Robotic Trajectories and Morphology Manipulation of Single Particle and Granular Materials by a Vibration Tweezer

Abstract

Robotic self-assembly of deformable materials holds potential for the automatic construction of complex robots. Current manipulation for deformable manipulation mainly focuses on a soft robot. It still remains a great challenge for morphology manipulation of a swarm of particles. Chladni patterns have raised great interest in the field of self-assembly for different materials. The formation of Chladni patterns is driven by the vibration process that involves the particles moving from disorder to order. Particles bounce randomly on the plate, and gradually accumulate along nodal lines, whereas the instantaneous random effect is inevitable, meaning that the trajectories of particles are uncertain. Here, the vibration tweezer is proposed by programmable two-frequency driving Chladni patterns. Different materials can be precisely and flexibly trapped to the vibration node. The vibration tweezer is further programmed for arbitrary positions by solving the vibration inverse problem. Then, different controllable trajectories “PKU” manipulation of particle can be achieved through switching the tweezer positions. Most importantly, the vibration tweezer exhibits the morphology of granular materials assemblages with collection, motion, and rotation. This work paves the way for the control of complex self-assembly, thereby enabling programmable manipulation of granular materials and micro robots.

Introduction

Widespread self-assembly phenomena can be seen in natural systems. The second law of thermodynamics states that disorder tends to increase unless there exist external forces.¹ These ordered processes have been modeled by the particles system.² Inspired by these phenomena, the self-assembly is manipulated with different materials such as particles, fluid, and soft materials.³

The manipulation of particles into ordered patterns and trajectories has important applications in many fields such as 3D printing,⁴ microfluidic chips,⁵ micro robots,⁶ medicine engineering,⁷ etc. Different kinds of manipulation methods have been widely studied, including optical tweezers,⁸ magnetic field,⁹ electric field,¹⁰ and acoustic method.¹¹ The optical tweezer, which uses an optical wave of a small wavelength, can only manipulate nanoscale particles. The magnetic and electric manipulation methods require that particles should have electric and magnetic properties,¹² whereas the acoustic method has advantages such as no selectivity to particles properties and being contactless for different particle sizes. Besides, acoustic waves with different modes possess excellent programmable capacity for manipulation.¹³

To get better manipulation, the acoustic method has been applied in different environments, involving the vibrating plate,¹⁴ the surface acoustic wave,¹⁵ and the fluid medium.¹⁶ Particles are generally trapped to regions near nodal lines or antinodes, where the mechanical equilibrium condition exists. Therefore, the manipulation comes down to the control of acoustic fields. The main control methods focus on the actuators and the acoustic structure design. By using multiple actuators,¹⁷ both of the phase and amplitude distributions can be controlled, and multiple particles can be individually manipulated. Thus, different particles patterns and trajectories can be manipulated.¹⁸ Besides, the wave-front can be modified by designing inhomogeneous structures,¹⁹ and desired patterns or trajectory of particles can be manipulated with an acoustic hologram.¹⁸ However, these structures are prefabricated and only suitable for fixed pattern or trajectory manipulation, which cannot be tunable for other trajectories.

The dynamic manipulation has been investigated on a rigid vibrating plate as well as a flexural vibrating plate. Different patterns of granular materials form through switching the driving frequency, amplitude, and particles number.²⁰ Also, Lynch and colleagues²¹ developed the robotic platforms by the rigid vibrating plate. The object is manipulated by the inducement of friction in a position-dependent manner. On the flexural vibrating plate, under the driving of an external harmonic force, simple patterns form for the resonant modes, which are called Chladni patterns.²² In 1787, Chladni patterns were originally applied by the German physicist Chladni to make the vibration visible.²³ Moreover, particles of size ranging from millimeters to nanometers can be manipulated.^24,25 Large particles bounce on the vibrating plate and accumulate onto nodal lines, forming Chladni patterns, whereas fine particles are dragged to antinodes due to the acoustic streaming, forming inverse Chladni patterns.²⁶

The particles move randomly on the Chladni plate, which means that the trajectory is uncontrollable. Besides, the motion before they settle onto nodal lines is still not very well understood. To explain the random effect, different hypothetical stochastic models were proposed, including the Brownian motion model²⁷ and the Gaussian distribution model.²⁸ To overcome the random effect, the machine vision feedback method has been applied for the motion control.¹⁴ However, the method is based on learning the displacement and uncertainty data for each frequency, which strongly depends on the complex feedback control system. The manipulation for this method is induced by the impulsive force far from nodal lines. The particle motion, therefore, is still dispersing due to the random effect.

More recently, Latifi et al.²⁹ have demonstrated the merging of a swarm of particles for inverse Chladni patterns in water. However, the merging is at a particular position by selecting a special frequency, which is induced by the combination of acoustophoretic effect with plate vibration. Up to now, the random effect is inevitable for the particle manipulation on the Chladni plate, for which the manipulation of a swarm of particles gets dispersed. Current manipulation for deformable manipulation mainly focuses on a soft robot.³⁰ The arbitrary morphology manipulation of a swarm of particles still remains a great challenge.

In this article, we will show the robotic manipulation by a vibration tweezer. The object can be trapped and manipulated, regardless of the random effects. The particle's motion is induced by the impulsive force, which is characterized statistically by correlation with the vibration amplitude. Moreover, we show that the manipulation is controllable in a converging impulsive force field. Further, the converging impulsive force field can be constructed by the intersection of two nodal lines, which are generated by a two-frequency signal. Therefore, the two-frequency driving vibration tweezer is proposed, where the impulsive force field is converging to a vibration node. The tweezer is controlled for arbitrary trajectories “PKU” by programming the two-frequency driving signal sequence. Moreover, the vibration tweezer exhibits morphology assemblages of granular materials with collection, motion, and rotation at arbitrary positions. Therefore, the vibration tweezer is able to manipulate not only a single particle of different properties but also the morphology of granular materials.

Methodology

The robotic manipulation platform

Experiments have been performed on an aluminum plate, which is centrally driven by a vibration generator, with the four edges serving as free boundaries. Therefore, the flexural vibration mode is driven on the plate to manipulate particles, as shown in Figure 1a. To manipulate particles to different positions, different vibration modes and the corresponding frequencies, which are simulated by a computer by using the finite-element solver COMSOL Multiphysics, are generated on the plate. Then, the voltage signal data are imported from a programmable signal generator (Agilent 33220A). The power amplifier (B&K 2721) is used to amplify the signal, which is preprogrammed as a sequence of two-frequency signal segments, as shown in Figure 1b. All the patterns and trajectories processes are imaged by a video camera (Canon EOS 80D).

FIG. 1.

Experimental setup for the programmable vibration tweezer. (a) The schematic of robotic manipulation on the Chladni plate. (b) The driving signal sequence is solved by the simulation method, preprogrammed by the signal generator, and amplified by the power amplifier. Color images are available online.

First, the Chladni experiment is performed on a rectangular plate with dimensions of 280 × 240 × 2 mm. Silica sand particles (nominal grain size 0.125 mm, black color) are uniformly sprinkled on the plate to form Chladni patterns, which are driven by the first five resonant frequencies of a rectangular plate (134, 192, 488, 758, 1042 Hz). The first five-order Chladni patterns match well with the vibration acceleration minimum regions, which are acquired by a Polytec Laser Doppler Vibrometer (Polytec PSV400), as shown in Figure 2a.

FIG. 2.

The particle motion mechanism on the Chladni plate. (a) The formation process of Chladni patterns (lines formed by black silica sands). Chladni patterns are overlaid on the experimental vibration modes (the normalized vibration velocity by the maximum acceleration). The trajectories are traced for four times repeated experiments with the same starting positions (yellow circles) and different ending positions (crosses and pluses). (b, c) Simulation for the two-dimensional particle collision process. The bounce amplitude has a statistically positive correlation with the vibration amplitude. (d) The particles vertical bounce experiment. The bounce amplitude has a statistically positive correlation with the vibration amplitude, and a fitted line $η {Φ_{0}}^{2}$ is proposed (the dashed line); the width of the image is $a 8$ . (e) The sketch of particles bouncing with the Chladni plate. Color images are available online.

Besides, the moving process is tracked by using a small steel cylinder (diameter 4 mm, and height 1 mm). Four times of repeated experiments are performed for the two same starting positions, which are marked as yellow circles in Figure 2a. The first five-order modes are each driven for 30 s, which provides enough time for the particle to finally settle on the nodal lines. The particle jumps further, and gradually the motion becomes more dispersing when driven by the next modes. The trajectories are different, and end on different positions, which are marked as pluses and crosses in Figure 2a. The instantaneous random effect is inevitable along the trajectory, whereas the particle on the nodal line is stable. Thus, the particle trajectory is uncertain, resulting in dispersing distribution of particles along the nodal lines.

The robotic manipulation mechanism on the vibrating plate

To understand the motion mechanism, we investigate the vertical bounce process by simulation and experiment. The hard sphere collision model³¹ is developed to simulate the collision and motion process on the vibrating plate (see the Method for Particles Collision Simulation section in Supplementary Data). Here, we simulate the two-dimensional collision process. The vibration amplitude of the plate is modeled as $Φ_{0} = sin (x)$ with driving frequency $f = 134 Hz$ , as shown in Figure 2b. On the other hand, the bounce process is performed experimentally in a two-dimensional xz plane, as shown in Figure 2d.

The red sand particles bounce near a node of $a 8$ width (a is the width of the plate, as shown in the Experimental Results for the Vertical Bounce Process of Particles section in Supplementary Data), where the vibration amplitude is also about $Φ_{0} = sin (x)$ .

The bounce height h is of positive correlation with the vibration amplitude $Φ_{0}$ in Figure 2c and d. The bounce height h is statistically plotted with mean value $\bar{h}$ and standard error $σ$ , as shown in Figure 2c. Therefore, according to the simulation and experimental results, we assume that the bounce height is position dependent, and a fitted line is proposed to fit with h and $\bar{h}$ , as follows: $\bar{h} = η {Φ_{0}}^{2},$ (1)

where $η$ is a fitted factor; we obtain $h = \bar{h} + σ$ .

Then, the particles fall freely from a height h, and they bounce onto the plate with velocity $v$ , with the vertical component: $v_{Z} = \sqrt{2 g h}$ . By considering the nearly vertical free-falling, v_Z is dominant compared with the horizontal component $v_{X Y}$ . For the two-dimensional vibrating plate, the flexural vibration displacement is $z (x, y, t) = Φ (x, y) sin (2 π f t)$ , where $Φ (x, y)$ and f are the vibration amplitude and excitation frequency, respectively. The motion is induced by the impulsive force, Figure 2e shows that the impulse vector can be split into the normal and tangential components $J = J_{n} n + J_{τ} b o l d τ b o l d$ , as follows: $\{\begin{matrix} J_{n} = - (1 + e) m v_{a b} \cdot n \\ J_{τ} = - \frac{2}{7} (1 + β_{0}) m v_{a b} \cdot τ \end{matrix},$ (2)

where e and $β_{0}$ are the coefficients of normal and tangential restitution, respectively. The relative velocity at the contact point is $v_{a b}$ , with $|v_{a b}| = v_{Z} - 2 π f Φ cos 2 π f t$ .

For the Cartesian coordinate system with basis vectors $i$ , $j$ , $k$ , the horizontal component of the impulse vector $J_{X Y} (x, y) = i (J \cdot i) + j (J \cdot j)$ can be obtained as $J_{X Y} (x, y) = - C (v_{Z} - 2 π f Φ (x, y) cos 2 π f t) \nabla Φ (x, y) sin 2 π f t,$ (3)

with $C = (1 + e) m - 2 (1 + β_{0}) m 7$ . As previously discussed, the bounce height (with mean value $\bar{h}$ ) is of positive correlation with the vibration amplitude $Φ (x, y)$ , and the draft direction is contrary to $\nabla Φ (x, y)$ . In addition, the collision time t is random during a vibration period. Therefore, we propose the impulsive force vector field to be $J_{X Y} (x, y) = - (B_{t} + β) α (f) Φ (x, y) \nabla Φ (x, y),$ (4)

where B_t is the standard random variable having a zero mean value, and $α$ is a variable of linear correlation with frequency f. According to the experiments that the particles accumulate to nodal lines, $β$ is a positive constant for this. Similarly, for the two-frequency driving plate, the flexural vibration as $z (x, y, t) = Φ_{1} sin 2 π f_{1} t + Φ_{2} sin 2 π f_{2} t$ , the impulsive force field can be obtained as follows: $J_{X Y} (x, y) = - (B_{t} + β) [α (f_{1}) Φ_{1} + α (f_{2}) Φ_{2}] \nabla (Φ_{1} + Φ_{2}) .$ (5)

Under the inducement of the impulsive force field, particles accumulate to regions near nodal lines, where there exists a mechanical equilibrium condition. For equilibrium positions, the maximum acceleration does not exceed the acceleration of gravity g, which means particles do not separate from the plate. Therefore, the mechanical equilibrium condition for maximum acceleration is obtained as

Particles bounce into this region, then settle down by the collision and friction on the plate.

The robotic manipulation by the vibration tweezer

The particle is induced by the impulsive force field; thus, it moves to nodal lines, where the impulsive force is 0. The manipulation, therefore, depends on how to control the nodal line's position. First, we perform the straight-line trajectory manipulation of the particle on a square plate with dimensions of 240 × 240 × 2 mm, as shown in Supplementary Movie S1. The manipulation is performed by using the first seven resonant modes, with the resonant frequencies such as 429, 795, 1186, 1953, 2329, 2579, and 3478 Hz, respectively. The resonant modes can be split into a sequence of nodal lines from $N_{1}$ to $N_{13}$ , which are marked as solid and dashed lines of different color in Figure 3a. The straight-line trajectory manipulation is performed by the nodal line induced method. The manipulation trajectory is forward by altering the position of the nearest nodal lines from $N_{1}$ to $N_{13}$ , and backward in the reverse order. The designated trajectory is manipulated from the green cross to the next cross, which are located on the interaction of nodal lines with the straight-line trajectory. Gradually, the particle moves along the straight-line trajectory.

FIG. 3.

Straight-line trajectory manipulation of single particle driven by single frequency. (a) The straight-line trajectory manipulation by altering the nodal line positions. The manipulation is induced by a sequence of nodal lines from $N_{1}$ to $N_{13}$ , which are split from the first seven resonant modes. (b) The dispersing force field at position $P 7$ for $N_{7}$ of the sixth mode. (c) The converging force field at position $P 10$ for $N_{10}$ of the first mode. Color images are available online.

For most part of the trajectory forward manipulation, the particle moves near these cross points. However, the particle begins to deviate markedly from $P 6$ to $P 7$ , when driven by the nodal line $N_{7}$ (the part of the sixth mode). Finally, the object converges back to the designed trajectory driven by $N_{10}$ (the part of the first mode), as shown in the dashed circle in Figure 3a. The deviation is attributed to two mechanisms. The first mechanism involves the wiggly motion due to the random effect. The second is that the impulsive force field is diverging for $N_{7}$ of the sixth mode, as shown in Figure 3b. It means that the particle from point $P 6$ will move dispersedly in uncertain directions when induced by the nodal line $N_{7}$ of the sixth mode. In contrast, it is observed that the particle will converge to a point $P 10$ in a certain direction when induced by the nodal line $N_{10}$ of the first mode, as shown in Figure 3c. Therefore, the controllable manipulation can be achieved for the converging field.

How to achieve the converged impulsive force field is further investigated on the rectangular plate. We perform the control of the nodal lines driven by two frequencies, which are visualized by multiple sand particles. The sandpile is initially located at the position $P_{A}$ $(0.385 a, 0.32 b)$ , with a and b as the dimensions of the plate. Then, the sand particles are manipulated by a single frequency (117 and 178 Hz) and two-frequency simultaneous driving the vibrating plate. The nodal lines of the two frequency, respectively, are located on the common point $P_{B}$ $(0.52 a, 0.46 b)$ . Figure 4a shows, for the single frequency excitation, that particles accumulate along nodal lines with a dispersing effect. In contrast, for the two-frequency driving vibration, particles directly accumulate onto the node $P_{B}$ . The converging impulsive force field is achieved for the two-frequency driving vibration plate, as shown in Figure 4b.

FIG. 4.

The dispersing and converging impulsive force field. (a) Sandpile manipulation by single-frequency Chladni patterns with a dispersing impulsive force vector field. (b) Two-frequency driving Chladni patterns with a converging impulsive force field. The impulsive force vector field is overlaid on the normalized maximum acceleration. The red solid circle indicates the initial position $P_{A}$ , and the scale bar represents 10 mm. Color images are available online.

Up to this point, we have demonstrated that the particles accumulate to regions where the vibration amplitude is 0, and further the vibration tweezer is achieved by the intersection of two nodal lines. The position of the vibration tweezer is based on controlling the nodal line positions. Here, we investigate the inverse problem based on variable frequencies. The flexural vibration displacement $z (x, y, t)$ satisfies the wave equation $D \nabla^{4} z + ρ h \partial^{2} z \partial t^{2} = F (x, y, t)$ , with D, $ρ$ , and h being the flexural rigidity, the density, and the thickness, respectively. With the eigenmodes and eigen frequencies obtained as $\{φ_{n} (x, y); f_{n}\}$ for free boundary conditions, the vibration amplitude $Φ (x, y)$ can be solved as the superposition of the eigenmodes²² $Φ (x, y) = \sum_{n = 1}^{\infty} \frac{A_{n}}{(f^{2} - {f_{n}}^{2})} φ_{n} (x, y),$ (7)

with $A_{n} = \int \int φ_{n} (x, y) F (x, y) d x d y = F_{0} φ_{n} (0, 0) .$

The centrally driven sinusoid function can be expressed as $F (x, y, t) = F_{0} δ (x) δ (y) s i n (2 π f t)$ , where F₀ is the amplitude of the centrally driving force with frequency f, and $δ (\cdot)$ is an impulse function. Here, how to achieve the nodal lines locating at arbitrary position $(x_{0}, y_{0})$ is studied. Considering the vibration amplitude being 0, the frequency f satisfies $Φ (x_{0}, y_{0}, f) = 0$ . There exist solutions of f_i between the adjacent resonant frequency $f_{n} < f_{i} < f_{n + 1}$ , and f_i is in the complex form. In practice, the frequency f should be real.

For a point such as $P_{B}$ , there are other real solutions besides 117 Hz and 178 Hz. The vibration tweezer is achieved by driving two of these frequencies simultaneously, as shown in Figure 5a. To solve the real solutions, $Φ (x_{0}, y_{0}, f)$ is plotted in Figure 5b, which displays the first eight solutions (A solving method is also proposed according to the function; see the Solutions for Vibration Tweezer on Arbitrary Points section in Supplementary Data). We further investigate the real solutions for arbitrary positions, and the existence condition is $Φ (x_{0}, y_{0}, {f_{n}}^{+}) Φ (x_{0}, y_{0}, {f_{n + 1}}^{-}) < 0$ for $f_{n} < f_{i} < f_{n + 1}$ . It is calculated that there exist at least two real solutions for $f < f_{10}$ , as shown in Figure 5c. Thus, the vibration tweezer can be solved at arbitrary positions. Therefore, the trajectory can be manipulated to arbitrary shape by altering the vibration tweezer at arbitrary positions.

FIG. 5.

The inverse problem for vibration tweezer. (a) Point $P_{B}$ located on multiple nodal lines for different frequencies. The vibration tweezer is indicated as the red point. (b) Multi-solutions of $Φ (x_{0}, y_{0}, f) = 0$ for point $P_{B}$ , which is marked as the green points. The red dashed lines indicate the resonant frequencies. (c) For arbitrary points on the plate, there exist at least two real solutions of $Φ (x_{0}, y_{0}, f) = 0$ for $f < f_{10}$ . Color images are available online.

Results

The robotic trajectory manipulation

Here, the vibration tweezer is used to manipulate particles with arbitrary trajectories. We split the designated trajectories into a set of points $P_{i}^{T}$ $(x_{i}, y_{i})$ . To achieve the vibration tweezer on these points, respectively, we preprogram a two-frequency voltage signal sequence with frequencies $(f_{1, i}^{T}, f_{2, i}^{T})$ and amplitudes $(A_{1, i}^{T}, A_{2, i}^{T})$ , respectively. The duration time is $Δ t = 5 s$ , which provides enough moving time, and a trapezoid envelope is used to guarantee no saltation for the input voltage signal.

In Figure 6a, different trajectories “PKU” manipulation is demonstrated (Supplementary Movie S2). The trajectory for each character is split into 50 points, which are marked as the red crosses. The particle moves from point to point, and it gets trapped to the vibration tweezer. Then, the particle settles steadily on the equilibrium point, till the next signal segment is driven. Now less affected by the random effect, the particle finally moves along the designated trajectories. Moreover, the hard sphere model is used for the simulation of the manipulation process, which is in good correspondence with the experimental trials and the programmable trials. The simulation also has programmable two-frequency waves of frequencies $(f_{1, i}^{T}, f_{2, i}^{T})$ and amplitudes $(A_{1, i}^{T}, A_{2, i}^{T})$ along the trajectory. The duration time is $Δ t = 5 s$ for each moving step. The zoomed view of trajectory “U” is shown in Figure 6b, which illustrates that the vibration well travels along the trajectories with the altering of the driving signals. The vibration tweezer corresponds to the position of the vibration node.

FIG. 6.

Programmable trajectory manipulation of single particles by vibration tweezer. (a) Trajectories manipulation of single particles: the programmable trials, experimental trials, and simulation trials. (b) The zoomed view of the trajectory “U.” The vibration well (the areas near simulation minimum of the normalized acceleration) travels along the designated trajectories. Color images are available online.

The morphology manipulation of granular materials

We further demonstrate that the vibration tweezer is able to manipulate not only a single particle but also granular materials. Similar to the previous manipulation of trajectory “P,” the sandpile is manipulated along the same trajectory by altering the vibration tweezer position, as shown in the Supplementary Movie S3. In Figure 7a, sandpiles of eight positions for the equal time interval are demonstrated. Sand particles are initially spread on the plate; then, they get trapped to a point by the vibration tweezer. The sand particles are trapped to each point along the trajectory. On the other hand, the sandpile exhibits the function of the visualization for the vibration tweezer transition process.

FIG. 7.

Programmable morphology manipulation of granular materials by vibration tweezer. (a) The sandpile manipulation of trajectory “P” transition. (b) The sandpile of length 20 mm rotates in different directions. (c) The vibration well (the areas near the simulation minimum of the normalized acceleration) is formed from different shapes by programming the two-frequency signals. Color images are available online.

At the same time, the vibration tweezer exhibits morphologies manipulation of granular materials. As discussed in Figure 5a, it is able to achieve variable directions of nodal lines for arbitrary positions, such as the point $P_{B}$ $(0.52 a, 0.46 b)$ . Here, we perform a uniform rotation by selecting three groups of frequencies $f_{1} = 117 Hz, f_{2} = 178 Hz$ ; $f_{3} = 1694 Hz, f_{4} = 891 Hz$ ; and $f_{5} = 1178 Hz, f_{6} = 3400 Hz$ , which correspond to $0^{\circ}, 9 0^{\circ}$ ; $3 0^{\circ}, 12 0^{\circ}$ ; $6 0^{\circ}, 15 0^{\circ}$ , respectively. Sand particles are uniformly sprinkled on the vibrating plate, and further trapped to form shapes such as a point or a line segment, as shown in Figure 7b.

The shape can be further manipulated with the altering of the frequencies and amplitudes. Taking the line segments of 0°, 90° for example, it is driven by frequencies f₁ and f₂. The tweezer exhibits particles assemblages with different morphologies for proper amplitudes ratio, such as a point and lines with different directions. The 0° line segment forms when the signal of f₁ is dominant. The 0° line segment stretches with the increase of the amplitude for f₁ signal. The case of 90° dominated by the f₂ signal is the same. By preprogramming a sequence of these two-frequency signals, the rotation with the same length of 20 mm is achieved (Supplementary Movie S4). According to Equation (6), particles settle on areas near the vibration node. In Figure 7c, the vibration well transforms with the altering of the two-frequency voltage signals, which can match with the morphology of the sandpile.

Conclusion

In this article, we have investigated the two-frequency-driven Chladni patterns. The vibration tweezer is achieved with the intersection of two nodal lines. Under the inducement of the converged impulsive force field, the particles can be trapped by the vibration tweezer, regardless of its properties. The robotic manipulation has been performed well for other shapes, including the polystyrene ball with different diameters (0.2–1.0 mm) and the deformable plasticine (cuboid, spherical, etc.). The trajectory manipulation can be realized from point to point by the preprogrammed two-frequency driving signal sequence. To achieve the vibration tweezer for diverse positions and shapes, the inverse problem is investigated based on variable frequencies, which are not limit to the particular frequencies. Further, by switching the driving frequencies and amplitudes, both the position and shape of the vibration tweezer can be controlled. Therefore, the vibration tweezer can manipulate not only a single particle but also the morphology of granular materials in the form of motion, rotation, and shape change.

Based on the vibration inverse problem for frequency, we proposed a new kind of tweezer. With diverse manipulation functions, the programmable vibration tweezer provides a powerful tool for the controlling of objects and the dynamic manipulation of granular media's position and morphologies. We believe that the vibration tweezer will show potential for applications such as micro robots, biomedicine, particles assembly, and material fabrication. In addition, the inverse problem for the vibration tweezer satisfies Chladni plate of different sizes. Therefore, the manipulation can be expanded to micro-scale and nano-scale robots.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

National Natural Science Foundation of China (Grant numbers: 11521202, 11890681, 11522214).

Supplementary Material

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Supplementary Material

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