with the upper portion of the scissor -jointed 287 mechanism hinged to the crease vertices of the arched variant panel. To ensure proper 288 coordination between the scissor -jointed arched mechanism and the upper arched 289 variant of the Miura -ori, two conditions below must be met : 290 (1) The distance between the ends of the upper bar s is equal to the distance between 291 the crease inte rsections of the upper arched variant of the Miura -ori. 292 (2) The arc radi us between the arched scissor -jointed mechanism and the upper 293 arched variant should maintain consisten cy throughout the un- and folding process. 294 Condition (1) necessitates that the dis tance between the crease intersections at the 295 lower end of the arched variant (denoted as l) must be equal to the distance between the 296 upper bar ends of the arched scissor -jointed mechanism (represented by m), as 297 expressed by Eq. (11) and (12), respectively. 298 2 sin2la= (11) 299 2 sin( ) 2 cos2 2 2m c c  = − = (12) 300 In Eq . (11), ξ repre sents the apex angle of the arched variant (Fig. 4d), while in Eq . 301 (12), α denotes the angle formed between the two bars of the arched scissor -jointed 302 mechanism (Fig. 4e). Condition (1) can be met when α and ξ satisfy Eq . (13): 303 sin cos22= (13) 304 Condition (2) requires that R = ρ+Δ, where R represents the arc radius of the top 305 arched variant of Miura -ori, and ρ+Δ represents the radius of the upper arc of the arched 306 scissor -jointed mechanism ( Fig. 4e). It can be derived from [38] that: 307 18 22 22 12 cos2sin2=11 si4os nca dc d cdc d cdc r +−+−+= +−−− (14) 308 In Eq. (14), introducing a new variable K as shown in Eq. (15), then substituting K 309 into Eq. (14) and conductin g simplification , we get Eq. (16). 310 = ( 1)c c dKc ac adad=−− (15) 311 2 2 2 2 2 1sin22)cos ( sin ( )4cd K c d + =  +− (16) 312 Eq. (16) features the constants a, c, and d, making K a constant. The un- and folding 313 of the entire mechanism are regulated by ξ, ξ1, and α. The right side of the equation is 314 a constant , while the left side is indeterminate , which suggests that the three - 315 dimensional scissor -jointed units , linked to the arched variant of the Miura -ori, cannot 316 create a mechanism with a single degree of f reedom. 317 We present an enhanced design for the arched variant of the Miura -ori, depicted in 318 Fig. 4c, in which the longitudinal arrangement remains unaltered, but the arched 319 scissor -jointed units in the transverse direction are substituted with prestressed steel 320 cables (or telescopic bars ). This modification facilitates unrestricted un- and folding in 321 both the x - and y -direction, while guaranteeing the coordinated motion of the lower 322 arched scissor -jointed mechanism and the upper arched variant of the Miura -ori. 323 3.3 Fan-shaped variant of retract able roof 324 During the un - and folding of the fan -shaped variant mechanism, the crease vertices 325 19 in the radial direction remain coplanar , while within the same plane, all mountain crease 326 vertices and valley crease vertices are collinear. The geometric propert ies of the fan - 327 shaped variant in the transverse direction make it convenient to arrange the scissor - 328 jointed mechanism. The distances between the mountain crease vertices (represented in 329 red) and valley crease vertices (shown in blue) in the radial directio n remain constant 330 during the folding process, by virtue of the identical basic unit s (Fig. 5a). As a result, 331 arched scissor joint units similar to those used in the longitudinal direction can be 332 utilized in the radial direction, allowing for the hinging and assembly of the upper 333 structure (Fig. 5b). 334 335 Fig. 5 The fan -shaped variant of the retractable roof. (a) The layout of the origami; (b) 336 The arrangement of the scissor join t units ; (c) The schematic diagram of the improved 337 fan-shaped variation . 338 20 Similarly, additional prestressed steel cables or rotatable tracks ( Fig. 5c) can be 339 added between the lower bar ends of adjacent scissor -jointed rectangular units in the 340 radial direction , which not only ensures coordination with the upper fan-shaped variant 341 mechanis m, but also helps to improve the overall integrity and stability of the structure. 342 4. Motion s imulation of retractable roof based on origami principles 343 In this section, the structure model s of the retractable roof structures based on 344 origami principles are simplified, and then SolidWorks , a three -dimensional simulation 345 software , is adopted to simulate its motion . After verifying that the roof panel with 346 thickness can un - and fold smoothly, the variation law of the structure during its motion 347 is analyzed. The kinematic analysis of the three new retractable forms is investigated 348 exhaustively. 349 4.1 Miura -ori retractable roof 350 We assume that , as shown in Fig. 6, the Miura -ori retractable roof model consists 351 of four basic units , with two units longitudinally and transversely each. The length of 352 each bar in the lower scissor joint unit is 2 c = 3100 mm and 2 d = 3570 mm. 353 Considering the thickness of the upper roof panel and the dimensions of the lower 354 scissor hinge on the structural motion, two angles, namely 𝜙1 = 20° and 𝜙2 = 120° , 355 which correspond to the fully folded and unfolded states, respectiv ely, are adopted to 356 prevent interference phenomena . Throughout the entire opening and closing process, 357 the angular variation between the roof panels totals 100°. Detailed parameters of the 358 model are presented in Table 1. 359 21 360 Fig. 6 Miura -ori retractable roof model. A, B, C, and D denote reference points ; point 361 O represents the motion center of the structure ; point M denotes the place whe re the 362 rotational driving motor is set. 363 Table 1 The detailed parameters of Miura -ori retractable roof 364 a (mm) b (mm) β (°) 𝝓𝟏 (°) 𝝓𝟐 (°) c (mm) d (mm) 1500 1732 60 20 120 1550 1785 Note: the parameters are illustrated in Fig. 3. 365 To simulate the motion of adjacent roof panels rotating 100 degree s using 366 SolidWorks, we incorporated a rotational driving motor at point M , the intersection of 367 the bottom edges of the two panels, as illustrated in Fig. 6. The angle between the panels 368 increases uniformly at a constant angular velocity of 10 °/s for 10 seconds. Our 369 simulation, depicted in Fig. 7, demonstrate s that the whole mechanism can successfully 370 un- and fold as an entirety. 371 22 372 Fig. 7 The motion of the Miura -ori retractable roof . (a), (b) , and (c) respectively 373 denote the 0th, 5th, and 1 0th seconds during the course of motion. 374 The displacement -time curves of reference points A, B, C, and D relative to the 375 center point O of the orbit are illustrated in Fig. 8. Upon comparative analysis, it can be 376 observed that the displacement of points B and D in x -direction is equivalent, as is the 377 displacement of points A and C in the same direction , which suggests that during the 378 process of un- and folding the roof, these two sets of points always remain coplanar 379 with the scissor joint units , ensuring coordinated motion of the structure. As the entire 380 structure unfolds, the displacement of all reference po ints in x-direction exhibits a 381 smooth and gradual increase, reaching its peak value at t = 10 s. The panel comprising 382 points O, B, and D rotates around the x -axis and y -axis, resulting in a relatively minor 383 overall displacement in the y -direction for point D, which is only 182.33 mm. Moreover, 384 the displacement in z-direction of point D showcases the gradual decrease in the overall 385 height of the structure as it unfolds. 386 23 387 Fig. 8 The displacement -time curves of the Miura -ori retractable roof. (a), (b), (c), and 388 (d) denote the displacement -time relationship of point s A, B, C, and D , respectively . 389 (x-, y-, and z -disp. represent the displacement components of a reference point in the 390 three principal axis directions while T-disp. is the total displacement of the 391 corresponding point. ) 392 Fig. 9 depict s the velocity -displacement curves of points A, B, C, and D. The upper 393 roof comprises fou r identical parallelograms, resulting in comparable velocities of 394 points A and C, as well as B and D, at the onset of the motion. Specifically, the velocity 395 of points A and B is roughly twice that of C and D ( more vividly demonstrate d in Fig. 396 9e). For a significant period, the velocity of point C in x-direction consistently exceeds 397 that in y-direction , although the disparity between the two gradually diminishes. At t = 398 8.36 s, the x - and y -direction of the velocity at point C are both 140.80 mm/s. 399 24 Subsequently, the y -direction of the velocity at point C overtakes the x -direction . This 400 indicates that, d uring the motion process, the structure initially moves predominantly 401 in x-direction and gradually shifts towards movement in y-direction. The velocity of 402 point D , as depicted in Fig. 9e, gradually decreases over time, reaching a minimum 403 value of 102.68 mm/s at t = 7.76 s. Afterward, it gradually increases and eventually 404 surpasses the velocities of points A and B. 405 406 Fig. 9 The velocity -time curves of the Miura -ori retractable roof . (a), (b), (c), and (d) 407 denote the velocity -time relationship of point s A, B, C, and D, respectively ; (e) 408 depicts the velocity comparison of four reference points. (x-, y-, and z -vel. represent 409 the velocity compone nts of a reference point in the three principal axis directions 410 while T -vel. is the total velocity of the corresponding point.) 411 4.2 The arched variation of retractable roof 412 The arched variant retractable roof model , as shown in Fig. 10, comprises three 413 arched scissor -jointed mechanisms, consisting of six basic origami units. The length of 414 each bar in the lower scissor joint unit measures 2100 mm (2 c). To ensure smooth and 415 25 uninterrupted movement of the entire structure, we have set the angles between the 416 retractable panels to be ξ1 = 45° and ξ2 = 180°, corresponding to the fully un- and folded 417 states of the arched variant of the retractable roof. For further details regarding the 418 specific parameters of the retractable roof model, kindly refer to Table 2. 419 420 Fig. 10 The Arched variation of the Miura -ori retractable roof model . 421 Table 2 The specific parameters of the arched variation of Miura -ori retractable roof 422 a (mm) h (mm) β1 (°) β2 (°) ξ1 (°) ξ2 (°) c (mm) 1500 1000 80 70 45 180 1050 Note: the parameters are illustrated in Fig. 4 and Fig. A. 1. 423 We utilize revolute connections in SolidWorks to link the bars of the transverse and 424 longitudinal scissor joint units. Through path coordination, the retractable roof moves 425 along a predetermined motion trajectory line (the dash line s in Fig. 10) and is propelled 426 by a motor. A rotational driving motor is positioned at point O, the center point of the 427 entire structure (as depicted in Fig. 10). This motor uniformly increases the included 428 angle between adjacent roof panels at an angular velocity of 9 °/s and for 15 seconds. 429 The simulation, presented in Fig. 11, demonstrate s that the entire mechanism can 430 smoothly achieve opening and closing. 431 26 432 Fig. 11 The motion of the arched variation of the Miura -ori retractable roof . (a), (b), 433 and (c) respectively denote the 0th, 5th, and 15th seconds during the course of motion. 434 Fig. 12 illustrates the displacement -time curves of reference points A, B, C, and D 435 concerning the center point O of the structure . As the structure is in motion , the 436 displacement of all reference points in x-direction increase s smoothly and attain s their 437 maximum values at t = 15 s. Point B follows circular motion around the center point O 438 in y-direction, with its displacement in this direction gradually decreasing over time. 439 Point D represents the intersection of the lower ends of two scissor joint units and 440 experiences no displac ement in y-direction. Nevertheless, its displacement in z- 441 direction gradually increases (i.e., moving upward) over time, then begins to decrease 442 slowly at t = 7.40 s, before moving upward again at t = 10.96 s. Notably, during the 443 period of t = 7.40 ~ 10.96 s, the structure primarily unfolds in x-direction, with a 444 relatively minor degree of unfolding in y-direction. Point D ultimately reaches its 445 highest point at -640.31 mm. Simultaneously, the displacements of points C and C' in 446 x-direction remain precisely the same, indicating that the entire structure moves in a 447 coordinated manner. 448 27 449 Fig. 12 The displacement -time curves of the arched variation of retractable roof. (a), 450 (b), (c), and (d) denote the displacement -time relationship of point s A, B, C, and D, 451 respectively . (x-, y-, and z -disp. represent the displacement components of a reference 452 point in the three principal axis directions while T -disp. is the total displacement of 453 the corresponding point. ) 454 The velocity -displacement curves of the key reference points , as shown in Fig. 13, 455 reveal that the motion process can be segmented into three stages. During the initial 456 stage (t = 0~4.76 s), the velocity components of each reference point in all directions 457 are relatively uniform. In the middle stage (t = 4.76~13.20 s), the total velocity and x - 458 component velocity of the four reference points are nearly equivalent, indicating that 459 the movement is primarily in x-direction. In the subsequent stage (t = 13.20~15.0 s), 460 the x -component velocities of points A, C, and D decrease rapidly to zero, 461 28 demonstrat ing that they move primarily in the z -direction. In contrast, point B 462 undergoes circular motion around the center point O, with a relatively lower velocity 463 compared to other points but with a smooth movement. 464 Points C and D exhibit a significant decrease in velocity, followed by a rapid 465 increase, with the times of minimum velocity bei ng identical at t = 4.96 s and t = 4.76 466 s, respectively. In the early stage, points A and B experience similar velocity changes, 467 with a substantial reduction in velocity during the initial stage (at t = 2.0 s and t = 3.4 s, 468 respectively). The decrease in v elocity for point A is more significant than that of point 469 B. Throughout the motion process, the velocity of point B remains the lowest among 470 the four key reference points. During the period of t = 0 to 4.12 s, vD is greater than vA, 471 but the difference gra dually decreases, and at t = 4.12 s, the instantaneous velocities of 472 these two points are the same. Subsequently, their velocities become closer and exceed 473 the velocity of point C at t = 11.68 s. 474 475 Fig. 13 The velocity -time curves of the arched variation of the Miura -ori retractable 476 roof. (a), (b), (c), and (d) denote the velocity -time relationship of point s A, B, C, and 477 29 D, respectively; (e) depicts the velocity comparison of four reference points. (x -, y-, 478 and z -vel. repres ent the velocity components of a reference point in the three principal 479 axis directions while T -vel. is the total velocity of the corresponding point.) 480 4.3 Fan-shaped variant of retractable roof 481 The fan -shaped variant retractable roof model (depicted in Fig. 14) comprises four 482 basic units, with two units oriented in the circumferential and radial directions, 483 respectively, and the length of the bars in the lower scissor joint units is l = 4500 mm. 484 To ensure smooth movement of the entire structure, the angles between the retractable 485 roof panels are θ1 = 30° and θ2 = 180° for the fully folded and unfolded states of the 486 fan-shaped variant, respectively. For additional information regarding the specific 487 parameters of the retractable roof model, kindly refer to Table 3. 488 489 Fig. 14 The fan-shaped variation of the Miura -ori retractable roof model . 490 Table 3 The specific parameters of the fan-shaped variation of the Miura -ori retractable 491 roof 492 b1 (mm) ac (mm) φ1 (°) φ2 (°) θ1 (°) θ2 (°) l (mm) 1500 2000 80 60 30 180 4500 Note: the parameters are illustrated in Fig. A. 2. 493 30 The simulation software employs revolute connections to link the bars of the 494 transverse and longitudinal scissor joint units at the connection points. By path 495 coordination, the retractable r oof moves along the trajectory line at the lower end of the 496 scissor joint units (the dash line s in Fig. 14) and is driven by a motor. In Fig. 14, a 497 rotational driving motor is installed at point M, which deno tes the center point of the 498 upper panel structure. This motor uniformly increases the angle between two adjacent 499 panels at an angular velocity of 10°/s and persists for 15 seconds. The simulation , 500 presented in Fig. 15, illustrate s that the entire mechanism achieves smooth opening and 501 closing. 502 503 Fig. 15 The motion process of the fan-shaped variation of the Miura -ori retractable 504 roof. 505 The displacement -time curves of the motion reference points A and B during the 506 movement process are shown in Fig. 16a-b. The displacements of the retractable roof 507 in y-direction undergo minimal change, with a displacement increase of only 6.68% as 508 the roof unfolds. During the motion process, point A follows a counterclockwise path 509 around the center point O, with its displacement in x-direction gradually decreasing as 510 time t increases. Since point B shares the same polar angle as point A in the projection 511 plane, their displacements and velocities are identica l in y-direction , as evident from 512 31 the trend of the curves in Fig. 16b. Nevertheless, due to the outward expansion 513 movement of the retractable roof, the displacement o f point B exhibits an opposite trend 514 to that of point A in x-direction . As the motion of the retractable roof continues, the 515 displacement of point B in x-direction gradually increases, but the rate of displacement 516 change remains comparable to that of point A. The displacement of point B in z- 517 direction reflects the variation pattern of the height of the retractable roof, which 518 decreases gradually with time t during th e unfolding process, exhibiting a relatively 519 minor rate of change, and its curve is approximately linear. 520 Fig. 16c presents the curve of the distance (polar radius) fr om point A to the polar 521 axis in the projection plane as a function of time. The polar radius of point A increases 522 from an initial value of 2678.54 mm to 4313.43 mm, and its rate of change gradually 523 decreases , which indicates that during the unfolding proce ss, the inner edge of the fan - 524 shaped variant roof panel gradually moves away from the center of the structure, and 525 the space in the middle increases as the structure unfolds. Fig. 16d depicts the variation 526 of the maximum angle ρ (as shown in Fig. 14) of the structure. The ρ gradually 527 decreases with time t, with its rate of change also gradually decreasing. Throughout the 528 unfolding process, the angle ρ decreases from 124. 57° to 80.15°, re presenting an 529 opening efficiency of 35.66% in the circumferential direction. Overall, the movement 530 of the entire structure is well -coordinated. 531 32 532 Fig. 16 Parameters at each reference point of the fan -shaped variant retractable roof. 533 (a) and (b) denote the displacement -time relationship of point s A and B, respectively; 534 (c) depicts the distance of point A from the polar; (d) is the maximum angle ρ of the 535 retractable roof. (x-, y-, and z -disp. represent the displacement components of a 536 reference point in the three principal axis directions while T -disp. is the total 537 displacement of the corresponding point. ) 538 5. Conclusion 539 Three different retractable roof structure systems , inspired by the concept of 540 origami , are proposed in this paper , and the process of conducting thick plate analysis 541 and parametric modeling , adopting Grasshopper, is elaborately described. Theoretical 542 analysis of the three retractable roof structures is then carried out using geometric 543 formulas, followed by a detailed motion simulation analysis. Our main contributions 544 33 and conclusions: 545 (1) T hree novel retractable roof structure systems are proposed. We propose t hree 546 new types of retractable roof structure systems by combining the geometric principles 547 of the Miura-ori mechanism and its arched and fan -shaped variations with the scissor 548 hinge rectangular unit. The necessary conditions for the coordinated motion of the three 549 new retractable roof structure systems in two directions are further derived, and the 550 compatibility solution is proposed for case s where coordinated motion cannot be 551 achieved between the scissor joint and origami mechanisms. 552 (2) T he effect of thickening on the maximum opening ratio of the structure is 553 presented. The thickening method based on the shape adjustment technique is proposed, 554 and the formula for the maximum opening ratio of the new retractable roof system is 555 derived accordingly . The analysis results demonstrate that the effect of panel thickness 556 on the maximum opening ratio shows different patterns as the value of the notch length 557 varies. Specifically, as the depth of the cut increases while keeping the panel thickness 558 constant, the maximum opening ratio also increases. When it comes to the case where 559 the panel thickness is fixed and the depth of the cut varies, a consistent pattern is 560 observed : the maximum opening ratio increas e as the length of the notch enlarges . 561 (3) Parametric modeling steps are elaborately describe d. The rigid origami truss 562 structure , based on the principles of origami mechanism , is established by adopt ing a 563 “two-step” approach that uses Grasshopper and isogeometric segmentation ). The 564 modeling steps and parameter control methods are described in exhaustive detail. 565 34 (4) The kinematic patterns of the new retractable roof structures are investigated. 566 The simulation demonstrate s that all three types of structures can be un- and folded 567 smoothly without any occurrence of motion interference : 1) During the motion process 568 of the Miura -ori retractable roof structure , the primary movement is transverse , which 569 gradually shifts to longitudinal movement. It is sufficient to control the movement 570 during the initial phase of motion a s the speed of motion s lows down in the later stages. 571 2) As to the arched variant, the velocity relationship between the four key reference 572 points is such that vA≈vD>vC>vB, which suggests that the overall motion speed of the 573 edge ends and scissor joint units is greater than that of the middle part of the structure. 574 Therefore, in practic al applications, it is crucial to prioritize control of the speed of the 575 far end. 3) As the fan -shaped variant retractable roof structure unfolds, the z -component 576 of point B reflects a nearly const ant rate decrease in the structure height. Additionally, 577 the included angle ρ of the structure gradually decreases with increasing time t, with a 578 progressively slowing rate of decrease. 579 We, in this paper, primarily focus on the development of new structura l systems and 580 the theoretical analysis and motion simulation of their coordinated motions . The 581 following aspects , nevertheless, require further research . Firstly, there is still much 582 exploration needed for retractable roof structure systems that draw inspiration from 583 origami while also incorporating more practical designs . Moreover, f uture research is 584 recommended to concentrate on refining the design of the nodes and examining their 585 impact on the structural motion and mechanical performance , for we streamlined the 586 35 design of connection nodes in the current study . Lastly, it is of utmost importance to 587 carry out practical experiments to verify the workability o f the structures and the 588 precision of the theoretical investigations, owing to the cross -disciplinary nature of this 589 research. 590 Declaration of competition interest 591 All authors have no conflict of interest to declare. 592 Acknowledgment 593 This work was supported b y the National Natural Science Foun - dation of China 594 (Grant no. 51778129) , High Level Talent Projects of Six Talent Peaks in Jiangsu 595 Province, China (Grant no. JZ -010), Young Academic Leaders of Qing Lan Fund 596 Project of the Jiangsu Higher Education Institu tions of China. 597