Modular multi-degree-of-freedom soft origami robots with reprogrammable electrothermal actuation

Papilio zelicaon is a common inhabitant of North America, consisting of 12 body segments with 3 pairs of thoracic legs and 4 pairs of prolegs (Fig. 1A). The 7 body segments marked in Fig. 1B show deformation modes of contraction and bending.

To mimic the deformation modes observed from the caterpillar segments, we designed a symmetric octagon Kresling unit consisting of two thermal bimorph actuators assembled on the two opposing sides of the unit (Fig. 1C). The octagon geometry is preferable as it requires less energy to deploy compared to the hexagon and square designs (SI Appendix, Fig. S1). The geometry of the Kresling origami is determined by four independent parameters, i.e., height ${\mathrm{H}}_0$ in the folded state of the unit, height ${\mathrm{H}}_1$ in the deployed (initial) state, number $\mathrm{n}$ of polygon edges, and the corresponding edge length $\mathrm{b}$ (see detailed geometrical representation of the Kresling origami in SI Appendix, section 1). The four geometry parameters, ${\mathrm{H}}_1$, ${\mathrm{H}}_0$, $\mathrm{n}$, and $\mathrm{b}$, need to satisfy the following design constraint to avoid the locking stage where valley folds intersect in the folded state, i.e.,

Fig. 1D shows the simulation of Kresling units connected in series mimicking the configuration of the caterpillar in Fig. 1B. The corresponding electric currents through all 14 actuators in the 7 units are shown in Fig. 1E (dimensions of the Kresling unit and the heating pattern are provided in SI Appendix, Fig. S2). For example, when both actuators of unit [1] ([1]-1 located on the outer side of the curved body unit and [1]-2 on the inner side) are powered with the same current, unit [1] shows a symmetric contraction just like the axial contraction of segment 1 in Fig. 1B. When the current inputs are applied differently on some units, e.g., unit [5], the actuator [5]-2 generates a larger deformation than the actuator [5]-1 resulting in bending deformation toward the actuator [5]-2. The bending angle θ is defined by the relative angle between the two octagon panels. Fig. 1F shows the bending angle θ of each Kresling unit extracted from Fig. 1D. It can be observed that with the increase of current input on both actuators, the contraction of the Kresling unit increases. With the increase of current difference ΔI on the two actuators, the bending angle increases. With the individual control of distributed actuators, the Kresling unit–enabled soft robot allows spatial and temporal programming of its deformation. Moreover, Fig. 1G (Top) shows that the strain energy stored in the origami unit for the bending deformation increases as the bending angle increases. In addition, the two energy minimums (Fig. 1 G, Middle) verify that the intrinsic Kresling origami design has two kinematic stable configurations. In practice, as the thermal bimorph actuators are fully integrated into the origami panels, the folding stiffness $\left(\overline{{\mathrm{k}}_{\mathrm{r}}}\right)$ increases; thus, the Kresling unit behaves monostably (43). This is verified by the numerical result in Fig. 1G (Bottom) showing that the strain energy monotonically increases with vertical displacement. A parametric study (SI Appendix, section 2) shows how the stiffness $\left(\overline{{\mathrm{k}}_{\mathrm{r}}}\right)$ influences the monostable and bistable behaviors, respectively.

To achieve precise actuation of the Kresling units, we designed and fabricated parallelogram-shaped thermal bimorph actuators, following the same shape as the side panel of the Kresling pattern, which can be directly integrated onto the Kresling pattern (Fig. 2). Silver nanowires (AgNWs) have been adopted as the heating material in the soft actuator due to their excellent electric conductivity and mechanical compliance (24, 44–47). In this work, we used AgNWs embedded just below the surface of a polyimide (PI) matrix (48, 49) (Fig. 2 A–D). The AgNWs (~30 μm in length and 100 nm in diameter) were in the form of a percolation network while the PI resin was spun-coated on top of the AgNW network and cured; the serpentine-shaped heating pattern was defined by laser cutting of the AgNW/PI composite film on a sacrificial substrate. The thermal actuator is a bimorph structure with a layer of commercial PI film (with an adhesive coating) and a layer of liquid crystal elastomer (LCE) ribbon laminated on the two sides of the AgNW/PI heater (Fig. 2E). The LCE ribbon was fabricated by mechanical stretching of a rectangular flat LCE strip synthesized by two-stage polymerization (50, 51). Plasma treatment and mechanical pressure were applied to form a strong bond between the AgNW/PI composite film and the LCE ribbon. The mesogen alignment of the LCE ribbon, laid perpendicular to the longer diagonal of the parallelogram, ensures that the bending deformation of the actuators can be converted to the folding of the Kresling origami. The Kresling origami pattern was made of a piece of laser-cut polyethylene terephthalate (PET) sheet (Fig. 2E). Two opposing panels of the PET Kresling pattern were replaced with the thermal bimorph actuators (SI Appendix, Fig. S2A). The new pattern with integrated actuators was assembled with another octagon end panel to form a 3D Kresling unit (Fig. 2F). To minimize the interference to the locomotion, we have designed a hollow cylindrical region in each Kresling unit so that the wires attached to all units are inside the hollow structure of the origami, coming out from the rear end of the robot. In this way, the wires do not interfere with the contact between the robot and the substrate. This simple actuator-integration approach introduces very little weight to the original origami design, leading to a compact and lightweight robotic system.

When electric current is applied to the AgNW network, heat is generated as a result of Joule heating and transferred to the PI layer and the LCE layer (Fig. 3A). An infrared (IR) image shows a uniform temperature increase along the serpentine pattern (with a line width of 1.3 mm). The sheet resistance of the AgNW/PI composite after annealing was 0.55 Ω/sq, resulting in a total resistance of 40.7 Ω for each actuator. The dimensions of the AgNW heating pattern are given in SI Appendix, Fig. S2B. Fig. 3B shows the heating performance of the AgNW/PI composite heater. With the increase of applied current, the maximum temperature increased. With 0.125 A current applied (5 V), the temperature of the heater rose from room temperature to 80 °C within 10 s. In the following experiments, the actuators and robots are operated in a constant room temperature environment (23 °C). The LCE ribbons yielded a large compressive strain (~30%) within the temperature window between 65 and 80 °C (SI Appendix, Fig. S3). The high heating efficiency of the AgNW/PI heater and the low responsive temperature of LCE together enabled the fast actuation of the thermal actuators. Of note is that most reported thermal bimorph actuators use two thermal-expanding materials with different CTEs while in this actuator one layer expands and the other layer shrinks. This opposite thermal response increases the bending amplitude and speed compared to traditional bimorph actuators. Our previous study of PDMS/LCE bimorph actuators provided a detailed analysis of the bending performance with respect to modulus ratio, thickness ratio, and heating temperature (8). A brief summary is provided in SI Appendix, section 3 and Fig. S4. Fig. 3C shows the folding angle of the side panel which is measured across the crease line as defined in the Kresling origami (Inset of Fig. 3C). With 0.125 A applied current, the effective folding angle reached 120° within 12 s. During the folding and unfolding, the resistance of the AgNW heater is fully recoverable, which enables the cyclic operation of the actuators (SI Appendix, Fig. S5). With a further increase of the current, the bending angle can go beyond 120°, which, however, could cause delamination between LCE and PI in the bimorph actuator. In the following discussion, we choose the current of 0.125 A for efficient and stable actuation.

Fig. 3 D and E show snapshots of the Kresling unit in the axial contraction mode, and in the bending mode (Movie S1), respectively. With the same current on both actuators, the lateral Kresling origami panels fold symmetrically along the predefined creases, causing a coupled rotation and contraction motion (no bending). Different currents on both actuators lead to a bending motion coupled with rotation and contraction. The contraction (H₁ − h)/H₁ and rotation angle α are defined by the relative distance and rotation of the top octagon panel with respect to the bottom panel. The bending angle θ is defined as the angle between the horizontal direction and the top octagon panel. Fig. 3F captures the contraction and rotation of the Kresling unit with the 0.125 A current on for 11 s and then turned off. A maximum height change of 25% and rotation angle of 22.4° were achieved. With the cooling of the heater and relaxation of the structure, the Kresling unit recovered within 10 s. Fig. 3G shows the minimum relative height (Left), the maximum rotation angle (Middle), and the maximum bending angle (Right) versus the applied current (or current difference). With 0.125 A on one actuator and 0 A on the counterpart, a maximum bending angle of 27.3° was achieved.

The crawling mechanism of caterpillars involves sequential activation and deactivation of multiple body segments as shown in Fig. 4A. The wave of cyclic crawling motion normally starts from the terminal segment and ends at the thoracic legs. A complete travel of contraction wave from tail to head is termed a “crawl.” To demonstrate the feasibility of mimicking the crawling motion, we built a caterpillar-inspired crawling robot consisting of an interlaced series of three active Kresling units (each with two actuators marked in yellow) and four passive units (same setup but without actuators, marked in white). Fig. 4B shows schematically one cycle of crawling motion with sequential activation of Kresling units 1, 2, and 3 (Movie S2). When unit 1 is activated in the contraction mode, the friction force generated by the far-left passive unit is less than that by the rest of the robot. As a result, the tail of the robot slides rightward on the ground. In the next step, unit 1 is deactivated and cools down while unit 2 is activated at the same time. During this step, unit 1 extends to the original length while unit 2 contracts, resulting in a net force to translate the passive unit between units 1 and 2 to the right without moving the tail or the head of the soft robot. Because the heating and cooling of the heater require roughly the same duration of time (~11 and 10 s), the deactivation of unit 1 and activation of unit 2 are programmed seamlessly. During the following step, unit 2 is deactivated, and unit 3 is activated, repeating the translation but for the next passive unit. In the final step, the deactivation of unit 3 pushes the far-right passive unit forward, completing a forward crawling cycle.

During the crawling motion, friction is a key factor that affects the locomotion of our crawling robot. The condition for the locomotion is that the output force of the active units is larger than the friction force,

where F_output is the output force of the active unit, W_passive is the gravity force of one passive unit, and μ is the friction coefficient between PET and the substrate material. The output force of the Kresling unit has been measured and shown in SI Appendix, Fig. S6. For example, with applied current of 0.125 A, F_output = 0.23 N, W_passive = 6.86 × 10⁻³ N, the critical coefficient of friction for locomotion μ_c = 33.5. In the experiments, we have shown the locomotion of our soft robot (SI Appendix, Fig. S7A) on various surfaces, including synthetic leather mat, nylon cloth, and marble surface (SI Appendix, Fig. S7B). We measure the static friction coefficients of PET (major component of the robot body) on all three substrates using the setup in SI Appendix, Fig. S7C. The measured friction coefficients are shown in SI Appendix, Fig. S7D.

It is worth noting that the soft robot itself is a symmetric structure without any preference for crawling direction. The only factor that determines the crawling direction is the activation sequence of units 1, 2, and 3 (Fig. 4C, ①–⑤). With a reversed activation sequence (3, 2, and 1), as shown in Fig. 4C, ⑥–⑧, the soft robot can crawl backward to the original position. Fig. 4D shows the programmed current applied on the active units. Fig. 4E shows the rigid-body displacement (translation) of the soft robot finishing a cycle of forward crawling followed by a cycle of backward crawling. The cycles of forward and backward crawling each take ~40 s, which translates to an average crawling speed of 0.195 mm/s. The capability of bidirectional locomotion can be useful, for example, in search and rescue operations. In the 7-unit crawling robot, we adopted a symmetric design of chirality (SI Appendix, section 4). If we note the first passive unit as “L” in chirality and “R” for the opposite, the soft robot possesses the sequence LRRLRRL.

In general, steering direction during locomotion has been challenging for soft robots. To navigate through complex tree branches and leaves, caterpillars have extremely flexible body segments between the thoracic legs and prolegs that can bend in any direction. Inspired by the caterpillar bending, we have demonstrated omnidirectional crawling with a programmable trajectory. The three active units of the robot are now delegated with different tasks—a “steering” unit (unit 2 in Fig. 5A) guiding the direction and two “driving” units (units 1 and 3 in Fig. 5A) for propulsion (locomotion). The schematics and photographs in Fig. 5 A and B, respectively, show the top–down snapshots of one cycle of the steering locomotion (Movie S3). The steering cycle starts by applying different currents on the two actuators in unit 2. The bending of unit 2 defines an initial curvature. Then, units 1 and 3 are activated sequentially both in the contraction mode. Like the peristaltic crawling locomotion, the tail is dragged forward in the third snapshot. With the deactivation of unit 1 and activation of unit 3, unit 2 and its two neighboring passive units travel forward following the curved direction. Finally, with the deactivation of unit 3, the head takes one step forward and turns away from its starting direction (Fig. 5B). With the local bending capability, the soft robot can be programmed to follow any in-plane trajectory while respecting a maximum curvature limit. For example, the robot can be programmed to follow an S-shaped trajectory (Fig. 5C and Movie S4) or even a tree crotch (Fig. 5D). The key to programming the S-shaped trajectory is to actively control and follow the real-time curvature of the locomotion. Fig. 5E shows the difference of input current ΔI on the two actuators of the steering unit 2 with respect to time. From 0 to 400 s, the snapshots ①, ②, and ③ show a gradual increase of the bending curvature (from ρ₁ = 25.8 cm to ρ₂ = 15.5 cm) due to the increasing ΔI (from 0.075 A to 0.125 A). Then, the steering unit is deactivated for 400 s to complete a period of rectilinear motion (snapshot ④). From 900 to 1,200 s, a negative ΔI with decreasing magnitude (from −0.125 to −0.075 A) is applied to the steering unit 2 for reversed bending. Due to the symmetric Kresling structure with two addressable bimorph actuators, the second half of the trajectory (from ⑤ to ⑦) resembles the first half with the negative curvature, which completes an S-shape. Fig. 5F shows the cyclic current input on the two driving units 1 and 3 for the S-shaped locomotion.

Modular robots are constructed from multiple interchangeable and customizable modules, each of which contributes to their overall functionality (52, 53). Incorporating the modular concept into soft robots, we achieve versatility, customizability, scalability, and even better adaptability to the environment (54). The Kresling origami units are excellent modular components due to their multiple deformation modes and symmetric design.

We propose a two-level module design for Kresling robots: single-unit modules and multi-unit modules (Fig. 6A). The single-unit modules can be divided into two types, the active units which are equipped with actuators, and the passive units without actuators. With active units and passive units connected through the Top/Bottom octagon panels, a multi-unit module can be constructed with properly designated functions. To name the modules, 1 active unit and 1 passive unit would form an AP module; and similarly, 1 passive unit, 1 active unit, and 1 passive unit in sequence would form a PAP module.

We demonstrate the module concept by having an APA module pick up a cargo P module and then assemble with a PAP module for obtaining steering functionality (Fig. 6B and Movie S5). To start, a PAP module sits on the table in “standby” mode. Note that this module has a symmetrical configuration with only one active unit in the center. The PAP module can be activated, but unable to locomote, unless an asymmetric “feet” design is introduced (55). An APA module, on the other hand, has two active units which can be activated sequentially to navigate either forward or backward. In the first stage, both active units of the APA module serve as driving units to navigate rightward to approach the cargo P module (a single passive unit). When the two modules are close enough, the opposing octagon panels with embedded magnets attract each other and assemble into one (Inset of Fig. 6B). The new APAP module now carries the cargo P module on the tail. By changing the actuation sequence of the two active units, the APAP module can switch locomotion direction and move left. Again, when the APAP module approaches the PAP module, they merge through the embedded magnets and become a combined PAPAPAP module. Compared with the APA module, the PAPAPAP module is capable of not only forward and backward locomotion but also steering motion. Movie S5 captures this transition from linear locomotion to steering locomotion (the two phases of Movie S5 were taken separately and joined together as one). Of note is that the middle active unit, which initially serves as a driving unit, now reprograms its role to guide the steering curvature of the combined PAPAPAP module for bending deformation. With the steering function, the crawling robot can carry the cargo deviating from its original 1-D space. This modular design concept can be extended by increasing the number of units and sequence of the active and passive units, and possibly 3D assembly with customized connectors.

The LCE ribbons were fabricated following the thiol-acrylate Michael addition reaction method (61). First, 2 g of the liquid crystal mesogenic monomer, 1,4-bis-[4-(3-acryloyloxypropyloxy)benzoyloxy]-2-methylbenzene (RM 257) was dissolved in 0.7 g of toluene at 85 °C and cooled down to room temperature. Then, 0.42 g of the chain extender 2,2′-(ethylenedioxy) diethanethiol (EDDET, Sigma-Aldrich), 0.18 g of cross-linker pentaerythritol tetrakis (3-mercapto propionate) (PETMP, Sigma-Aldrich), and 0.012 g of the photoinitiator (2-hydroxyethoxy)-2-methylpropiophenone (HHMP, Sigma-Aldrich) were mixed and dissolved at 85 °C. Finally, 0.288 g of catalyst (dipropyl amine, Sigma-Aldrich) solution (2 wt%, in toluene) was added to the solution. After fully mixed and degassed, the solution was poured in a mold and cured for 24 h. The dried LCE sample was uniaxially stretched to 100% strain. The stretched sample was exposed to ultraviolet (UV) light (365 nm) at an intensity of 20 mJ/cm² for 10 min.

First, 60 mL of a 0.147 M polyvinylpyrrolidone (PVP) (MW ~ 40,000, Sigma-Aldrich) solution in ethylene glycol (EG) was added to a flask, to which a stir bar was added; the solution was then suspended in an oil bath (temperature 151.5 °C) and heated for 1 h under magnetic stirring (150 rpm). Then, 200 µL of a 24 M CuCl₂ (CuCl₂ · 2H₂O, 99.999+%, Sigma-Aldrich) solution in EG was injected into the PVP solution. The mixture solution was then injected with 60 mL of a 0.094 M AgNO₃ (99+%, Sigma-Aldrich) solution in EG (62).

The AgNW solution was drop-cast on a sacrificial glass slide, which was then placed onto a hot plate at 50 °C to evaporate the solvent. After the solvent was evaporated, PI resin (NeXolve) was drop-cast and spin-coated on top of the AgNW network. After curing at 140 °C for an hour, the AgNW/PI composite film (10 μm thick) was then laser cut into designed serpentine patterns. After laser cutting, the AgNW/PI heater was transferred to the commercial PI tape (60 μm thick, Kapton) on the side with adhesive. Then, the heater/PI tape composite and synthesized LCE film (300 μm thick) were put in the plasma chamber (Plasma Cleaner PDC-32G, HARRICK PLASMA) for 30 s. After the plasma treatment, the treated heater/PI tape and LCE were laminated together with 50 kPa pressure for 2 min. The Cu wires were attached to the two ends of the conductive patterns by silver epoxy (MG Chemicals). The fabrication of the LCE/AgNW/PI bimorph actuator was finalized by trimming it into the parallelogram shape from the octagon Kresling origami design. Of note is that the mesogen alignment of the LCE layer is perpendicular to the longer diagonal line of the parallelogram, i.e., the folding crease. The remaining part of the Kresling unit was folded and assembled with a piece of laser-cut PET sheet (50 μm thick). The two parallelogram side panels, in opposing directions, were replaced by two LCE/AgNW/PI actuators.

We used a reduced-order model to simulate the multi-degree-of-freedom of the robot segment involving contraction, twist, and bending. Our analysis was based on the nonlinear bar-and-hinge method (63, 64). As shown in SI Appendix, Fig. S8, we utilized bar elements along mountain and valley creases to model axial deformation, and rotational springs along actuators to capture folding. We described the kinematics by the total Lagrangian approach, taking reference to the initial configurations. Additionally, we used the standard linear elasticity as the constitutive relationship. The nonlinear equilibrium equation is formulated as follows:

Here, $\mathit{R}$ denotes the residual force vector, $\mathit{u}$ refers to the displacement vector, the vector $\mathit{F}$ contains the forces applied to the nodes of the bar-and-hinge system, $\mathit{T}$ denotes internal force vector, $\chi$ is known as the load factor that controls the magnitude of the external loads, and $\mathit{f}$ is a unit reference load vector. We adopted an incremental-iterative procedure to solve Eq. 3. The nonlinear solution scheme is detailed in SI Appendix, section 2, and the algorithm containing the pseudocode of the scheme is provided in SI Appendix, Table S3. Our implementation was performed using an in-house MATLAB code inspired by the open-source software MERLIN.

S.W., T.Z., Y.Z., and G.H.P. designed research; S.W. and T.Z. performed research; S.W. and T.Z. contributed new reagents/analytic tools; S.W., T.Z., Y.Z., and G.H.P. analyzed data; and S.W., T.Z., Y.Z., and G.H.P. wrote the paper.

The authors declare no competing interest.