Focused ultrasound enables selective actuation and Newton-level force output of untethered soft robots

Untethered miniature soft robots have significant application potentials in biomedical and industrial fields due to their space accessibility and safe human interaction. However, the lack of selective and forceful actuation is still challenging in revolutionizing and unleashing their versatility. Here, we propose a focused ultrasound-controlled phase transition strategy for achieving millimeter-level spatially selective actuation and Newton-level force of soft robots, which harnesses ultrasound-induced heating to trigger the phase transition inside the robot, enabling powerful actuation through inflation. The millimeter-level spatial resolution empowers single robot to perform multiple tasks according to specific requirements. As a concept-of-demonstration, we designed soft robot for liquid cargo delivery and biopsy robot for tissue acquisition and patching. Additionally, an autonomous control system is integrated with ultrasound imaging to enable automatic acoustic field alignment and control. The proposed method advances the spatiotemporal response capability of untethered miniature soft robots, holding promise for broadening their versatility and adaptability.

As shown in Supplementary Fig. 2, the signal generator produces a continuous sinusoid wave (that matches with the resonant frequency  of the transducer), which drives the transmitter to generate a mechanical vibration that travels through the material between two transducers and thus converted into electrical signal by the receiver.By reducing the distance of the sample between the transmitter and receiver (∆), the phase difference between transmitted signal and received signal will shift by ∆, thus the sound speed (v) in the sample can be calculated as follows: The measured density and sound speed for Ecoflex 00-30 and Fe3O4NPs dopped Ecoflex 00-30 can be found in Supplementary Table 1.
Supplementary Table 1: Acoustic impedance of different materials.

Material
Density (10 3 kg m -3 ) Velocity (m s -1 ) Impedance (MRayl) transmitter, which produces mechanical vibrations that pass through the sample and is converted into electrical signals by the receiver.The voltages of the receiver and transmitter are recorded to calculate the ratio of their signal amplitudes (Supplementary Fig. 4b).The voltage ratio of the receiver to the transmitter decreases with the thickness increase of the sample film.Additionally, the doping of Fe3O4NPs also decreases this voltage ratio.
The acoustic attenuation () of the Ecoflex 00-30 film with and without Fe3O4NPs doping at 1.7 MHz can be calculated using the following equation: Here,  1 and  2 are the received pressure amplitude of samples with thicknesses When the actuators are actuated by the acoustic field, the heat source  of the material heated can be calculated as 2 : where  is the magnitude of the acoustic intensity,  is the acoustic pressure, and  is the acoustic impedance.Based on the measured parameters, the introduction of Fe3O4NPs into the polymer increases heat absorption under identical acoustic fields.
Combining this with the heat transfer equation, the temperature field distribution can be obtained using equation ( 4): where  represents the temperature field,  is the material density,   is the specific heat of the material, and  is the thermal conductivity.
At the initial stage of heating, the temperature of the material is close to the environment, heat dissipation due to the gradient is negligible (as shown in equation ( 4), ∇ ⋅ (∇) ≈ 0).Therefore, most of the heat that induces the temperature change in the actuator originates from the acoustothermal effect.Moreover, since the attenuation of the Fe3O4NPs doped Ecoflex 00-30 is more considerable, more heat is generated, resulting in a faster temperature rise (Fig. 2d).Similarly, when the material reaches the steady-state temperature, most of the heat produced by the acoustothermal effect is used to balance the heat dissipation, causing the slope of the temperature curve to decrease until a steady state is reached.Furthermore, a larger input heat () caused by a larger voltage induces a stronger acoustothermal effect, leading a higher steadystate temperature (Equation ( 4) and Fig. 2e).
The thermal conductivity capability of the materials embedded within the soft robot also plays a role in the cooling speed.For instance, the inclusion of Fe3O4NPs in the structure improves the thermal conduction, enabling faster cooling rates.
Additionally, the size of the actuator has a direct impact on the heating and cooling time.
According to the scaling law, smaller actuators exhibit faster response rates.This is because the amount of heat required for actuation is proportional to the volume, which scales with the cube of the characteristic size.In contrast, the heat dissipation area is proportional to the square of the characteristic size.Therefore, smaller actuators dissipate heat more efficiently and experience faster heating and cooling rates.Our experimental results confirm this trend, as the 4 mm-sized actuator (Fig. 4h) demonstrated significantly shorter actuation and recovery times (within several seconds).
To summarize, the recovery phase of the soft robot can be adjusted by considering several factors.Immersion in a liquid environment, such as water, accelerates the cooling process, while the choice of materials, thickness of the structure, and size of the actuator also influence the heating and cooling rates.By optimizing these parameters, the soft robot can be tailored to meet the requirements of time-sensitive applications.
The finite element method was applied to predict the deformation and motion of the actuators/robots 4 .Specifically, according to the continuum mechanics, the deformation of a solid can be described: 5  = (, ) =  + (, ) where the coordinates  and  are denoted as the locations of the material particle before and after deformation, respectively.(, ) is the displacement vector pointing from  to .
The deformation gradient is given by:  = (, ) =  + (, ) where  is the Jacobian matrix of the transformation, for incompressible hypothesis, the corresponding determinant () = 1,  is the identity matrix.By making use of the polar decomposition theorem,  can be decomposed into a product of a pure rotation matrix () and a right stretch tensor () given in the material frame: The right Cauchy-Green deformation tensor  is defined by: By subtracting the identity tensor from the right Cauchy-Green deformation tensor , the Green-Lagrange strain tensor  is defined: which can then be written on component form as: For the strain tensor, we have three fundamental invariants: Using the principle stretches , which is given by the ratio of deformed length (  ) to undeformed length (  ), the principal invariants reduce to: For incompressible material ( 3 = 1), the stress-stretch relations are obtained from the strain energy function by virtual work considerations: where  is the principal Cauchy stress,  is the hydrostatic pressure which can be determined from the equilibrium equations and boundary conditions, the strain energy density function  is the amount of energy stored elastically in the material under the state of stretch.For Ogden hyperelastic materials, 6,7 the stored energy of the compressible material is given in terms of principal stretches: where  =  1  2  3 = 1 for the material is incompressible, and   ,   ,  are the material constants.The axial component of stress is given as: For the uniaxial tensile test, the axial stress in the compressible Ogden model is reduced to: When an incompressible condition ( = 1) is imposed, the axial stress in the direction of the load is given by: where the stretch can also be replaced by the strain: Which gives the strain-stress relationship under axial stress condition, using a Levenberg-Marquardt solver, we have fitted the Ogden model with the experimental results (the order N of the Ogden model is selected to be 3), the fitted curves are provided in Supplementary Fig. 10 with the material parameters shown in Supplementary Table 2.

Supplementary Fig. 10 :
The stress-strain curves of Ecoflex 00-30 and Fe3O4NPs doped Ecoflex 00-30 films.Where the scatters were obtained from experiments (EXP) and the lines (Ogden) represent the fitted model.