Pulse-driven robot: Motion via solitary waves

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).

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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 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 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 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 u_h suddenly drops from uhmax=0.8L to about uhcycle=0.32L (Fig. 2B). To limit such backsliding, we increase the mass at the head, m_h. In Fig. 2D, we report the average value of uhcycle over three cycles as a function 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) uhcycle=0.32L, we find that for 0.26<mh/mtot<0.39 uhcycle increases to ∼0.45L. However, if m_h/m_tot> 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 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 uhcycle monotonically increases with A_in. However, more insight into the role of A_in can be gained by calculating the efficiency η of such pulse-driven locomotionη=WreqWin(1)where 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 substrateWreq=μ mtotguhcycle(2)andWin=μ mhguhmax+na∫0AinnaF(Δu)dΔu(3)where 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 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 n_a loops, which is eventually carried by the excited pulse. For the tests considered in Fig. 2, n_a = 10 and we measure μ = 0.17. Moreover, we obtain the F-Δu 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_a. In particular, as shown in Fig. 2H, we find that for a wide range of n_a, the efficiency is maximum at A_in = 100 mm. However, such maximum decreases if n_a < 7 or n_a > 16. For large values of n_a, 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 n_a 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 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).

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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 asm∂2ui∂t2=F(ui+1−ui)−F(ui−ui−1)(4)where u_i represents the displacement of ith mass and F(u_i+1 − u_i) 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+1−ui)=k[(ui+1−ui)+α (ui+1−ui)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., u_i+1 − u_i ≥ 0) and therefore limits us to investigate rarefaction wave solutions.

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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 equation1c02∂2u∂t2=∂2u∂ξ2+112∂4u∂ξ4+α∂∂ξ(∂u∂ξ)3(6)where 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 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 (ξ−ξ0−c tW)]+π423α(7)withW=13(c2/c02−1)(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., A_in/na)Ainna=max (∂u∂ξ)=2α(c2c02−1)(9)and in this way obtain a relation between propagation velocity c and the applied input A_in. 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 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.