Simple and Fast Locomotion of Vibrating Asymmetric Soft Robots

Abstract

To be fully integrated into the activities of our daily lives, robots need to be capable of traversing unstructured environments and interacting safely with their surroundings. Soft robots are perfect candidates since they can adapt to their surroundings through passive material compliance, rather than relying on complex control. However, the same compliance hinders the generation of propelling forces, and current approaches face a trade-off between traveling speed, action range, and control complexity. We overcome this trade-off by developing a locomotion mechanism based on the synergistic interaction between symmetric vibrations, elasticity, and asymmetric morphology. We then realize a rapid soft locomotor using inexpensive off-the-shelf components and requiring only elementary actuation and control. A single robotic unit can travel at speeds up to 100 mm/s when tethered and 35 mm/s when untethered. We derive a model that predicts the speed of the robot as a function of several design parameters and physical properties, highlighting the role of geometric asymmetries in the resulting anisotropic motion. Moreover, these elementary units can be added together to create more complex behaviors. By adding 2 units in parallel, the assembly can locomote and be steered following nonholonomic constraints. Our approach opens the door to a new class of low-cost soft robots that can travel fast and far with elementary fabrication and control, and which can be combined to achieve complex functions without compromising their essential simplicity.

Introduction

Although machines excel at repetitive tasks in preprogrammed situations, they are greatly inferior to organic life when functioning in unstructured environments, such as a busy street or a littered garden, considerably limiting their inclusion in daily life activities. The field of soft robotics promises to bring machines closer to the innate versatility of biological systems by combining simple actuation with the complex behavior of compliant materials, which allows conforming to surrounding shapes.

The ability to conform to external objects could also bring great benefits to locomotion: the natural asperities of a terrain could be “absorbed” by the material, rather than requiring balance and force adjustments to be performed at a computational level, potentially reducing cost and complexity of fabrication and control of these machines. In fact, several strategies have already been explored to generate propulsion in soft machines, including animal-mimicking approaches,^1–10 peristalsis,^11–15 traveling waves,^2,16–22 elasticity-enabled hopping,^23–28 snap-through elastic instabilities,^29,30 rolling,²¹ vibrating tensegrity structures,^31,32 and shape-shifting materials.^33,34

However, all these strategies face a trade-off between the resulting locomotion speed and its range, which is often limited by tethers or the extent of the external fields; moreover, the simplicity granted by exploiting material compliance for adaptability is often counterbalanced by the complexity of the propulsion mechanisms, as they can require multiple joints, actuators, and signals to act concertedly.

In recent years, both rigid and soft robots have explored the generation of propulsion by combining symmetric vibrations with asymmetric contact forces, achieving net forward motion.^35–41 Combining vibration with asymmetric contact provides a simple way to achieve locomotion that only requires a single active element (a vibration source) driven by a single electrical sinusoidal signal. However, the anisotropic contact is typically achieved by adopting semirigid tilted features (e.g., bristles or beams) as feet, often resulting in suboptimal gaits and back-slippage, leading to relatively low speeds and a steep inverse proportionality between speed and external loads. As a consequence, even sub-gram external loads can cause drastic speed reduction,³⁵ whereas simultaneously requiring high driving voltages (>150 V) to achieve locomotion, effectively forcing most vibration-based robots to tethered function and, consequently, to a limited range of action.

To this date, achieving simultaneously simple, fast, and long-ranged locomotion remains a major challenge for soft robots.

In this study, we tackle this challenge by combining elastic and frictional properties of soft matter to further develop the vibration-based approach to locomotion. We do so by introducing a soft silicone-based robot with a sawtooth-patterned surface and an embedded source of one-dimensional vibrations (Fig. 1a). During vibration, the sawtooth teeth undergo cycles of elastic extensions and compression, respectively, decreasing and increasing the effective friction and causing an overall net forward movement (Fig. 1b). Our robot achieves recorded speeds as high as 100 mm/s (Fig. 1c and Supplementary Video S1), among the highest values reported in the literature, whereas maintaining elementary simplicity in fabrication, actuation, and control.

FIG. 1.

Locomotion of a CT unit. (a) Photographs and schematic representations of a CT unit; the arrow indicates the direction of vibration. (b) Schematic representation of the working mechanism of vibrating robots. (c) Snapshots of a CT unit traveling forward. CT, “Christmas Tree.”

In addition, compared with soft robots of equal simplicity, it is the second fastest robot in absolute terms, and the fastest robot per input voltage, requiring <20 V for peak performance (an order of magnitude less than the fastest vibration robot previously reported³⁷). The robot can carry loads in the order of tens of grams, making it possible to equip off-the-shelf sensors and energy sources. Without adding complexity, the robot demonstrates untethered locomotion as well, reaching speeds up to 35 mm/s, approximately an order of magnitude higher than the highest speed previously reported for an untethered vibrating robot.

Finally, this study shows that frictional dynamics of soft ratcheted surfaces, whose physical behavior remains an open topic of research,⁴² can be leveraged to design fast and simple soft locomotors.

Throughout the remainder of this article, we will refer to a single robotic unit, that is, a combination of soft ratcheted surface and vibration motor, as a “Christmas Tree” (CT) unit, due to its resemblance to the silhouette of the popular holiday pine tree.

Materials and Methods

Experimental setup

The samples were fabricated by curing a silicone rubber solution (EcoFlex 050; Smooth-On, Inc., TX) into a 3D printed mold for ∼3 h. As a vibration source, we used a HapCoil One (Tactile Labs, Montreal, Canada), which generates one-dimensional vibrations exploiting a coil–magnet interaction with a resonant frequency of 65 Hz and peak acceleration of 8 G when applied to a 100 g mass; when the source was embedded within the silicone body, we observed an upward shift in resonant frequency, resulting in peak vibration amplitudes at ∼100 Hz. The signal for the vibration was provided by a Digilent Analog Discovery (National Instruments, TX) wave generator and it was amplified using a PD200 amplifier (PiezoDrive, NSW, Australia). During untethered experiments, the signal was generated on a laptop by a commercial audio streaming service and then transmitted through Bluetooth to a battery-powered 15 W Bluetooth stereo amplifier board (Walfront, DE).

We acquired the high-framerate recordings using a PCC Phantom camera VEO-E 310L and used the associated software to perform automated tracking of a tooth tip. During the measurements of frictional properties, a custom-fitted 3D-printed holder was embedded in the surface, allowing us to fix it in place. The 6-axis force sensor Nano43 (ATI Industrial Automation, Incorporation, NC) was attached to a 5 mm thick plexiglass plate, and we affixed the plate itself to a motorized linear guide. The experiments were conducted by commanding the linear guide to slide back and forth along the long axis of the surface, moving 20 mm in one direction at 5 mm/s, then reversing the motion. A single acquisition procedure took ∼16 s. Five of these sequences were performed and averaged.

Finite elements

The finite elements model was set up in COMSOL Multiphysics 5.6 (COMSOL Group, Sweden), using the Solid Mechanics module and performing a static analysis. A single tooth was modeled as a right triangle with a 5 mm base and 12 mm height. The material considered in this model was the built-in solid silicone rubber, and the tooth was meshed with triangular elements and 0.5 mm spaced nodes.

The coordinates of three points of a tooth in the CT unit, corresponding to the vertices of the modeled triangle, were tracked from the high-framerate recordings. The spiral motion of the points was simplified as an ellipsis for the duration of a single cycle and used as input for the displacement of the vertices in the COMSOL model. The pressure inside the modeled tooth was then computed throughout a cycle.

Results

Propulsion and working principle

We fabricated the robot by casting a silicone-based rubber in a 3D printed mold with a sawtooth-patterned bottom surface and embedded in it a solenoid-based vibration source; based on research published on asymmetric semirigid feet^35–41 and on soft ratcheted surfaces,⁴² we hypothesized that the broken geometric symmetry would introduce anisotropic friction, leading to stick-slip motion. To first test that this design can generate propulsion, we fixed the CT unit to a custom-fitted holder and used a triaxial force sensor to measure the force it exerted on a plane (Fig. 2a), observing oscillating values with a nonzero average equal to 0.2 N (Fig. 2b) and thus demonstrating a net propelling force.

FIG. 2.

Forces exerted and experienced by a CT unit. (a) Setup schematics for normal and tangential force measurements. (b) Force exerted by a vibrating CT unit in a direction parallel to the contact plane during nine cycles and its time average, showing a net average propulsion. (c) Effective normal force of a sliding CT unit during forward and backward movements. Arrows indicate the direction of sliding (forward and backward). (d) Effective frictional force experienced by a CT unit during forward and backward movements. Arrows indicate the direction of sliding (forward and backward). (e) Zoomed-in snapshots of the teeth of a CT unit during a single cycle of vibration. The arrows indicate the instantaneous direction of motion of the three points their base is placed on, whereas the gray shades indicates positive and negative pressures (inwards and outwards), respectively.

Frictional anisotropy

We further characterized the frictional properties of the CT unit by performing a series of sliding experiments in the two directions, recording the normal and lateral forces with the same 6-axis sensor of the previous experiment. These tests showed a complex relationship between normal and tangential forces: while the total frictional force measured was indeed higher during backward sliding (i.e., against the orientation of the teeth), the normal force exerted on the sliding surface was also higher, and the opposite was true during forward sliding (Fig. 2c, d).

To further investigate these results, we recorded the motion of a traveling CT unit at 1500 frames per second (Supplementary Video S2) and analyzed the videos to reconstruct the trajectory of different points of a single tooth (yellow arrows in Fig. 2e). Moreover, we used these trajectories as an input to a finite element model to reconstruct the internal pressure of a tooth during vibration (color overlays in Fig. 2e). These results suggest that the asymmetric frictional behavior is, in fact, the result of an asymmetric elastic dynamics at the tooth level during sliding: during backward motion, the tooth is compressed and elastic forces increase the pressure on the ground, leading to higher friction; as the source force switches sign, the tooth extends, jumping forward and reducing the contact pressure (see Fig. 2d for a schematic representation).

This behavior is comparable with the stick-slip effect, which is typically observed in bodies under the time-varying load of eccentric-rotating masses. In contrast, this explanation for the asymmetric forces observed is in contrast with previous work on soft ratcheted surfaces under linear vibrations,⁴² where it was hypothesized that friction anisotropy arises from the interplay of adhesion and mechanical interlocking.

Model of vibration-based locomotion

The average traveling speed of a CT unit essentially depends on the difference between the distance moved forward and backward within one vibration cycle. In general, we can expect this difference to depend on several parameters, such as the force of vibration and the mass of the robot: a stronger vibration force will increase the speed, whereas a larger mass might decrease it. Moreover, assuming for simplicity only right-angle triangles in the design space of the sawtooth pattern, it must be true that there exists an optimal tilting angle θ⋆, since both θ = 0 (flat surface) and θ = π/2 (vertical teeth) should result in null traveling speed, as they do not break the frictional symmetry. To maximize the traveling speed, then, it is important to identify this optimal angle θ⋆. In particular, we can expect θ⋆ to be the angle for which the (e.g.) forward motion is maximized, whereas the backward motion is minimized; in the ideal case, there is no backward motion (v_b = 0 mm/s).

To find and study θ⋆, let us start by analyzing how the geometry of a single tooth reorients the forces introduced by the vibrating element. First, we observe that in the finite element model presented in Figures 2 and 3a, the forces are mainly distributed along the edges of the triangle. Since the tilted edge is the only one to break geometric symmetry, it is likely to be the factor creating the asymmetric behavior. Therefore, we consider only the deformation of the tilted edge, using it as an approximation of the behavior of a whole tooth. We can then approximate its elastic behavior by modeling it as a single linear spring, connected to a point-mass m at the bottom vertex.

FIG. 3.

Conceptualization of the mechanics of a CT unit. (a) Schematic representation of the equivalent spring-mass system of a single tooth. (b) Schematic representation of the forces acting on the mass during different phases of motion of the oscillating extremity. (c) Tangential and normal forces acting on the mass as an effect of the oscillation of the extremity of the spring, assuming quasi-static conditions and no sliding.

The vibrating element is modeled by a horizontally oscillating point connected to the other end of the spring (Fig. 3a). We can derive the stiffness to be associated with the spring based on the Young's modulus of the body of the robot, while we can consider m to be equal to its mass. It is useful to point out here that it is irrelevant, for the purposes of this model, to match the elastic properties of a CT unit exactly: although elastic stiffness will likely have a large impact on the overall velocity of the robot, which a static model will fail to capture in any case, it will not have a significant impact on the symmetry breaking we seek to investigate; in other words, we do not expect the Young's modulus (or its equivalent spring stiffness) to affect θ⋆.

For the motion of the oscillating point, we assume it to be restricted to the horizontal plane, and to correspond to the motion imparted by the vibrating source, found by integrating twice its acceleration: x(t) = L₀ cos(θ₀) + (A/ω²)sin(ωt), with A being the amplitude of the acceleration of the vibration source, x₀ the coordinate at rest of the top point, and ω = 2πf its angular frequency. As this point moves, it will cause an elongation or a compression of the spring (Fig. 3b): assuming that the bottom point is located at x_b = 0, we can write the elongation as $Δ L = L - L 0 = \frac{L_{0} cos (θ_{0})}{cos (θ)} + \frac{A cos (ω t)}{ω^{2} cos (θ)} - L_{0},$ (1)

where $θ = {tan}^{- 1} (\frac{y}{x (t)})$ is the new inclination due to the oscillation of the top point, with θ₀ being the spring inclination at rest.

The point mass (representing the tooth tip) will then move if the horizontal component of the elastic force acting on it is larger than the static friction that opposes the motion, and will otherwise stay stuck in place (Fig. 3c). We can write this balance of forces as $k Δ L c o s (θ) - μ (m g - k Δ L s i n (θ)) = 0,$ (2)

where k is the elastic constant of the spring, g is the gravitational acceleration, μ is the coefficient of static friction, and the normal force term (mg − kΔLsin(θ)) is considered to be either positive or null (Fig. 3c), since as a “negative” weight would correspond to the mass-point breaking contact with the ground and entering ballistic motion, which we assume to be friction-less.

We can now introduce a function X (θ, F_e, m, μ, t) representing the total distance traveled during a single vibration cycle: $X (θ, F_{e}, m, μ, t) = \int \int k Δ L c o s (θ) - μ (m g - k Δ L s i n (θ)) d t,$ (3)

where F_e represents the magnitude of the restoring elastic force KΔL.

We used numerical integration to study how the angle θ⋆ depends on different physical or design parameters, namely the mass of the robot (Fig. 4a, b), the coefficient of friction between the robot and the ground (Fig. 4c), and the total elastic force acting on the mass representing the tip of the tooth (Fig. 4d). This last term encompasses the effects of the elastic and geometric properties of the robot, as well the source vibration amplitude and frequency.

FIG. 4.

Modeled parametric dependency of the average traveling speed. (a) Normalized average traveling speed as a function of the external mass loading a 20 g robot and the tilting angle of the teeth. The black points indicate the front of maximum speed. (b) Curves of normalized speed as a function of tilting angle for a few exemplary values of external mass load. (c) Normalized average traveling speed as a function of the coefficient of friction between the robot and the ground, and the tilting angle of the teeth. The black points indicate the front of maximum speed. (d) Normalized average traveling speed as a function of the elastic forces acting on the tip of the tooth and the tilting angle of the teeth. The black points indicate the front of maximum speed.

In Figure 4, we can observe that, for certain combinations of parameters, there is a discontinuity in the predicted average velocity. This discontinuity is the result of a transition from a regime in which the robot moves forward and backward during each oscillation (forward motion being in any case larger in amplitude), to a regime in which backward motion is entirely prevented by static friction. The transition corresponds to the transition of the tangential forces gradually slipping below the frictional threshold.

Based on the results of the experiments and of this basic model, we can expect the mean traveling speed of a CT unit to take the general form of $\bar{υ} = α \frac{ω (θ - θ^{*}) F_{υ} n E A}{m f L_{0}} + C,$ (4)

where α and C are constants, F_v is the vibration force and f its frequency, m is the mass of the robot, nEA/L0 represents the elasticity of the robotic body, with E being its Young's modulus, A the area of the robot, n the number of teeth and L₀ the length at rest of their tilted side; θ is the tilting angle of a tooth with respect to the ground plane, and w(θ − θ⋆) is a function defined for θ ∈ [0,π] with a maximum for θ = θ⋆. Based on the results of our model, we predict that the speed of a CT unit can be further increased by increasing the vibration amplitude, the number of teeth, and/or the elastic stiffness of the material of the body.

The size of each tooth and the overall mass of the robot both have a negative impact on performance. Furthermore, we note that, in principle, a lower frequency should be expected to increase the robotic speed, since a greater vibration period would lead to greater accelerations during the positive phase of each cycle; however, in resonant systems (such as the vibration source employed in this work), we expect the dependency of vibration amplitude on resonance to be the dominant effect.

Experiments on locomotion performance

To validate the model and test the performance of the CT unit, we performed a series of locomotion experiments under different conditions, whereas maintaining the robot tethered to a signal and power source.

First, without any external load, we varied the input power between 4 and 10 W (corresponding to vibration forces between 2 and 5 N, based on the datasheet of the vibration source) and recorded the corresponding locomotion (Supplementary Video S2). These experiments were performed in front of a high-speed camera, and the speed was estimated five times from different sections of each recording, based on the distance traveled during 10 periods of vibration. The results are consistent with the predictions of the model, showing a linear proportionality between input power (or vibration force) and locomotion speed (Fig. 5a).

FIG. 5.

Locomotion experiments. (a) Measurements of tethered traveling speed of a CT unit as a function of the vibration force. The dashed line corresponds to a linear fit of the data points. (b) Measurements of tethered traveling speed of a CT unit under different loading conditions; a load of 0 g corresponds to the unit carrying only its own weight (20 g). The dashed line corresponds to an exponential fit of the data points. (c) Snapshots of the CT unit on planes with different inclinations.

We then recorded the locomotion speed of the robot under a constant input power of 15 W (corresponding to a vibration force of ∼7 N), while it carried loads between 5 and 55 g (Supplementary Video S3); for each loading condition, we acquired five normal-speed video recordings of the locomotion. Consistently with our predictions, we observe an inverse proportionality between speed and external load, although the experiments show a more complicated nonlinear dependence between these two variables (Fig. 5b).

Finally, we tested the ability of the CT unit to travel over inclined terrain under an input power of 15 W and without external loading (Supplementary Video S4). We observed speeds of several centimeters per second for slopes as steep as 30° before a sharp decrease in mobility takes place; we found the limit slope to be inclined by 40°, observing that the CT unit remains stationary when vibrating (Fig. 5c).

Untethered locomotion

Both numerical and experimental results show that the CT unit is capable of transporting external load in the order of the tens of grams while still being able to move at considerable speed. We, therefore, built an untethered version of the CT unit by connecting the vibration source to a battery-powered Bluetooth receiver. The robot was then driven remotely by a smartphone playing a 100 Hz tone, on average reaching speeds ∼25 mm/s (Fig. 6a, b), with a maximum speed observed at 35 mm/s. The speed itself can be controlled directly by varying the volume on the phone, whereas the trajectory is, in principle, fixed; however, we observed that small left/right imbalances in the distribution of mass (e.g., due to the positioning of the battery) resulted in curved trajectories (Supplementary Video S5).

FIG. 6.

Untethered locomotion. (a) Untethered CT unit. A plastic frame fixes to the unit the Bluetooth receiver, to which are connected the vibration source and a lithium battery. (b) Locomotion example of the untethered CT unit; due to weight imbalances, the trajectory is skewed compared with the initial orientation of the unit. (c) The double-unit untethered robot, with each CT unit connected to a side of the Bluetooth receiver. (d) Example of rectilinear locomotion of the double-unit robot. (e) Example of the circular motion of the double-unit robot. (f) Example of 2D steering of the double-unit robot, which is driven to approach and circumnavigate an obstacle.

This unbalancing effect suggests the possibility of 2D steering by controlling the spatial distribution of forces acting between the robot and the ground. We investigate this possibility by combining two CT units into a single robot, with the two vibrating motors parallel to each other and placed symmetrically at the sides of the Bluetooth receiver (Fig. 6c), achieving two important results: first, the two off-centered units allow for a more stable distribution of weight and contact, so that the impact on the trajectory of small variations of mass distribution (e.g., due to the battery placement) is drastically reduced; furthermore, by controlling the relative intensity of the vibration force of the 2 U it becomes possible to achieve both rectilinear and circular motion, and any combination thereof.

Our experiments showed that the rectilinear speed of the double-unit robot is slightly higher than that of the untethered single CT unit, being on average ∼30 mm/s (Fig. 6d), whereas the circular motion covers approximately π/2 rad/s (Fig. 6e). Finally, we tested the full extent of the 2D locomotion possibilities by steering the double-unit robot to approach, move around, and leave behind an obstacle (Fig. 6f). The obstacle was positioned at a distance of 250 mm from the robot, and this track was completed in 24 s. The robot was controlled again through a 100 Hz tone signal sent wirelessly through a smartphone, and the left/right steering was achieved by changing the left/right balance of the audio signal (Supplementary Video S5).

Discussion

By combining vibration-based approaches to locomotion, broken symmetries, and elasticity, we have demonstrated a new class of fast soft robotic locomotor that could find direct application in remote exploration scenarios. Without the need for large amplitude vibrations, high voltages, or custom-made electronics, our prototype can travel at speeds of up to 100 mm/s while tethered and 30 mm/s while untethered. These speeds are among the highest reported in the literature for soft robots, while simultaneously maintaining the lowest level of complexity reported (Fig. 7), where complexity is intended here as a combination of the number of separate body parts and signals required to achieve locomotion.

FIG. 7.

Speed and complexity in vibrating soft robotic locomotors. Complexity is defined here as C = S + P − 1, where S is the number of signals and P is the number of body parts necessary for locomotion. Gray shade represents the input voltage at which the robots are driven. The black markers represent the two fastest nonvibrating robots we are aware of. Squares represent tethered robots, whereas circles represent untethered ones. The stars mark the CT unit. Dashed lines connect the tethered and untethered realizations of the same robot.

Moreover, compared with robots of equal complexity,^37,40 the CT unit is the only design shown to be capable of untethered locomotion. This efficient locomotion is enabled by the out-of-plane, asymmetric redirection of forces achieved by the tilted elastic elements. This same principle could be implemented in more complex machines, such as legged robots, where the actuation of limbs could be simplified and reinforced by the introduction of similarly symmetry-breaking elastic elements combined with a simple vibration source.

Equation (4) predicts the effects of scaling the robot up or down, highlighting, in particular, the trade-off between tooth size, number of teeth, and mass. It should be noted, however, that the model makes a few key simplifications, neglecting the effects of viscous damping, adhesive frictional forces, and dynamic effects (e.g., resonances). Experimental validation or correction of the scaling relations will be an important step toward widening the range of applications for this locomotion principle.

Although we have observed that the robot is capable of overcoming minor asperities in the ground (e.g., cracks between floor planks), the locomotion possibilities of our robot are still largely limited by challenges common to most robots: the teeth of the CT units need to be free to oscillate and to find purchase on the ground, that is, the locomotion is impaired by slipping conditions (e.g., on wet surfaces) and obstacles (e.g., pebbles or branches). In particular, the frictional properties of the ground surface can play a large role in the performance of the robot. As shown in Figure 4, there exists an optimal strength of frictional forces for locomotion, depending on the tilting angle of the asymmetric features, such that the difference between backward and forward locomotion is maximized within each cycle.

In a similar manner, the presence of elements that affect the coupling between the robot and the ground can deteriorate the resulting locomotion, if it causes parts of the vibrating surface to lose traction against the ground. Furthermore, the weight distribution of the robot can affect the locomotion direction: since all points in contact with the ground contribute to the locomotion, and the resulting speed of each section depends on its load, an uneven distribution (caused e.g., by tilted ground or uneven body density) can cause the robot to steer in the direction of the heavier side.

Despite these current limitations, the capability of our robot to travel untethered at significant speeds makes a significant step toward realistic applications for remote and autonomous exploration. In addition, the low cost and simplicity of manufacturing and controlling our robot suggest applications for swarm robotics, which, to date, have been dominated by rigid machines.

Footnotes

Acknowledgments

The authors acknowledge funding received within the context of the Soft Robotics Consortium financed by the 4TU Federation Project.

Authors' Contributions

Conceptualization, visualization, and project administration by A.S. and M.W. Methodology and writing—original draft by A.S. Investigation by A.S. and M.A.A. Funding acquisition and supervision by M.W.

Data and Materials Availability

All data are available in the main text or the Supplementary Videos S1–S5.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This study was supported by the 4TU Soft Robotics Program.

Supplementary Material

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