Origami Frustration and Its Influence on Energy Landscapes of Origami Assemblies

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 Origami Frustration and Its Influence on Energy Landscapes of Origami Assemblies Shixi Zanga, Tuo Zhaoa, Diego Misseronib,2, and Glaucio H. Paulinoa,c,2 This manuscript was compiled on September 25, 2025 Harnessing instabilities of multicomponent multistable structural assemblies can potentially lead to scalable and reversible functionalities, which can be enhanced by exploring frustration. For instance, standard Kresling origami cells exhibit non-tunable intrinsic energy landscapes determined by their geometry and material properties, limiting their adaptability after fabrication. To overcome this limitation, we introduce frustration to enable fine-tuning of the energy landscape and resulting deformation states. By prestressing the Kresling cell by means of special springs with individual control, we induce either global or localized (i.e., crease level) frustration, which allows changing the energy barrier (cell or assembly). We investigate the mechanical behavior of frustrated Kresling assemblies, both theoretically and experimentally, under various loading and boundary conditions. Our findings reveal that changing the frustration state leads to precise control of folding sequences, enabling previously inaccessible folding paths. The proposed concept paves the way for innovative applications in mechanical metamaterials and other fields requiring highly programmable and reconfigurable systems. Geometrical frustration |Origami |Kresling pattern |Energy landscape R econfigurable assemblies consist of engineered macroscopic nonlinear structures undergoing large deformations. The behavior of the assemblies depends on the material properties and geometrical nonlinearity of the local unit cells. Classic nonlinear cells, such as snap-through beams ( 1 , 2 ), and buckling-driven continuum elements( 3 , 4 ), have been explored in applications, including energy absorption ( 5 , 6 ), soft robotic actuators ( 7 , 8 ), non-commutative response ( 9 ), wave propagation ( 10 ), acoustic metamaterials ( 11 , 12 ), and soft matter undergoing dramatic shape changes ( 13 , 14 ). More recently, origami-inspired geometry has enriched the design space of the nonlinear unit cells such as Kresling ( 15 – 20 ), squaretwist ( 21 , 22 ), Waterbomb ( 23 , 24 ), and Miura-Ori variations ( 25 – 27 ), and curved folds ( 28 , 29 ). Furthermore, the non-rigid origami assemblies have enabled finite deformation for applications involving shape morphing ( 30 – 32 ) and controllable energy landscape ( 32 – 34 ). The aforementioned reconfigurable assemblies assume a pre-defined deformation path; thus, the nonlinear properties, e.g., the shape of the energy landscape and the instability behavior, are non-tunable after fabrication. On the other hand, reprogrammable structures enable continuously variable elastic modulus via changing the configurational state of local units ( 35 ), e.g., heights of the elastic shells ( 36 ), rotation angles of the gears ( 37 ), and rolling motion of the cams ( 38 ). A limited number of studies have applied this reprogrammability concept for assemblies with tunable instability behaviors ( 39 , 40 ), e.g., switch between monostable and bistable responses. The switching behavior is achieved by actuating two distinct topological states of a local unit. However, the limited local states restrict the number of global deformation paths, which makes it difficult to reprogram the energy barrier of the assembly continuously. Here, we introduce geometrical frustration ( 41 ) into the origami-inspired assemblies, together with novel experimental fixtures (e.g., free-rotating and freetranslating), to achieve continuous energy landscape reprogrammability. The geometrical frustration is embedded within the origami cells by means of three mechanisms: global stretch, global rotation, and crease (local) stretch (Fig. 1A). Each mechanism integrates shell-based origami with special spring elements, which introduce prestress into the frustrated origami cell. The prestress level of the frustrated model is continuously adjustable by controlling the spring properties, i.e., stretching/rotating direction and magnitude. The frustrated assemblies, with tunable prestresses, enable one to engineer the energy landscape of multiple stable states on the fly. We achieve unprecedented folding paths, otherwise infeasible (Fig. 1B, Movie S1). This finding paves the way for potential Significance Statement Frustration: detrimental or desirable? Sometimes detrimental, but sometimes desirable to achieve new functionalities of nonrigid multistable origami structures. It provides the means to augment the energy landscape of such structures as it can be tailored to the features of the geometry of the origami unit cell and the frustration type. By equipping the cell with special, controllable, elastic springs, fine-tune control over energy barriers is enabled, facilitating precise folding sequences. Experiments demonstrate that activation or deactivation of the frustration can enhance the programmability of a multicell origami array, unlocking otherwise unfeasible folding paths. With potential impact in fields such as mechanical computing and noncommutative state transition, this approach offers possibilities for scalable and adaptable structures with high tunability. Author affiliations: a Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, 08544, USA; b Department of Civil, Environmental and Mechanical Engineering, University of Trento, Italy; c Princeton Materials Institute (PMI), Princeton University, Princeton, NJ 085444, USA Author contributions: S.Z., T.Z., D.M., and G.H.P. designed and performed research; S.Z., T.Z., and D.M. performed experiments; S.Z. implemented the computational approach; and S.Z., T.Z., D.M., and G.H.P. analyzed data, developed the theory, and wrote the paper. The authors declare no competing interest. 2 To whom correspondence should be addressed. E-mail: diego[email protected] or [email protected] www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS — September 25, 2025 — vol. XXX — no. XX — 1–12 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 tunable energy landscape intrinsic energy landscape u φ spring spring U u or φ l η l e e e U u or φ energy barrier (ΔU) base energy U u or φ U u or φ Standard Kresling cell assembly Frustrated Kresling cell assembly torsional spring global rotation global stretch crease stretch infeasible ΔU1 ΔU1 ΔU2 ΔU1 ΔU2> feasible ΔU1 ΔU1 ΔUfru ΔU1 ΔUfru < 0 80 0 T (N•mm) -20 40 φ (deg) φ u F=0 T A B =ΔU2-Uspr b δ Freetranslating fixture a b c δa c δ Fig. 1. Standard and geometrically frustrated origami assemblies with tunable energy landscapes and folding paths. (A) Top: Schematic of the standard Kresling origami (brown) and its intrinsic energy landscapes. The symbols u and φ denote displacement and twist angle, respectively. Bottom: Schematics of the frustrated models (orange) with three types of prestress: (i) global stretch, (ii) global rotation, and (iii) crease stretch (local) and their tunable energy landscapes. The symbols ℓe and ηe denote the length of the axial spring and the rotating angle of the torsional spring, respectively. (B) Top: an infeasible folding path using the standard Kresling assembly. Here, ∆ U1 and ∆ U2 denote the intrinsic energy barriers of the origami cells made of different materials. Bottom: frustrated assembly achieving an unprecedented folding deformation. Here, ∆ Ufru and Uspr denote the energy barrier of the frustrated model and the elastic energy stored in the torsional spring element, respectively. Insets show the rotational test setup with the free-translation fixture and experimental data of twist angle φversus torque T. applications in reconfigurable mechanical metamaterials and non-commutative state transitions. Results Theory of geometrical frustration. Starting from the theoretical modeling of the Kresling origami, which describes its mechanics through an elastic energy dependent on two independent variables, displacement u and twist angle φ , we develop an enhanced model for the frustrated system. We write the total elastic energy of the frustrated model, Ufru(u, φ), as follows: Ufru(u, φ) = Uspr(u, φ) + U(u, φ),[1] where U ( u, φ ) is the elastic energy of the standard Kresling cell ( 42 ), and Uspr ( u, φ ) denotes the elastic energy stored in the prestressed springs, which embed frustration into the origami cell. The five-term elastic energy of the standard cell is expressed as: U(u, φ) = 1 2nbks,b(b(u, φ)−b0)2 +1 2ncks,c(c(u, φ)−c0)2 +1 2nakr,a(δa(u, φ)−δa0)2[2] +1 2nbkr,b(δb(u, φ)−δb0)2 +1 2nckr,c(δc(u, φ)−δc0)2. We offer a detailed explanation of the material parameters ks,i ( i = band c ) and kr,i ( i = a, b, and c ), the geometric parameters b , c , δi ( i = a, b, and c ), and the parameters ni ( i = a, b, and c ) in the SI Appendix, section 1, Fig. S1, and Table S1. Using the principle of minimum total potential energy, the equilibrium conditions for axial and torque loading can be derived with Eq.1. Specifically, the axial force and torque can be calculated as follows: Ffru(u, φ) = ∂Uspr(u, φ) ∂u +∂U(u, φ) ∂u ,[3] Tfru(u, φ) = ∂Uspr(u, φ) ∂φ +∂U(u, φ) ∂φ .[4] Here, the expression of Uspr ( u, φ ) depends on the prestressed spring mechanism of interest. We present three types of frustrated models, i.e., global stretch, global rotation, and crease stretch, in the following sessions (see details of theoretical formulation in SI Appendix, section 1). Global stretch. This model involves a single deformed spring element inserted in the origami and aligned with its central axis. In the theoretical analysis, the initial state of the spring can be either extended or compressed, providing the origami with tunable prestress properties. The extended spring deforms the origami cell in the folding direction, while the compressed spring stretches the unit in the deploying direction. The prestressed spring is coupled with the origami 2— www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Zang et al. 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 cell to achieve a new equilibrium stage, denoted as the frustrated mode. The resulting energy landscape of the frustrated mode is programmable by adjusting the elastic energy stored in the spring element, defined as follows: Uspr(u, φ) = 1 2ks,e(∆ℓe−u)2,[5] where ks,e and ∆ ℓe are the stiffness and length change of the spring element embedded in the frustrated Kresling cell. Global rotation. This model takes advantage of the rotational degree of freedom of the Kresling origami and embeds torsional prestress into the unit cell. The prestress level is controlled by a torsional spring integrated with the origami cell. The spring rotates the undeformed cell and reaches a new equilibrium state with prestresses. The spring enables two types of prestresses, which are defined as positive and negative. The positive prestress rotates the cell in the same direction as its intrinsic twisting direction, while the negative prestress rotates the unit in the opposite direction. The elastic energy functional for the torsional springs is defined as follows: Uspr(u, φ) = 1 2kr,e(∆ηe−φ)2,[6] where kr,e and ∆ ηe are the stiffness and rotating angle change of the torsional spring integrated in the frustrated model. Crease (local) stretch. This model embeds prestressed springs along the mountain creases (local) of the origami cell to frustrate the system. Those local spring elements deform the unit into a new stable state with a non-zero base energy. The magnitude of the base energy and energy barrier of the frustrated model are tunable by controlling the elastic energy stored in the springs, defined as follows: Uspr(u, φ) = 1 2neks,e(b(u, φ)−b0+ ∆ℓe)2,[7] where ne is the number of the stretching springs along the mountain creases ( ne = 3 in this paper), b ( u, φ ) is the length of the mountain crease, b0 is the initial length of the mountain crease at the undeformed state of the Kresling cell, and ∆ ℓe is the length change of the spring element. Parametric study. The frustrated cells integrate three types of prestressed models into the standard Kresling origami. Eqs. 57show that the elastic energy stored in the prestressed springs is controlled by its deformation. Thus, we can navigate the energy landscape by varying the length change ∆ℓeand rotating angle change ∆ ηe of the spring elements, respectively (Fig. 2). We refer to SI Appendix, Table S2 for the selection of parameters. Recall that we denote the positive prestress as deforming the origami cell in the folding direction, while the negative prestress deforms the unit in the deploying direction. Both positive and negative prestress drive the undeformed origami into new equilibrium states with nonzero base energy, and the corresponding energy landscapes are tunable. For instance, the global stretch model enables two frustrated modes: one with negative prestress and the other one with positive prestress (Fig. 2A). Both modes have nonzero base energy at the initial stable state; however, in the negative mode, the energy at the second stable state always increases, while for the positive mode, the base energy of the 2nd stable state can either increase or remain unchanged depending on the given length change in the prestressed spring. Note that the base energy of the 2nd stable state for the positive mode is similar to the base energy of the standard Kresling. This behavior is due to the frustrating setup used in the positive mode, where the elongated spring becomes invalid after returning to its original length. By contrast, the springs are always compressed in the negative mode. Thus, the resulting energy landscapes for the two modes, negative and positive, can be quite different. This finding holds for results obtained using two independent loading conditions: compression with free-rotation (Fig. 2B-left) and torsion with free-translation (Fig. 2B-right). Notably, by controlling the level of prestress in the negative mode, we can switch the instability behavior from bistable to monostable, as shown in Fig. 2B (Top: two orange curves). Another unique feature is the capability to continuously program the energy barrier of the cell with frustration. Here, the energy barrier is defined as the difference between the local maximum on the landscape and the initial base energy. Fig. 2C shows the theoretical energy barrier as a function of the stretching length of the prestressed spring. Moreover, the curve shows a smooth transition between the negative mode and the positive mode. This result highlights the capability of the frustrated model to achieve fine-tuned control over energy barriers. Another frustration model we investigate is the global rotation (Fig. 2D), which also has two modes similar to the global stretch, i.e., negative mode and positive mode. The negative mode integrates a prestressed torsional spring rotating the origami opposite to the twisting direction of the cell while it folds. The positive mode embeds a torsional spring that rotates the origami along the same direction while the cell folds. By controlling the prestress level of the torsional springs, we can achieve non-zero bases for the initial stable states of the cell. Further, the shape of the energy landscape can be programmed as shown in Fig. 2E. As a result, the global rotation-induced frustrated cell has switchable instabilities, i.e., monostable or bistable behavior. Notably, the energy barrier between the local maximum and the initial base energycan be continuously tunable as shown in Fig. 2F. The last frustration model we investigate is the crease stretch shown in Fig. 2G. The prestressed springs are located along some mountain creases due to fabrication considerations, as shown in the later experimental validation sessions. The crease stretch has two prestressed modes, i.e., positive mode with elongated springs and negative mode with compressed springs. The two prestressed modes enable the frustrated cell with programmable energy landscapes (Fig. 2H). The stored elastic energy is contributed by the stretch of prestressed springs and the deformation of the origami panels and creases. The mountain creases of the Kresling cell shortens when folding is initiated, but the creases return to the initial length at the folded stable state. On the other hand, the deformation of the prestressed springs behaves differently under the two modes. For the negative mode, the prestressed spring has similar kinematics to the mountain crease. The compressed spring is further shortened while the cell is folding. Then, the spring returns to its initial compressed length at the folded stable state. For the positive mode, the spring is elongated at the initial stable state. As Zang et al. PNAS — September 25, 2025 — vol. XXX — no. XX — 3 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 the cell folding initiates, the amount of elongation reduces. At the folded stable state, the spring returns to its initial elongated state. We can see that the deformation history of the prestressed spring in the positive mode and negative mode is quite different. As a result, the two modes lead to distinct shapes of the energy landscape. This observation enriches the programmability of the frustrated cells by means of local crease control. Moreover, we present an additional local frustrated model, i.e., crease rotation, in the SI Appendix, section 1, Fig. S2, and Table S3. The aforementioned theoretical analysis considers linear spring mechanisms with constant stiffnesses, i.e., ks,e = constant and kr,e = constant . However, the theoretical framework (Eqs. 5-7) can be generalized to incorporate nonlinear springs in the frustrated model. We can replace ks,e and kr,e with expressions describing the nonlinear behavior of the spring elements. More details of the theory for nonlinear spring modeling are shown in SI Appendix, section 2, Fig. S3, and Table S4. Experimental study on unit cells. Experiments involve the three frustrated models theoretically investigated in the previous session. We develop a modular fabrication solution to realize the global stretch model, including the spring element, the wire connector, 3D-printed frames and handles (Fig. 3A, Movie S2). The handle controls the level of prestress in the spring element. The spring extension ∆ ℓ is a function of the radius of the handle rhandle and the number of interval turns n , defined as follows: ∆ ℓ = n (2 πrhandle ) / 12. The prestressed spring deforms the standard Kresling into a new equilibrium state configuration. This new stable state is frustrated as it stores elastic energy in both panels and the spring. The amount of energy stored in the frustrated model is tunable by controlling the spring extension. In the compression experiments with the free-rotating fixture, we test three different spring extensions and compare the results with those of the standard origami cell (Movie S2). Figure 3B reports the experimental results and theoretical prediction. For the standard Kresling, we conduct tests on five specimens and calculate the mean value (solid curves) and the corresponding standard deviation (shaded regions) using the formulas in SI Appendix, section 3. For the frustrated Krelsing, we conduct tests on three specimens for each prestressed model. The error bar at the first stable state is the standard deviation calculated from the measured displacement. The strain energy plots (Fig. 3B top-left) confirm that the frustrated models have non-zero base energy at the initial stable states, and the amount of the base energy depends on the spring extensions. Note that we assume that the spring elements do not contribute to the energy formulation anymore if they are deactivated (i.e., zero prestress). Consequently, the behavior of the frustrated cell becomes identical to that of the standard cell (Fig. 3B bottom-left). The plots in Fig. 3B (right) illustrate the force-displacement relationship obtained from both experiment and theory, respectively. We zoom in on the initial loading region to show the shift of force curves. Both experimental data and theoretical analysis verify that the starting point of the force curve shifts as the prestress increases. The amount of shift corresponds to the height change at the first stable state in the frustrated models. Note that the magnitude of the peak force decreases as more prestress is applied in experiments. In theoretical analyses, the peak forces are similar. This discrepancy is caused by the panel buckling at initial loading with activated frustration (Fig. 3B bottom-right), which is not considered in the theoretical analysis (see more discussion in SI Appendix, section 4). We design a specific mechanism to apply positive prestress in the global rotation model. This mechanism, which behaves like a torsional spring, involves the customized inclined component, the spring element, the wire connector, and 3D-printed frames and handles (Fig. 3C, Movie S3). The handle controls the amount of prestress applied along the torsional direction. The torsional angle ∆ η and torsional stiffness kr,e are defined as: ∆ η = n (2 πrhandle/rframe ) / 12 and kr,e = T/ ∆ η , where T is the reaction torque, and rframe is the radius of the frame. Compared with the standard cell, the prestressed cell is deformed in its initial stable configuration with a non-zero base energy. The elastic energy is stored in both the deformed panels and the springs. Given a constant spring stiffness, the torsional angle controls the magnitude of the energy stored in the initial configuration of the frustrated model. We test three different angles under torsional loading with the free-translating fixture (Movie S3). Both theoretical and experimental results (Fig. 3D-left) verify the capability of the global rotation model for tuning base energy at initial stable states. In addition, the twist angle-torque curves for deactivated and activated frustration are different as shown in Fig. 3D (right). The zoomed-in plots further illustrate the different initial loading points. The shift of those points is related to the amount of prestress applied in the global rotation frustrated model. The two aforementioned global frustrated models incorporate only positive prestresses. In contrast, the crease (local) stretch with negative prestress further enhances the energy landscape programmability of the frustrated model. The crease stretch prototype involves compressed springs inserted in 3D-printed cases aside from the mountain creases of the cell (Fig. 3E, Movie S4). Due to the prestressed spring elements, the frustrated cell gets extended and then stays in a new equilibrium state. The new state has a non-zero elastic base energy contributed by the deformed origami cell and the prestressed spring element. Tuning the magnitude of the prestress leads to controllable base energy at the initial stable states, as shown in Fig. 3F (top-left). Moreover, the experimental results show that the shape of the energy landscapes depends on prestress levels of the springs. The more the spring element is compressed, the higher energy barrier is achieved for the frustrated cell. This behavior agrees with the theoretical analysis shown in Fig. 3F (bottom-left). Figure 3F (right) verifies that the initial loading position of samples with deactivated and achieved frustration are different. The initial loading position is related to the configuration of the Kresling origami at the first stable state. The negative displacement u indicates increased height of the origami sample in the frustrated model. On the other hand, we validate the crease (local) stretch with positive prestress. We refer to SI Appendix, section 5 and Fig. S4 for more details. Experimental study on assemblies. Beyond the studies at the unit cell level, we explore the mechanical behavior of origami assemblies composed of standard cells and frustrated 4— www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Zang et al. 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 cells. The cells are modular, and they can be connected by miniature neodymium magnets embedded in the frames. Standard Kresling assemblies. For an origami array with three standard cells, there are, in total, eight stable states as shown in Fig. 4A. In theory, fifty-six folding paths connect any two stable states. In practice, sixteen out of fifty-six folding paths are unachievable without transiting through other stable states (Fig. 4B). For instance, state I (with all three cells deployed) cannot deform directly to state VIII (with all three cells folded) unless passing through the other states where one or two cells have been folded. Note that eight additional paths in Fig. 4C are infeasible due to the intrinsic energy barrier ∆ U built in the three cells, i.e., ∆ U (red cell) > ∆ U (yellow cell) ≈ ∆ U (blue cell), which are differentiated by controlling the panel thickness (see details in Materials and Methods). Moreover, state VIII cannot be pulled to state V because lower-energy barrier cells (blue or yellow) must deploy before the higher-energy barrier cell (red). As a result, only thirty-two out of fifty-six folding paths are feasible (Fig. 4D). Eight feasible paths have been verified experimentally under two types of loading conditions: axial loading and torsional loading (Movie S5, details in Movie S5 are shown in SI Appendix, section 6 and Fig. S5). The axial loading condition involves two fixtures, i.e., one is rotationally constrained, and the other one displays free rotation (see more details in SI Appendix, section 7). The corresponding testing results are presented in Fig. 4E. On the other hand, the torsional loading condition is equipped with the axially constrained fixture and the free-translating fixture, respectively. The corresponding experimental data are shown in Fig. 4F. According to the experimental data, we calculate the energy landscapes of eight feasible paths in SI Appendix, section 8 and Fig. S6. Frustrated Kresling assemblies. The 3-cell origami assembly shown in Fig. 5A is prestressed such that the top two cells include springs on the mountain creases (local level) and the bottom cell includes a rotational spring (global level). Each of the frustrated cells has two stable states with tunable energy barriers enabled by the prestressing level. As a result, the assembly has, in total, eight stable states, like Fig. 4; however, the behavior is quite different. The frustrated assembly can be continuously reprogrammed by means of local cell control. This reprogrammability leads to precise control of the folding sequences, and enables the eight previously infeasible folding paths of Fig. 4C to be achieved as feasible folding paths in Fig. 5B. For example, although the intrinsic energy barrier of the red cell is bigger than that of the blue cell, the global rotation mechanism actively lowers the red cell energy barrier. Therefore, when state I deforms to state VII following path 1  , the frustrated red cell folds while the blue cell remains deployed during the process. The loading condition for this deformation is axial compression with a rotationally constrained fixture, as shown by the green curve in Fig. 5C (Movie S6). The plot has an orange curve, which corresponds to the feasible deformation path 2  in Fig. 5B, under axial loading with free-rotating fixture (Movie S6). Other feasible folding paths, i.e., path 3  and path 4  , are achieved by torsional loading conditions with the axially constrained fixture and the free-translating fixture, respectively, as shown in Fig. 5D (Movie S6). In addition, energy landscapes corresponding to feasible paths in Fig. 5C and D are presented in SI Appendix, section 8 and Fig. S7. More details of the reprogrammability of the feasible folding paths and energy landscapes are shown in SI Appendix, section 9 and Fig. S8. The results in Fig. 5demonstrate that the ability to switch among different states, by controlling the prestressing levels, enables the 8 feasible folding paths of Fig. 5B, something not achievable with the non-frustrated Kresling array (Fig. 4). While these folding paths could also be enabled by designing new cells with predefined energy barriers, through material and/or geometry selections, this discrete approach inevitably introduces new infeasible folding paths and does not support in-situ reconfiguration. In contrast, the continuous approach of geometric frustration, combined with a finely tuned spring mechanism, enables the elimination of infeasible folding paths. This allows for dynamic reprogramming of folding behavior within the same array, leading to adaptability and control. Scope of frustration. Though our designs of frustration is created based on Kresling origami structure, they can be used to embed frustration into other origami structures. For instance, Figure 6A gives an example to explore the influence of prestressing on a curved-crease origami tube. The states at the red and blue points as well as the force curves indicate that the mechanism with stretching spring causes a shift of initial state of the origami tube. Under axial compression loading, the frustrated curved-crease tube exhibits significant panel buckling (see the state at the blue-star point in Fig. 6A). Moreover, detailed comparison between the frustrated and non-frustrated curved-crease origami is shown in SI Appendix, section 10 and Fig. S9. The second application relates to programmable noncommutative behavior of Kresling arrays (Fig. 6B). We consider a Kresling array consisting of two lower energy barrier cells at the top and one higher energy barrier cell at the bottom, and consider two examples, one that does not involve frustration and one that does. In both examples, we apply counterclockwise twist followed by clockwise twist, and then we reverse the actuation sequence (i.e., clockwise twist followed by counterclockwise twist), always resulting in a total zero net twist at the end of each actuation sequence. For the first example, the array shows history-dependent behavior in the sense that the deformed configuration depends on the sequence of the twist actuation. We demonstrate this feature by experiments on a reference configuration under the actuation sequences described above (Fig. 6B-top). Note that details of the jagged portion on the unloading curve are elaborated in SI Appendix, section 11 and Fig. S10. The two different end configurations demonstrate the relevance of twisting history, which indicates non-commutative behavior. For the example involving frustration, the top unit cell (blue) is the same as before; however, the bottom two cells are frustrated. The middle cell has linear (local) springs (with induced negative prestress) providing it with a higher energy barrier than the top cell. The bottom cell has an especially designed torsional (global) spring (with induced positive prestress) providing it with the highest energy barrier of the assembly. We observe non-commutative state transitions as well (Fig. 6B-bottom); however, comparatively, the two final states in the frustrated Kresling array are different from those in the non-frustrated Kresling array, resulting in Zang et al. PNAS — September 25, 2025 — vol. XXX — no. XX — 5 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 a programmable (frustration-dependent) non-commutative state transition. Furthermore, we create a shape-morphing metamaterial by embedding the spring mechanism into a 3D-printed truss prototype involving multiple Kresling columns ( 43 ) (Fig. 6C). The level of prestress in each Kresling column depends on the stiffness of springs. Here, the column 0 has no spring, while the stiffness of the springs in column 1 and column 2 is 0 . 11 N/mm and 0 . 35 N/mm , respectively. All the springs are connected with a handle through wires. Rotating the handle applies a constant stretch to all the springs. Since the spring of column 2 has the highest stiffness, it deforms more than column 1 and column 0 (see Fig. 6C-bottom). As a result, the varying deformation in different columns allows the metamaterial to achieve shape-morphing behavior. Discussion We present geometrically frustrated Kresling assemblies with tunable energy landscapes and folding paths. The assembly is modular, and it consists of both standard origami cells and frustrated cells. We introduce frustrated modules with three types of prestress, i.e., global stretch, global rotation, and crease (local) stretch. The prestress of the frustrated model is continuously adjustable by controlling the special springs, which allows for changing the energy barrier for the cell. The theoretical analysis verifies that the energy landscape for the frustrated cells is programmable and the corresponding energy barrier is continuously tunable. We prototype all three types of frustrated origami cells and use four loading and boundary conditions to validate the behavior of the cells experimentally. Experiments demonstrate that activating and deactivating frustration can dramatically enhance the programmability of the origami assembly, unlocking otherwise infeasible folding paths. The present concept can be implemented widely in reconfigurable systems where in-situ programmability is needed in the application, e.g., adaptable metamaterials for shape morphing. Materials and Methods Formulation of the frustration theory. Since the frustrated cell can be controlled by both axial force, F , and torque, T , the work done on the cell is calculated by W ( u, φ ) = RFdu + RTdφ . The total potential energy of the frustrated cell, Π, can be expressed using the total elastic energy, Ufru, and work, W, i.e., Π(u, φ) = Ufru(u, φ)−W(u, φ).[8] Notice that Π( u, φ ) is a function of two independent variables, u and φ . Based on the principle of minimum total potential energy, equilibrium is achieved when ∂ Π /∂u = 0 and ∂ Π /∂φ = 0. Thus, the axial force, F, and the torque, T, are calculated by: F(u, φ) = ∂Ufru(u, φ) ∂u , T(u, φ) = ∂Ufru(u, φ) ∂φ .[9] Fabrication of Kresling origami cells. Both standard and frustrated origami cells were fabricated by a material composed of muti-layer origami papers (Tant) and adhesive tapes in between (3M 9474LE, 0.17mm-thick).The crease patterns of the blue and yellow cells include two layers of origami papers and one layer of adhesive tape. The crease pattern of the red cell is made of three layers of origami papers and two layers of adhesive tapes. We cut the mountain creases for all the cells. Additionally, for the crease (local) stretch (Fig. 3E) design, we cut a trapezoid hole on the panel to avoid interaction between the prestressed element and the origami cell. More details are shown in SI Appendix, section 12 and Fig. S11. Fabrication of 3D-printed truss. The 3D-printed truss model in Fig. 6C consists of two components: rods and soft joints, which are fabricated using a Stratasys J55 Prime polyjet printer. The rods are printed using the VeroWhite material, and the joints are printed using FLXA9950 (Shore-A 50), with a mix of VeroUltraClear and ElasticoClear. Fabrication of prestressed elements. The 3D printed prestressed components are fabricated using a Stratasys J55 Prime polyjet printer. For the global stretch model, the frames and handle are printed by the VeroWhite material, and the markers are printed by the VeroMagenta material. For the global rotation model, the top frame and markers are fabricated by the same material as the global stretch design, while the bottom frame and handle are printed by the VeroUltraClear material. For the crease (local) stretch model, the frames and markers are printed by the same material as the global stretch one, while the case assembled on the top and bottom frames printed by the VeroYellow and VeroMagenta materials,respectively. The pulley used in the global rotation model is a ball bearing pulley (MiSUMi, SZV3-12). The spring used in the global stretch model is an extension spring (McMaster-Carr, 9065K566). The spring used in the global rotation model is an extension spring (McMaster-Carr, 5108N036). The spring used in the crease (local) stretch model is a compression spring (McMasterCarr, 9657K641). The springs in the global models are connected to the 3D-printed components using a 0.3 mm diameter fishing wire. More detailed parameters are provided in SI Appendix, section 12. Note that the spring system can be manufactured by standard 3D printers, which enhances the practicality of the frustration concept. For instance, we show an illustration of integrating 3D-printed springs into cell origami in SI Appendix, section 12 and Fig. S12. Experimental setups. In Fig. 3, we conduct both compression tests and torsion tests on an Instron loading frame machine (Model 68SC5 Single Column Testing System), equipped with free-rotating and free-translating fixtures, respectively (see more information in SI Appendix, section 12 and Fig. S13). The applied axial load and torque have been measured with a force/torque sensor (Biaxial Load Cell ±445 N, ±5.65 Nm). The compression experiments are conducted at a speed of 0.25 mm/s, while the torsional experiments are performed at 0.5 deg/s. In Figs. 4and 5, we conduct compression tests using both rotationally constrained and free-rotating fixtures. 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XX — 7 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 B U (mJ) 0 50 0 0.5 1 u/um U (mJ) 0 50 φ/φm 0 0.5 1 100 100 U/φ = 0 U/u = 0 A 0 0.5 1 0 U (mJ) 50 u/um U (mJ) 0 50 φ/φm 0 0.5 1 100 100 global stretch global rotation crease stretch h0hfru u l e l e,0 Δl e l e Δl e negative positive hfru u 0 0 Δl =0 e negative positive 012 0 20 40 60 ΔU (mJ) Δl e/um negative positive Ufru I Ufru II Umax ΔU=U - max Ufru I E U (mJ) 0 50 0 0.5 1 u/um U (mJ) 0 50 φ/φm 0 0.5 1 100 100 D 0 0.5 1 0 U (mJ) 50 u/um U (mJ) 0 50 φ/φm 0 0.5 1 100 100 negative positive negative positive 012 0 20 40 60 ΔU (mJ) Δη e/φm negative positive Ufru I Ufru II Umax H U (mJ) 0 50 0 0.5 1 u/um U (mJ) 0 50 φ/φm 0 0.5 1 100 100 G 0 0.5 1 0 U (mJ) 50 u/um U (mJ) 0 50 φ/φm 0 0.5 1 100 100 fru l e l e,0 Δl e l e Δl e negative positive Δl =0 e negative positive 0 0.75 0 20 40 60 ΔU (mJ) Δl e/um negative positive Ufru IUfru II Umax Δη e Δη e Δη e=0 θfru θ0 φ 0 θfru φ 0 b0b fru b 80 global stretch ( -0.6 -0.3 0.5 1Δle/um): crease stretch ( -0.12 -0.06 0.3 0.6Δle/um):global rotation ( -1 -0.5 0.6 1.2):Δηe/φm 1 stable state (I) st 2 stable state (II) nd standard Kresling -0.44 C F I -0.79 -0.71 80 Fig. 2. Theoretical model of geometrically frustrated Kresling origami cells. (A) Global stretch feature. Standard Kresling origami (top-middle) and two frustrated models (top-left and top-right). The left model has a compressed spring (negative prestress), while the right one has an extended spring (positive prestress). Here, h0 denotes the height of the standard Kresling cell, hfru is the height of the frustrated cell, and u0 is the height difference. (B) Reprogrammable energy landscapes. Top: intrinsic energy landscape (black) versus tunable landscapes (orange) with the negative prestress model. Bottom: intrinsic energy landscape (black) versus tunable landscapes (orange) with the positive prestress model. Left: the elastic energy U versus the normalized axial displacement u/um under axial loading with free-rotation. Right: the elastic energy U versus the normalized twist angle φ/φm under torsional loading with free-translation. (C) Continuously tunable energy barrier with the global stretch feature. Normalized length changes of the spring ∆ ℓe/um versus the energy barrier ∆ U , which is defined as the maximum energy of a frustrated model Umax minus the energy at the first stable state UI fru . (D) Global rotation feature. Standard Kresling origami (top-middle) and two frustrated models (top-left and top-right). The left model has a deformed torsional spring that rotates the origami clockwise. The right model has a deformed torsional spring that rotates the origami counterclockwise. Here, ∆ ηe denotes the rotating angle of the torsional spring. (E) Reprogrammable energy landscapes. (F) Continuously tunable energy barriers for the frustrated model with global rotation. (G-I) Crease (local) stretch feature and the corresponding energy solutions. The symbols b0and bfru denote the lengths of mountain creases in the standard and the frustrated model, respectively. 8— www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Zang et al. 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 A Δl rhandle n interval F wire handle spring global stretch (positive spring) B u0 Δl1=5.24mm stable states T=0 F u φ U (mJ) 0 15 30 0 U (mJ) 20 F (N) 0 1.5 30 3 ExperimentTheory u (mm) 0 8 0 20 10 45 U (mJ) u (mm) 0 8 -0.5 0 0.5 u (mm) -1.5 0 15 30 45 F (N) u (mm) Δl2=6.55mm Δl3=7.85mm standard Kresling 0 20 0 8 0 20 30 10 U (mJ) u (mm) 0 15 30 u (mm) 45 u (mm) 0 15 30 45 F (N) 0 1.5 3 -1.5 0 8 0 0.5 F (N) u (mm) -0.5 s0s1 Freerotating fixture 40 40 C Δl rhandle n interval T Δη rframe wire handle spring φ0 global rotation (positive spring) D U (mJ) 0 60 90 U (mJ) 0 30 φ (deg) F=0 T φ u ExperimentTheory φ (deg) 0 6 0 10 0 20 30 10 30 Δη1=6.86ostable states Δη2=11.43oΔη3=13.72ostandard Kresling U (mJ) φ (deg) 0 10 -10 0 10 -30 -60 60 0 60 9030 φ (deg) 0 20 30 10 0 60 90 φ (deg) 30 0 30 φ (deg) 0 6 0 10 U (mJ) φ (deg) 0 10 -10 0 10 -30 -60 60 0 60 9030 φ (deg) T (N mm) T (N mm) T (N mm) T (N mm) s0s1 Freetranslating fixture F U (mJ) F (N) 0 30 45 0 U (mJ) 40 F (N) -3 0 60 3 crease stretch (negative spring) Δl F restriction spring E s0 bl b0 b0 T=0 F u φ ExperimentTheory u (mm) u (mm) 15 0 30 4515 20 0 40 60 20 Δl1= -2mm stable states Δl2= -3mm standard Kresling 6 -3 0 3 6 0 30 45 u (mm) u (mm) 15 0 30 4515 -2 2 0 1.5 F (N) u (mm) -1.5 -2 0 0 1.5 F (N) u (mm) -1.5 -2 2 u (mm) 0 10 U (mJ) -2 2 u (mm) 0 10 U (mJ) s1 2 0 0 0 Freerotating fixture Fig. 3. Experiments involving the three frustrated models of the previous figure. (A) Global stretch with positive prestress. Top-left: illustration of axial loading with a free-rotating fixture. Top-right: schematic of the design enabling prestress in the axial direction. Bottom-left: photo of an undeformed origami cell. Bottom-right: photo of a prestressed unit in its initial stable state. (B) Experimental results and theoretical predictions. Top: experimental results comparing the behavior of the frustrated models with the standard origami cell. Solid lines represent the mean value and shade regions represent the standard deviation of the experimental data. Bottom: corresponding theoretical predictions. Left: axial displacement u versus stored elastic energy U . Right: displacement versus applied forces F . The insets highlight the early stages of the test. (C) Global rotation with positive prestress. Top-left: illustration of torsional loading with a free-translating fixture. Top-right: schematic of the design with torsional prestress in the cell. Bottom-left: photo of an undeformed origami cell. Bottom-right: photo of a prestressed cell in its initial stable state. (D) Experimental results and theoretical predictions. (E) Crease (local) stretch with negative prestress. Top-left: illustration of the loading with a free-rotating fixture. Top-right: schematic of the compressed springs strategically located along the mountain creases. Bottom-left: photo of an undeformed origami cell. Bottom-right: photo of a prestressed cell in its initial stable state. (F) Experimental results and theoretical predictions. Zang et al. PNAS — September 25, 2025 — vol. XXX — no. XX — 9