Abstract
The unique properties of nonlinear waves have been recently exploited to enable a wide range of applications, including impact mitigation, asymmetric transmission, switching, and focusing. Here, we demonstrate that the propagation of nonlinear waves can be as well harnessed to make flexible structures crawl. By combining experimental and theoretical methods, we show that such pulse-driven locomotion reaches a maximum efficiency when the initiated pulses are solitons and that our simple machine can move on a wide range of surfaces and even steer. Our study expands the range of possible applications of nonlinear waves and demonstrates that they offer a new platform to make flexible machines to move.
INTRODUCTION
Flexible structures capable of sustaining large deformations are attracting increasing interest not only for their intriguing static response (1–3) but also for their ability to support large amplitude elastic waves. It has been shown that by carefully controlling their geometry, the elastic energy landscape of these highly deformable systems can be engineered to enable the propagation of a variety of nonlinear waves, including vector solitons (4–6), transition waves (7–9), and rarefaction pulses (10, 11). As such, the dynamic behavior of these structures not only displays a very rich physics but also offers new opportunities to manipulate the propagation of mechanical signals, enabling unidirectional propagation (5, 8), mechanical logic (7), wave guiding (12, 13), focusing (14), energy trapping (15), and mitigation (10) and damping of nonlinear periodic vibrations (16).
Here, inspired by both the retrograde peristaltic waves observed in earthworms (Fig. 1A) (17, 18) and the ability of linear elastic waves to generate actuation in ultrasonic motors (19–22), we show that the propagation of nonlinear elastic waves in flexible structures provides opportunities for locomotion. To demonstrate the concept, we focus on a Slinky (Fig. 1B) (23–26)—an iconic stretchable toy that has captivated children and adults all over the world—and use it to realize a pulse-driven robot capable of propelling itself. Our simple machine is built by connecting the Slinky to a pneumatic actuator and using an electromagnet and a plate embedded between the loops to initiate nonlinear pulses that propagate from the front to the back. Notably, we find that the directionality of these pulses enables our simple robot to move forward. Moreover, our results indicate that the efficiency of such pulse-driven locomotion is optimal when the initiated waves are solitons—large amplitude (nonlinear) pulses with stable shape and constant velocity along propagation (4, 5, 7, 8, 10, 27–30). As such, our study expands the range of possible applications of solitary waves and demonstrates that they can also be exploited as simple underlying engines to make flexible machines move.
(A) Schematic showing the locomotive mechanism of an earthworm based on retrograde peristaltic waves (i.e., waves that propagate in the opposite direction to locomotion). (B) Picture of metallic Slinky used in this study. (C and D) Pictures of our Slinky-robot (C) before and (D) after the pneumatic actuator is elongated. (E) Front view of the Slinky-robot showing the electromagnet. Note that several red plastic spheres are glued on to the Slinky to prevent it from rolling. Photo credit: Bolei Deng, Harvard University.
RESULTS
Our Slinky-robot
We consider a metal Slinky with a length of 50 mm, an outer radius of 46.58 mm, and 90 loops, each with mass of m = 1.01 g (Fig. 1B and section S1) and investigate how to exploit its intrinsic flexibility to realize a simple machine capable of rectilinear locomotion. To this end, we connect two Slinkies in series (for a total of 180 loops with length L = 100 mm) and implement a simple actuation strategy based on a pneumatic actuator with a stroke of 140 mm (realized using a plastic syringe and a pump), an electromagnet (12V 20N, UXCELL), and three acrylic plates: a front plate inserted between the 20th and 21st loops of the Slinky, a loading plate with an embedded metallic nut fitted between the 30th and 31st loops, and a back plate used to support the tail (Fig. 1, C to E). To construct the robot, we take the pneumatic actuator, connect one of its ends to the front plate, and glue the electromagnet directly on the actuator (Fig. 1D and section S2). When the magnetic field is on, the loading plate remains in contact with the electromagnet, and the 10 loops of the Slinky between the front and loading plates can be stretched and shortened using the pneumatic actuator (Fig. 1E). We test the response of our simple machine by placing it on a smooth and flat surface (Canson Bristol paper) and repeatedly extending the 10 loops to Ain (Ain being the maximum distance between the loading and front plates; Fig. 2A). We monitor the tests with a high-speed camera (SONY RX100) and extract the displacement of the head, uh, by tracking the position of the front plate via a superpixel-based method (31).
(A) Snapshots taken during a test in which we extend the front 10 loops to Ain = 100 mm while keeping the electromagnet on. (B) Displacement of the head during three cycles for tests in which (i) mh/mtot = 0.23 and we keep the electromagnet on (red line), (ii) mh/mtot = 0.23 and we turn off the electromagnet after stretching (green line), and (iii) mh/mtot = 0.32 and we turn off the electromagnet after stretching (blue line). (C) Snapshots taken during a test in which Ain = 100 mm and we turn the electromagnet off after stretching the front loops. (D) Evolution of
We start by actuating the Slinky while keeping the electromagnet on. In Fig. 2A, we report snapshots taken during a test, in which we repeatedly extend the front 10 loops to Ain = 100 mm. Although the head of the robot reaches
Although our results indicate that the propagation of elastic pulses can be exploited to make our flexible machine crawl, they also reveal that the conditions used in our experiments are not optimal because there is noticeable backsliding immediately after the electromagnet is turned off. Specifically, if we focus on the first cycle, we find that uh suddenly drops from
Having identified an optimal range for mh, we choose mh/mtot = 0.32 (resulting in mh = 78 g and mtot = 246 g) and investigate the effect of Ain on the ability of our robot to crawl (Fig. 2E). As expected, we find that
Propagation of nonlinear waves
To understand why the efficiency of our robot is maximum for Ain = 100 mm, we carefully investigate the propagation of large-amplitude pulses through the Slinky. In these tests, we focus on a single Slinky (with N = 90 loops) and monitor the position of green markers located at every other loop. Moreover, to minimize the effect of friction, we lift the Slinky from the substrate and use a plastic rod to support it (Fig. 3A). As in the tests conducted on our Slinky-robot, we find that by prestretching 10 loops near the front and turning off the magnetic electromagnet, we can initiate elastic waves that propagate toward the back (Fig. 3B). Furthermore, these tests enable us to get deeper insights into the propagation of the pulses because we monitor the displacement of each individual loop (movie S4). In particular, two important features emerge from these tests. First, we find that the backward-propagating waves move the center of mass of the Slinky forward (Fig. 3C)—an observation that further explains how the pulses make our Slinky-robot move. Second, we find that for Ain = 100 mm, the excited waves propagate while maintaining their shape at a constant velocity of 279 loop/s, with only slight acceleration near the end due to boundary effects (Fig. 3D). This suggests that the Slinky supports the propagation of large-amplitude solitary waves. To further confirm this observation, we calculate the cross-correlation of the velocity signals measured at the 10th and 80th loops (Fig. 3E). We find that the cross-correlation is maximum and approaches unity for Ain = 100 mm (movie S4), the same amplitude that maximizes the efficiency of our Slinky-robot (see Fig. 2G).
(A) Experimental setup used to test the propagation of pulses in the metallic Slinky. At t = 0 s, na = 10 loops between the loading plate and the front of the Slinky are stretched to Ain = 100 mm. (B) Snapshots of the propagation of the pulse in the Slinky at t = 0.10, 0.17, 0.24, and 0.34 s. The circular markers indicate the positions of the center of mass of the Slinky. (C) Displacement of the center of mass of the Slinky, uCM, as a function of time. Circular markers correspond to the time points considered in (A) and (B). (D) Spatiotemporal displacement diagram of the propagating pulse. (E) Velocity signals measured at the 10th and 80th loops. (F) Evolution of the cross-correlation of v10 (t) and v80 (t) as a function of the input amplitude Ain. The triangular marker corresponds to Ain = 100 mm. The green dashed line corresponds to the amplitude of the supported soliton, As, predicted by Eq. 10. Photo credit: Bolei Deng, Harvard University.
To provide deeper insight into these experimental results, we developed a mathematical model based on a one-dimensional array of concentrated masses m connected by nonlinear springs, which represent the mass and elasticity of an individual loop, respectively (Fig. 4A). The governing equations for such discrete system can be written as
(A) Schematic of our model. Each nonlinear spring represents the elastic response of an individual loop, whereas each concentrated mass represents its mass. (B) Spatial displacement profiles at t = 0, 0.10, 0.17, and 0.24 s as measured in our experiments (circular markers) and predicted by our analytical solution given by Eq. 7 (solid lines).
Next, because in our experiments the wavelength of the propagating waves is much wider than a single loop, we take the continuum limit of Eq. 4 and retain derivatives up to fourth order to obtain the continuum governing equation
DISCUSSION
To summarize, we have shown that backward propagating solitons can be harnessed to efficiently make a Slinky-robot move forward. Although limbless organisms have recently inspired the design of a variety of robots (33–40), to the best of our knowledge, this is first robotic system that exploits elastic pulses to move. It is also worth noting that the principles presented in this study are different from those used by ultrasonic motors (19). Our Slinky-robot is flexible and uses nonlinear pulse waves to change the position of the center of mass. By contrast, ultrasonic motors are powered by linear sinusoidal waves, induce local microscopic displacements (eventually adding up over many periods to produce a large motion), and modulate friction by exerting an oscillating normal force between stiff surfaces.
It is important to point out that, while in this study we have focused on rectilinear forward crawling, the flexibility of the Slinky can be exploited to expand the range of achievable motions. For example, we can make the robot steer by twisting the last loop at the back of the robot by an angle β before initiating the wave (Fig. 5A and movie S5) and easily control the steering angle θ by tuning both the direction and magnitude of β (Fig. 5B).
(A) Snapshots taken during a test in which we twist the last loop by β = 180° before initiating the wave. The Slinky-robot steers by an angle θ = 13° when the pulse reaches its tail. (B) Evolution of the steering angle θ as a function of the applied twist β. (C) Evolution of
We also want to emphasize that our robot can move over a broad range of surfaces. To demonstrate this point, in Fig. 5C, we report
Last, while in this study we have used a Slinky to realize such pulse-driven locomotion, the principles are general and can be expanded to a broad range of stretchable systems across scales, opening avenues even for microscale crawlers suitable for medical applications.
MATERIALS AND METHODS
The basic properties of the metallic Slinky are provided in section S1. Details on fabrication, testing, and additional results of the Slinky-robot are described in section S2. Details on the experiments conducted to characterize the propagation of nonlinear waves are presented in section S3. Details on the model established to characterize the propagation of nonlinear waves are presented in section S4.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/18/eaaz1166/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).