Fast‐Moving and Highly Adaptable Hollow Piezoelectric Miniature Robot

Fast motion and high adaptability are crucial for the practical applications of miniature robots. However, combining these capabilities into a single robot skillfully is challenging, particularly in the domain of miniature structural design. Inspired by the amphibious motion of crocodiles in nature, a hollow piezoelectric miniature robot (HPMR) is proposed. The key benefit of the HPMR is to concurrently output the oscillating motion of the driving foot (oscillating mode) and the rotating motion of the internal rotor (rotating mode). That is, the mode of single‐energy‐input‐multimotion output allows the robot to balance the requirements of speed and adaptability. In the oscillating mode, HPMR can run at speeds of up to 12.3 BL s−1 and realize a displacement resolution of 0.3 μm. Following the conical rotor assembly, the rotor can operate at a speed of 890 rpm, with a displacement resolution of 7.28 mrad. In the rotating mode, the robot demonstrates its multitasking ability in moving amphibiously, carrying loads, running on surfaces with varying roughness, crossing obstacles, and resisting shocks. The results show that the HPMR is capable of fast, high‐resolution moving combined with high adaptability, opening up the possibility of using it for mobile detection tasks in real‐world settings.


Introduction
[3] They can access confined spaces or harsh conditions that conventional actuators cannot do due to their compact size, lightweight, straightforward design, and great adaptability. [4,5]As can be seen, there is a wide range of potential applications for miniature robots, including disaster relief, [6] remote reconnaissance, [7] and medical treatment. [8]There have been advances in the development of micro crawling robots, [9] microflying machines, [10] and micro underwater robots, [11] however, the focus of this work is on millimeter to centimeter-scale mobile robots.
Traditional mobile robots are typically powered by electromagnetic motors, [12][13][14] which offer the benefits of great drive efficiency and motion precision.However, the complex control and transmission system hinder their miniaturization.In this way, miniature robots have been developed to utilize smart materials as drive sources, representative of which include dielectric elastomers, [15,16] shape memory alloys, [17,18] magnetostrictive materials, [19,20] and piezoelectric materials. [21,22]The first three types of robots, however, have issues with complicated manufacturing procedures, poor motion stability, and the requirement for high-power loading devices, respectively.25][26] Piezoelectric miniature robots can be categorized into nonresonant [27,28] and resonant types [29,30] depending on whether they operate in their resonant states.In nonresonant piezoelectric robots, a piezoelectric beam robot proposed by Hariri [31] was able to realize 1D motion with the aid of structure-borne traveling waves. [32]The motion flexibility was further increased by extending traveling waves to 2D surfaces. [33]Subsequently, based on the θ-based traveling wave control approach developed by Zhao, [34] the piezoelectric beam robot moved more smoothly and had a higher velocity.The aforementioned robots can achieve stable motion, but further miniaturization is prevented by their special requirements for body length.Some other nonresonant piezoelectric robots utilize the deformation of piezoelectric stacks and move by accumulating small steps for instance, the planar piezoelectric actuator using bending-bending hybrid transducers proposed by Deng [35] or the hexapod piezoelectric robot developed by Yu. [36] All of them exhibit good motion resolution, but at the expense of a significant motion loss rate, which is unfeasible for the motion efficiency.Resonant robots, on the other hand, have facile structural miniaturization [37,38] and strong vibration displacement output in the resonant state. [39,40]his fulfills the requirement for fast motion, which is then configured and optimized to obtain specific motion capabilities.For example, the insect-scale [41] or spiral soft robots [42] manufactured by PVDF exhibited exceptional speed performance adopting the jumping gait mechanism; the arthropod-metamerism robot [43] or MQPR, [44] proposed by Liu, could guarantee both fast motion and high displacement resolution, as well as weight bearing and climbing; the TMMR designed by Wang [45] could swim on water, but this required extra oscillating paddles on the driving legs.All of the aforementioned robots, despite their impressive performance, operate in the single-energy input, one motion-output mode.This results in them being less environmentally adaptable and generally moving only in a particular environment, such as the smooth glass substrate.If it were feasible to produce several motions from single-energy input, the capacity of the robot to adapt to its surroundings could be enhanced significantly.In particular, the issue of amphibiousness, which cannot be avoided in real environments, is critical to its applications.
Inspired by the fast amphibious motion of crocodiles in nature, this study proposes a hollow piezoelectric miniature robot that can operate under two modes, that is, the oscillating and rotating modes.The robot is capable of simultaneously oscillating the driving foot and rotating the internal rotor.This performance gives it the potential for multitasking with the same energy input and fewer external components.Specifically, the proposed hollow-beam piezoelectric miniature robot has the following key advancements.1) In the oscillating mode, the robot (body length of 13 mm) can achieve a high speed of 160 mm s À1 (12.3 BL s À1 ) under the voltage of 200 V p-p , a minimum speed of 7.8e À2 mm s À1 (6e À3 BL s À1 ) at 10 V p-p , and a high displacement resolution of 0.3 μm (2.3e À5 BL).After assembling the rotor, the motion speed of HPMR and rotor speed are 95 mm s À1 (7.3 BL s À1 ) and 890 rpm, respectively, and the corresponding displacement resolutions are 0.84 μm (6.5e À5 BL) and 7.28 mrad, respectively.2) In the rotating mode, the robot (body length of 60 mm) can move at speeds of 396 mm s À1 (6.6 BL s À1 ) and 156 mm s À1 (2.6 BL s À1 ) on the glass substrate and the water surface at 400 V p-p , respectively.3) Load capacity of up to 90 g (12.72 times its own weight) in the oscillating mode and 20 g (1.19 times its own weight) in the rotating mode allow the robot to carry micro detection devices.4) The robot can run on surfaces with varying roughness and climb slopes; especially in the rotating mode, the HPMR can resist impacts, cross obstacles, and jump down the step, showing high adaptability.

Structural Designs
Crocodiles are well renowned for having superior locomotor abilities, especially their capacity for the amphibious motion on land and in water. [46]Figure 1a shows the body posture of the crocodile during a motion cycle.Overall, the forelegs and hind legs swing in the same pattern, both simultaneously rather than alternately.The difference in the motion mode is that the forelegs oscillate a quarter of a cycle earlier; therefore, we focus on the hind legs.The rhythm of this motion has four stages, as shown in Figure 1a for states I-IV.In state I, the hind legs remain stationary on the ground; upon reaching state II, the hind legs start to pedal the ground and approximate leaving it, causing the crocodile to move forward under the force of friction; in state III, the hind legs are vacated to keep the movement state and gradually approach to the ground; after reaching state IV, the hind legs return to the ground and adjust their angles to begin the following movement cycle.The motion stages of the hind legs can be combined into a circular trajectory, as depicted in Figure 1a.In addition, crocodiles can also swim in the water by swinging their legs in this manner.
Inspired by this motion mechanism, we propose a hollow piezoelectric miniature robot, as shown in Figure 1a.In the process of configuration design, the body and driving feet of the robot are simplified to resemble the body and legs of crocodiles.The hollow design allows the robot to simultaneously oscillate the driving feet and rotate the internal rotor, so we configure the oscillating mode and rotating mode depicted in Figures 1b,c.In the oscillating mode, a hollow square beam with eight PZT elements glued on it and two driving feet make up the robot.The centrosymmetric configuration of the robot can couple orthogonal bending vibrations to generate elliptical driving trajectories of the foot end and bore edge. [43,47]The added components after assembling the rotor are two conical rotors, springs, threaded rods, and locknuts.As such commercially available micro detectors have not been discovered, the conical rotor is utilized in their place.The taper of the rotor provides for better contact with the inner bore and prevents eccentricity during rotation, while the spring is used to adjust the preload and thus the speed or torque.In contrast, the rotating mode aims to investigate the aquatic or amphibious qualities of HPMR, but three considerations need to be taken into account during the design process.First, replacing the conical rotor with propellers for propulsion is an efficient solution due to the small oscillation of the driving foot.Then, second, the cabin needs to provide enough buoyancy to keep the robot from submerging.Finally, the film is pasted on the exposed area of the robot as a waterproof measure while guaranteeing its lightweight design (More details on structural and dimensional parameters are shown in Section S2, Supporting Information).
Overall, the structural design advantages of HPMR are as follows.1) Simple and compact configuration, all motion modes are direct-driven without additional transmission mechanism.
2) The axisymmetric structural properties directly eliminate the tedious process of adjusting structural dimensions during the frequency tuning process.Despite the asymmetric configuration of the driving foot results in a slight difference between the bending vibration modes in the y and z directions; this process is mostly unaffected due to the light weight of the driving foot.3) As a result of the hollow configuration, the robot can output the oscillating motion of the driving foot and the rotational motion of the conical rotor (or propeller), giving it a variety of operating modes.Specifically, the robot may be capable of mobile detection in the oscillating mode if the rotor is switched out for a micro camera; rotating mode enables the robot to operate in complex scenarios on land or in the water.

Working Mechanism
The proposed robot combines vibrations in orthogonal bending directions (y and z directions), resulting in an elliptical trajectory at any point in its body.In this process, the friction force is utilized to drive the robot and the conical rotor, where the contact points with the ground and the rotor are denoted as P f and P c , respectively.Note that all points on the edge of the inner bore drive the rotor at different moments during the operation of the robot, respectively, but we conduct targeted analysis on the point P c . Figure 2a shows the arrangement and excitation scheme of the PZT elements, labeling them as P 1 -P 8 in a counterclockwise orientation.Applying voltages V y and V z to the PZT elements causes the robot to vibrate both vertically and horizontally.(Actuating mechanisms of PZT elements and piezoelectric laminated structure are shown in Section S1, Supporting Information).The displacements u, v, and w in the x, y, and z directions at an arbitrary point can be expressed as where, u m , v m , and w m are the maximum amplitudes; λ u , λ v , and λ w are the corresponding initial phase angles; f is the excitation frequency.During the vibration of the robot, the displacements u at both ends are canceled out, but the trajectory formed in the plane o-yz can be represented as [23] At the condition of jλ w À λ v j ¼ 90°, the trajectory takes the shape of an ellipse with the expression as Note that adequate torque is needed for the rotor or propeller to rotate.As the excitation frequency rises, the amplitude decreases significantly.The high amplitude assures the pressure and driving force on the rotor; hence, the first-order bending frequency is temporarily chosen as the excitation frequency to analyze the driving mechanism of the robot.Then, under the excitation of harmonic voltages Þ , and the deformation of HPMR, the elliptic trajectories of P f and P c are shown in Figure 2b.In the schematic diagram of the driving principle shown in Figure 2c, F f and F c denote the frictional forces on the driving foot and rotor, respectively, while F nf and F nc are the pressures on them.Specifically, the runtime history of the robot is divided into four stages, each of which can be summarized as follows:1) t = 0 À T/4: When the sinusoidal voltage V z decreases from the peak to zero, P f and P c vibrate from the maximum displacement to the initial position (the position at the undeformed moment) along the oz direction; then V y increases from zero to the peak, and the above points vibrate along the oy direction from the initial position to the maximum displacement.Given that the displacements of P f or P c are orthogonal in the oy and oz directions, they are synthesized as one-quarter of the elliptical trajectory in Figure 2b.In this state, the pressure F nf gradually increases, and the point P f begins to contact the ground, so the robot is driven by the friction F f .The conical rotor is separated from the point P c and cannot be driven.However, the spring provides the contact pressure between the rotor and the beam, and the bore of the latter is circumferential rather than planar, so the point P c3 can drive the rotor.2) t = T/4 À T/2: When V z increases from 0 in the reverse direction to the negative peak, P f and P c vibrate along oz in the negative direction from the initial position to the reverse maximum displacement; after that, they return from the maximum displacement to the initial position along the oy negative direction when V y drops from the peak to zero.At this point, F nf steadily declines and P f recedes from the ground; nevertheless, the robot continues to be driven.Similarly, the conical rotor is still not in contact with P c , but the point P c4 now provides the rotational force F c . 3) t = T/2 À 3 T/4: As V z decreases from the negative peak to 0, P f and P c return from the reverse maximum displacement to the initial position along the oz direction; they synchronously vibrate from the undeformed position to the reverse maximum displacement along the oy negative direction while V y increases from zero to the negative peak.The driving foot continues to move away from the ground and is not propelled by the frictional force F nf , but the robot advances due to inertia.Instead, the pressure F nc steadily grows as the point P c approaches the conical rotor, and the friction force F c causes the rotor to begin rotating.4) t = 3 T/4 À 2 T: In the last time period, V z increases from 0 to the peak, and P f and P c vibrate from the initial position to the maximum displacement along the oz direction; the above points return to the initial position from the reverse maximum displacement along the oy direction after V y decreases from the negative peak to 0. The robot is not being driven at this moment by frictional force F f .Despite separating from the point P c , the rotor can keep rotating under the friction F c generated by the point P c2 .
After going through stages ( 1)-( 4), the intended function of HPMR is realized by depending on the closed elliptical trajectory consisting of the four-segment circular arc portion.(The animated presentation is shown in Movie S1, Supporting Information).Furthermore, the motion speed and direction are controlled by the voltage amplitude and phase difference, respectively.According to related research works, it is known that similar types of robots also utilize elliptical trajectories in conjunction with friction to generate driving forces. [3,31,39,43,44]owever, in the elliptical trajectories characterized in Figure 2c, the majority of research works essentially use the displacement trajectories of the stages S 3 -S 4 to drive the robot.The trajectories in stages S 1 and S 2 separate the robot from the ground and overcome its roughness, with practically no driving effect.As a result, only 50% of the trajectory components play a driving role throughout the entire motion cycle.Contrarily, the closed circular cross section formed by the hollow structure allows the conical rotor or propeller to be driven by the entire elliptical trajectory, which includes the trajectory portion that has no driving effect on the robot, such as the trajectory of the point P c .This approach can endow the robot with unique locomotion abilities, such as aquatic motion.

Simulation Analysis
As shown in Figure 3, the finite-element method is utilized to obtain the vibration mode and transient responses of the HPMR based on the commercial ANSYS software.We simulate only the assembly of the hollow piezoelectric beam and the driving foot in the finite-element model displayed in Figure 3a, in which the adhesive layer with a thickness of 0.05 mm is considered in the modeling process to approximate the realistic physical model.The material parameters to be defined during the simulation calculations can be found in Section S3, Supporting Information.
As illustrated in Figure 3b, the bending vibration frequencies and modal shapes of the HPMR in the y and z directions are first obtained by modal analysis.The results show that the first-order bending frequencies along the y and z directions are 10942.40and 10894.37Hz, with a frequency difference of 48.03 Hz and a relative error of 0.44%.The driving foot is a nonaxisymmetric structure, which contributes to the frequency inaccuracy, but due to its tiny mass, it has little impact on the numerical computations.Thus, it can be approximated that the two bending vibration modes can be coupled to generate an elliptical trajectory for driving the robot.The transient analysis results are utilized to analyze the working mode of the robot, as shown in Figure 3c.Since the vibrational deformations are symmetrical, the displacements of the driving foot or inner bore in the x direction are canceled out.The bending vibrations in the y and z directions are coupled in the plane o-yz, and their trajectories are periodic circular motions, which are consistent with the anticipated motions in Figure 2c.
Further, the transient response extraction results of P f and P c are shown in Figures 3d,e.After a period of abrupt fluctuations, it is evident that their response enters a steady-state phase.In the partially enlarged view of response given in Figure 3f, the amplitudes of P f along the y and z axes are 2.3 and 1.88 μm, whereas those of P c are 4.14 and 3.37 μm, respectively.Their amplitudes in the same direction are orthogonal to one another and those in different directions are in phase, which meet the demand for simultaneous output of oscillating and rotating motions.The amplitude of P c is significantly higher than that of P f , because the former is closer to the beam boundary in the first-order resonance state.However, it makes sense to place the driving foot farther from the boundary to avoid the interaction between it and the rotor.Furthermore, the displacement of P f or P c along the y-axis is larger, but the excitation frequency is the z-direction bending frequency, which implies that the physical model has a greater impact on the amplitude than the excitation frequency.Figure 3g plots the trajectories of P f and P c in the o-yz plane, both of which exhibit an elliptical shape with a clockwise direction.
Since the amplitudes v and w are higher than the 1 μm utilized to drive the robot, [43,44] the HPMR is able to operate in the oscillating and rotating modes.

Vibration Mode and Elliptical Trajectory Tests
Before testing the motion performance of the HPMR, it is necessary to obtain its vibration mode and displacement response.The prototype of the HPMR is shown schematically in Figure S3, Supporting Information, and the test procedure is depicted in Figure S4, Supporting Information.Sweep signals in the frequency range of 8-12 kHz are used to excite the PZT elements, and the test results are demonstrated in Figures 4a,b.It can be seen that the bending vibration frequencies in the y and z directions are 10856.5 and 10769.8Hz respectively.Due to the nonsmoothness of the curves, the results utilizing the Lorentz fitting method are 10874.9and 10779.1 Hz.Compared with the simulation results in Section 2.3, the relative errors of the frequencies are 0.62% and 1.07%, respectively.The variances in material properties, assembly errors, and uneven adhesive layer thickness are to blame for the discrepancy between the test and simulation results.However, the error values are within acceptable ranges and can prove the correctness of the finite-element model.In addition, the frequency error between the two bending directions is 0.89%, which indicates the bending vibrations in orthogonal directions can be well coupled.
After determining the modal frequencies of the prototype, the vibration displacements of P f and P c are tested.The test procedure is similar to that in Figure S4, Supporting Information, with the exception that the controller is utilized to directly output the vibration displacement measured by the laser head.Two sinusoidal signals with a phase difference of 90°, excitation frequencies of 10.8 kHz, and amplitudes of 100 V p-p are applied to the PZT elements.The displacement curves of the P c along the y and z directions collected by laser heads are displayed in Figures 5a,  b, respectively.Be aware that the displacement of locations on the inner bore is difficult to measure, so spots on the square surface of the beam are chosen for testing instead.The vibration displacements within 0-1 ms are described in Figure 5c, and the curves have a sinusoidal shape.Specifically, the displacement amplitudes in the y and z directions are 3.19 and 3.33 μm, respectively, resulting in a relative error of 4.4%.The reason for the error comes from the asymmetric structure of the driving foot and the z-direction vibration displacement of the prototype being constrained by the fixture.Figure 5d shows the synthesized trajectory for 0-5 ms, whose trajectory shape is approximately elliptical.The synthesized trajectories show that the test results are in good agreement with the simulation results.The reason for the slight difference comes from the difference in the material between the prototype and the finite-element model, the assembly error, and the clamping effect on the prototype.
In Figure 6a,b, the test results for point P f on the driving foot are displayed.Since the driving foot has an angle at the end, the arc position parallel to the o-xy plane is chosen as the measuring point for its z-direction displacement.Then, the partially enlarged view of the displacement curves at 0-1 ms is displayed in Figure 6c.The vibration displacements of the driving foot along the y and z directions reach 2.07 and 2.55 μm, respectively, with a relative error of 23%.The reason for the significant error and the nonsmooth displacement curve is that the driving foot surface manufactured by 3D printing technology is rougher than the beam surface, which is not conducive to the acquisition of accurate displacement by the laser heads.In Figure 6d, the synthesized trajectories of the driving foot are also approximated as an ellipse, which deviates from the simulation results in Section 2.3.It can be observed that the tested displacements of the driving foot in the y and z directions are lower and higher than the simulated values, respectively.The former is due to the clamping action of the fixture hindering the vibration of the prototype, while the latter results from the inability to precisely find the measuring point for z-displacements at the driving foot end.Overall, the errors are acceptable and the modal vibration behavior of the experimental prototype agrees well with the simulation results.

Performances in the Oscillating Mode
First, the optimal frequency of robot operation should be determined.(The test procedure is shown in Figure S5, Supporting Information).Therefore, the relationship between the motion speed of the robot and the excitation frequency in the oscillating mode is depicted in Figure 7a.We set the excitation voltage as 200 V p-p and gradually increase the frequency from 10 to 11.4 kHz.The results indicate that the maximum forward and backward speeds of the robot are 160 and 150 mm s À1 at 10.7 kHz, respectively (Movie S2, Supporting Information).Compared to the simulation and the frequency test results, the relative errors of the frequencies corresponding to the maximum speed are 2.27% and 1.63%.The errors arise due to the interaction of the robot with the glass substrate.Then, we measure the bidirectional motion speeds versus the excitation voltages, as shown in Figure 7b.As the voltage increases, the speed of the robot gradually increases, and their relationships are approximately linear.This is because an increase in voltage causes the PZT elements to expand further linearly, which rises the displacement of the driving foot.(Figure S6a.1-a.3,Supporting Information, show the images extracted from the experimental video at voltages of 120-200 V p-p ).In addition, the forward and backward motions of the robot have good consistency; therefore, we only measure and discuss its forward motion performances during the following study.It can be observed that 0-40 V p-p in Figure 7b is the dead zone of the macroscopic motion, but the robot can still operate at low voltages, as shown in Figure 7c.The reason is that the robot can still be driven by the oscillations of the driving foot at low voltage, but its motion trajectory cannot be distinguished and recorded by the camera.Therefore, we utilize the laser displacement sensor to record the position of the robot at various moments to determine the speed.(The test procedure is shown in Figure S5, Supporting Information).The relationship between the microscopic motion speed and voltage remains linear; in particular, the robot can run at a speed of 0.078 mm s À1 with an excitation voltage of 10 V p-p , demonstrating its capacity to accomplish highprecision continuous motion.Miniature robots usually need to carry some micro detection devices to handle their tasks, so we test the load-carrying performance of the robot, as shown in Figure 7d.At a voltage of 200 V p-p , the speed of the robot increases and then decreases as the load increases (Movie S2, Supporting Information).The maximum speed of 178 mm s À1 is achieved with a load of 5 g, an increase of about 7.8% in comparison to the unloaded speed; the maximum load carried is 90 g (12.72 times its weight), and the speed drops to 41.2 mm s À1 (about 25.7% of the unloaded speed).This is because a slight increase in load improves the driving force generated by friction, but an excessively heavy load weakens the ability to overcome the roughness of the substrate.In Movie S2, Supporting Information, the robot undergoes a large motion tilt relative to the straight path direction with a load of 50 g.One reason is that the asymmetric mass distribution of the load with respect to the centroid of the robot results in unequal driving frictions of the driving foot on both sides.For another reason, the collision of the guiding scale against one of the driving feet weakens its speed.Furthermore, the robot at no load also exhibits sensitivity to the motion direction due to the assembly errors or variations in the roughness of the substrates, which can be mitigated by adjusting the voltages applied to the unilateral PZT elements (P 1 , P 3 , P 5 , and P 7 ).
After that, we measure the motion characteristics of the robot when assembling the rotor in the oscillating mode.shows the variation of the motion speed and rotational speed with respect to the excitation frequency at 400 V p-p , respectively.In Figure 8a, the motion speed of the robot increases and subsequently drops with frequency for various preload pressures, and 10.6 kHz is the optimal frequency.Without the spring being compressed, the robot can run at a speed of 258.3 mm s À1 , which is a slight drop of about 1% from 260.9 mm s À1 without the rotor (Movie S3, Supporting Information).Comparing the frequency test results without rotor, the optimal excitation frequency shifts forward.This is because the rotor components add more mass to the prototype than stiffness. [47]In addition, the increase in preload force reduces the motion speed of the robot; at a preload force of 1.18 N, its speed dropped by 42.5% to 150 mm s À1 .The reason is that the preload force limits vibration of the hollow beam and thus reduces its amplitude.In Figure 8b, the maximum rotational speed is 3.015 krpm at a preload force of 0-0.59 N and the corresponding frequency is 10.4 kHz.There is no doubt that the optimal frequency has shifted forward, due to high amplitudes causing the vibrations of the rotor.However, the preload of 1.18 N returns the optimal frequency to 10.6 kHz, as the high preload force eliminates the vibration of the rotor.Further, we evaluate the combined performance of the two motions (Movie S3, Supporting Information), defining the speed ratio S r as follows S r ¼ Rotational speed Linear speed (4)   where S r is the the ratio of the rotational speed to the linear speed and indicates the output efficiency of the rotor motion during the operation of the robot.When the excitation frequency is 10.2 kHz and the preload force is 1.18 N, the speed ratio reaches 27.98.This means that the robot runs for 1 mm and the rotor rotates for about 28 revolutions; therefore, it has a relatively efficient motion of the rotor and is defined as the scan mode.With an excitation frequency of 10.4 kHz and a preload force of 0.59 N, the speed ratio is 13.25, which is defined as a high-performance mode due to its fast rotational and motion speeds.At a frequency of 10.6 kHz and without the preload force, the speed ratio is 9.76, and because of the rather fast motion, this is considered to be the motion mode.To summarize, we can configure the mobility mode of the robot according to the tasks it needs to perform. Figure 8d shows the motion speed and rotational speed versus voltage.In this instance, the preload force provided to the rotor is adjusted at 0.59 N because the motion of the rotor will be unstable if there is no preload applied, whereas excessive preload necessitates a high power input to overcome the friction, which will induce the heat generation and wear problems.As the voltage increases, the motion speed and rotational speed rise To investigate the transient response characteristics of the robot and its ability to achieve high displacement resolution, we use a sinusoidal pulse excitation scheme as shown in Figure 9a.First, the pulse frequency is set to 1 Hz, the excitation frequency in a single pulse is 10.7 kHz, the factor is 20%, and the voltage is 40 V p-p .The transient characteristics results of the robot in the oscillating mode are shown in Figure 9b.Wherein the displacement values are collected by the laser displacement sensor, the velocity is obtained by differentiating the displacements.The velocities are calculated by differentiating the displacements with time.It can observed that the starting and braking time of the robot is about 15 and 54 ms.The millisecond starting times enable the robot to respond quickly and reduce the energy consumption required for startup.Following that, we set the pulse frequency, the excitation frequency, and the voltage to 10 Hz, 10.7 kHz, and 20 V p-p , respectively; once the rotor is assembled, the frequency and voltage are set as 10.4 kHz and 50 V p-p , respectively.It should be noted that the voltage is the value gained through multiple experimental attempts to get the smallest step displacement, while the excitation frequency is the optimal value obtained in the previous experiments.Figure 9c,d show the step displacement of the robot versus the duty factor in the absence and presence of the rotor, respectively.At a duty factor of 0.5%, we measure their minimum step displacements of 0.3 and 0.84 μm, respectively.In Figure 9e, the step displacement of the robot shows an upward trend as the duty factor is increased, and its displacement resolution without the rotor is smaller at the same duty factor.However, the robot achieves a submicrometer resolution in both states, demonstrating its ability in precise localization.To evaluate the stability of the step displacement, we perform five replicate experiments at each duty factor and calculate their relative random deviations, as shown in Figure S7, Supporting Information.The relative random deviation is calculated as the ratio of the standard deviation to the mean value acquired from five displacement periods chosen at random, where the displacement for each period is the average of the displacements after ten pulse cycles.The results show that the maximum relative random deviation is 3.7% with a standard deviation of 268 nm when the duty factor is 5%; especially at the duty factor of 0.5%, the relative random deviation and standard deviation are 1.8% and 5 nm, respectively, which is acceptable and indicates a good stability of the step displacement.During this process, the robot may be slightly offset by the roughness of the substrate and thus tilts, but it can also be corrected by making minute modifications to the voltages of the unilateral PZT elements.The relationship between the step rotation angle of the rotor and the duty factor is depicted in Figure 9f.The rotor has a minimum step rotation angle of 7.28 mrad at a duty factor of 0.5%, and it can be observed that the milliradian deflection angle enables the robot to accurately manipulate the sensor during mobile detection.

Performances in the Rotating Mode
The mobility capabilities of the HPMR in the rotating mode are the main subject of this section.After the rotor is replaced with the propeller, the robot is capable of aquatic motion, but the driving foot is left inoperative.Since the rotational speed of the propeller determines the motion speed, we measure its relationship with the excitation frequency, as shown in Figure 10a.The width of the propeller blade is 3 mm and the voltage is set as 400 V p-p .The increased preload force causes the rotational speed of the propeller to decrease, also because the high preload force weakens the amplitude of the hollow beam.Under the preload force of 0.59 N, 1.18 N, and 1.77 N, their corresponding optimal frequencies are significantly backward shifted to 10.4, 10.6, and 10.7 kHz, where the maximum speed is 2.3 krpm at 0.59 N. The reason is that the preload force rises the stiffness of the prototype more than it increases the mass.Figure 10b describes the rotational speed of the propeller as a function of the voltage.We observe that the increase in preload force rises the starting voltage of the robot.At the preload forces of 1.18 N and 1.77 N, the rotational speed and voltage are linearly correlated, while at 0.59 N, the rotational speed initially rises linearly and then fluctuates when the voltage exceeds 400 V p-p .This phenomenon stems from the high amplitude of the hollow beam, causing the propeller to vibrate and thus destabilize its rotation.The relationship between the speed and the voltage for the aquatic motion of the HPMR is depicted in Figure 10c.The aquatic speed rises with the increasing voltage, with a maximum speed of 102 mm s À1 at 600 V p-p (Movie S4, Supporting Information).Furthermore, the small preload force produces fast rotational speed whereas the large preload force results in high torque.Figure 10c demonstrates that the rotational speed is the primary factor influencing the aquatic speed since the smaller the preload force, the faster the robot swims.Subsequently, the effects of propellers with different widths on the speed are described in Figure 10d.The robot can swim faster with the paddle width of 4 mm because it produces more propulsion.
To broaden the application scenarios of the HPMR and to give it the capability of amphibious movement, we utilize the propeller with T-shaped blades, which enables the robot to gain thrust in the forward direction on the ground.The motion speed of the robot on the glass substrate and the water surface as a function of the voltage is shown in Figure 10e.At this stage, the prototype weighs 16.82 g, the excitation frequency is set to 10.6 kHz, and the preload force is set as 1.18 N in order to provide a powerful torque to overcome the friction of the ground.The experimental results indicate that the robot moves faster with the increase of voltage, and at a voltage of 600 V p-p , the robot can reach the maximum motion speeds of 666.7 and 203.3 mm s À1 on the ground and water surface, respectively (Movie S4, Supporting Information).The speed ratio of ground to aquatic motion is roughly 2.3-3.27times, indicating that the resistance of water to the propeller is roughly three times that of the ground, which arises from the fluid friction against the blades and cabin.The experimental recorded images of the amphibious motion at different moments under 400 V p-p and 10.6 kHz are depicted in Figure 10f.In contrast to the oscillating mode, the motion in this mode has an acceleration stage and a constant speed stage.This is because the propeller is an actuator between the ground and hollow beam, and it takes some time to accelerate to its peak speed.
The adaptability of a robot is an important indicator for evaluating the scope of its future applications.Figure 11a characterizes the climbing ability of the robot, with the excitation frequencies all set to the optimal values from the prior research and the voltage set as 400 V p-p .The robot runs slower as the climbing angles rise, but regardless of the motion mode, it can climb an 8°slope at 67% (in the oscillating mode without the rotor), 34% (in the oscillating mode with the rotor), and 23% (in the rotating mode) of their respective maximum speeds (Movie S5, Supporting Information).Figure 11b shows the motion speeds of the robot on substrates with different roughnesses (Movie S6, Supporting Information).In the oscillating mode, the robot runs fastest on the smooth glass substrate and slowest on the roughest paper substrate, but it is immobile on the foam board with the larger damping.This phenomenon occurs because the higher friction or damping prevents the generation of elliptical motions and thus reduces the speed.Comparing the case with the presence of the rotor, the slight increase in rotor mass improves the contact with the driving foot and the substrate, so the robot runs faster on each substrate.At the same voltage, the robot in the rotating mode runs significantly faster than it does in the oscillating mode.On the comparatively rough paper substrate, the robot in the rotating mode moves the fastest, whereas on the glass substrate, it runs the slowest.This is because, at high rotational speeds, the smooth substrate exacerbates the slipping of the propeller against it.
In the actual motion environments, the robots should have the ability to resist the obstruction and water impacts (Movie S7, Supporting Information).As shown in Figure 11c, when the robot moves to the position I at 0.68 s, it is pushed back to the position II by the obstruction and held there until 5.0 s; after removing the obstruction, the robot can move to the position III at 5.84 s.In Figure 11d, a water pump is added to generate the water impact, yet the robot is still able to move to the opposite side of the container under the impact of water.Besides, the robot should be able to carry the probing equipment for detecting the surroundings, where the commercially available microcameras have a mass of about 20 g (Movie S7, Supporting Information).Figure 11e demonstrates that the robot is able to carry a 20 g weight to achieve motion.Finally, the adaptability of the robot on the ground is characterized as shown in Figure 11f-h (Movie S8, Supporting Information).In Figure 11f, the robot can cross a gully with a width of 35 mm and a height of 5 mm; in Figure 11g, the robot can continuously cross obstacles with heights of 1 and 3 mm, as well as the hole traps with diameters of 6 mm; in Figure 11h, the robot can jump down a step with a height of 44 mm and maintain running.From the above results, the HPMR possesses high adaptability, which demonstrates its potential ability to realize multi-scene motion.

Performance Comparisons with Existing Robots
To characterize the advantages of the proposed HPMR, we conduct a comprehensive comparison of its performance with several representative piezoelectric robots and plot a performance hexagon figure of crucial parameters, as shown in Figure 12.The performances of these robots are compared from six aspects: body length, speed range, load performance, displacement resolution, agility, and adaptability.Among them, body length is the length of the robot in the direction of motion; speed range is the minimum to maximum speed that the robot can achieve; resolution is the minimum step displacement; load capacity is the maximum weight that the robot can carry; agility refers to the degrees of freedom of motions realized; and adaptability is the ability of the robot to achieve specific motion properties.The performance testing characteristics of several representative resonant piezoelectric robots can be found in Table S2, Supporting Information.As shown in Figure 12, the HPMR performs more comprehensively.Traditional mobile robots are driven by electromagnetic motors, but the additional transmission mechanism is required during the design process, such as the robot in another study. [10]In particular, the incorporation of the reduction mechanism to attain adequate torque for manipulating the robot results in a complicated robot configuration and demands high assembly.However, the proposed robot is simple in configuration and easy to assemble.Whether utilizing the oscillating feet or propellers, the robot is driven directly by contact friction, eliminating the need for an additional transmission mechanism and facilitating speed regulation.Following that, the HPMR outperforms the comparable motor-driven robot in another study, [48] exhibiting greater velocity, displacement accuracy, and load-carrying but its agility still has to be improved.Compared to the motor-driven robot that can operate in real-world surroundings in the ref.[49], the HPMR has remarkable advantages in terms of miniaturization and amphibious speed, but its exceptional environmental adaptability is worth emulating.Furthermore, we conduct a performance comparison between our HPMR and some superiorperforming piezoelectric robots.Compared to the robots with similar mass in the refs.[29,39] our robot has a faster motion speed.Despite being marginally slower than the robots with comparable body lengths in the refs.[25,41] our robot exhibits a better load-carrying capability.Besides, the HPMR is capable of achieving the submicrometer displacement resolution, even better than that of robots in the ref.[43].More importantly, the HPMR in the oscillating mode has superior applicability, especially for the realization of aquatic motion.Compared to the piezoelectric robot with amphibious capability in the ref.[45], the proposed robot is comprehensively leading in terms of amphibious speed, load capacity, and adaptability.Despite the excellent motion performance of our robots, their inability to achieve turning or steering motions limits the applications, so this is a future focus.In addition, the development of an untethered HPMR is necessary; therefore, the Cost of Transport of the HPMR is evaluated in Section S4, Supporting Information.The results show that the power consumption of the HPMR is relatively high; therefore, future work also concentrates on reducing it by optimizing the configuration design and assembly process to achieve untethered operation.

Conclusion
Inspired by the amphibious motion of crocodiles in nature, we present a hollow piezoelectric beam miniature robot for potential applications of mobile detection in real environments.The innovation of the robot lies in the single-input, multiple-output mode of operation, which can simultaneously oscillate the driving foot and rotate the rotor in a single vibration cycle.In the oscillating mode, the robot is able to run at speeds of up to 12.3 BL s À1 , while has a displacement resolution of 0.3 μm, demonstrating its ability for fast and precise motions.The additional rotor reduces the linear speed to 7.3 BL s À1 but achieves a rotational speed of 890 rpm.In addition, their motion resolutions remain on the order of submicrometer and milliradians, respectively, which is further evidence of the superior motion performance exhibited by the hollow design.In the rotating mode, the running capacity at speeds of 6.6 and 2.6 BL s À1 on land and water, respectively, is rarely reported in robots of this kind.Further, high adaptability is a key indicator toward practical applications.The robot shows exceptional agility in carrying load, climbing slopes, and running on substrates with varying roughness.In particular, the rotating mode enables the robot to resist shocks, cross obstacles, and jump steps.In summary, the proposed HPMR meets the requirements of miniaturization, fast and precise motion, and high adaptability to a certain extent, especially the amphibious motion broadens the range of scenarios.Overall, the robot in the oscillating mode can carry sensors for rapid mobility and high-accuracy localization, which shows potential for application in the field of precision inspection.The rotating mode brings strong environmental adaptation and amphibious capacity, rendering it a viable option for mobile exploration and rescue missions.In our future work, we will focus on improving motion agility, real-world applications, and untethered operations.

Experimental Section
Materials and Fabrication: The hollow beam was made of aluminum and processed by CNC (Shenzhen Junkai Precision Machinery Co., Ltd, China).TJ-47 (TJ Piezo Specialties, Ltd, China) was chosen as PZT element due to its low mechanical loss and high Curie temperature. [50]The driving foot, conical rotor (propeller), cover, and cabin were made by 3D printing technology, which used photosensitive resin for their materials (Shenzhen Jialichuang Technology Group Co., Ltd, China).The prototype of HPMR is shown in Figure S3, Supporting Information.In the oscillating mode, the robot weighed 7.07 g.During assembly, the PZT elements and the driving foot were respectively pasted to the surface and both ends of the hollow beam using epoxy resin (DP460, 3M Company, America) and then cured for 24 h.PZT elements on the same surface were welded together using wires with a diameter of 0.28 mm to achieve an electrical parallel connection.In addition, the enamel wires with a diameter of 0.1 mm were used to power the PZT elements to minimize their drag forces.Finally, the enameled wires were pasted on the beam edges as the electrode utilizing the insulation tape with a thickness of 0.13 mm.The partially enlarged view in Figure S3a, Supporting Information, shows the structure of the additional rotor (weight 1.9 g), which includes a conical rotor, a spring, a locking nut, and a threaded rod.In Figure S3b, Supporting Information, the prototype in the rotating mode had a weight of 16.58 g and the cabin had a volume of 60 Â 52 Â 10 mm 3 .During the assembling process, the prototype was positioned within the chamber, secured with the cover, and ultimately locked with bolts.In this instance, the conical rotor was swapped out for the propeller which had eight blades, each measuring 8.5 mm in length, 1 mm in thickness, and 3-4 mm in width.Finally, the exposed areas were waterproofed using the PVC transparent film with a thickness of 48 μm.
Vibration Tests: The vibration mode testing procedure of the prototype is shown in Figure S4, Supporting Information.The experimental setups, shown in Figure S4a, Supporting Information, included a PC, a data acquisition card (NI-9263/9234, National Instruments Corporation, America), power amplifiers (ATA-62082, Aigtek, China), and a laser displacement acquisition system (LK-H020, Keyence, Japan).The test procedure is shown in Figure S4b, Supporting Information, where the data acquisition card was used to generate sinusoidal voltage signals with phase difference.Then, the voltage signals were amplified by the power amplifiers and excite the PZT elements.A controller and two laser heads (Lh-1/2) were included in the laser displacement acquisition system.The laser heads 1 and 2 were utilized to obtain the vibration displacements of the prototype in y and z directions, respectively; the controller and PC were used to receive and process the data returned by the laser heads.To avoid inaccurate measurements caused by the rigid body motion of the prototype, we positioned the prototype in a cabin created using 3D printing technology and fixed it with a jaw vice.In addition, all testing equipment was placed on a precision optical platform to minimize the effects of environmental vibration.
Motion Performance Tests: The experimental setups and procedure for testing the kinematic characteristics of the robot are shown in Figure S5, Supporting Information.The power amplifier has the ability to generate excitation signals with a frequency of 0-500 kHz and voltages up to 800 V p-p to power the robot.Two types of motion characteristics needed to be tested.One was the macroscopic linear motion of the robot recorded by a camera (Redmi Note 8 pro) at a frame rate of 30 frames per second.Then, the video was imported to the PC and the motion speed of the robot was calculated based on its coordinates at different moments in time.In addition, the microscopic linear displacement of the robot was captured by the laser displacement sensor (LK-H020), and the speed was the derivative of the collected displacement with respect to time.The other was the step displacement of the robot or the rotational angle of the rotor captured by the laser displacement sensor.In this case, three layers of insulating tape were pasted on the conical surface of the rotor (the thickness of each layer is about 0.13 mm).One revolution of the rotor was followed by a step in the captured signal, and its rotational speed was obtained by calculating the time difference.In addition, the robot ran on a glass substrate with coordinate paper attached to the back to assist in recording the positional coordinates.Two scales were arranged on both sides of the robot to calibrate its coordinates and avoid orientation shifts due to the unevenness of the substrate.To evaluate the Cost of Transport, we used a digital power meter (WT310E, Yokogawa, Japan) to test the power consumption of the robot during operation.

Figure 1 .
Figure 1.Structural designs of the hollow piezoelectric miniature robot.a) Inspired by crocodiles in nature that run fast by swinging their forelegs and hind legs periodically, a hollow piezoelectric miniature robot is proposed, with the main components of PZT elements, a hollow beam, and driving feet.b) Structure of the robot in the oscillating mode: the rotor components are utilized to output rotating motion, and red circles indicate the contact of the robot and its inner bore with the ground and rotor, respectively.c) Structure of the robot in the rotating mode: the conical rotors are replaced with propellers and a cabin is added for aquatic motion.

Figure 2 .
Figure 2. Working mechanism of the hollow piezoelectric miniature robot.a) Arrangement and excitation scheme of the PZT elements.b) The deformation of the HPMR at a phase difference of 90°for the excitation signal.Elliptical trajectories of contact points P f and P c .c) Driving principle of the robot.

Figure 3 .
Figure 3.The finite-element simulation results of HPMR.a) Finite-element model of the robot.b) Modal shapes of bending vibration along the y and z directions.c) Structural deformations calculated by transient response.d,e) The partially enlarged view of response.f ) The synthetic elliptical trajectory of P c and P f .

Figure 4 .
Figure 4. Vibration mode test results of the HPMR.a,b) Testing and fitting results of first-order bending vibration modes in y and z directions.

Figure 5 .
Figure 5. Vibration displacement test results of the point P c .a,b) The displacement curves of the P c along the y and z directions collected by laser heads 1 and 2, respectively.c) The partially enlarged view of the curves at 10-11 ms.d) The synthesized trajectories, where the red and blue lines are the test and simulation results, respectively. Figure8a,b

Figure 6 .
Figure 6.Vibration displacement test results of the point P f .a,b) Displacement curves of the P f along the y and z directions collected by laser heads 1 and 2, respectively.c) The partially enlarged view of the curves at 10-11 ms.d) The synthesized trajectories, where the blue and red lines are the test and simulation results, respectively.

Figure 7 .
Figure 7.The motion performance characterization of the HPMR without the conical rotor in the oscillating mode.a) Relationship between the bidirectional linear motion speed and the excitation frequency under 10-11.4kHz.b) Relationship between the speed and the voltage under 40-400 V p-p .c) The motion displacement of the robot in 0-2 s at low voltages of 10, 12, 16, and 20 V p-p , where the values in brackets are its speeds.d) Relationship between the speed and carrying load of the robot.

Figure 8 .
Figure 8.The motion performance characterization of the HPMR with the conical rotor in the oscillating mode.a) The motion speed of the robot versus excitation frequency in the range of 10.2-11.0kHz.b) The rotational speed of the robot versus excitation frequency.c) The combined performance of the linear motion and rotational motion.d) The motion speed of the robot and the rotational speed of the rotor versus voltage in the range of 60-400 V p-p .

Figure 9 .
Figure 9.The transient response characteristics and displacement resolution of the HPMR in the oscillating mode.a) The excitation signal in the sinusoidal pulse mode.b) The transient response characteristics of the robot in the sinusoidal pulse mode, where the pulse frequency is 1 Hz, the duty factor is 20%, and the voltage is 40 V p-p .c, d) In the absence and presence of the rotor, the step displacement of the robot under the duty factors of 0.5%, 1.0%, 3.0%, and 5.0%, respectively.e) The step displacement of the robot versus the duty factor, and the values in the graph are the average of the displacements after ten pulse cycles.f ) The step rotation angle of the rotor at the duty factors of 0.5%, 1.0%, and 3.0%.

Figure 10 .
Figure 10.The motion performance characterization of the HPMR in the rotational mode.a) The relationship between the rotational speed of the propeller and excitation frequency under the preload forces of 0.59 N, 1.18 N, and 1.77 N, respectively.b) The rotational speed of the propeller versus the voltage.c) The motion speed of the robot versus the voltage.d) Influence of the blade width of a propeller on the motion speed.e) After replacing the propeller blades with T-shaped ones, the relationship between amphibious motion speed and voltage.f ) The experimental recorded images of the amphibious motion of the robot at different moments under 200 V p-p and 10.6 kHz.

Figure 11 .
Figure 11.Adaptability of the robot in rotating mode.a) Relationship between the speed of the robot and the climbing angles.b) The speeds of the robot on the substrates with different roughnesses.c,d) The robot keeps moving after resisting the obstruction and water impacts.e) The robot can carry a load of 20 g for motion.f,h) The robot can cross the gully, obstacle, and uneven terrain, as well as jump down the step.

Figure 12 .
Figure 12.Comprehensive performance comparison with representative piezoelectric robots.The performances include six aspects: body length, speed range, load, resolution, agility, and adaptability.