Mobility and robustness are two important features for practical applications of robots. Soft robots made of polymeric materials have the potential to achieve both attributes simultaneously. Inspired by nature, this research presents soft robots based on a curved unimorph piezoelectric structure whose relative speed of 20 body lengths per second is the fastest measured among published artificial insect-scale robots. The soft robot uses several principles of animal locomotion, can carry loads, climb slopes, and has the sturdiness of cockroaches. After withstanding the weight of an adult footstep, which is about 1 million times heavier than that of the robot, the system survived and continued to move afterward. The relatively fast locomotion and robustness are attributed to the curved unimorph piezoelectric structure with large amplitude vibration, which advances beyond other methods. The design principle, driving mechanism, and operating characteristics can be further optimized and extended for improved performances, as well as used for other flexible devices.

The locomotion mechanisms of animals continue to inspire the design of soft robots ( 4 , 35 ). In particular, arthropods show how rapid, cyclic locomotion at high frequencies at this scale is possible without compromising robustness and survivability in harsh conditions ( 36 , 37 ). Flying mosquitos can oscillate or vibrate their wings at more than 800 Hz ( 38 ), and 1-mm mites attain relative ground speeds exceeding 200 body lengths per second (BL/s) ( 39 , 40 ). In this work, we introduce fast and robust insect-scale soft robots based on a curved piezoelectric PVDF unimorph structure to achieve several key advancements: (i) Under an alternating current (AC) driving power near the resonant frequency (850 Hz) of the structure, a prototype 10-mm-long robot (0.024 g) attained a relative speed of 20 BL/s—the fastest among published reports of insect-scale soft ground robots; (ii) after stepping on the robot with an adult human’s full body weight (59.5 kg, about 1 million times heavier than the robot), the robot could still move afterward, demonstrating exceptional robustness; (iii) the robot could move smoothly carrying a load weighing 0.406 g, which is six times heavier than that of the robot; (iv) further enhancement of agility was demonstrated by designing the moving mechanism to emulate features of galloping-like gaits using a two-leg prototype robot.

Mobility and robustness are two engineering challenges for robots. Unlike large-scale robots based on materials of high stiffness/density and powered by bulky actuators/motors, small-scale soft robots are often restricted to small actuators with low output power based on materials of low stiffness/density. Hence, insect-scale soft robots are known to be easily damaged, exhibit poor control of locomotion, or are slow moving due to the nature of their small structures. Improving the mobility, efficiency, and robustness of soft robots made of a deformable body with the capability to carry extra weights to perform various functions has been challenging ( 1 – 4 ). Researchers have tried to develop soft robots that negotiate complex environments by taking advantage of soft matter physics in the interdisciplinary field termed “robophysics” ( 5 , 6 ). Recent advances include micro/millimeter-scale robots with good mobility, such as crawling robots ( 7 – 11 ), hopping robots ( 12 , 13 ), and multi-legged robots ( 14 – 18 ). However, these robots are made of rigid or partially rigid parts, resulting in poor robustness and low adaptability to shape changes and/or external perturbations. On the other hand, soft robots actuated by humidity ( 19 – 21 ), light ( 22 – 24 ), heat ( 25 ), and magnetic force ( 26 – 28 ) have been demonstrated but have slow responses, whereas others require bulky setups to generate the external power sources such as magnetic fields. Robots using thin film–based actuators based on lead zirconate titanate (PZT) have been successfully developed ( 17 , 18 , 29 – 31 ), but PZT is a brittle material containing poisonous lead. Polyvinylidene difluoride (PVDF) is soft, flexible, and lightweight, making it suitable for potential soft robot applications ( 32 , 33 ), but one key challenge has been to generate fast, effective movement and even locomotion ( 34 ).

RESULTS

Structure and working mechanism A prototype 3 cm–by–1.5 cm robot, consisting of a curved body and a leg-like structure at the front, is pictured alongside a U.S. quarter in Fig. 1A. A cross-sectional view scanning electron microscopy (SEM) image shows the unimorph structure made of an 18-μm-thick PVDF layer, two 50-nm-thick palladium (Pd)/gold (Au) electrodes (top and bottom of the PVDF film), a 25-μm-thick adhesive silicone, and a 25-μm-thick polyethylene terephthalate (PET) substrate at the bottom. The PVDF layer can produce periodic extension and contraction by the piezoelectric effect under an AC driving voltage to change the shape of the robot; the details of the actuation mechanism are explained in section S1 and fig. S1. This results in an oscillatory center of mass (COM) trajectory pattern (see movie S1) similar to many running animals (36). Figure 1B compares the COM motion of a cockroach and our prototype robot. Although the robot has a unique morphology compared with many animals, it also showed a wavelike path. We developed a two-segment mass-spring model to best predict the robot’s dynamic movements. Fig. 1 The prototype and working mechanism. (A) Optical photo showing a robot connected with two electrical wires to the top and bottom electrodes, respectively, alongside a U.S. quarter coin. The inset SEM image shows the cross-sectional view of the prototype robot with different layers of materials. (B) Comparison of the wavelike running paths showing the movements of the COM of a cockroach (41) and a prototype robot (from movie S1). (C) Series of optical images recording the movements of a prototype robot in one driving cycle. (D) Applied driving signal (black line) and vertical (blue lines) and lateral displacements (red lines) of a prototype robot, where the bold solid lines are the average movements for 20 cycles. (E) Two-step cycles of the vertical displacement of the COM during cockroach running [red line for a period of 60 ms, (41)] compared with a prototype soft robot (blue line for a period of 100 ms). High-speed videos (sampling rate of 2000 frames per second) with a sequence of optical images seen in Fig. 1C have been used to record the postures and positions of the prototype robots (under −60 to 60 V of sinusoidal driving voltage at 200 Hz, which generates nonmaximal running performance; movie S2) running on a standard printing paper substrate. Within one cycle of the applied sinusoidal voltage, one set of the corresponding successive postures is depicted as states (I) to (V) in Fig. 1C as an example. In state (I) under −60 V, the body is extended and the front leg of the robot is in the ground-touching posture, whereas the abdomen is in the aerial posture. After 1.1 ms at state (II) under an applied voltage close to 0 V, the body recovers its initial shape, whereas the front leg of the robot is still in the ground-touching posture and the abdomen is in the aerial posture with a shorter distance to the ground as compared with that in state (I). In state (III) under +60 V, the body is contracted, and both the front leg and abdomen of the robot are in the ground-touching posture. In the first-half driving cycle from state (I) to state (III), the body transitions from extended near-flat shape, to the initially curved shape, and then to the contracted shape. These shape changes cause the front leg to strike against the ground and produce a forward-pushing ground reaction force. In the second-half driving cycle from state (III) to state (V), the body goes through the similar shape changes, with the reverse order from the contracted shape to near-flat shape, which could cause the front leg to produce a backward-pushing ground reaction force to slow down the forward moving speed of the robot. Hence, we designed the bending angle of the front leg to be less than 90° to enhance the forward movements and reduce backward movements. Driven under high-frequency actuation coupled with various ground impact conditions and manufacturing variations, the exact shape changes and movements of the robot are rather complex. However, by varying design and operation parameters, the forward moving speed of the robot could be optimized. For example, Fig. 1D shows the experimental results of the lateral displacement (red lines) and vertical displacement (blue lines) of a prototype robot under a driving voltage between −60 and 60 V (black line). In this analysis, the lateral/vertical displacement is defined as the lateral/vertical movements of the COM of the robot with respect to the original position. The randomness of the individual cycle is clearly observed in both displacements, although the average vertical COM positions follow the driving patterns and the average lateral COM positions show incremental forward movement. Figure 1E compares two-step cycles of the vertical movement of a cockroach (41) and the prototype soft robot (movie S1) with respect to time. Animals appear to use resonant frequencies to oscillate their muscles and segments (42), with the flight muscles and thorax of flying insects serving as an example in the higher frequency range (43). We found that it was also desirable to drive the prototype robots near their resonant frequencies for largest deformations. To constrain the running direction of the robot so as to characterize their relative running speed, we used a transparent quartz tube with an inner diameter of 1 inch, as shown in fig. S2 (A and B). In this case, a 10-mm-long prototype (0.024 g) robot was used to achieve a relative running speed up to 20 BL/s driven near its resonant frequency at 850 Hz. In comparison, under driving frequencies of 800 and 900 Hz, lower relative running speeds of 13 and 3.6 BL/s were recorded, respectively (movie S3).

Locomotion analysis We observed four main postures during the operation of the robot: aerial, front-touching, back-touching, and both-touching. In each posture, the robot’s body can be either expanding or contracting depending on the applied driving signal at that instant. Hence, there are eight possible configurations, as shown in Fig. 2 (A to D), where gray dashed lines indicate the previous shapes and red solid lines are the immediate current shapes. In this illustration, G, F f , and F a are the gravitational force, ground reaction force at the front leg, and ground reaction force at the end of the abdomen, respectively. The shape of the robot changes periodically based on the PVDF actuation force to excite elastic oscillations for the curved unimorph structure. The directions of the ground reaction force at the front leg and abdomen (blue arrows in the figure) will change depending on the posture and shape changes of the robot. The front leg construction of the robot is important because it produces anisotropic lateral forces to generate forward locomotion. The most effective configuration for forward motion, a contracting touchdown of the front leg (contraction of the front-touching posture), greatly increased the lateral component of the ground reaction force and the vertical component for taking off. The abdomen functions to keep the balance of the dynamic system by adjusting the pitch of the body when it taps the ground, preventing the robot from flipping over backward. The details for qualitative introduction of the locomotion mechanism are discussed in section S2 and figs. S3 and S4. Fig. 2 Locomotion gait analysis. (A to D) Optical photos from the high-speed camera (top), corresponding contracted configurations (middle), and corresponding expanded configurations (bottom) of a prototype robot showing different gaits in the cross-sectional views. (E) Simplified dynamic model based on two rigid bodies joined by a pin joint (both-touching posture as an example) with a torsional spring-damper system. (F) Duty cycles in different gaits of both experimental and simulation results for a 25-mm-long prototype robot driven at its fastest speed at resonance of 200 Hz. (G) Relationships between the vibration amplitude and moving speed as well as aerial duty cycle for driving frequencies of 170, 190, 200, 210, and 230 Hz. Error bars indicate mean ± 1 SD. Using a high-speed camera, we found that the first vibrational mode is easily excited and dominates the shape change, whereas higher-order modes can be neglected. A dynamic mass-spring model consisting of two rigid bodies (m 1 , m 2 ) joined by a pin joint was analyzed in MATLAB to model the curved robot, as shown in Fig. 2E (both-touching posture as an example). A torsional spring-damper (k θ -d θ ) at the pin joint is excited by a sinusoidally varying torque source (τ m ) to represent the mechanical motions of the PVDF layer under the AC excitation. We modeled the ground contact at the front and back as a vertical spring-damper (k-d) with a normal force in the vertical direction (F n ) and a friction force in the lateral direction (F f ). The values of material parameters used in the model and simulation can be found in table S1, and the simulation details are discussed in section S3 and fig. S5. A 25-mm-long prototype robot was driven on a paper substrate and recorded under different driving parameters. The fastest running speed was 8.7 cm/s (movie S1) under 200 V at 200 Hz. The running speed reduced under the same applied voltage of 200 V at varying frequencies of 170, 190, 210, and 230 Hz, respectively (movie S4). To further study the operation of the robot, we statistically characterized the duty cycles for the eight configurations (Fig. 2, A to D). We plotted the results in Fig. 2F (shaded columns) for the trial with the fastest speed at 8.7 cm/s at 200 Hz (see fig. S6 for other frequencies with slower speeds). We also compared these results with the simulation data from the mass-spring model running at 200 Hz (Fig. 2F, hatched columns, and movie S5). We observed that a large percentage of aerial duty cycles were required to generate fast running speeds for the robot. For example, in this trial, the aerial cycles for the contracted and expanded configurations are about 36 and 43%, respectively, whereas all the other configurations have the duty cycles of less than 10%. In principle, large-amplitude oscillation driven at the resonant frequency should result in large deformation and greater forces to induce longer aerial duty cycles and higher foot velocities for faster speeds. Figure 2G compares the average vibration amplitudes (measured when the robot is in the aerial posture) under the driving frequencies of 170, 190, 200, 210, and 230 Hz and their corresponding running speeds and aerial duty cycles. As expected, the large vibration amplitude due to the structural resonance resulted in faster running speed as well as longer aerial duty cycles (the combination of contracted and expended configurations). Again, we note that although the morphology and motion of our robot do not mimic any specific animal, small runners, such as cockroaches (41) and desert ants (44), also use aerial phases to attain their fastest speeds.

Parameter optimization and performance characterization Geometric parameters play an important role in the performance of the robot. To simplify the structure and identify appropriate configurations, we defined the geometric parameters as shown schematically in Fig. 3A, where L is the body length of a robot, α is the body curvature, l is the length of the front leg, λ is the distance between the front leg and the head, and β is the contact angle between the front leg and the ground. Using a prototype robot of 10 mm (length) by 15 mm (width) by 3 mm (height) as an example, we first selected 25 combinations (Fig. 3B, gray dots) of the above geometric parameters to fabricate prototypes and conducted experiments to plot the normalized running speed map as a function of relative leg position (λ/L) and relative leg angle (β/π) in Fig. 3B. The color bar shows the magnitudes and directions of the normalized speed of the robot, with the red color areas representing the fastest running speed. We found that the values of λ/L and β/π near 0.1 and 0.4, respectively, resulted in robots with the fastest running speeds. Fig. 3 Geometric parameter optimization and performance characterization. (A) Side view of a robot with the definitions of geometric parameters. (B) Experimental results (gray dots) for normalized running speeds from a 10-mm-long robot with different geometric combinations used to plot a color map as a function of relative leg position (λ/L) and relative leg angle (β/π). The color map is interpolated by the thin-plate spline interpolation scheme for surface fitting. Original data can be found in table S2. (C) Relative running speeds of robots versus the driving frequencies for robots with lengths of 10, 15, 20, 25, and 30 mm. Shaded bands represent 90% confidence limits. (D) Relative running speeds (under the resonant frequency) of robots versus the driving voltages for robots with lengths of 10, 15, 20, 25, and 30 mm. Error bars indicate mean ± 1 SD. We then fabricated prototype robots with different lengths ranging from 10 to 30 mm at an interval of 5 mm using the map of λ/L and β/π of 0.1 and 0.4 for guidance. The resonant frequencies of robots with different lengths were approximately evaluated both analytically and experimentally and discussed in section S4 and figs. S7 and S8. In general, robots with smaller lengths have higher resonant frequencies and faster relative speeds. The relationships between driving frequency and relative speed for robots with lengths of 10, 15, 20, 25, and 30 mm are shown in Fig. 3C driven by a peak-to-peak voltage of 200 V to achieve measured maximum speeds of 20 BL/s (20 cm/s), 8 BL/s (12 cm/s), 4.05 BL/s (8.1 cm/s), 2.4 BL/s (6 cm/s), and 1.33 BL/s (4 cm/s), respectively. In Fig. 3D, the amplitude of the driving voltage versus the relative speed of robots with different lengths was measured near their resonant frequencies. As expected, larger driving voltages result in faster running speeds. For a 10-mm-long robot, as shown in fig. S9 (A and B), we observed noticeable motion even under an AC drive voltage as low as 8 V peak to peak (movie S6), which is a relatively low voltage requirement among insect-scale piezoelectric actuators (45). Using a prototype 30-mm-long robot operating at 140 Hz as an example, we measured the voltage and current simultaneously using a data acquisition system for five cycles (fig. S10). The power consumption could be estimated as 0.343 mW by the integral of the voltage-current measurement results. When the robot operated near its resonant frequency, the cost of transport (COT) of the robot was about 14 (section S5), the lowest reported COT for robots and insects below 1 g (fig. S11) (46–59) with a relative speed of 4 BL/s under a power of 0.343 mW.

The comparison of relative moving speed The relative moving speed is very important for animals because they often depend on fast locomotion to hunt for food, escape from predators, and/or compete for mating partners (60, 61). Researchers have shown that animals with high relative speeds are less likely to be caught and that relative speed may be more “ecologically relevant” than the absolute speed in various performance characterizations (61, 62). Figure 4 shows a comparison of relative running speeds with respect to body weights including our robots (red stars) and living animals, such as terrestrial mammals (purple) and running arthropods (orange), along with reported artificial soft robots or actuators (blue). For mammals, the trend in the elliptical and purple color shaded area indicates that the relative speed decreases as the body mass increases due to the scaling of mechanical constraints on the locomotive performance (61). However, small-size arthropods outperform larger animals in terms of their relative moving speeds. For example, a small mite, Paratarsotomus macropalpis, is now the world’s fastest known running animal, with a relative speed at several hundred body lengths per second (39). An opposite trend exists for soft robots, as shown in the elliptical and blue color shaded area, which suggests that the relative speed increases as the body mass increases (19) except for recent robots driven by an external magnetic force (26–28). The robots presented in this work (five red stars with the body lengths from 30 to 10 mm; Fig. 4) have sizes similar to those of arthropods with a similar performance trend where the relative speed increases as the body mass decreases. As discussed in section S4, the relative running speed of our prototype robots had a positive correlation with the resonant frequency, so our smaller robots operated at higher resonant frequencies to achieve faster relative running speeds. The working efficiency of our prototype robot is high because the simple structure contains no redundant energy-consuming components. Although some soft robots driven by magnetic fields, humidity, or heat or light sources can have fast instantaneous running speeds, slow responses and a bulky setup to generate the external power, such as the magnetic field, are among the limitations. Fig. 4 Relative running speeds of some mammals, arthropods, soft robots, and actuators versus body mass. For animals including both mammals (purple) and arthropods (orange), relative speeds show a strong negative scaling law with respect to the body mass, showing that relative running speeds increase as body masses decrease. However, for soft robots, the relationship appears to be the opposite, where the relative running speeds decrease as the body mass decrease. The performances of the prototype robots (red stars with body lengths from 30 to 10 mm) follow a scaling law similar to that of living animals: Higher relative running speed was attained as the body mass decreased, with the fastest measured running speed at 20 BL/s among reported insect-scale robots and actuators (blue). For data, see table S3.

Robustness Robustness is essential for the survival of animals displaying both fail-safe and fault-tolerant behavior. For example, a cockroach can withstand a load 900 times its own body weight without injury because of its soft and shape-changing exoskeleton (37). The robot presented here also has exceptional robustness characteristics resulting from the assembly of soft materials with simple structures. Experimentally, the robustness of the prototype soft robot was demonstrated by applying a 100-g mass (1500 times its own body weight) with little change in its speed after the mass was removed, as shown in movie S7. Moreover, the soft robot could continue to function (one-half of the original speed) after being stepped on by an adult human (59.5 kg), a load about 1 million times its own body weight (Fig. 5, A to C, and movie S7). We systematically tested the robust performance of a 3 cm–by–1.5 cm prototype robot (fig. S12). The prototype robot was driven with the same condition (200 V and 140 Hz) before and after the applied load. In fig. S12, we report the speeds of a prototype robot after applying and removing different loads with magnitudes ranging from 10 to 59,500 g. We define the compressibility as the ratio of the vertical height change to the original height of the robot. We observed that if the applied load is below 100 g, then the robot can recover back to the original shape and maintain greater than 88% of its original speed. As the applied load increased, the moving speed decreased. When the applied load was above 10 kg, the compressibility of the robot increased and saturated at about 0.95, while the speed reduced and saturated at near 50% of the original speed. Even under heavy applied loads that flatten the robot initially, the robot could still partially recover after the removal of the applied load. Fig. 5 Weight-bearing, slope-climbing, and load-carrying capabilities. (A to C) Soft robot can continue to function (one-half of the original speed) after being stepped on by an adult human (59.5 kg), a load about 1 million times its own body weight. Scale bars, 3 cm. A robot climbs a slope (D) of 7.5° with a relative speed of 7 BL/s and a slope (E) of 15.6° with a relative speed of 1 BL/s. Scale bars, 1 cm. (F and G) A robot (0.064 g) carries a peanut (0.406 g), which is six times its own body weight, to show the load-carrying capability. The speed with the peanut on top is about one-six of the original speed without the peanut. Scale bars, 1 cm.

Climbing and carrying loads Animals and robots often need to do work such as climbing and carrying loads. The slope climbing capability of the robot is demonstrated in movie S7, in which the robot reached 7 BL/s while climbing a slope with an angle of 7.5° (Fig. 5D) and 1 BL/s while climbing a slope with an angle of 15.6° (Fig. 5E). Our soft prototype robot could also carry loads equal to the weight of a peanut (0.406 g) (Fig. 5, F and G). The robot was able to move smoothly while carrying a load that is six times its own weight at about one-sixth of the original speed (movie S7).