Pulse-driven robot: Motion via solitary waves

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Science Advances  01 May 2020:
Vol. 6, no. 18, eaaz1166
DOI: 10.1126/sciadv.aaz1166


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.


Flexible structures capable of sustaining large deformations are attracting increasing interest not only for their intriguing static response (13) 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 (46), transition waves (79), 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 (1922), 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) (2326)—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, 2730). 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.

Fig. 1 Our Slinky-robot.

(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.


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).

Fig. 2 Performance of the Slinky-robot.

(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 uhcycle as a function of mh/mtot for tests in which Ain = 100 mm. The square and triangular markers correspond to mh/mtot = 0.23 and 0.32, respectively. (E) Evolution of uhcycle as a function of Ain for tests in which mh/mtot = 0.32. The triangular marker corresponds to Ain = 100 mm. (F) Static response of the Slinky as measured in a uniaxial test. (G) Evolution of η as a function of Ain for tests in which mh/mtot = 0.32. The triangular markers correspond to Ain = 100 mm. The green dashed line corresponds to the amplitude of the supported soliton, As. (H) Evolution of η as a function of na and Ain for tests in which mh/mtot = 0.32. Photo credit: Bolei Deng, Harvard University.

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 uhmax=max(uh)=80 mm when the actuator is fully extended, because of the symmetric frictional properties it goes back to the initial position at the end of each cycle (Fig. 2B); therefore, no forward motion is achieved (Fig. 2A and movie S1). Next, in an attempt to break symmetry and make our machine crawl, we turn off the magnetic field after stretching the front loops to Ain = 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 uh at the end of each cycle—a clear indication that our robot moves forward. Specifically, if we denote with uhcycle the difference between the displacement of the head at the end and at the beginning of a specific cycle, we find that uhcycle/L=0.32±0.02 (green line in Fig. 2B). Therefore, our results indicate that the directionality introduced by the elastic waves can be exploited to make the robot move even in the presence of identical friction coefficients in backward and forward directions. Furthermore, we also find that the phenomenon is robust as the robot behaves almost identically during various cycles (Fig. 2B).

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 uhmax=0.8L to about uhcycle=0.32L (Fig. 2B). To limit such backsliding, we increase the mass at the head, mh. In Fig. 2D, we report the average value of uhcycle over three cycles as a function of mh/mtot (mtot being the total mass of the Slinky). While for the robot considered in Fig. 2C (for which mh/mtot = 0.23) uhcycle=0.32L, we find that for 0.26<mh/mtot<0.39 uhcycle increases to ∼0.45L. However, if mh/mtot> 0.4, uhcycle then suddenly drops as the weight prevents the head to move forward when the pneumatic actuator is extended (movie S2).

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 uhcycle monotonically increases with Ain. However, more insight into the role of Ain can be gained by calculating the efficiency η of such pulse-driven locomotionη=WreqWin(1)where Wreq denotes the work required for the robot to move on the substrate against friction and Win is the work supplied to initiate the pulse. Specifically, if we denote with μ the friction coefficient between the Slinky and the substrateWreq=μ mtotguhcycle(2)andWin=μ mhguhmax+na0AinnaF(Δu)u(3)where g is the gravitational constant, na 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 Δu. 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 na loops, which is eventually carried by the excited pulse. For the tests considered in Fig. 2, na = 10 and we measure μ = 0.17. Moreover, we obtain the Fu 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 Ain obtained using Eqs. 1 to 3. Notably, we find that the efficiency of our machine is maximum for Ain = 100 mm. While in Fig. 2G we focus on a robot in which 10 loops are prestretched, this result persists also when we vary na. In particular, as shown in Fig. 2H, we find that for a wide range of na, the efficiency is maximum at Ain = 100 mm. However, such maximum decreases if na < 7 or na > 16. For large values of na, 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 na 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, uhcycle saturates (see fig. S7) and the efficiency drops because part of the energy carried by the pulse is lost through collisions between the loops.

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).

Fig. 3 Wave propagation in the Slinky.

(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 asm2uit2=F(ui+1ui)F(uiui1)(4)where ui represents the displacement of ith mass and F(ui+1ui) denotes the force in the spring connecting the ith 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 byF(ui+1ui)=k[(ui+1ui)+α (ui+1ui)3](5)with a stiffness constant 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., ui+1ui ≥ 0) and therefore limits us to investigate rarefaction wave solutions.

Fig. 4 Mathematical model.

(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 equation1c022ut2=2uξ2+1124uξ4+αξ(uξ)3(6)where u(ξ, t) is a continuous function that interpolates ui(t), i.e., u(ξ = i, t) = ui(t), ξ denotes the loop number, and c0=k/m=280.0 m/s is the velocity of linear waves in the long-wavelength regime. Equation 6 has been shown to be equivalent to the generalized Boussinesq equation and admits an analytical solution in the form of a rarefaction solitary wave (32)u=23αarctan [tanh (ξξ0c tW)]+π423α(7)withW=13(c2/c021)(8)where ξ0 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., Ain/na)Ainna=max (uξ)=2α(c2c021)(9)and in this way obtain a relation between propagation velocity c and the applied input Ain. Last, the amplitude of the supported soliton can be calculated asAs=u(ξ)u(ξ)=π6α(10)and is found to be an intrinsic property of the Slinky, because it only depends on α (i.e., on the strength of the nonlinear term in its static force-displacement response; see Eq. 5). For the Slinky considered in this study, using Eqs. 8 to 10, we obtain As = 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 AinAs = 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 Ain = As; 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.


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 (3340), 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).

Fig. 5 Steering and moving over different surfaces.

(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 uhcycle as a function of the friction coefficient between the Slinky and the substrate, μ. The triangular marker corresponds to Bristol paper—the substrate considered in Fig. 2. Photo credit: Bolei Deng, Harvard University.

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 uhcycle recorded when our Slinky-robot (with mh/mtot = 0.32 and Ain = 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, uhcycle remains almost constant (uhcycle0.45L). For larger values of μ, the frictional forces prevent propagation of the pulses and uhcycle drops significantly—an additional demonstration of the important role played by the wave propagation in achieving locomotion (movie S6).

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.


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 material for this article is available at

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.


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.
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