A Biomimetic Piezoelectric Composite‐Driven Undulation Underwater Robotic Structure: Characteristic Analysis and Experimental Investigation

Numerous proposals and recommendations have been made in the development of underwater undulation robots, but very few studies have focused on the undulation characteristics of undulation robots. The undulating mode of the clearnose skate utilizing the activation of the pectoral fins in undulatory bursts is particularly suitable for slow and efficient swimming. Inspired by the working principle of the clearnose skate, herein, an undulation robotic structure is presented. Considering the hydrodynamic skeleton and propulsion mechanism of the clearnose skate, the solution is based on a piezoelectric composite for realizing undulatory propulsion. This approach leverages the converse piezoelectric effect and modal coupling to produce the desired undulating motion, thereby achieving the undulation design of the robot. The proposed robotic structure can realize the propulsion mode using piezoelectric composites to undulate in the same direction and can use the clockwise or counterclockwise undulation of piezoelectric composites to achieve the steering mode. In addition, finite element analysis is employed as an approach to investigate the undulation characteristics and transient vibration characteristics of the robot. The operational efficiency of the robotic structure is assessed through experiments. The results show that the propulsion mode achieves the speed of 3.33 body‐length s−1. The steering mode exhibits the steering velocities of 17.4 deg s−1.


Introduction
Over millions of years of evolutionary adaptation via natural selection, skates and the majority of rays have developed highly effective mechanisms for thriving in water environments, exemplified by their utilization of fins to achieve propulsion through water. [1,2]Locomotion traits that rely on undulating fins are frequently observed in benthic species.Undulation is defined as more than one propulsive waves in the fin at the same time during steady motion, which constitutes a thrusting pattern that transfers momentum from the moving body to the fluid, as shown in Figure 1.[5] Some scientists and engineers have used the undulatory propulsion mode employed by skates and rays using fins in biomimicry research.Recently, researchers have explored the propulsion mechanism based on undulation through studies on morphology, kinematics, hydrodynamics, muscle activity, and energetics of skate marine organisms.8][9][10][11] In 2006 and 2011, Thekkethil et al. quantitatively analyzed the swimming kinematics of stable undulations generating thrust using the fluid dynamics visualization method. [5,12]In 2017, Di Santo et al. experimentally investigated the three-dimensional (3D) kinematics of an undulation fin during stable swimming and concluded that the formation and control of waves are important for efficient swimming, maneuvering propulsion, and stable propulsion. [13]he artificial undulatory robot primarily relies on electromagnetic motors and smart materials.Wei et al. developed a large undulatory robot that utilizes servo motors to generate a propulsion-driven traveling wave of one wavelength on each fin. [14]Similarly, Liu et al. and Uddin et al. proposed flexible fins controlled autonomously through multiple sets of servo motors. [15,16]19][20][21] By employing smart materials, undulatory robots can function as propulsion units and facilitate elastic energy storage and exchange, leading to improved efficiency. [22][25] Overcoming these challenges, piezoelectric driving emerges as a viable solution.Shintake et al. devised circular and square undulatory robots using piezoelectric composites embedded in a flexible membrane. [26]Furthermore, Behlert and Schrag introduced a novel integrated undulatory robot based on piezoelectric micro-electromechanical systems employing aluminum nitride thin film technology. [27]Piezoelectric actuators possess an unenclosed configuration, rendering them impervious to water pressure.They seamlessly operate in aquatic environments and eliminate the need for intricate transmission mechanisms. [19][35] However, there is currently a lack of simulation and analysis methods to investigate the undulation characteristics and transient time-domain behavior of piezoelectric bimorphs, presenting an ongoing challenge. [36]39][40] Typically, piezoelectric actuators employ single-mode or multimode-coupled vibration, and modal harmonic response analysis is commonly employed to determine the required vibration mode and study the vibration characteristics in the frequency domain.Certain researchers, such as Zhang et al. and Liu et al., have utilized commercial FEA software to investigate the steady-state vibration characteristics of piezoelectric actuators, including modal resonant frequency and amplitude responses. [41,42]owever, these studies lack a comprehensive examination of the vibration characteristics of undulation robots in the time domain.Recently, Gohari et al. applied piezoelectric linear theory and classical plate theory to develop an analytical solution for the bending response of smart piezoelectric laminates with flexible spring boundary structures.[45][46][47] While FEA-based analyses and finite element software have been widely used to study the transient vibration characteristics of complex piezoelectric actuators, [48,49] existing studies primarily focus on simple vibration of simple piezoelectric composite plate structures, lacking analysis of the undulation characteristics of complex piezoelectric structures in the time domain.
Given the aforementioned challenges, there is a clear lack of an effective simulation analysis method and evaluation criteria to assess the wave performance and predict the dynamic behavior characteristics of piezoelectric bimorph wave robots.This research article aims to address this gap by introducing a time-domain transient FEA method for the structural design and prediction of dynamic behavior in piezoelectric wave robots with multiple working modes.The primary approach employed in this study is FEA, which allows for the investigation of undulation characteristics and time-domain vibration behavior of the piezoelectric undulatory robot.Specifically, we present a summarized FEA transient analysis method for the undulation robotic structure composed of piezoelectric bimorphs, along with an FEA-based approach for analyzing the undulation characteristics of the piezoelectric bimorph group.This approach involves three criteria: wave node recognition, determination of the traveling wave coefficient, and symmetry determination.Based on these criteria, a general performance evaluation guideline is proposed for structural-functional integration of piezoelectric undulatory robots with multiple operating modes.To validate the effectiveness of the proposed FEA-assisted design method for undulation characteristics of piezoelectric-driven robots, vibration measurements are conducted.Additionally, we propose a new structural-functional integrated piezoelectric undulatory miniature robotic structure based on double-mode vibration inspired by fin undulation.The proposed piezoelectric undulatory robotic structure consists of four groups of eight piezoelectric bimorphs, with each group serving as a structural component in the framework of the undulatory robot.The power source for undulatory propulsion and steering is provided by the superposition of vibration modes excited by alternating current, resulting in a traveling wave.To verify the validity of our proposed propulsion method and analyze the undulation and transient timing characteristics of the piezoelectric bimorphs, a transient simulation of the proposed robotic structure is performed using FEA.Furthermore, vibration measurements and mechanical performance tests are designed as experiments to validate the accuracy of our proposed simulation analysis and the effectiveness of the proposed robotic structure.
The subsequent sections of the manuscript are structured as follows: In Section 2, the working principle of the piezoelectric bimorph is introduced, and a corresponding structural design method for the structure-function integrated piezoelectric wave robotic structure is proposed.Section 3 presents a general performance evaluation guideline for structural-functional integration of piezoelectric undulatory robots with multiple operating modes.This section also explores the wave characteristics and time-domain vibration characteristics of piezoelectric wave robots.Sections 4 and 5 describe the prototype proposed in this study.The prototype undergoes wave measurement experiments and motion performance tests to validate the effectiveness of the proposed simulation method and the feasibility of the performance evaluation guideline.The experimental results are discussed in detail.Fin-based undulation images of Raja eglanteria fish during locomotion.Reproduced with permission. [1]Copyright 2001, The Company of Biologists.

Configuration of the Proposed Piezoelectric Undulatory Robot
The proposed piezoelectric-driven robotic structure consists of link plates, four-link blocks, and eight piezoelectric composite actuators (hereinafter we refer to them as piezoelectric bimorphs).As shown in Figure 2a, the robotic structure has a square frame structure, and flexible membranes are covered on both sides of the frame.The piezoelectric bimorphs are divided into two orthogonal groups: y and z, which are connected via link plates.The link block was used to link the two piezoelectric bimorphs in each group.Each piezoelectric bimorph receives an external alternating electric field from Ports 1 to 4. The y piezoelectric bimorphs receive the electric field from Ports 1 and 3, and the z piezoelectric bimorphs receive the electric field from Ports 2 and 4, as shown in Figure 2b.
The piezoelectric bimorph is a composite structure, and piezoelectric ceramics (i.e., PZTs) in the d31 mode are printed on the upper and lower surfaces of a metal substrate.As shown in Figure 3a, PZTs on the piezoelectric bimorph are polarized along the 0 to 3 and À3 to 0 axes and have different polarization directions.As shown in Figure 3b, the d31 mode means that the electric field of PZT is polarized and received along the 3-axis, and deformation is generated along the 1-axis.
The excitation of bending vibrations in piezoelectric bimorph actuators involves the utilization of the d31 mode exhibited by piezoelectric ceramics.Exciting bending vibrations in piezoelectric bimorph actuators require employing the d31 mode present in piezoelectric ceramics.Figure 4a illustrates that the application of identical electric fields to the piezoelectric ceramic layers on both sides of the bimorph results in opposing deformations,  where one side contracts and the other side extends.The deformation of PZTs induces the bending of the piezoelectric bimorphs.The piezoelectric bimorphs can generate periodic bending vibrations through periodic alternating current signal excitation, and the piezoelectric bimorph group receiving the same electrical signals can also generate periodic bending vibrations, as shown in Figure 4b.The periodic vibration of the bimorph group yields a traveling wave that serves as the fundamental mechanism for the proposed piezoelectric undulatory robot.This robotic structure design features a fully exposed architecture without any enclosed compartments.This design can reduce the constraints related to the water environment on the robot.The use of a piezoelectric bimorph effectively simplifies the structure of the robot, reduces the weight, and effectively helps the miniaturization of the proposed undulation robot.

Operating Principle of an Undulatory Robot
The proposed undulation robotic structure exhibits dual functionality, with the capability of functioning in both propulsion and steering modes.The propulsion mode depends on the undulating motion generated by the y piezoelectric bimorph groups excited by the electrical signal.The forward direction of the robotic structure is opposite to the undulating direction of the piezoelectric bimorph group.The flexible membranes arranged on the piezoelectric bimorph groups are used to amplify the amplitude of the traveling wave generated by the piezoelectric bimorph group; in the propulsion mode, the z piezoelectric bimorph groups are excited by the electrical signal to generate standing waves that do not affect the robotic structure propulsion, as shown in Figure 5a.The steering mode of the undulatory robotic structure is realized by the undulation of the four groups of piezoelectric bimorphs in different directions.A clockwise or counterclockwise traveling wave around the x-axis on the robotic structure frame enables the robotic structure to steer in the opposite direction of the traveling wave, as shown in Figure 5b.

Validation of the Operating Principle of the Robotic Structure via Finite Element Simulation Analysis
The proposed robotic structure has an irregular structure, and the behavior of the traveling wave generated by the piezoelectric bimorph group exhibits nonlinear motion characteristics;  therefore, the FEA can be used as a numerical calculation tool for analyzing the aforementioned requirements.The finite element simulation analysis of the undulatory robotic structure was performed using the commercial finite element software ANSYS Workbench 2021R2.We performed finite element simulation verification of the two working modes of the wave robot, including the use of modal analysis to determine the resonant modal used by the undulation robotic structure to generate traveling waves and to analyze the characteristics of the traveling waves in the frequency domain.The type of traveling wave generated by the piezoelectric bimorph group and the undulation characteristics of the traveling wave in the time domain are verified via transient analysis.The feasibility of using the undulation of the piezoelectric bimorph groups to realize the two working modes of the robotic structure is also verified.

Finite Element Model
The piezoelectric bimorph used in the undulatory robotic structure is a standard-sized industrial product (PZT-5, Haiying, Wuxi), and its metal substrate has a size of 50 mm Â 10 mm Â 0.2 mm.The size of the piezoelectric ceramics printed on both sides of the metal substrate is 50 mm Â 10 mm Â 0.3 mm.For meshing the eight-piece piezoelectric bimorph having a regular hexahedral structure, we used the SOLID186 element with three degrees of freedom per node (UX, UY, and UZ).SOLID186 is a 20-node uniform structural solid element having high-order 3D capabilities that exhibits quadratic displacement behavior.The four-link blocks on the robotic structure responsible for connecting the piezoelectric bimorphs are regular hexahedral structures, and the link plates are regular structures; thus, we also used the SOLID186 elements to build finite element models.Because flexible membranes are not responsible for motion, we did not consider adding flexible membranes for establishing the finite element model, as shown in Figure 6.Moreover, because the overall structure of the robotic structure is relatively regular, we used the multizone division method provided by the ANSYS Workbench to divide it.We employed four metrics, including the Jacobian ratio, skewness, orthogonal quality, and mesh metric, to evaluate the quality of meshing.Our assessment criteria stipulate that the quality of the mesh improves as the values of the Jacobian ratio, orthogonal quality, and mesh metric approach 1, whereas a skewness value closer to 0 indicates better mesh quality.Finally, referring to the aforementioned criteria, we determined that the size of the divided finite element unit is 1 Â 10 À3 m.The mesh quality of the finite element model is shown in Table 1.The robotic structure finite element model includes three subparts, of which the piezoelectric ceramic part has 8000 units, the metal substrate part has 4800 units, the link plates and link blocks have 8200 units, and the finite element model includes all 21 000 units.
The piezoelectric bimorph is composed of PZT-5, an anisotropic material with a piezoelectric modulus.The parameters of PZT-5 are listed in Table 2.The FE model cannot define Poisson's ratio for piezoelectric ceramics due to the lack of numerical input of Poisson's ratio for anisotropic materials in the simulation software.Stainless steel was utilized as the metal substrate for the piezoelectric bimorph, and resin was employed as the material for the link plates and link blocks.Their respective material properties are described in Table 3.These materials, including PZT, a metal substrate, link plates, and link blocks, were incorporated in the ANSYS Workbench finite element software to simulate the behavior of the undulation robot.

Modal Analysis
The undulation robotic structure has two operational modes that rely on the first-order and second-order out-of-plane bending resonance modes of the robotic structure frame.These modes are excited using the piezoelectric bimorph vibration.The piezoelectric actuator is devised to synchronize the resonant frequencies of the two modes, thereby minimizing interferences from adjacent modes and optimizing the functionality of the two modes. [33,50,51]The geometric finite element model of the robotic structure (Figure 7), along with the corresponding material parameters, is imported into the ANSYS Workbench.A modal analysis is conducted to optimize the geometric parameters of the piezoelectric actuator based on the design objectives, and the resultant optimized dimensions are tabulated in Table 4.
As shown in Figure 8, the results of the FEA indicate that the resonant frequency for the required first-order out-of-plane bending resonance mode is 135.81Hz and that for the required second-order out-of-plane bending resonance mode is between 127.01 and 139.93 Hz, as illustrated in Figure 8.The differences between the first-order and second-order resonant frequencies corresponding to the two operating vibration modes are 4.12 and 8.08 Hz, respectively, which meet the design criteria and represent the maximum frequency difference of 5.95%.

Transient Analysis of the Propulsion and Steering Modes of the Robot
The propulsion mode of the proposed undulatory robotic structure is realized using the traveling wave generated by the y piezoelectric bimorph groups.Therefore, the traveling wave generated by the y piezoelectric bimorph groups is formed via the modal coupling of the first-order and second-order resonant modes.Furthermore, the transient analysis method can be used to analyze the undulation characteristics of the traveling waves generated in opposite directions by the piezoelectric bimorph groups on both sides, as shown in Figure 9a.When the steering mode is implemented, the four y/z piezoelectric bimorph groups simultaneously generate undulation, and the overall movement presents a clockwise or counterclockwise rotation around the x-axis, as shown in Figure 9b.Thus, the transient analysis method is used to verify the undulation characteristics when the y and z piezoelectric bimorph groups undulate simultaneously.
In the propulsion mode, we used Ports 1 and 3 to apply the same sinusoidal electrical signal as the electrical load to the y piezoelectric bimorph groups and Ports 2 and 4 to apply the positive and negative cosine electrical signals, respectively, as the electrical load to the z piezoelectric bimorph groups.We only needed to adjust the electrical signals of the three phases by 180°in the same direction to realize the reverse undulation of the y piezoelectric bimorph groups.Two electrical signals with different phases were applied to piezoelectric bimorph groups through Ports 2 and 4 (i.e., sine signal) and Ports 1 and 3 (i.e., cosine signal) to realize the steering mode.To reverse the steering mode, the sine and cosine signals need to be adjusted to the negative sine and cosine signals.In addition to different electrical signal loads, the robotic structure model is always set to a free boundary in the two modes, and the contact state between the piezoelectric bimorph group of the robotic structure and the link plate and link block is a fixed contact, as shown in Figure 10.The calculation time step of the transient analysis is set as 1/20 of the piezoelectric bimorph frequency, where the total calculation period is 60 piezoelectric bimorph vibration periods, and the initial time is set as 0 s.
The maximum and minimum values of the three resonant frequencies obtained via the modal analysis are used as the upper and lower limits of the frequency range for transient calculations.
In this interval, we selected five resonant frequencies: 121, 128, 130, 132, and 135 Hz, for transient analyses.As shown in Figure 11 and 16 equally spaced elements located on the y and z piezoelectric bimorph groups are selected as the reference points, and the displacement time information of the reference points can be obtained and counted.Because of the axisymmetric structure, we only needed to select either of the y and z piezoelectric bimorph groups.

Transient Analysis Results of the Propulsion and Steering Modes
The transient vibration data for a total of 32 reference points on the y and z piezoelectric bimorph groups acquired at each frequency were aggregated, plotted, and interpolated with respect to the y and z piezoelectric bimorph groups.As shown in Figure 12, the x-axis represents the absolute position of the reference point, and the y-axis represents the relative displacement of the reference point at a certain moment relative to the initial       The analysis results shown in Figure 12 indicate that when the robotic structure is in the propulsion mode, reference points 5 and 13 of the z piezoelectric bimorph group exhibit no obvious relative displacement at the five excitation frequencies.Hence, reference points 5 and 13 can be defined as the node points, and their maximum relative displacement occurring at 132 Hz is only 1.32 mm.We determined that, at five excitation frequencies, the undulation behavior of the z piezoelectric bimorph group is a standing wave; that is, no undulatory wave is generated, which is consistent with our design ideas.According to this criterion, the undulation behavior of the piezoelectric bimorph group in the propulsion mode can be inferred; that is, the existence of node positions on the piezoelectric bimorph group can be determined.Figure 13a,e shows that the undulation behavior of the y piezoelectric bimorph group at the excitation frequencies of 121 and 135 Hz is a standing wave, which does not meet our design ideas.Calculating the traveling wave coefficients of the y piezoelectric bimorph group at the excitation frequencies of 128, 130, and 132 Hz can help in selecting the optimal excitation frequency for the undulation motion of the piezoelectric bimorph group.The formula for calculating the traveling wave coefficient is expressed in Equation (1) as follows: where K is the traveling wave coefficient, jdj min is the amplitude of the node position reference point, and jdj max is the maximum amplitude of the traveling wave position reference point.The values of traveling wave coefficients at the excitation frequencies of 128, 130, and 132 Hz are listed in Table 5.
With the central position of the y piezoelectric bimorph group along the y and Ày directions taken as the reference axis, the undulation symmetry of the amplitude of each position of the piezoelectric bimorph group centered on this axis is also an important evaluation criterion for analyzing its undulation characteristics.We selected four sets of data: reference points 1 and 16, reference points 3 and 14, reference points 5 and 12, and reference points 7 and 10, to determine the undulation symmetry.
The amplitudes of the symmetry points in each set were divided to determine the symmetry of each set.The average value of the four sets of data can be used to estimate the undulation symmetry of the entire piezoelectric bimorph group at different excitation frequencies, as shown in Table 6.
After the assessment of the three criteria, we selected 130 Hz as the optimal excitation frequency.When the robotic structure is in the propulsion mode, the y piezoelectric bimorph group produces an incomplete traveling wave with a traveling wave coefficient of 0.225, undulation symmetry of 84.7%, a maximum amplitude of 3.48 mm, and a minimum amplitude of 0.87 mm.Moreover, the z piezoelectric bimorph group exhibits a standing wave characteristic.
When the robotic structure is in the steering mode, as shown in Figure 13, the y piezoelectric bimorph group produces an obvious node at reference point 7 at an excitation frequency of 121 Hz, and the amplitude at this point is only 0.83 mm.The z piezoelectric bimorph produces two distinct nodes with an amplitude of 0.37 mm at reference points 5 and 14.At the excitation frequency of 135 Hz, the undulation characteristics of the y and z piezoelectric bimorph groups are opposite to those at the excitation frequency of 121 Hz.Moreover, two nodes with a maximum amplitude of 0.19 mm appear in the y piezoelectric bimorph group, and one node with an amplitude of 0.21 mm appears in the z piezoelectric bimorph group.We determined that the y and z piezoelectric bimorph groups exhibit standing wave undulation characteristics at two excitation frequencies.At the excitation frequency of 132 Hz, the y piezoelectric bimorph group produces a node at the five selected reference points; thus, the y piezoelectric bimorph group generates a standing wave, which is consistent with our design ideas.
As shown in Figure 13, the traveling wave coefficient and undulation symmetry of the y and z piezoelectric bimorph groups are analyzed at the excitation frequencies of 128 and 130 Hz, and the analysis results are listed in Table 7.
The transient simulation results shown in Table 7 indicate that the excitation frequency of 130 Hz can be used as the optimal excitation frequency for the robotic structure in the steering mode.The y and z piezoelectric bimorph groups exhibit traveling wave undulation characteristics, and the y piezoelectric bimorph group produces an incomplete traveling wave with a traveling wave coefficient of 0.294, an undulation symmetry of 81.3%, a maximum amplitude of 3.12 mm, and a minimum amplitude of 0.92 mm.The z piezoelectric bimorph group produces an incomplete traveling wave with a traveling wave coefficient of 0.22, an undulation symmetry of 89.9%, a maximum amplitude of 3.44 mm, and a minimum amplitude of 0.74 mm.
The undulation state of the piezoelectric bimorph groups under the two working modes of the robotic structure is analyzed transiently via FEA, and the undulation characteristics of the piezoelectric bimorph groups excited by electrical signals at different frequencies are calculated.We used the three criteria introduced previously to determine the optimal excitation frequency that can obtain the optimal undulation characteristics in the excitation bandwidth obtained through modal analysis.
At the optimal excitation frequency, we selected reference point 9, located at the center of the y piezoelectric bimorph group, to analyze the transient time-domain vibration behavior in the propulsion and steering modes.This study shows the various characteristics of the displacement of the reference point from the initial vibration moment (time = 0 s) until it reaches steady-state vibration, as shown in Figure 14.By analyzing Figure 14a,b, we determined that reference point 9 can reach steady-state vibration within 200 ms in both working modes.Reference point 9, which vibrates stably in the propulsion mode, exhibits the characteristic of having different amplitudes in two adjacent vibration cycles, and this characteristic appears regularly.This characteristic can be attributed to the standing wave vibration generated by the z piezoelectric bimorph group that interferes with the traveling wave vibration generated by the y piezoelectric bimorph group.When we analyzed the steady-state vibration behavior of reference point 9 in the steering mode, we observed that this phenomenon disappeared because the z and y piezoelectric bimorph groups generate undulation time-domain vibrations in the steering mode.Thus, this interference phenomenon does not occur.and link block, and silicon rubber was employed to fabricate the flexible membrane.To ensure electrical insulation in the water environment, the piezoelectric ceramic part of the piezoelectric bimorphs was covered with a thin layer of epoxy resin glue. [17,33,50,52]

Vibration Measurement of the Robotic Structure Prototype
A 3D laser Doppler vibrometer (PSV-500-3D-M, Polytec, Germany) was employed to perform a vibration measurement experiment aimed at determining the vibration characteristics of the piezoelectric actuator prototype.As shown in Figure 16, four groups of piezoelectric bimorphs were selected as the measurement area, reflective tapes were pasted on them to enhance laser reflection, and reflective paint was applied to the link plate to avoid errors caused by laser penetrating through the resin. [51,53]We used the main control computer to manipulate three laser scanning heads that cooperate to evenly mark the scanning points on the structural-functional integration frame of the robot.To meet the experimental requirements, a bench clamp was employed to secure the measured surfaces during the experiment, providing support and stability to ensure that accurate results were obtained.
The scanning frequency bandwidth of the vibrometer was adjusted to 90 Hz, within the frequency range of 90-180 Hz, based on the frequency range of the two resonance modes obtained from the modal analysis simulation (i.e., 127-140 Hz).As depicted in Figure 17, the frequency magnitude curve of the first-order bending vibration mode and two second-order bending vibration modes are acquired through experimentation.The analysis of the results revealed that the resonant frequencies of the two second-order bending vibration modes were 120 and 136 Hz, whereas the resonant frequency of the first-order vibration mode was 133 Hz.The frequency differences between the first-order and second-order vibration modes were 13 and 3 Hz, respectively.The obtained frequency changes from the vibration measurement experiment are deemed acceptable, with rates of change of 10% and 2.3%.
Based on the results of the finite element transient simulation analysis, we used the specified frequency measurement mode of the Doppler vibrometer to measure the undulation characteristics of the y and z piezoelectric bimorph groups at a specified frequency to determine the optimal undulation excitation frequencies of the robotic structure in the propulsion and steering modes.Because of the axial symmetry of the robotic structure frame, the undulation state of any y or z piezoelectric bimorph group needs to be observed.We used five groups of frequencies: 120, 125, 130, 135, and 140 Hz, to observe the undulation states of the piezoelectric bimorph groups in the propulsion and steering modes.By analyzing the obtained undulation data, we determined the undulation characteristics of the y and z piezoelectric bimorph groups in the two operating modes and their optimal excitation frequency of 130 Hz.As shown in Figure 18, in the propulsion working mode, using 130 Hz as the excitation frequency and 50 V pp as the excitation voltage, we determined that  the undulation characteristics of the y piezoelectric bimorph group are traveling waves with a traveling wave coefficient of 0.441, an undulation symmetry of 93.7%, a maximum amplitude of 3.34 mm, and a minimum amplitude of 1.29 mm.The vibrational behavior of the z piezoelectric bimorph group is a standing wave with a maximum amplitude of 5.85 mm and a minimum amplitude of 0.42 mm.Table 8 shows the comparison of the simulation and experimental results at the optimal excitation frequency.
As shown in Figure 19, in the steering working mode, using 130 Hz as the excitation frequency and 50 V pp as the excitation voltage, we determined that the undulation characteristics of the y piezoelectric bimorph group are traveling waves, with a traveling wave coefficient of 0.455, an undulation symmetry of 93.2%, a maximum amplitude of 3.35 mm, and a minimum amplitude of 1.61 mm.The vibrational behavior of the y piezoelectric bimorph group is a traveling wave with a traveling wave coefficient of 0.417, an undulation symmetry of 88.3%, a maximum amplitude of 4.05 mm, and a minimum amplitude of 1.69 mm.
By comparing Table 8 and 9, we determined that the undulation characteristics of the piezoelectric bimorph group obtained through the finite element transient simulation can be verified via vibration testing.Whether in the propulsion or steering mode, the traveling wave coefficient and undulation symmetry of the undulation characteristics of the piezoelectric bimorph group obtained in the experiment are better than those obtained in the simulation, which increased by 42.5% and 9.50% in the propulsion mode, respectively.In the propulsion mode, the traveling wave coefficient and undulation symmetry of the y piezoelectric bimorph group increased by 29.5% and 7.93%, respectively, and those of the z piezoelectric bimorph group increased by 52.5% and 3.54%, respectively.We further determined that the transient simulation of the finite element of the proposed robotic structure is effective, whereas the finite element simulation is instructive for the structural design and undulation performance prediction of the piezoelectricbimorph-driven undulation robot.Because the 3D laser Doppler vibrometer cannot provide the time-domain vibration displacement data of the measurement point, we used a laser displacement sensor (optoNCDT 2300, Micro-Epsilon, Germany) to obtain the time-domain vibration displacement data of a specific point on the robot.To prevent the robotic structure from changing its position because of vibrations, we used a bench clamp to fix it, as shown in Figure 20.We selected a point located in the y piezoelectric bimorph group and close to reference point 9 in the simulation, irradiated the laser of the laser displacement sensor on this point, and started the laser displacement sensor and robotic structure as simultaneously as possible.We collected the vibration data of this point in the propulsion and steering modes for the first 450 ms, as shown in Figure 21.We used an excitation voltage of 50 V pp in the experiment.
As shown in Figure 21, compared with that in the simulation analysis results, the scanning point can reach a stable vibration state faster during the experiment in the propulsion mode, taking approximately 150 ms.However, in the steering mode, the scanning point reaches a stable vibration state only 350 ms after startup during the experiment, which is 100 ms slower than that in the simulation analysis results.Similar to the simulation analysis results, regular vibration changes at the scanning point are detected in the propulsion mode during the experiment.Compared with that of the simulation analysis results, in which each regular periodic change includes two vibration periods with different amplitudes under steady-state vibration, the number of vibration periods with different amplitudes in the regular periodic changes increased to eight during the experiment.The steady-state vibration of the scanning point in the steering mode during the experiment is consistent with the simulation analysis results and does not produce the aforementioned regular periodic changes.The reason for this difference is that the clamping method that we used interferes with the overall vibration state of the robot.This interference is unavoidable during the measurement because it is difficult to accurately position the laser displacement sensor without the clamping method.

Mechanical Performance Analysis of the Robotic Structure Prototype
The maneuverability of the robotic structure prototype is a key parameter for evaluating its steering characteristics.Meanwhile, the forward and backward speed characteristics, as well as the start-stop working characteristics of the robotic structure prototype, serve as important parameters for assessing its propulsion performance.Figure 22 illustrates the diverse underwater experimental platforms developed to examine the mechanical features of the prototype in water environments.To conduct this research, we established several experimental setups.

Investigating the Mechanical Behavior of the Robotic Structure in the Propulsion Mode
During the underwater experiments, a motion camera with a frame rate of 60 fps is utilized to capture the lateral movement of the prototype, and the motion video is analyzed frame by frame, with each frame lasting 1/60 s, as depicted in Figure 23.An electro-optical sensor is employed to determine the average motion speed of the robotic structure prototype.The sensor initiates timing when the robotic structure crosses the first laser beam and stops timing when the robotic structure crosses the second laser beam.
Figure 24a,b illustrates the mechanical characteristics of the underwater propulsion of the piezoelectric-driven robot.The propulsion characteristic data of the robotic structure were obtained by modulating the excitation frequency and voltage.Our findings indicated that the optimal working frequency for the forward propulsion motion of the robotic structure was 126 Hz, resulting in an average velocity of 60 mm s À1 (3.33 BL s À1 ) under an excitation voltage of 105 V pp .Similarly, for the backward propulsion motion characteristics of the robot, the optimal excitation frequency was 126 Hz, resulting in the highest velocity of 56.7 mm s À1 (3.15 BL s À1 ) under an excitation voltage of 105 V pp .
To determine the start and stop times of the robotic structure in the propulsion mode, we employed a laser displacement sensor that emits a laser onto the float of the robotic structure above the water.As depicted in Figure 25, we established an  underwater mechanical measurement platform consisting of a laser displacement sensor and a 126 Hz-frequency excitation robotic structure with an excitation voltage of 105 V pp and an input current of 109 mA.Specifically, we utilized the laser displacement sensor model LK-H020 purchased from Keyence Corporation in Japan.
The time-displacement data of the robotic structure while in operation were recorded and analyzed, and a scatter plot of the underwater start-stop motion characteristics was generated.Subsequently, we performed a partition analysis of the collected dataset.The sampling period for the laser displacement sensor was 10 ms (50 sampling periods), and 100 sampling instances were obtained after the robotic structure prototype ceased operation.
As shown in Figure 26, the start-stop motion of the robotic structure can be decomposed into five stages: slow acceleration

Investigating the Cost of Transport of the Robotic Structure
Cost of transport (CoT) is calculated as the ratio of power consumption to the swimming speed of an object with a specific mass, expressed as energy consumed to travel a unit distance.Therefore, CoT scales proportionally to the speed and inversely to the energy efficiency of the object.CoT then scales as [54] CoT ¼ where m is the mass of the robotic structure, and v is the steady average velocity.For the robotic structure, the power consumption P is the total input power from the AC supply to the piezoelectric bimorphs.Therefore, the unit of CoT can be expressed as mW (kgm s À1 ) À1 is also J kgm À1 .P can be derived by Equation (3).
Among them, V i and I i are the input voltage and input current, respectively, θ is the power factor of the circuit, and is the phase difference between the input voltage and the input current.The input voltage is the excitation voltage, its unit is V, and its value is half the peak-to-peak value of the voltage in unit V pp .An ammeter (PT710-D, INTECH, China) can record the input current, and the recording unit is mA.We adjusted the excitation voltage, recorded the current, and found that the input current showed  a nearly linear increase in the voltage range of 75-165 V pp with the increase of the input voltage.The phase difference between the input current and voltage was maintained at 89.47°, which could be monitored by an oscilloscope (DPO2014, Tetronix, USA).At an excitation voltage of 105 V pp , the power consumption of the robotic structure is 26.5 mW.

Discussion
A simulation analysis method for analyzing the undulation characteristics of the piezoelectric bimorph group is proposed herein.This method can determine the undulation characteristics of the piezoelectric bimorph group that composes the robotic structure and the undulation characteristics of the robot.The method of judging the undulation characteristics through the simulation analysis results includes three criteria: 1) analyze the existence of wave nodes in the plotted interpolated graph of the displacement time of each reference point on the piezoelectric bimorph group.If obvious wave nodes exist, it can be deduced that the undulation behavior of the bimorph group is a standing wave.If obvious nodes are absent, it can be deduced that the undulation behavior of the bimorph group is a traveling wave; 2) to analyze the undulation characteristics of traveling waves, distinguishing between complete and incomplete traveling waves is necessary.If the traveling wave coefficient is 1, then it is a complete  traveling wave; otherwise, it is an incomplete traveling wave; and 3) for the incomplete traveling wave generated by the bimorph group, its undulation symmetry needs to be assessed.Moreover, when the undulation symmetry value is 100%, the undulation in the incomplete wave is completely symmetrical.This method can guide the design of the piezoelectric composite-driven robotic structure and can be used to guide the study of the transient time-domain vibration characteristics of the piezoelectric bimorph.
According to the results of the finite element simulation, we proposed a structural-functional integration miniature robotic structure that is driven by a piezoelectric bimorph group to verify the effectiveness of our proposed performance evaluation criteria.Moving forward, our research will focus on improving the working efficiency of the collaborative working mode, optimizing the fluid-structure design of the prototype, and designing a suitable power supply system to ensure the normal operation of the robotic structure under hydrostatic pressure.Our research will place particular emphasis on exploring the potential of the balanced hydrostatic method as a key component of the underwater robotic structure design.The findings of our research may inspire innovative approaches to simplify and miniaturize underwater robots.
A performance comparison table is created to evaluate the proposed robotic structure in terms of length, number of driving mechanisms, propulsion speed, steering velocity, and structural-functional design in comparison with recently published undulatory robots, as shown in Figure 30.Propulsion speed and steering velocity are used to describe the basic motion characteristics of the undulatory robot.The propulsion speed unit is BL s À1 because of the differences in thrust size.The number of driving mechanisms is used to assess the complexity of the robotic structure, and the criterion for judging structuralfunctional design is whether the robotic structure can provide power while serving as a functional component.The results showed that the proposed robotic structure exhibits superior mechanical characteristics compared with the recently reported small underwater robots, with the second-fewest driving mechanisms and the second-shortest structural length.

Conclusion
In this study, our primary approach was to utilize FEA to investigate the undulation characteristics and time-domain vibration characteristics of the piezoelectric undulatory robot.We propose an FEA-based approach specifically designed for analyzing the undulation characteristics of the piezoelectric bimorph group that comprises the robot.This approach enables us to determine both the undulation characteristics of the piezoelectric bimorph group and the overall undulation characteristics of the robotic structure itself.Building upon this foundation, we have developed a comprehensive performance assessment guideline for structural-functional integrated piezoelectric undulatory propulsion systems encompassing multiple working modes.To validate the effectiveness of our FEA-based approach for analyzing the undulation characteristics of the piezoelectric-driven robot, we conducted vibrational measurements.These measurements not only confirm the effectiveness of the robotic structure but also allow for experimental evaluation of the primary mechanical characteristics of the prototype.In the propulsion mode, the y piezoelectric bimorph group of the robotic structure produces an incomplete traveling wave with a traveling wave coefficient of 0.441 and an undulation symmetry of 93.7%, they have changed by 42.5% and 9.5%, respectively, compared to the simulation results.Moreover, the z piezoelectric bimorph group produces a standing wave with two wave nodes.In the steering mode, the y and z piezoelectric bimorph groups produce traveling waves with traveling wave coefficients of 0.455 and 0.417 and undulation symmetries of 93.2% and 88.3%, the maximum gradient in the experimental data relative to the simulation results was 52.5% and 7.93%, respectively.Vibration measurements confirmed the effectiveness of the robot, and the main mechanical characteristics of the prototype were evaluated experimentally.A novel undulatory robotic structure driven by a piezoelectric bimorph with structure-function integration for both propulsion and steering modes is presented.The working principle and modes of the proposed robotic structure are described in detail, with the undulation characteristics of the piezoelectric bimorph group and the robotic structure determined using FEA-based approach.The proposed robotic structure has several advantages over miniature robots using electromagnetic motors, including lower manufacturing costs, simpler driving methods, quietness

Figure 2 .
Figure 2. The configuration of the piezoelectric undulatory robotic structure and its details: a) parts of the robotic structure; b) polarization direction of piezoelectric bimorphs.

Figure 3 .
Figure 3.The configuration of the piezoelectric bimorph: a) structural details of the piezoelectric bimorph; b) the d31 mode of piezoelectric ceramics.

Figure 4 .
Figure 4. a) Sketch of piezoelectric bimorph bending under excitation of piezoelectric ceramics; b) sketch of piezoelectric bimorph group bending.

Figure 5 .
Figure 5. Operating principle of the undulatory robot: a) propulsion mode and b) steering mode.

Figure 6 .
Figure 6.Types of elements used in each model in the finite element analysis.

Figure 7 .
Figure 7. Major geometrical parameters of the proposed robot.
location of the reference point at time equal to 0 s of the transient simulation).One of the curves shown in the legend to the right represents the relative position of all reference points on the same piezoelectric bimorph at a certain moment in an undulation period.Therefore, we displayed all the curves shown in the legend of the chart at the same time to obtain the image of the one-period undulation of the piezoelectric bimorph group at a fixed frequency.The undulation figure at each frequency is drawn along with 40 times interpolation curves, and the time interval between each curve is 1/40 period.The interpolation image intuitively draws the vibration displacement characteristics of the reference points in one cycle, and this image can help us effectively evaluate the undulation characteristics of the y and z piezoelectric bimorph groups.

Figure 8 .
Figure 8. Modal simulation of the proposed robot.

Figure 9 .
Figure 9. Operating principle of the undulatory robot: a) propulsion mode and b) steering mode.

Figure 10 .
Figure 10.Loads and boundary conditions of the finite element analysis model: a) propulsion mode and b) steering mode.

Figure 11 .
Figure 11.Selected reference points for transient analysis.

Figure 12 .
Figure 12.Propulsion mode: undulation characteristics of the y and z piezoelectric bimorph groups at excitation frequencies of a) 121 Hz, b) 128 Hz, c) 130 Hz, d) 132 Hz, and e) 135 Hz.

Figure 13 .
Figure 13.Steering mode: undulation characteristics of the y and z piezoelectric bimorph groups at the excitation frequencies of a) 121 Hz, b) 128 Hz, c) 130 Hz, d) 132 Hz, and e) 135 Hz.

4 .
Experimental StudyUsing a comprehensive methodology that involved finite element modeling and geometric parameter optimization of the piezoelectric actuator, we designed a novel piezoelectric-driven robotic structure prototype.The resulting product, which weighs 9.53 g and has dimensions of 18 mm Â 17 mm Â 0.8 mm, is depicted in Figure15.The selection of materials was guided by the considerations of functionality, durability, and cost-effectiveness.Stainless steel was employed as the metal substrate of the piezoelectric bimorph, resin was utilized for fabricating the link plate

Figure 14 .
Figure 14.Transient time-domain vibration characteristics from the initial instant to the steady state at reference point 9: a) propulsion mode and b) steering mode.

Figure 15 .
Figure 15.A prototype of the proposed piezoelectric undulatory miniature robotic structure exhibiting structural-functional integration.

Figure 16 .
Figure 16.Experimental setup for measuring the vibration characteristics of the prototype.

Figure 17 .
Figure 17.Vibration frequency measurement results of the proposed robot.

Figure 18 .
Figure 18.Undulation characteristics of the y and z piezoelectric bimorph groups in the propulsion mode under vibration testing.

Figure 19 .
Figure 19.Undulation characteristics of the y and z piezoelectric bimorph groups in the steering mode under vibration testing.

Figure 20 .
Figure 20.The measurement method of the transient time-domain vibration characteristics of the scanning point of the robot.

Figure 21 .
Figure 21.Transient time-domain vibration characteristics from the initial instant to the steady state at the scanning point: a) propulsion mode; b) steering mode.

Figure 22 .
Figure 22.Mechanical characterization of the robotic structure prototype: experimental procedures.

Figure 23 .
Figure 23.The measurement method of the propulsion mode of the prototype.

Figure 24 .
Figure 24.An investigation of the mechanical properties of the robotic structure in the propulsion mode: a) correlations between propulsion speed and excitation frequencies and b) correlations between propulsion speed and excitation voltages.

Figure 25 .
Figure 25.The measurement method of the start-stop characteristics of the prototype.
www.advancedsciencenews.com www.advintellsyst.comstage, fast acceleration stage, stable working stage, deceleration stage, and stop stage, according to time.The slow acceleration stage lasted approximately 0.11 s, resulting in a linear displacement of 2.44 mm, during which the average speed of the propeller was approximately 22.18 mm s À1 (1.23 BL s À1 ).During the period of 0.11-0.19s, the propeller produced a more obvious acceleration than during the slow acceleration stage.The average propulsion speed during the fast acceleration stage was 74.87 mm s À1 (4.15 BL s À1 ), and the total average speed during the entire acceleration process of the propeller was 44.37 mm s À1 (2.46 BL s À1 ).The stable working stage lasted 0.41 s and produced a displacement of 24.11 mm.The average speed during this stage was 58.8 mm s À1 (3.27 BL s À1); compared with the data obtained by the forward propulsion motion characteristic experiment, the average speed was 1.6% slower and within the reasonable error range.The stage from the moment the propeller is powered off till it stops is called the deceleration stage.This stage lasts 0.17 s, and the displacement during this stage is 2.99 mm.Subsequently, during the stop stage, the propeller is in a static state, with only marginal undulations under the action of water.4.2.2.Investigating the Mechanical Behavior of the Robotic Structure in the Steering ModeDuring the mechanical characteristic experiment of the robotic structure in the steering mode, we used video analysis software to analyze the clockwise and counterclockwise steering processes of the robotic structure frame by frame.The average rotational angular velocity of the robotic structure in the steering mode can be obtained through a simple calculation, as shown in Figure27.Figure28a,b presents the underwater steering characteristics of the piezoelectric-driven robot.To explore the influence of excitation frequency and voltage on steering behavior, an experimental investigation was conducted.Optimal performance parameters were identified for both counterclockwise and clockwise steering movements.Notably, the highest average counterclockwise angular velocity of 17.4 deg s À1 was achieved at a frequency of 126 Hz, an excitation voltage of 105 V pp, and an input current of 109 mA.Meanwhile, a frequency of 126 Hz and an excitation voltage of 105 V pp yielded the maximum angular velocity of 15.3 deg s À1 under clockwise steering.

Figure 26 .
Figure 26.Start-stop characteristics of the robotic structure in the propulsion mode.

Figure 27 .
Figure 27.The measurement method of the steering mode of the prototype.
Figure 29a,b presents the CoT of a robotic structure in propulsion and steering mode, respectively.The experimental data on mechanical performance show that the average propulsion and steering velocity of the robotic structure have a positive correlation with the excitation voltage.The vertical axis of the CoT represents the average propulsion or steering velocity.The analysis indicates that the CoT of our proposed robotic structure increases with the increase of velocity under the excitation voltage in the range of 75-165 Vpp.Under the excitation voltage of 105 Vpp, the CoT of the robotic structure is 5.16 and 5.18 J kgm À1 when moving forward and backward, respectively.Also, the CoT is 16 and 18.08 J kg À1 °when steering clockwise and counterclockwise, respectively.

Figure 28 .
Figure 28.An investigation of the mechanical properties of the robotic structure in the steering mode: a) correlations between propulsion speed and excitation frequencies and b) correlations between propulsion speed and excitation voltages.

Figure 29 .
Figure 29.Investigating the cost of transport for a robotic structure: a) in propulsion mode and b) in steering mode.

Figure 30 .
Figure 30.Performance of the proposed robotic structure compared with those reported in recent studies based on the performance data.Note: the structural-functional integration design of the robotic structure is indicated by Y/N.The steering velocity is marked as N/A if the robotic structure is not maneuverable.
, and compact structures.The open structure of the proposed robotic structure enables adaptation in the underwater environment.The maximum propulsion speeds of the forward and backward propulsion of the prototype with an excitation voltage of 105 V pp and an excitation frequency of 126 Hz were 60 mm s À1 (3.33 BL s À1 ) and 56.7 mm s À1 (3.15 BL s À1 ), respectively.The maximum clockwise and counterclockwise steering angular velocities of the prototype with an excitation voltage of 105 V pp and a driving frequency of 126 Hz were 15.3 and 17.4 deg s À1 , respectively.

Table 1 .
Mesh quality of the finite element model of the robot.

Table 3 .
Material parameters for stainless steel and resin.Materials Young's modulus [GPa] Poisson's ratio Density [kg m À3 ]

Table 2 .
Piezoelectric materials used in the piezoelectric bimorphs.

Table 5 .
Traveling wave coefficients of the piezoelectric bimorph group at the three excitation frequencies.

Table 6 .
Undulation symmetry of the piezoelectric bimorph group at the three excitation frequencies.

Table 7 .
Traveling wave coefficients and undulation symmetries of the piezoelectric bimorph groups at two excitation frequencies.

Table 8 .
Comparison of the simulation and experimental results of the undulation characteristics of piezoelectric bimorph groups in the propulsion mode.

Table 9 .
Comparison of the simulation and experimental results of the undulation characteristics of piezoelectric bimorph groups in the steering mode.