Vibration Propulsion in Untethered Insect-Scale Robots with Piezoelectric Bimorphs and 3D-Printed Legs

Abstract

This research presents the development and evaluation of a miniature autonomous robot inspired by insect locomotion, capable of bidirectional movement. The robot incorporates two piezoelectric bimorph resonators, 3D-printed legs, an electronic power circuit, and a battery-operated microcontroller. Each piezoelectric motor features ceramic plates measuring 15 × 1.5 × 0.6 ${\mathrm{m}\mathrm{m}}^3$ and weighing 0.1 g, with an optimized electrode layout. The bimorphs vibrate at two flexural modes with resonant frequencies of approximately 70 and 100 kHz. The strategic placement of the 3D-printed legs converts out-of-plane motion into effective forward or backward propulsion, depending on the vibration mode. A differential drive configuration, using the two parallel piezoelectric motors and calibrated excitation signals from the microcontroller, allows for arbitrary path navigation. The fully assembled robot measures 29 × 17 × 18 ${\mathrm{m}\mathrm{m}}^3$ and weighs 7.4 g. The robot was tested on a glass surface, reaching a maximum speed of 70 mm/s and a rotational speed of up to 190 deg./s, with power consumption of 50 mW, a cost of transport of 10, and an estimated continuous operation time of approximately 6.7 h. The robot successfully followed pre-programmed paths, demonstrating its precise control and agility in navigating complex environments, marking a significant advancement in insect-scale autonomous robotics.

1. Introduction

Miniature robots have the potential to transform multiple areas of science and technology by enabling access to confined environments [1]. They could allow engine inspections without disassembly, repair satellite computer boards, assess pollution in confined spaces, collect essential information from uncertain environments or disaster areas, manipulate objects, modify the environment, and perform surgical procedures in hard-to-reach areas of the human body [2]. However, the manufacturing, energy sources, power consumption, all-terrain locomotion, and control of these robots for real-world applications remain substantial obstacles [3].

Insects consistently inspire scientists; notable examples include the inchworm [4], the bee [5], and the cockroach [6]. However, these biological machines outperform their robotic counterparts in aspects such as power conversion, actuation, sensing, speed, and control [7]. While large-scale robots and vehicles have achieved impressive speeds, replicating nature’s efficiency at the insect scale remains challenging. Organism speeds are often measured in body lengths per second (BL/s), ranging from typical speeds to exceptional rates like the 323 BL/s observed in the mite Paratarsotomus macropalpis. Current technology achieves high-speed movement primarily through large-scale machines (BL > 100 mm) and powerful engines, as seen in vehicles like Formula One cars (50 BL/s) and quadrupedal robots reaching 9.1 BL/s. Nevertheless, the development of small-scale robots (1 mm < BL ≤ 100 mm) capable of high speeds remains daunting due to the complexities in miniaturizing traditional high-performance motors and transmission systems [8].

The design of insect-scale robots must account for the electronics, sensors, actuators, power supplies, and tools required for specific tasks. During the initial design phase, designers need to balance size with functionality [9]. Moreover, miniature robots must use fewer actuators with lower energy consumption, often incorporating one actuator per limb or just one or two in a coupled drive [3]. However, a current challenge for millimeter-scale robots (less than 40 mm in length) is that the power sources, such as batteries and capacitors, are not efficient due to their limited specific energy and power ratings. Additionally, these power sources can constitute 30% to 70% of the robot’s volume and total mass [10]. This challenge represents a major bottleneck for the performance of untethered robots [11]. Therefore, low-power electronics are essential to integrate onboard power supplies. Furthermore, low-power electronics still require a high-voltage boost converter circuit to efficiently transform onboard energy into signals that can drive most actuators [12]. The cornerstone of autonomous untethered operation lies in the battery life; for example, the batteries selected in the designs presented in [6,7,8,11,12,13,14] allow continuous operation for periods shorter than an hour. In hazardous environments, frequently replacing the batteries would be impractical. Therefore, advancements in battery technology and energy-efficient components are crucial in extending the operational time and enhancing the practicality of these robots in real-world applications [15].

To address the aforementioned challenges, this work presents the design of an insect-scale robot based on standing wave locomotion [16], using vibration modes produced by the inverse piezoelectric effect. This work builds on the advances presented in [17,18,19]. In the referenced work [19], a unidirectional robotic system was introduced, which operates on the principle of piezoelectric unimorph plates resonating at a singular frequency. The current study advances this concept by introducing several innovative elements. Firstly, we have developed a bidirectional locomotion mechanism that operates based on two sequential resonant modes for each piezoelectric element. This enhancement allows for more versatile movement patterns. Secondly, we have optimized the electrode layout to facilitate the generation of flexural standing waves, thereby improving the locomotion efficiency of the robot. Thirdly, we have designed an interface circuit that is capable of driving these two resonant frequencies, further enhancing the operational flexibility of the robotic system. Lastly, we have implemented new 3D-printed robot legs and supports, which allow for rapid prototyping and customization. These advancements collectively contribute to the development of a more efficient and versatile piezoelectric robotic system.

Among the advantages of standing wave locomotion are its low production cost, suitability for miniaturization, and simpler structures [20]. It requires only a piezoelectric element with uniform poles, some conductive cables, and a single power source. Additionally, it boasts high efficiency (theoretically up to 98%) [21]. The contributions of this work to energy efficiency include (1) an innovative electrode layout design for the efficient actuation of two consecutive vibration modes; (2) the design of support columns and passive legs to minimize vibratory coupling effects that directly decrease the quality of the resonance of the vibration modes; and (3) the retuning and improvement of the piezoelectric drive high-voltage circuit for two different and consecutive frequencies. All these contributions aim to achieve high agility through bidirectional movement and maintain very fast linear movement speeds while retaining the ability to make turns. Specifically, a rapidly moving robot could lose its stability while executing turns due to inertia [14]. Both agility and trajectory manipulation are important characteristics for practical robots. Robots that run fast but lack the ability to avoid obstacles by coordinating rapid translational and rotational movements would not be able to perform even common tasks in complex environments.

The structure of this work is organized as follows. Section 2 details the materials and methods, divided into two subsections. Section 2.1, Device Design, describes the components of the robot, focusing on the operational principles. This subsection provides a comprehensive explanation of each robot component, including the power source, locomotion system, control signals, piezo drive circuit, and 3D-printed structure. Section 2.2, Fabrication, explains the construction techniques employed. Section 3 presents the results, beginning with Section 3.1, Electrical Characterization, which focuses on the resonance peaks of the bimorph piezoelectric plates. Section 3.2, Kinetic Characterization, evaluates the robot’s translational and rotational movements, including the measurement of speeds, distances, and more complex combined motion trajectories. This section also includes a comparative analysis of key performance parameters with similar-sized robots. Finally, Section 4 concludes the work by summarizing the principal findings and proposing avenues for future research.

2. Materials and Methods

2.1. Device Design

A 3D schematic of the fully assembled robot is shown in Figure 1. The parts that compose the robot include a battery, a microcontroller board, an efficient drive circuit, and the locomotion system. The locomotion system consists of two parallel rectangular plates composed of lead zirconate titanate (PZT) piezoelectric bimorphs. Each plate has four legs attached, each with four claws.

The principle of locomotion involves the generation of standing waves (SW) on both bimorph plates, combined with the strategic placement of the legs along the shape of the standing wave [22]. This approach has been successfully applied in previous works [17,23], but, in this case, each bimorph plate was designed as a fully independent SW motor.

For the design of the locomotion system, two important criteria were considered when choosing a pair of vibration flexural modes (two standing waves for bidirectional movement): the first was the frequency and the second was the locations of the external nodes of each mode. The locations of these nodes are crucial in supporting the full weight of the robot because the loss of vibration energy is minimal [24], and higher frequencies result in smaller electronic components. According to [25], we used a two-index naming convention for bending vibration modes: the first index denotes the number of nodal lines along the length of the bimorph plate, and the second indicates those along the width. After studying several pairs of bending modes, such as (20)–(40), (30)–(40), (40)–(50), and finally the pair (50)–(60), the flexural modes (50) and (60) were selected to balance the node proximity while avoiding excessively high modes that could complicate the driving electronics, fabrication, or actuation. Figure 2 shows a 2D representation of the considered mode shapes. This approach ensured that the external nodes were sufficiently close to meet the fabrication tolerances (see the purple lines in Figure 2 that define the semi-node areas $N_{1-2}$), minimizing the vibration losses in both modes. Additionally, modes (50) and (60) are consecutive, implying that the resonance peaks are closely spaced in frequency. This is important to simplify the piezo drive circuit mentioned above.

In Figure 2, two areas of each wave crest (the region between two nodes) are clearly differentiated in both modes. The left side of the wave crest is represented in blue, while the right side is depicted in red. These areas are crucial for standing wave-based locomotion; the legs must be attached to the bimorph in these locations. Placing the legs in the blue zones will produce forward movement, while placing the legs in the red zones will result in backward movement [22]. Therefore, achieving bidirectional movement requires placing the legs in the blue zone for mode (50) and in the red zone for mode (60). It is also valid to place the legs in the red zone for mode (50) and in the blue zone for mode (60). The green lines in Figure 2 represent the first case, where the blue area corresponds to mode (50) and the red area to mode (60), implying that mode (50) moves forward and mode (60) moves backward when tuning the frequency.

Figure 3 shows the detailed design of the locomotion system. The moving part of the system consists of a pair of rectangular bimorph plate resonators with a length of $L_B=15 \mathrm{m}\mathrm{m}$, a width of $W_B=1.5 \mathrm{m}\mathrm{m}$, and a thickness of $T_B=0.6 \mathrm{m}\mathrm{m}$. The other two components are the supporting columns and four passive legs. All dimensions shown in Figure 3 are compiled in

Table 1. Dimensions of the components of the locomotion system.

Component	Dimensions (mm)
Supporting columns	$L_C=2.8$$, D_c=0.6$
Bimorph resonator	$L_B=15$$, W_B=1.5$$, T_B=0.6$
Leg	$L_{TL}=1$$, T_p=0.6$$, L_p=0.5$$, L_{mc}=0.5$$, D_{mc}=0.1$

.

The dimensions of both the supporting columns and the passive legs were designed considering their first resonance frequency, which should be much higher than the frequencies of modes (50) and (60) of the bimorph resonator to avoid bimorph–leg coupling effects [18].

Another crucial consideration was the leg stiffness. When the leg establishes contact with the surface, it experiences a compressive force due to the weight of the robot, as well as a shear force due to the frictional force. However, it must remain perpendicular to the contour of the standing wave (without bending deformation). If the legs are stiff enough, they will stay perpendicular. This was essential to ensure that the tip of the leg followed a very narrow elliptical trajectory for effective locomotion. The physical properties of the chosen 3D printing material, including its Young’s modulus, will be detailed in the next section.

Once the specific vibration modes were selected, the subsequent step involved achieving the efficient actuation of these modes, where the layout of the electrodes played a crucial role in this part of the design. The effective performance of a vibration mode depends on the distribution of the acting forces on the bimorph, specifically on the areas where the electric field is applied through the electrodes. According to [26], optimizing the performance of a vibration mode requires maximizing its displacement, which is equivalent to maximizing the collected charge. Considering the plane stress approximation and the fact that the piezoelectric coefficients satisfy $d_{31}=d_{32}$ in PZT (the piezoelectric material of the bimorph), the total charge collected $q$ by an electrode over the bimorph surface can be expressed by

$$q\approx d_{31}{\iint }_{{\mathsf{\Omega }}_e}\left(S_{xx}+S_{yy}\right)dxdy,{\mathsf{\Omega }}_e\subset \mathsf{\Omega }={\mathsf{\Omega }}_{+}+{\mathsf{\Omega }}_{-},$$

(1)

where $S_{xx}$ and $S_{yy}$ are the components of the stress tensor, and ${\mathsf{\Omega }}_e$ is the total surface area of the electrode. The region $\mathsf{\Omega }$, where the charge collected is the maximum, can be split into two subregions, ${\mathsf{\Omega }}_{+}$ and ${\mathsf{\Omega }}_{-}$, defined as the areas within $\mathsf{\Omega }$ where the function inside the integral is positive and negative, respectively. To identify these regions, stress and strain simulations for the (50) and (60) vibration modes of the bimorph were conducted using COMSOL Multiphysics 5.5. In Figure 4, subregions ${\mathsf{\Omega }}_{+}$ and ${\mathsf{\Omega }}_{-}$ for each vibration mode are depicted, with subregion ${\mathsf{\Omega }}_{+}$ in red and subregion ${\mathsf{\Omega }}_{-}$ in blue.

To reach an optimized response and filter out unwanted modes, subregions ${\mathsf{\Omega }}_{+}$ and ${\mathsf{\Omega }}_{-}$, shared by both modes, were selected for the final electrode layout. These common subregions are depicted in Figure 5, where the subscripts + and − indicate the direction of the electric field in the common subregions of the bimorph.

This electrode layout ensured the effective operation of both modes. The following section shows in detail the layout of the electrodes and their manufacturing.

This design performs four types of motion: forward (FW) and backward (BW) bidirectional translation, as well as clockwise (CW) and counterclockwise (CCW) bidirectional rotation. Figure 6 illustrates these four motions schematically. Each bimorph can be operated in either the (50) or (60) mode by applying periodic signals at the corresponding resonance frequencies. By combining different modes on each of the two bimorph plates, four distinct motions can be achieved through differential drive [18]. Typically, one of the two modes moves faster than the other. Therefore, for forward and backward movements, different frequencies were applied to achieve better balance in both directions. For clockwise and counterclockwise movements, equal frequencies were applied.

For the robot to follow a specific path, it is crucial to effectively combine the four different types of motion. Achieving these motions depends on the microcontroller’s ability to generate periodic signals tuned to the required frequencies for the vibration modes. While most microcontrollers can generate low-voltage pulse-width modulation (PWM) signals for the control of DC motors, generating sufficient force and displacement in piezoelectric actuators requires high voltages. Consequently, the microcontroller board must include an interface circuit to convert and boost the voltages of the drive signals, particularly in systems operating from low-voltage power sources like batteries. Additionally, this interface circuit should be capable of recovering unused capacitive electrical energy from the bimorph resonators to reduce the power consumption [27].

Figure 7a shows the schematic of the high-voltage piezo drive, an idea adapted from reference [28]. This simple circuit consists of a resistor, a transistor, a diode, and an inductor. In simple terms, the transistor functions as a switch controlled by signals from the microcontroller. When the switch is closed (i.e., $V_{CE}=0$), the inductor stores energy in the form of a magnetic field, and the voltage across its terminals is zero. When the switch is abruptly opened (i.e., $I_C=0$), the stored energy is released, causing the voltage across the inductor’s terminals to rise significantly. The resistor and diode serve as protective elements for the transistor, preventing damage from high currents and voltages. Additionally, the inductance value must be selected so that the resonance frequency of the LC circuit (where C represents the electrical capacitance of the bimorph) is twice the resonance frequency of the mode of interest, matching with the second harmonic of the vibration mode’s mechanical frequency.

This circuit (Figure 7a) was successfully applied in [19]. However, the challenge in the case of bidirectional movement was working with two different frequencies, which implied having two circuits for each frequency. Since we also had two bimorph resonators, this resulted in a total of four circuits. One way to minimize this problem was to select consecutive vibration modes to ensure a minimal frequency difference, allowing the inductor to work with both frequencies if they were not too far apart. Therefore, for the particular case of modes (50) and (60), the inductor must be designed to resonate at twice the frequency of the highest mode, $f_{60}$. This condition ensured that it would also resonate correctly for $f_{50}$. Additionally, tuning the duty cycle to eliminate current spikes at turn-on time increases the electrical efficiency for both modes. For our design, the duty cycle was adjusted to 64% to maximize the voltage in both modes (see Figure 7b).

Finally, to regulate the amount of energy delivered to the locomotion system, two burst-type signals were programmed on the microcontroller, one for each bimorph plate. A burst signal consists of a finite number of square signal cycles, where we control the signal supply time ($T_{on}$), the turn-off time ($T_{off}$), and the period of the burst signal ($T_b$), similar to a pulse width modulation signal (PWM). Figure 8 shows this type of signal for forward motion. By tuning $T_{on}$, we controlled the energy supplied to the bimorph plate to correct the trajectory of the robot. By tuning $T_{off}$ and $T_b$, we controlled the speed of the robot [19].

2.2. Robot Fabrication

The main element of the robot propulsion system was the piezoelectric bimorph actuator. We chose a commercial piezoelectric bimorph element (LFS Piezo Bimorph Vibration Sensor, RS PRO, Japan [29]) as the resonator plate. In a previous publication, we used thin unimorph plates vibrating in the extensional mode and capable only of unidirectional motion [19]. The bimorph structure exhibited greater mechanical strength and reliability due to the inclusion of a metallic sheet (called a metal shim or elastic shim), which is sandwiched between the two piezoceramic plates (see Figure 9). By using this metallic shim, the structure can be preserved even if the piezoelectric ceramic fractures [30].

To obtain the efficient resonance of modes (50) and (60), it was necessary to modify the electrodes of the bimorph according to Figure 5. The electrodes were separated into eight regions by mechanical engraving. These regions can be classified into three types, (1) regions with a positive electric field ($E_{+}$), (2) regions with a negative electric field ($E_{-}$), and (3) regions without an electric field applied, as shown in Figure 10.

Following the design guide in Figure 5, we made the electrical connections of the common regions using 20-µm-thick coated copper wire, as shown in Figure 10. This design reduced a four-region electrode to only two wires, avoiding the need for additional wires and voltage signal inverters to reverse the electric field.

Next, the dimensions of the supporting legs and columns were designed to move their first resonance frequency as far as possible from the (50) and (60) modes of the bimorph resonator. Additionally, a 3D printing material with very high rigidity was chosen to further increase its resonance frequency and avoid strain due to the weight of the robot. Therefore, a highly glass-filled resin (Formlabs Rigid 10K resin [31]) was chosen as the 3D printing material. In addition, locomotion through vibration modes requires passive legs, which means that they must be rigid and hard. These characteristics are more suited to claws. In nature, claws are among the most common attachment structures found on the feet of climbing mammals, birds, lizards, and arthropods [32]. Therefore, a novel approach was adopted to enhance the attachment: where there was originally a single leg, it was replaced with a structure composed of four miniature legs or claws. Figure 11 shows the new 3D-printed leg design. This design also included sufficient thickness for the mini claws to minimize the risk of fracture. All dimensions are compiled in Table 1 and Figure 3. The legs and supporting columns were glued to the modified bimorph plate with cyanoacrylate adhesive (Loctite, Düsseldorf, Germany), following the criteria described in the design section (see Figure 3).

Moreover, a 3D-printed supporting structure was designed with two purposes: first, to distribute the weight of the robot in a balanced manner; second, to accommodate the rest of the robot hardware in a compact scheme. The robot components to be accommodated included a 3.7 V, 90 mAh Li-ion battery (LIR1654 from RS PRO [33]) as the DC power supply, the Trinket microcontroller board (3V Trinket from Adafruit, New York City, NY, USA), and an auxiliary PCB featuring the high-voltage piezo drive circuit (see Figure 7). The final fully assembled robot can be seen in Figure 12. Additionally,

Table 2. Mass distribution of the robot.

Component	Mass (g)
3D-printed legs and supporting columns	0.008
Wires and welding	0.026
Pin header connectors	0.076
Two bimorph plates	0.210
Electrical contact for battery	0.596
High-voltage piezo drive PCB	0.916
3D-printed supporting platform	1.121
Microcontroller	1.197
Battery	3.270
Total Mass	7.42

shows the masses of the parts of the robot, and the total mass of the system was 7.42 g. Observe that the locomotion system is capable of carrying 33 times its weight.

3. Results

3.1. Electrical Characterization

The electrical characteristics of the resonators were assessed using impedance spectrum analysis. An Agilent 4294A impedance analyzer, Keysight Technologies, Santa Rosa, CA, USA was employed to measure the conductance across various vibration modes for each of the bimorph plate resonators. Key electrical parameters, including the quality factor, resonant frequency, and peak conductance, were determined by fitting the impedance data to a modified Butterworth–Van Dyke equivalent circuit model [34]. Figure 5 and Figure 10 show the four common regions for the efficient performance of modes (50) and (60). To determine the contribution of each region, they were connected one by one to observe how the resonance peaks of modes (50) and (60) improved with each new electrical connection. Figure 13 and

Table 3. Q-factor and ΔG determined by fitting the impedance data shown in Figure 13 to an adapted Butterworth–Van Dyke equivalent circuit model.

	Resonant Mode
	Mode (50)			Mode (60)
Electrode Layout	Frequency (kHz)	Q-Factor	ΔG (µS)	Frequency (kHz)	Q-Factor	ΔG (µS)
${\mathsf{\Omega }}_{1+}$	69.6	69	76	100.6	60	68
${\mathsf{\Omega }}_{1+}\cup {\mathsf{\Omega }}_{2-}$	68.8	65	216	99.6	58	159
${\mathsf{\Omega }}_{1+}\cup {\mathsf{\Omega }}_{2-}\cup {\mathsf{\Omega }}_{3+}$	68.7	67	299	99.5	61	245
${\mathsf{\Omega }}_{1+}\cup {\mathsf{\Omega }}_{2-}\cup {\mathsf{\Omega }}_{3+}\cup {\mathsf{\Omega }}_{4-}$	68.5	65	327	99.2	59	273

show the enhancement process of the resonance peaks of modes (50) and (60). Note how this electrode layout design process also eliminates unwanted resonance peaks.

However, the final resonance peaks in Figure 13 were slightly modified when the robot was completely assembled. There was a decrease in the peak height due to the weight of the robot, and an unwanted peak also appeared near mode (60). Figure 14 shows the final resonance peaks of each bimorph when the robot was fully assembled. The resulting quality peak (Q) and peak height (ΔG) are also shown in

Table 4. Calculated Q-factor and ΔG of the bimorph when the robot is fully assembled.

	Resonant Mode
	Mode (50)			Mode (60)
Bimorph	Frequency (kHz)	Q-Factor	ΔG (µS)	Frequency (kHz)	Q-Factor	ΔG (µS)
1	69.7	58	279	99.8	56	203
2	69.2	54	255	97.4	60	239

.

3.2. Kinetic Characterization

The kinetic performance of the robot was recorded using a digital camera (Logitech StreamCam, Lausanne, Switzerland). This camera covered a 29 × 21 ${\mathrm{c}\mathrm{m}}^2$ area, with a spatial resolution of 200 microns at 60 frames per second (fps). The videos were then processed with MATLAB (version R2023b, application DLTdv digitizing tool [35]), which tracked a pattern of five points on a white piece of paper attached to the robot. The information obtained from these points allowed us to calculate both the speed and orientation of the robot. To control the robot, burst-type signals were programmed (see Figure 8) into a Trinket microcontroller board (3V Trinket from Adafruit, New York City, NY, USA). The control approach was open-loop, meaning that pre-programmed trajectories were used without any orientation or position sensors.

Then, the first important step was the fine-tuning of the driving signal frequencies, implemented in the microcontroller, at which the bimorph moves at maximum speed. It should be noted that when a single bimorph is activated, the robot executes a rotational movement. To determine the speed, we measured the arc length (distance) traced by the point closest to the active bimorph and divided it by the elapsed time. To accomplish this, a frequency sweep was programmed for each bimorph and each mode, and its speed was measured to ensure that the robot operated at maximum speed. Figure 15 shows the graphs of the distance traveled versus time for each bimorph and each mode. The slopes of the lines on each graph represent the average speed of the bimorph at the programmed frequency; the maximum slope corresponds to the maximum speed. Video S1, included in the Supplementary Materials, shows the movement curve for the (50) mode from bimorph 2 in Figure 15.

Table 5. Working frequencies of the bimorphs obtained for maximum speed.

Bimorph 1				Bimorph 2
Mode (50)		Mode (60)		Mode (50)		Mode (60)
Frequency kHz	Speed mm/s	Frequency kHz	Speed mm/s	Frequency kHz	Speed mm/s	Frequency kHz	Speed mm/s
69.6	5.2	100.9	23.2	67.7	31.1	98.5	37.1

shows the maximum speeds of both bimorphs in each vibration mode. As expected, one mode exhibits a higher speed than the other; in this case, for both bimorphs, $v_{mode \left(60\right)}>v_{mode \left(50\right)}$. However, the $v_{mode \left(50\right)}$ of bimorph 2 is significantly higher than that of bimorph 1. This large difference may be due to several factors, including the lack of perfect symmetry between the two bimorphs. For instance, the 3D-printed joining structures differ, and while the bimorphs are parallel, they are positioned at 180° relative to each other. Additionally, manufacturing errors, particularly in the manual aspects of the process, could contribute to this discrepancy.

Once the robot’s working frequency was determined (see Table 5), we began with controlled movement. Previously, in the design section, the two types of movements that the robot could perform (rotation and translation) were shown. First, the characterization of the rotary movement was carried out. To record the rotational movement of the robot, a longitudinal central axis was defined using three points. Additionally, an important objective of this research was achieving bidirectionality in both rotation and translation. Therefore, the robot was programmed to execute a clockwise rotation (0 to 1.8 s) immediately followed by a counterclockwise rotation (1.8 to 4 s), resulting in 339-degree and 408-degree rotations, respectively. Figure 16 shows, in detail, the trajectories of the three points representing the longitudinal axis of the robot for both clockwise and counterclockwise movements. Additionally, Figure 17 shows the frames of the longitudinal axis at the times indicated in Figure 16.

As shown in Figure 16, the robot did not execute a purely rotary movement; however, the translational deviation was small. Using the information from Figure 16, the angular velocity and the deviation from the center point were calculated and presented in

Table 6. Parameters for bidirectional rotational movement.

	Clockwise Rotation	Counterclockwise Rotation
Angular velocity	${\omega }_{CW}=186 {\displaystyle \frac{\mathrm{d}\mathrm{e}\mathrm{g}.}{\mathrm{s}}}$	${\omega }_{CCW}=190 {\displaystyle \frac{\mathrm{d}\mathrm{e}\mathrm{g}}{\mathrm{s}}}$
Positional deviationper rotation	$\mathsf{\Delta }x=2.5 \mathrm{m}\mathrm{m}$ $\mathsf{\Delta }y=-0.7 \mathrm{m}\mathrm{m}$	$\mathsf{\Delta }x=1.6 \mathrm{m}\mathrm{m}$ $\mathsf{\Delta }y=1.6 \mathrm{m}\mathrm{m}$
Angle	${\theta }_{CW}=339°$	${\theta }_{CCW}=408°$

. Video S2, included in the Supplementary Materials, shows the movement of Figure 17.

In the case of the rotational movement, it was not necessary to reduce the speed of the bimorphs (i.e., $T_{off}=0$) through the scheme presented in Figure 8 to obtain the desired trajectory. In contrast, for the translational movement, it was necessary to reduce the speed and balance the force (varying $T_{off}$) in each bimorph resonator to achieve good forward and backward movement. The first experiments on translational movement without force balance in the bimorphs resulted in very large deviations and strayed far from following a straight line, but the speeds achieved were up to 70.2 mm/s (4 BL/s). Burst-type control signals were programmed to compensate for trajectory deviations (see Figure 8). These signals work in a similar way to the traditional pulse width modulation (PWM) signals used to control direct current motors. Figure 18 shows the trajectory of the robot following a straight-line movement both forward and backward. The robot was also programmed to execute bidirectional movement, changing from forward to backward instantly.

Table 7. Parameters for bidirectional straight-line movement.

	Forward	Backward
Speed	$v_{fwd}=56.1{\displaystyle \frac{\mathrm{m}\mathrm{m}}{\mathrm{s}}}$	$v_{bwd}=59.7{\displaystyle \frac{\mathrm{m}\mathrm{m}}{\mathrm{s}}}$
Deviation	$\mathsf{\Delta }x=0.3 \mathrm{m}\mathrm{m}$	$\mathsf{\Delta }x=6 \mathrm{m}\mathrm{m}$
Distance	67.4 mm	74.6 mm

shows the speed of the robot in both forward and backward movements. Additionally, Video S3, included in the Supplementary Materials, depicts the trajectory shown in Figure 18.

Once the four movements (forward, backward, clockwise, and counterclockwise) were characterized, a more complex trajectory was programmed into the robot. It was required to follow a bidirectional L-shaped path that started and returned at the same point. Figure 19 depicts the bidirectional L-shaped trajectory, and

Table 8. Speed and deviation of the robot following a pre-programmed complex trajectory.

	Complex L-Shaped Trajectory
Speed	$v_{traj}=50{\displaystyle \frac{\mathrm{m}\mathrm{m}}{\mathrm{s}}}$
Deviation	$\mathsf{\Delta }x=-6 \mathrm{m}\mathrm{m}$ $\mathsf{\Delta }y=-3 \mathrm{m}\mathrm{m}$
Distance	323 mm

details the speed and deviation from the starting point. Video S4, included in the Supplementary Materials, shows the movement of the robot as depicted in Figure 19. One observation was that the speed decreased from 70 to 50 mm/s for a more complex and controlled open-loop movement. If the control of the robot improves, it is very likely that the robot will maintain a speed of 70 mm/s, and the deviation will also decrease.

Another control test involved programming a trajectory with the sequence straight line–deviation–straight line to clearly observe the changes in the robot’s path and its ability to redirect its course. Figure 20 shows the trajectory, clearly illustrating the changes in the robot’s direction. Video S5, included in the Supplementary Materials, shows the movement of the robot as depicted in Figure 20.

3.3. Power Consumption, Autonomy, and Cost of Transport

To calculate the power consumption, the RMS current was measured while the robot was in maximum-speed motion. The measured current was 13.3 mA, and the battery voltage was 3.8 V. Based on these measurements, the power consumption was calculated to be 50.5 mW. The Li-ion battery used has a capacity of 90 mAh. With this information, the calculated autonomy of the robot in motion is approximately 6.7 h.

Finally, energy efficiency in legged robots is crucial for their design and autonomous operation. Unlike conventional power-based efficiency metrics, it is best evaluated using the cost of transport (COT), a dimensionless metric that describes the specific energy economy of locomotion across varied terrains [36] The value of the COT is determined by the power consumption (P), mass (m), gravity (g), and speed of the robot (v).

$$\mathrm{C}\mathrm{O}\mathrm{T}={\displaystyle \frac{\mathrm{P}}{\mathrm{m}\mathrm{g}\mathrm{v}}}$$

(2)

The cost of transport (COT) of this robot on glass was as low as 10 (the robot performed all tests on glass).

Table 9. Performance comparison with other untethered insect-scale microrobots with a body length of less than 50 mm.

Microrobot Description	Size (mm)	Total Mass (g)	Speed (BL/s)	Cost of Transport	Power Consumption (mW)	Autonomy (min)
This work	17	7.42	4.1	10	50.5	${406}_{\mathrm{e}}$
BHMbot [13]	20	1.76	17.5	304	1770	$3_{\mathrm{m}}$
HARM-F [6]	45	2.8	3.8	84	600	${4.5}_{\mathrm{m}}$
DEAnsect [11]	40	1	0.3	1670	188	${14}_{\mathrm{e}}$
S²worm [12]	41	4.34	6.7	52	610.5	${13}_{\mathrm{e}}$
PVDF robot [14]	24	1.9	1.2	887	397	${19}_{\mathrm{m}}$
RoBeetle [7]	15	0.088	0.05	-	-	-
SEMR UR1 [8]	20	2.2	2.1	-	638	${4.5}_{\mathrm{e}}$

m: measured time; e: estimated time.

summarizes the aforementioned performance indicators and provides a performance comparison of recent publications on insect-scale robots.

4. Conclusions

This work has demonstrated a miniature autonomous robot inspired by insect locomotion, capable of bidirectional movement based on propulsion using the consecutive vibration modes of a bimorph piezoelectric structure with passive legs, significantly reducing the power consumption and thereby increasing the battery lifetime in insect-scale untethered robots. The results showed power consumption as low as 50 mW and an estimated continuous operation time of 6.7 h.

These findings suggest that (1) achieving the efficient actuation of two consecutive vibration modes using the electrode design proposed in this work, (2) minimizing the vibrational coupling effects of the passive legs and support columns, and (3) an efficient piezoelectric high-voltage drive circuit have important implications for the performance of insect-scale untethered robots, such as agile bidirectional motion in both translation and rotation.

However, this study has some limitations, particularly regarding the surfaces on which this robot can move. The robot was tested on glass and paper, but it could only move effectively on glass. These limitations should be taken into account when interpreting the results and can serve as a starting point for future research. We recommend that future research explore other vibration modes and different materials for the passive legs and support columns, which could yield better results.