## 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.

## 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 *A*_{in} (*A*_{in} 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, *u*_{h}, by tracking the position of the front plate via a superpixel-based method (*31*).

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 *A*_{in} = 100 mm. Although the head of the robot reaches *A*_{in} = 100 mm. As soon as the electromagnet is turned off, the stretch in the front loops results in the excitation of a wave that propagates backward and reaches the back end of the Slinky (Fig. 2C and movie S1). Note that no reflected wave is observed in the Slinky because of the large energy dissipation upon collisions of the loops. We find that the propagation of this unidirectional pulse results in a nonzero *u*_{h} at the end of each cycle—a clear indication that our robot moves forward. Specifically, if we denote with

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 *u*_{h} suddenly drops from *m*_{h}. In Fig. 2D, we report the average value of *m*_{h}/*m*_{tot} (*m*_{tot} being the total mass of the Slinky). While for the robot considered in Fig. 2C (for which *m*_{h}/*m*_{tot} = 0.23) *L*. However, if *m*_{h}/*m*_{tot}> 0.4,

Having identified an optimal range for *m*_{h}, we choose *m*_{h}/*m*_{tot} = 0.32 (resulting in *m*_{h} = 78 g and *m*_{tot} = 246 g) and investigate the effect of *A*_{in} on the ability of our robot to crawl (Fig. 2E). As expected, we find that *A*_{in}. However, more insight into the role of *A*_{in} can be gained by calculating the efficiency η of such pulse-driven locomotion*W*_{req} denotes the work required for the robot to move on the substrate against friction and *W*_{in} is the work supplied to initiate the pulse. Specifically, if we denote with μ the friction coefficient between the Slinky and the substrate*g* is the gravitational constant, *n _{a}* is the number of loops between the loading and front plates, and

*F*denotes the force required to stretch one loop of the Slinky by Δ

*u*. Note that the first term of Eq. 3 stands for the work done to move the head of the Slinky, while the second one represents the elastic energy stored in the front

*n*loops, which is eventually carried by the excited pulse. For the tests considered in Fig. 2,

_{a}*n*= 10 and we measure μ = 0.17. Moreover, we obtain the

_{a}*F*-Δ

*u*relation by conducting a quasistatic uniaxial tension test on a portion of the Slinky (see Fig. 2F and section S1 for details). In Fig. 2G, we plot the evolution of η as a function of

*A*

_{in}obtained using Eqs. 1 to 3. Notably, we find that the efficiency of our machine is maximum for

*A*

_{in}= 100 mm. While in Fig. 2G we focus on a robot in which 10 loops are prestretched, this result persists also when we vary

*n*. In particular, as shown in Fig. 2H, we find that for a wide range of

_{a}*n*, the efficiency is maximum at

_{a}*A*

_{in}= 100 mm. However, such maximum decreases if

*n*< 7 or

_{a}*n*> 16. For large values of

_{a}*n*, the energy carried by the excited pulses is small (see fig. S7) and not enough to make them propagate in the presence of friction (see movie S3). Differently, when

_{a}*n*is too small, the energy carried by the pulses is large, making the pulse to reach the back of the Slinky (see movie S3). At this point,

_{a}### Propagation of nonlinear waves

To understand why the efficiency of our robot is maximum for *A*_{in} = 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 *A*_{in} = 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 *A*_{in} = 100 mm (movie S4), the same amplitude that maximizes the efficiency of our Slinky-robot (see Fig. 2G).

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*u _{i}* represents the displacement of

*i*th mass and

*F*(

*u*

_{i+1}−

*u*) denotes the force in the spring connecting the

_{i}*i*th and (

*i*+ 1)th masses. For the Slinky considered in this study,

*m*= 1.01 g and the elastic response of an individual loop can be nicely fitted by

*k*= 79.2 N/m and a cubic nonlinearity parameter α = 155.1 m

^{−2}(see Fig. 2F and section S3 for details). Note that Eq. 5 only captures the response of the Slinky under tension (i.e.,

*u*

_{i+1}−

*u*≥ 0) and therefore limits us to investigate rarefaction wave solutions.

_{i}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*u*(ξ, *t*) is a continuous function that interpolates *u _{i}*(

*t*), i.e.,

*u*(ξ =

*i*,

*t*) =

*u*(

_{i}*t*), ξ denotes the loop number, and

*32*)

_{0}identifies the position of the wave at

*t*= 0 and

*c*and

*W*are the velocity and the characteristic width of the pulse, respectively. Because solitary waves propagate maintaining their shape, we require the maximum extension in the Slinky [i.e., max (

*∂u*/∂ξ)] to be equal to that imposed to the 10 loops before turning off the electromagnets (i.e.,

*A*

_{in/na})

*c*and the applied input

*A*

_{in}. Last, the amplitude of the supported soliton can be calculated as

*A*

_{s}= 103 mm,

*c*= 282.2 loop/s, and

*W*= 4.62 loop—values that match extremely well with the experimental results (Fig. 4B). Note that the model also confirms the experimental observations reported in Fig. 3, as it predicts that only pulses with amplitude

*A*

_{in}∼

*A*

_{s}= 103 mm propagate in solitary fashion through the Slinky. Last, our analysis reveals that the efficiency of the Slinky-robot is maximum when the initiated waves are solitons (i.e., when

*A*

_{in}=

*A*

_{s}; see green dashed lines in Fig. 2, G and H). The nondispersive nature and compactness of solitary pulses make them extremely efficient in transferring the energy provided by the pneumatic actuator to motion, ultimately resulting in the most efficient pulse-driven locomotion.

## 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).

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 *m*_{h}/*m*_{tot} = 0.32 and *A*_{in} = 100 mm) moves on surfaces characterized by a broad range of roughness (so that μ ∈ [0.1,1]). Notably, we find that for a μ < 0.3,

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

**Acknowledgments:**

**Funding:**K.B. acknowledges support from the NSF under grant no. DMR-1420570 and from the Army Research Office under grant no. W911NF-17-1-0147. V.T. acknowledges support from project Acoustics hub funded by Region Pays de la Loire. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. government. The U.S. government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.

**Author contributions:**B.D., V.T., and K.B. came up with the initial idea. B.D. and K.B. designed and supervised the research. B.D. and L.C. performed the experiments. B.D. and D.W. analyzed the experimental data. B.D. developed the theoretical model. All authors contributed in writing the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- 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).