Automatic Holonomic Mobile Micromanipulator for Submillimeter Objects Inspired by the Rhinoceros Beetle

Holonomic mobile micromanipulators driven by piezoelectric actuators offer precision and compact design. However, attaining both high speed and positioning repeatability with sufficient load capacity poses challenges, given that lightweight robots are susceptible to nonuniform driving surfaces and external forces. A holonomic beetle (HB) is proposed to realize a wide range, compact size, precise holonomic motion, automatic micromanipulation, and high‐speed movement simultaneously. Inspired by the biological makeup of a Rhinoceros Beetle, a length, weight, and load capacity of 8 cm, 74 g, and of 2000 g, respectively, are used. The HB is a two‐legged robot with a pseudoalternating tripod gait locomotion principle, with five piezoelectric actuators for XYθ displacement and switching of the contact leg. It achieves maximum walking velocities of 11.6 mm s−1 for translational motion and 19.3 deg s−1 for rotation, with a displacement resolution of 12 nm. Notably, under open‐loop control, the HB demonstrates high repeatability with a coefficient of variation of 0.3%. It exhibits the capability to climb 30°‐inclined surfaces, traverse flat surfaces with 0.1 mm bumps, and navigate 30 mm pits. Furthermore, the HB showcases practical automatic micromanipulation through visual servo control and machine learning, presenting potential applications in biomedical and microassembly fields.


Introduction
The recent advancements in the miniaturization of portable electronic devices, such as smartphones and PCs, alongside progress in the biological and medical fields, have been highly significant.This has led to an increased demand for miniaturized and flexible multiaxial micromanipulation technologies.Research into flexible and self-reconfigurable robotic factories has also been performed, [1,2] including the development of swarm miniature robotics. [3]umerous holonomic mobile robots have been subject to study. [4]For instance, omniwheeled robots [5][6][7] and ball-wheeled robots [8][9][10][11] have been investigated as types of wheeled mobile robots.Multilegged robots [12][13][14][15][16] excel at stable walking on uneven terrain and rubble due to the independent movement of multiple legs.Although these robots can achieve relatively high speeds, exceeding 100 mm s À1 , they are not suitable for precise positioning due to their intricate structure and significant mechanical backlash in gears and rotational axes.
30][31] Precision stages have undergone extensive research to enhance positional accuracy.34] However, their driving range is constrained to the submillimeter order, and even stages with expanded driving capabilities through piezoelectric motors are limited to ≈10 cm. [35]n contrast, mobile robots utilizing piezoelectric elements have demonstrated an increased driving range.Soft robots harnessing piezoelectricity have the potential to operate in challenging environments, such as within the human body.Notably, a piezoelectric soft robot emulating a cockroach exhibits high speed and a 15.6°hill climbing capability. [36]Another piezoelectric soft robot utilizes magnetic effects to achieve remote movement, environmental monitoring, and wireless communication, all without the need for an internal battery or external wired power source. [37]Nevertheless, it is unsuitable for precision positioning due to its restricted movement direction and lower accuracy when compared to precision stages.
][40][41][42] Such robots should be developed to combine the DOI: 10.1002/aisy.202300517Holonomic mobile micromanipulators driven by piezoelectric actuators offer precision and compact design.However, attaining both high speed and positioning repeatability with sufficient load capacity poses challenges, given that lightweight robots are susceptible to nonuniform driving surfaces and external forces.A holonomic beetle (HB) is proposed to realize a wide range, compact size, precise holonomic motion, automatic micromanipulation, and high-speed movement simultaneously.Inspired by the biological makeup of a Rhinoceros Beetle, a length, weight, and load capacity of 8 cm, 74 g, and of 2000 g, respectively, are used.The HB is a two-legged robot with a pseudoalternating tripod gait locomotion principle, with five piezoelectric actuators for XYθ displacement and switching of the contact leg.It achieves maximum walking velocities of 11.6 mm s À1 for translational motion and 19.3 deg s À1 for rotation, with a displacement resolution of 12 nm.Notably, under open-loop control, the HB demonstrates high repeatability with a coefficient of variation of 0.3%.It exhibits the capability to climb 30°inclined surfaces, traverse flat surfaces with 0.1 mm bumps, and navigate 30 mm pits.Furthermore, the HB showcases practical automatic micromanipulation through visual servo control and machine learning, presenting potential applications in biomedical and microassembly fields.[48][49][50] Among these, some nonholonomic mobile robots incorporating piezoelectric elements [51][52][53][54] exhibit remarkable performance, including high speed and excellent mobility across challenging terrain.However, their suitability for precision tasks in confined spaces is limited due to their restricted mobility, stemming from their nonholonomic nature.Consequently, there has been a focus on studying holonomic mobile robots to address this limitation.
Holonomic mobile robots driven by resonance mechanisms [55][56][57] and based on inchworm principles [58][59][60][61] have also been devised.Stick-slip holonomic mobile robots have been reported as well, [38,41] influenced by the kinetic friction between legs and surfaces.However, this introduces random positioning errors and leg wear.Moreover, these systems require specific surface conditions, such as a smooth board for the resonance drive or a finely polished ferromagnetic surface for inchworm motion.The coefficient of static friction on these surfaces should be ≈0.2 to enable precise movements.
Extensive research has been conducted on arthropod mobility as a solution to overcome the challenges associated with kinetic friction.The mobility of arthropods (flying and jumping) [62][63][64] has been the focus of considerable research.67][68] This study developed a new arthropod-mimicking robot, "holonomic beetle (HB)", which combines the strength of the rhinoceros beetles (Dynastinae) and the holonomic mobility of spiders.As shown in Figure 1, a maximum length of 18 cm, weight of 60 g, and load capacity of 2000 g were targeted. [69,70][73][74] Table 1 presents a comparison of precise holonomic mobile robots.As shown in Table 1, the HB simultaneously realizes holonomic motion, high speed, high resolution, high load capacity, and high repeatability, even on commonly used desks and plastic boards with a coefficient of static friction greater than 0.3.These properties are also designed to be balanced to contribute to precise, versatile, automatic, and multiscale micromanipulation in the electronic, microelectromechanical systems (MEMS), biomedical, and micropart assembly fields.

Results
We recently developed a novel and precise holonomic mobile robot called the HB, designed for automatic and versatile micromanipulation.The first prototype (HB-1) is depicted in Figure 2A. Figure 2B shows the mechanism which comprises an XYθ stage, two sets of aluminum alloy disk-shaped tripod units (upper outer and lower inner legs) and a z-axis piezoelectric actuator connecting the XYθ stage and the lower inner leg.Three ball gauge heads are attached to both the upper outer and lower inner legs.A z-axis piezoelectric actuator switches between the supporting and moving legs by adjusting the height between the two legs.Importantly, the mechanical amplifier mechanism of the actuator also serves as a suspension, providing alternating three-point contacts against rough surfaces.Many of the presently documented precision holonomic mobile robots utilizing piezoelectric actuators rely on "sliding" movements, such as stick-slip [38,41] or sliding feet movements. [55,60]In contrast, the HB has the theoretical capability to walk without slipping.This is achieved by raising and lowering its legs through a specially designed vertical actuator, distinguishing it significantly from other precision mobile robots driven by piezoelectric actuators.The absence of sliding motion enables this robot to navigate heterogeneous frictional surfaces with high repeatability, a feat challenging for other robots.We anticipate substantial enhancements in accuracy and driving performance, particularly during continuous walking.
The XYθ stage consists of four piezoelectric actuators (PZTF, PZTB, PZTL, and PZTR) on each side of the square, four flexure hinges made of polycarbonate, and two duralumin frames arranged diagonally (Figure 2C).The pseudoalternating tripod gait principle was realized by combining the XYθ displacement and alternative switching of the contacting leg (Figure 2D).Initially, with one leg immobilized, the other leg is elevated and precisely positioned in the direction of travel using the XYθ stage.Subsequently, the roles of the fixed and movable legs are alternated.By repeating this sequence, the robot achieves continuous walking, as depicted in Figure 2D.The MTDD08S350F18 (Mechano Transformer) served as the four actuators for the XYθ stage, while the Z-axis actuator employed was the APA120S (CEDRAT Tech.).Refer to Table 2 for detailed specifications of both actuators.
We developed the second prototype HB-2 with tweezers, Z stage, and a flat integrated XYθ stage for demonstrating automatic micromanipulation (Figure 2E).The tweezers consisted of a multilayer piezoelectric actuator, a displacement magnification mechanism made of duralumin, and fingertips made of polyoxymethylene (POM).The maximum gripping distance of the tweezers was ≈1.6 mm, which was sufficient for gripping a single-millimeter to submillimeter object.An adjustment mechanism was also incorporated into the connecting part such that the distance can be changed according to the size of the object to be gripped.The XYθ stage was integrated into a flat structure with the displacement magnification mechanisms of four multilayer piezoelectric actuators.A Z-axis linear stage driven by a stepper motor and feed screw mechanism was mounted on top, enabling the robot to lift a small object grasped by the tweezers.The mechanical model, driving principle, and performance are identical to those of HB-1.

Resolution Experiments
Experiments were conducted to investigate the minimum displacement resolution using a laser Doppler vibrometer (VibroOne; Polytech).We measured the resolution of the robot within a one-step motion and the displacement of the outer leg when the inner leg functioned as the contacting (supporting) leg. Figure 3A shows the X-axis displacement of the moving leg versus time when the voltage was changed in a staircase fashion with a step voltage of 0.0167 V.The figure shows that, on average, a fine movement of 12 nm was achieved.We considered that the bottle limitation of the resolution was determined by the uncertainty of the measurement conditions, rather than the specifications of the actuators and structure.The investigation of the displacement resolution and measurement method will be performed in our future work.Physical Quantity HB USM [55] Inchworm [60] MyCRoN [41] PER-hexpod [38] Principle

Walking Experiments
We investigated the continuous walking characteristics through several experiments.We describe them separately in terms of walking accuracy, driving performance, and applications on typical flat surfaces, such as commonly used office desks, microscopic tables, laboratory tables, plastic boards, and protective display films.

Continuous Walking Accuracy
We measured the trajectories of the mechanical center for translational motions in eight directions (45°intervals).Additionally, we measured the trajectories of the rotational motion.The walking frequencies were set to 1 and 50 Hz.The walking durations were 10 and 1 s at 1 and 50 Hz, respectively.Measurements were performed five times in each direction.The experimental conditions are listed in Table 3.The coefficient of variation (CV) of the travel length L was used to evaluate the accuracy of repetitive motions.
We define azimuthal stability P as the ratio of the average path length L ave to the average azimuth error Δϕ ave .
where φ is the angle of the straight line connecting the origin and the final destination in the translational motions, g Δϕ is the deviation of the azimuth from the ideal.Table 4 shows the average path length L ave , standard deviation σ, and CV for the translational motions in eight directions.Table 5 lists the average rotational angle θ ave , axial runout, and angular velocity for the rotational motion.
Figure 3B shows the trajectories of the translational motions in the eight directions.As we overwrote the five trajectories in the figures, there was a high repeatability at both 1 and 50 Hz.
Figure 3C illustrates the plots of the CV for travel length (L) and azimuth stability (P) across various moving directions.While there was a notable distinction in the distributions of CV at 1 and 50 Hz, a clear trend emerged: CV at 1 Hz was consistently smaller than CV at 50 Hz.This suggests a trade-off between speed and repeatability.We attribute the significantly larger CV at 180°of 50 Hz to the directional specificity of unexpected vibrations caused by backlashes and assembly errors.In contrast, azimuth stability (P) was influenced by both the structural asymmetry of the mechanism and its dynamic properties, as no distinct trend was observed between the distributions of P at 1 and 50 Hz.
Figure 3D shows the plots of the Xand Y-displacements over time for ϕ = 0°at 1 and 50 Hz, respectively.Ideal curves were calculated by substituting the experimental one-step displacements.
At 1 Hz, almost no deviation from the ideal curve was observed.In contrast, at 50 Hz, the experimental value was smaller than the ideal value.The one-step width at 1 and 50 Hz were 0.257 and 0.184 mm, respectively.There are two reasons for the decrease in displacement: 1) the slip of the leg and 2) a returning displacement due to the phase shift between the XYθ stage displacement and Z-axis displacement.
We conducted another experiment to measure the coefficient of static friction between the floor and legs using the inclined method.Consequently, the coefficient of static friction μ 2 = 0.31 AE 0.021.
Figure 3E shows the trajectories of center points O and A, 40 mm away from O in the positive X-axis direction at 1 Hz under open-loop control.The axial run-out was ≈0.05 mm during 4 degrees of rotation.Compensation for the axial run-out error will be the subject of future studies. [61]e conducted an orbital motion experiment to demonstrate that the robot accomplished flexible and smooth holonomic motion.Figure 3F shows sequential photographs captured during one revolution of the orbital motions with rotational radii of 58 and 110 mm, respectively.Experiments were performed at 10 Hz, and the robot was found to operate with good accuracy.(See the attached movie of S1 to S6, Supporting Information).

Driving Performance
In actual environments, flattening a driving surface perfectly so that it would not have any defects is difficult, and the robot must have the ability to run over bumps, slopes, and gaps.
First, we tested the ability to walk over bumps, and as shown in Figure 3G, the robot was able to walk over bumps as small as 0.1 mm; the robot walked over usual obstacles, such as hair and dust, and on the usual flat surfaces of commonly used desks.
Next, we tested the climbable angles.As shown in Figure 3H, the robot could climb a slope with an inclination of 30°.Additionally, we assessed the robot's capacity for bearing weight.As illustrated in Figure 3I, the robot successfully carried a 2 kg load with an average velocity of 1.86 mm s À1 at a 90 degree angle of ϕ when supplied with input voltages of 150 V and a driving frequency of 50 Hz.In further experiments, we verified the robot's ability to achieve translational motion in various directions and rotate around multiple axes, leading us to the conclusion that the robot maintains holonomic mobility even under the load of 2 kg.Finally, we tested a passable ditch.As shown in Figure 3J, the robot could walk over a ditch less than 30 mm wide; moreover, it could walk over most of the cracks and gaps on usual surfaces.We verified that HB-1 could walk without surface angle adjustment over typical surfaces such as office desks, microscopic tables, and laboratory tables.

Application
We used a visual feedback control with machine learning while observing the top and side universal serial bus (USB) cameras (HOZAN L-836) to enable the robot to automatically recognize, pick, and place arbitrarily shaped submillimeter objects.We demonstrated automatic micromanipulation using the HB-2 as a versatile and precise mobile manipulator.
Figure 4A shows a typical demonstration of the automatic recognition and arrangement of number-printed cubic beads of 5 mm in length in ascending order.We confirmed that HB-2 can automatically rearrange small objects using machine learning-based image recognition.
In Figure 4B,C, we demonstrate the automatic arrangement of tiny bricks and chip capacitors in a fan shape to demonstrate the  usefulness of HB-2 as a holonomic mobile manipulator.HB-2 could arrange millimeter-scale objects at different angles using the circular motions around the fingertip of the tweezers.Table 6 and 7 show the success rates of the automatic arrangements and the average working time.The success rate of a single arrangement was 100% over 65 trials.In these experiments, we set the walking speed of the HB-2 to 1.2 mm s À1 with a walking frequency of 5 Hz for accurate manipulation, although the maximum speed was potentially over 10 mm s À1 with a frequency of over 50 Hz.
Figure 4D shows the automatic color recognition and insertion of millimeter-sized cylindrical pipes into the trapezoid cones according to the predesigned color sample.In this experiment, HB-2 was used to perform a dense assembly of millimeter-sized    parts that required delicate manipulations.We verified the feasibility of the dense assembly of millimeter-sized objects.
Figure 4E shows the automatic piling up of two types of chip capacitors with widths of 2.5 mm and 0.5 mm.In these experiments, we confirmed that HB-2 could perform micromanipulation over the XYΘZ axes required for not only 3D implementation of the electronic chip parts and decorations of tiny accessories, but also biomedical processing such as artificial insemination and transgenic animals.
Figure 4F depicts the plots of clamping force versus the tweezer's displacement ratio.To measure the clamping force, we affixed a force sensor (Minebea, MMS101) with a resolution of 0.001 N. The graphs illustrate the relationship between clamping force and the ratio of displacement to the maximum displacement when a 100 V input was applied.It was observed that as the displacement increased, the clamping force decreased.Under zero displacement conditions, the maximum force recorded was 0.117 N, while the maximum displacement reached 0.70 mm under zero force conditions.

Discussion
We propose a new mechanical design for a precise holonomic mobile device driven by five piezoelectric actuators with displacement amplification mechanisms inspired by the rhinoceros beetle.
We succeeded in realizing holonomic motion, high velocity, high load capacity, and high repeatability simultaneously when moving on commonly used flat surfaces such as office desks, laboratory tables, microscopic tables, and sheets of cover glasses.
In experiments, the insect-sized robot achieved velocities of 11.6 mm s À1 for the translational motion and 19.3°s À1 for the rotation motion with CVs of 2.72% and 0.76% at 80 Hz, respectively, and velocities of 0.26 mm s À1 and 0.76°s À1 with CVs of 0.48% and 0.30% at 1 Hz with a precise motion.To demonstrate the feasibility of practical precise mobile manipulation, we mounted the manipulators on the robot and conducted automatic pick-and-place, insertion, and pile-up operations of submillimeter objects under visual feedback control with machine learning for the automatic recognition of the objects.
][75][76][77][78][79][80][81][82][83] The horizontal axis shows their weight, and the vertical axis shows their body speed.Evidently, among living organisms, the heavier the organism, the lower its speed, for arthropods, and the proposed robot follows this trend, with characteristics similar to those of beetles.][86][87][88][89][90][91] The horizontal axis represents the displacement resolution, and the vertical axis represents the body speed.The proposed robot has a moderate body speed and high resolution.
To improve the robot's ability to expand its applications to actual micromanipulation, the following issues must be addressed.To increase the speed, we studied the dynamic locomotion principle and an additional function for reducing the frictional force between the legs and surface.To improve the repeatability of positioning, we reduced the slippage of the legs by improving the input signals of the XYθ and Z-axis actuators.To control a mobile robot more precisely and accurately, we have developed precise multiaxial displacement sensors, [60] motion compensation, [61] and closed-loop control.To expand the versatility of manipulation, we plan to develop various compact manipulators for multiaxial, multiscale, and heterogeneous operations in the electronic, MEMS, biomedical, soft robot,  and microassembly fields.The study of collaborative work organized by multiple robots is one of the most important issues in automatic flexible mobile robotic factories.

Experimental Section
Input Value: The input voltages applied to the piezoelectric actuators were sine waveforms with an amplitude of 150 Vpp (0-150 V).They are expressed as follows.
where V F , V B , V R , V L , and V z are the input voltages; v F , v B , v R , v L , and v z are half amplitudes of input signal voltages; G and G Z are the voltage amplification rates; ω is the angular frequency; τ is the offset phase; and V offset and V Zoffset are the offset voltages.
To execute translational and rotational operations, the input signal voltages must to be regulated as described in a study. [59]F A B A R A L where A F , A B , A R , and A L represent the displacement of each actuator; v is a constant proportionality factor between voltage and displacement; ϕ is the azimuth; r is half the distance between flames, as shown in Figure 2C and 6A; W is the stride length during a half-step motion; θ is the posture angle; and Δθ is the change in posture angle during a half-step motion.The constant of proportionality v is assumed to be approximate, ensuring that the signal voltage and displacement of the piezoelectric actuator maintain a proportional relationship.
Figure 6A shows the displacements of the orthogonal, diagonal, and rotational motions.Table 8 and 9 compare the analytical and experimental displacements.We analyzed the displacement of the XYθ stage when the maximum voltages were applied.
Mechanics: Figure 6B shows the mechanical model of a pseudoalternating tripod gait.These parameters are defined as follows.
x 1 , x 2 : X-direction displacement of each leg  :: m 1 z :: m 2 x :: m 2 z :: Subscript 1 indicates the inner leg, and subscript 2 represents the outer leg.The forced displacement of each actuator is defined as follows.
For simplicity, the input waveform along the X-axis was sinusoidal, while the input waveform along the Z-axis was square.
As the forced displacement in the z-direction has a constant value at every half cycle, it can be divided into half-cycles.
When 0 ≤ t ≤ 1 2 T, the inner leg moved, and the outer leg was fixed to the floor.This model is shown in Figure 6C.
Because the outer leg was fixed, x 2 ¼ z 2 ¼ 0. Assume that the inner leg moves close to the ground (z 1 ¼ 0), and that the friction force acting on the inner leg f 1 ¼ n 1 ¼ 0. Substituting Equation (12a) and (12b) into Equation ( 8) and ( 9), we obtain m 1 x :: From Equation ( 14) Equation ( 17) expresses the displacement of the actuator required to lift a leg.
If ω 1 is defined as the natural frequency of the inner leg, Equation ( 13) can be written as follows.
x :: Solving Equation ( 18) for x 1 ðtÞ when x 1 ð0Þ ¼ A x 2 and ẋ1 ð0Þ ¼ 0 Substituting Equation (19) into Equation ( 15), we have From Equation ( 16) The condition of no slip can be written as follows.
Substituting Equation ( 21) and ( 22) into Equation ( 23) Here, ω s is defined as the maximum angular velocity without slip.Equation (24) indicates that the higher the coefficient of friction μ 2 between the fixed leg and floor, the closer is the maximum drive frequency ω s to the natural frequency ω 1 of the moving leg.
Assuming that there is no decrease in the rigidity of the XYθ stage in the moving direction, k x A x and k z A z can be replaced by the maximum blocked forces of the actuators F x and F z , respectively.
Here, we define f s as the maximum frequency without a slip.Experimental Setup: HB-1: An integrated 2-DoF glass scale was placed on top of the robot as a measurement target.Four optical linear encoders (TA-200, Technohands) with a resolution of 0.1 μm were installed at the bottom of the plate just above the scale.The displacement was measured by reading the engraved line on a scale.
We used a field programmable gate array (FPGA) (LabVIEW cRIO 9049, National Instruments) to generate the control signals to the robot and convert the transistor-transistor logic (TTL) signals from the encoders to the XYθ displacements with a sampling rate of 2.8 MHz.The control signals were magnified 30-fold using an amplifier circuit.Table 10 lists the performances of the measuring instruments. [60]o ensure simultaneous three-point contact with the floor, the scale stand, made of duralumin, weighed approximately 80 g.In addition, among the several floor materials tested, we chose a protective film on the tablet screen cover glass to achieve optimal walking performance.
Experimental Setup: HB-2: Figure 6D shows a photograph of the experimental setup for an automatic pick-and-place by HB-2 with visual feedback control under the monitoring of the top and side USB cameras (HOZAN L-836).FPGA (cRIO-9049) generates the input voltages of HB-2. Figure 6E shows a flowchart of the automatic manipulation during pickup.Figure 6F shows the procedure of automatic manipulation with automatic detection of objects of various shapes by machine learning and measurement of the position and attitude angle of the objects and tips of the tweezes by image analysis.
The measuring resolution of the distance, Δl, was estimated as the standard error of 7 μm when averaging the process for the barycentric coordinates of 1 mm objects with 50 Â 40 mm image size and 1.2 M pixels (1280 Â 960) for image resolution, which was calculated as follows.
Here, σ is the measuring resolution for 1 pixel, and N is number of pixels for a length of an object.
The measuring resolution of the angle, Δθ, was estimated as 2°, which is the acute angle of right triangle with the base of N pixel and the height of 1 pixel.
In this study, YOLOv7 was used as the algorithm for object detection.We prepared sufficient image data by taking 30-50 pictures of each object in various patterns and padding the images by rotating them, changing the contrast, and changing the gamma value.For training, a minibatch learning method was used, in which the entire dataset was divided into small groups.By inputting images of objects into the model trained by machine learning, the mask images of the detected objects were output as inference results.Because the position and attitude information of an object are necessary for automatic alignment, the object position and attitude angle were detected by performing image processing on the output mask image using OpenCV, an image processing library.As in the previous model, LabVIEW was used to control HB-2, and Python was used for machine learning and image analysis.

Figure 1 .
Figure 1.Schematic of a versatile precise mobile robot "HB".A) Photograph of a Hercules beetle (Dynastes hercules) as a concept animal.B) Alternating tripod gait with nonholonomic mobility of beetles.C) Comparison of specifications between HB and rhinoceros beetles (*maximum size of Dynastes hercules.† maximum weight of Megasoma elephas.).D) Photograph of a HB with precise tweezers driven by a piezoelectric actuator and Z-axis positioner driven by a stepper motor.E) Design of ring-shaped outer and inner legs for uniform holonomic mobility.F) Holonomic motion patterns; translation in any direction and rotation around any point on a plane.

Figure 2 .
Figure 2. Design of a prototype of HB.A) Photograph of the first prototype "HB-1".B) Assembly drawing of HB-1.C) Structure of the XYθ stage with four piezoelectric actuators with flexure hinges.D) Principle of "pseudoalternating tripod gait"; alternately switching contact legs by the Z-axis actuator synchronized with XYθ axes actuators).E) Computer aided design model of the second prototype "HB-2", consisting of the XYθ mobile mechanism and a tool with a tweezer and height positioning mechanism.The XYθ mobile mechanism for "HB-2" is similar to HB-1, consisting of a Z-axis piezo actuator for leg lift and an integrated XYθ stage for horizontal holonomic movement.Rubber sheets are attached to the tips of the legs.The height positioning mechanism is driven by a linear-guided stepper motor, and the tweezer mechanism magnifies the displacement of the piezoelectric element to open and close its tweezers with polyoxymethylene (POM) finger tips.

Figure 3 .
Figure 3. Results of versatile mobility of HB.A) Plots of displacements versus time of fine motion with voltage resolutions of 0.0167 V measured by a laser doppler vibrometer (Polytec, VibroOne).B) Five-time overlapping trajectories of translational motion in eight directions for 1 and 50 Hz.C) Plots of repeatability and azimuth stability versus moving direction of ϕ.The repeatability is defined as a ratio of root mean square of standard deviations of final points in X and Y axes to the average travel length.D) Plots of displacements versus time along ϕ = 0°for 1 and 50 Hz.E) Trajectories of mechanical center point of O and reference point of A during the pivot turn for 1 Hz.F) Sequential photograph of orbital motions with radius of 58 mm and 110 mm for 10 Hz.G) Photograph of traveling over 0.1 mm bump.H) Photograph of climbing a 30°-inclined surface.I) Photograph of carrying a wight of 2 kg.J) Photograph of traveling over a 30 mm ditch.

Figure 4 .
Figure 4. Demonstrations of automatic micromanipulations.A) Automatic recognition and arrangement of cubic number beads with 5 mm in length in ascending order B) Circular arrangement of tiny bricks with a length of 8 mm.C) Circular arrangement of 2030 chip capacitors with a length of 3 mm.D) Automatic color recognition and insertion of mm-sized cylindrical pipes into trapezoid cones according to the predesigned color sample.E) Automatic pile up operations for sub-mm chip capacitors using visual feedback control with top and side USB cameras.F) Plots of clamping force versus ratio of displacement to maximum displacement of 0.70 mm for applied voltage of 100 V measured by a six-axis force sensor (Minebea, MMS101).

Figure 5 .
Figure 5.Comparison with animals and holonomic robots.A) Plots of body speed versus weight.Among living organisms, the heavier the organism, the lower its speed tends to be for arthropods, and the proposed robot is within this trend, with characteristics similar to beetles among them.B) Plots of body speed versus resolution for representative holonomic robots.HB potentially has moderate body speed with high accuracy.

z 1 , z 2 :
Z-direction displacement of each leg m 1 , m 2 : Mass of each leg k x : Equivalent spring constant of the XYθ stage k z : Z-axis actuator spring constant d x ðtÞ, d z ðtÞ: Forced displacement of the actuators l x , l z : Natural length of the actuators A x , A z : Displacement amplitudes of the actuators F x , F z : Blocked force.T: Input wave period ω: Input wave frequency n 1 , n 2 : Normal force f 1 , f 2 : Frictional force μ 1 , μ 2 : Coefficient of friction The equations of motion for the inner and outer legs are expressed as follows.

Figure 6 .
Figure 6.Materials and methods.A) Finite element method (FEM) analysis of deformation of the XYθ stage and timing chart during orthogonal, diagonal, and rotational motions.B) Simplified mechanical model of the inner and outer legs.C) Reaction while moving inner leg and fixing outer leg.D) Experimental setup with top and side USB cameras with the measuring resolutions of AE7 micrometer and AE2 degrees for manipulating mm to sub-mm sized objects in experiments.E) Flowchart of automatic manipulation.F) Schematic diagram of the sequence.

Table 1 .
Comparison of the holonomic mobile robots driven by piezoelectric actuators.
Working tableGlass film with protective seal Initial attitude angle of the robot θ ¼ 0

Table 4 .
Experimental results of translational motion.

Table 5 .
Experimental results of rotational motion.

Table 6 .
Summary of automatic pick and place operations for cubes.

Table 7 .
Summary of automatic fan-shaped arrangements for rectangular bricks and chip capacitors.