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Abstract

Piezoelectrically actuated flapping-wing micro air vehicle (MAV) is an emerging technology. This paper discusses limitations of presently used voltage drivers for the piezoelectric actuators used in such applications and suggests an active filter-based unipolar high-voltage driver. A Chebyshev polynomial has been used to extract a unipolar sinusoidal high voltage signal from a unipolar square wave signal for driving the piezoelectric actuators at low frequency in flapping-wing MAV applications. The filter is based on Sallen–Key topology and is suitable for connecting a piezo-bimorph in series or parallel electrical connection. We have considered an application-specific piezoelectric bimorph actuator of 40 Hz flapping frequency and 32 nF capacitance in each piezoelectric layer to design a driver. The driver circuit in verified by simulation results of OrCAD® software. The driver circuit can control the frequency of the sinusoidal driving voltage, which causes wing-flapping, by radio control communication.

Keywords

Piezoelectric driver circuit active filter Chebyshev filter Sallen–Key topology piezoelectric actuator flapping wing micro air vehicle

Introduction

Flapping-wing micro air vehicles (MAVs) have attracted the attention of many research groups. Singh and Chopra¹ proposed hover capable flapping-wing MAVs. They have pointed out that recent advances in microelectromechanical systems have led to the development of miniature cameras, infrared sensors, and chip size hazardous substance detectors. MAVs can act as platforms for these micro-sensor systems. Typically, flapping-wing MAVs seek inspiration from biological systems such as insects or birds. Jung et al.² considered a bird-mimicking flapping wing. A simplified driving mechanism was used to implement the flapping motion in the vertical plane. The hawkmoth, Manduca sexta, has inspired several researchers in this area.^3–5 DeLuca et al.⁴ considered piezoelectrically actuated, biomimetically designed Manduca sexta type wings, at various amplitudes and flapping frequencies. The Lead Zirconate Titanate (PZT) power consumption studies confirmed that the flapping system needs less power at resonance compared to at off-resonance frequencies.

MAV, driven by piezoelectric actuators, has given a new direction to micro-robotic technology. Several researchers have investigated piezoelectric actuator-driven flapping-wing mechanisms.^6–9 These researchers used piezoelectric bending actuators and emphasized the need for high tip deflection, which is challenging to produce due to low electromechanical coupling coefficient of piezoelectric materials. A high electric field is an elegant approach to increase the tip deflection in such actuators. Other approaches involve mechanical amplification mechanisms. However, in contrast to the considerable research on structural and aerodynamic aspects of flapping-wing MAVs,^10,11 the electrical aspects have received less attentions.

Due to the polarity of piezoelectric materials, a piezo-bimorph should to be driven by a unipolar signal at high electric field.¹² In flapping-wing MAV applications, the electric field inside a piezoelectric layer is typically increased by increasing the voltage across it, instead of reducing its thickness. Average wing flapping frequency for a butterfly is less than 20 Hz,¹³ for a hawkmoth is around 25 Hz,¹⁴ and for a dragonfly is around 27 Hz.¹⁵ Naturally, in a bio-mimicked flapping-wing MAV of similar scale, a piezoelectric actuator is driven by a unipolar sinusoidal high electric field of very low frequency. In literature, we find mainly two methods to produce such a unipolar sinusoidal high electric field: a direct amplifier-based method and a switching amplifier-based method. For instance, Main et al.,¹⁶ Wang et al.,¹⁷ and Wallenhauer et al.¹⁸ used direct amplifiers to produce a unipolar high-voltage signal for driving piezoelectric actuators. On the other hand, Janocha and Stiebel¹⁹ reported a complex programmable logic-driven inductor-based high-frequency switching amplifier to produce such type of driving voltage. The switching amplifier-based method was also reported by Wurtz et al.²⁰ and Karpelson et al.²¹ to drive flapping-wings. The direct amplifier-based method consumes a lot of power for the biasing circuit, which is used to maintain a unipolar signal. On the other hand, in switching amplifier-based method, dc voltage samples of very high frequency are integrated to produce a unipolar high-voltage sinusoidal signal of low frequency. The dc samples are processed through a $L C$ circuit, controlled by a complex programmable logic device, Read Only Memory (ROM) and Analog to Digital Converter (ADC) to produce a unipolar sinusoidal high-voltage signal of low frequency. The switching amplifier-based method requires many additional components, such as an inductor, complex programmable logic circuits, timer circuits, ROM, and ADC. This technique also introduces considerable switching loss during high-frequency operation.

In this paper, we propose an active low-pass filter-based voltage driver to produce a unipolar sinusoidal high-voltage signal, which can drive piezoelectric actuators in flapping-wing MAV. A Chebyshev polynomial is used to get a sharp transition near the cut-off frequency to reject higher-order harmonics. The present design reduces the number of circuit components, design complexity, cost, discards biasing circuits, inductor-based circuits, and high-frequency operations, which are present in earlier drivers. The present driver circuit is synthesized by using Sallen–Key topology considering the specifications of a piezoelectric actuator. The design is verified by the OrCAD® simulation results. We have also discussed compensation technique to attain the specifications of piezoelectric actuators. The driving circuit also provides compatibility with radio control communication to adjust the frequency of the sinusoidal driving voltage, which is used to flap the wings of a MAV.

Background

Figure 1(a) represents a piezoelectric actuator-driven flapping-wing MAV. To produce a large tip deflection, a unipolar high electric field is applied across the piezo-actuator. Figure 1(b) shows a piezoelectric element. The Van Dyke circuit model of a piezoelectric element, adopted by the IEEE Standard on Piezoelectricity,²² is a well-known equivalent circuit. Figure 1(c) shows a modified Van Dyke circuit model of a piezoelectric element reported by Guan and Lao.²³ In Figure 1(c), C_V is modeled for the primary capacitance, resistance R_P is modeled for the dielectric loss, resistance R_S is modeled for the hysteresis loss, and each resonant branch corresponds to each mechanical vibration mode of a piezoelectric element. In each branch, $L_{i}$ , R_i, and C_i represent the equivalent mass, damping coefficient, and spring constant of respective vibration mode. In a flapping-wing MAV design (Figure 1(a)), a piezoelectric actuator is driven below its first resonant frequency. Neglecting dielectric loss R_p and hysteresis loss R_s, the equivalent circuit becomes a capacitor, which is equal to C_V (Figure 1(d)). As reported in earlier literature, Figure 2(a) shows a unipolar sinusoidal high-voltage signal produced by a direct amplifier driver, and Figure 2(b) shows a unipolar sinusoidal high-voltage signal produced by a switching amplifier-based driver. Complex programmable logic-driven switching amplifier-based voltage driver¹⁹ uses complex programmable logic circuit, ADC, ROM, $L C$ circuit, and high-frequency operation circuit to generate a low-frequency unipolar sinusoidal voltage (Figure 2(b)). In addition this, difficulty to fabricate inductors using Very Large Scale Integration (VLSI) technology, increase in mass for using discrete metal inductors, and susceptibility of inductor to an external magnetic field introduce additional limitations to this type of existing drivers. Overcoming the limitations of existing drivers and focusing on the requirements of a flapping-wing MAV, Figure 2(c) shows the technique of the present driver to produce a unipolar sinusoidal high-voltage signal. The following section contains analysis and synthesis of a Chebyshev active low-pass filter for driving a piezoelectric actuator in flapping-wing MAV application.

Figure 1.

(a) Piezoelectric actuator-driven flapping-wing MAV, (b) a piezoelectric element, (c) an equivalent electrical circuit model of the piezoelectric element, and (d) an effective electrical circuit model of the piezoelectric element for flapping-wing MAV application.

Figure 2.

Generating a high unipolar sinusoidal voltage using (a) a direct amplifier, (b) a switching amplifier, and (c) a low-pass filter.

Chebyshev active low-pass filter to drive piezoelectric actuators

Electrical capacitance of a piezoelectric flapping actuator

A piezoelectric bimorph actuator, operated at a high electric field in flapping-wing MAV application, is discussed by Chattaraj and Ganguli.^24,25 Figure 3 shows such a piezoelectric bimorph actuator connected in a series and parallel electrical connection.²⁵ For such a piezoelectric actuator, we find the effective dielectric constant in series electrical connection as²⁵

ɛ_{eq}^{s} = ɛ_{33}^{σ} (1 - κ_{31}^{2} (1 - \frac{6 c_{11}^{E} t_{p} {(t_{m} + t_{p})}^{2}}{{EI}_{eq}^{L}} - \frac{3 c_{11}^{E} {(t_{m} + t_{p})}^{2} d_{331} | V |}{2 d_{31} {EI}_{eq}^{L}} + \frac{d_{331} | V |}{4 t_{p} d_{31}}) + \frac{ɛ_{333}^{σ} | V |}{4 t_{p} ɛ_{33}^{σ}})

(1)

and in parallel electrical connection as²⁵

ɛ_{eq}^{p} = ɛ_{33}^{σ} (1 - κ_{31}^{2} (1 - \frac{6 c_{11}^{E} t_{p} {(t_{m} + t_{p})}^{2}}{{EI}_{eq}^{L}} - \frac{3 c_{11}^{E} {(t_{m} + t_{p})}^{2} d_{331} | V |}{d_{31} {EI}_{eq}^{L}} + \frac{d_{331} | V |}{2 t_{p} d_{31}}) + \frac{ɛ_{333}^{σ} | V |}{2 t_{p} ɛ_{33}^{σ}})

(2)

Figure 3.

Piezoelectric bimorph bending actuator for flapping-wing MAV application in (a) series/antiparallel connection and (b) parallel connection.

Here, $ɛ_{33}^{σ}$ is the dielectric constant along z direction under constant stress ( $σ_{1} = 0$ ), $c_{11}^{E}$ is the linear elastic stiffness coefficient of the piezoelectric material along x direction under constant electric field ( $E_{3} = 0$ ), d₃₁ is the piezoelectric strain constant, $κ_{31}^{2} = \frac{c_{11}^{E} d_{31}^{2}}{ɛ_{33}^{σ}}$ , which is the electromechanical coupling coefficient of the piezoelectric element, t_p is thickness of each piezoelectric layer, t_m is thickness of the support layer, $c_{11}^{m}$ is the linear elastic stiffness coefficient of the support layer along x direction,

{EI}_{eq}^{L} = (c_{11}^{m} t_{m}^{3} + c_{11}^{E} (6 t_{m}^{2} t_{p} + 12 t_{m} t_{p}^{2} + 8 t_{p}^{3})),

d_331, and

ɛ_{333}^{σ}

are the nonlinearity constants, and V represents electric voltage. In series electrical connection, the electric field is

E_{3} = - \frac{V}{2 t_{p}}

inside each piezoelectric element. In parallel electrical connection, electric field in top piezoelectric element is

E_{3} = - \frac{V}{t_{p}}

and in bottom piezoelectric element is

E_{3} = \frac{V}{t_{p}}

. Equivalent capacitance of the piezoelectric bimorph actuator in series electrical connection is

C_{V}^{s_L} = \frac{ɛ_{eq}^{s} L B_{L} (γ + 1)}{4 t_{p}}

and in parallel electrical connection is

C_{V}^{p_L} = \frac{ɛ_{eq}^{p} L B_{L} (γ + 1)}{t_{p}}

. Total mass of the actuator is

m_{a} = \frac{1}{2} L B_{L} (1 + γ) (ρ_{m} t_{m} + 2 ρ_{p} t_{p}) + L_{ext} B_{L} ρ_{m} (t_{m} + 2 t_{p}) .

As shown in Figure 3, L denotes length of each piezoelectric element, B₀ and B_L denote widths of each piezoelectric element at x = 0 and x = L, and

γ = \frac{B_{0}}{B_{L}}

, which is the width ratio. ρ_p is the density of the piezoelectric material, and ρ_m is the density of the support material. Considering the capacitance

C_{V}^{s_L}

of the bimorph actuator in series electrical connection, and

C_{V}^{p_L}

parallel electrical connection, we can express the mass m_a of the actuator as

m_{a}^{series} = \frac{2 t_{p} C_{V}^{s_L}}{ɛ_{eq}^{s}} (ρ_{m} t_{m} + 2 ρ_{p} t_{p}) + L_{ext} B_{L} ρ_{m} (t_{m} + 2 t_{p})

(3)

m_{a}^{parallel} = \frac{t_{p} C_{V}^{p_L}}{2 ɛ_{eq}^{p}} (ρ_{m} t_{m} + 2 ρ_{p} t_{p}) + L_{ext} B_{L} ρ_{m} (t_{m} + 2 t_{p})

(4)

Equations (3) and (4) show that for constant thickness of piezoelectric layers, any increase in capacitance in the actuator also increases the mass of the actuator, which is undesirable in flapping-wing MAV design. Now, considering the expressions of capacitance in series and parallel electrical connection, we discuss the design of a Chebyshev active filter driver in the following section.

Chebyshev active filter driver for flapping piezoelectric actuators

Figure 4(a) represents a Chebyshev filter driver operating a piezoelectric actuator. Comparing the functionalities of an active filter-based driver with a switching amplifier-based driver, we notice the simplicity and elegance of the present piezoelectric driver compared to the earlier driver \cite{Janocha98}. Figure 4(a) shows that voltage V_in, which is used to control the flapping frequency by radio-controlled communication, generates a high-voltage unipolar square wave signal at that specified frequency. Then these square waves are processed through a low-pass Chebyshev filter to generate a unipolar high-voltage sinusoidal signal. A Chebyshev polynomial is used to achieve a sharp transition near the cut-off frequency for rejecting the high-frequency components of the square waves as much as possible. Figure 4(b) shows the frequency band of a Chebyshev low-pass filter.

Figure 4.

(a) Chebyshev filter driver for piezoelectric actuators in flapping-wing MAV applications, (b) frequency band of the driver, and (c) a third-order Chebyshev active low-pass filter using Sallen–Key topology.

The Chebyshev transfer function is defined as

G_{N} (ω) = | H_{N} (j ω) | = \frac{1}{\sqrt{1 + λ^{2} Ψ_{N}^{2} (\frac{ω}{ω_{p}})}}

(5)

Here, Ψ_{N} (\frac{ω}{ω_{p}}) = \cos (N \cos^{- 1} (\frac{ω}{ω_{p}})) : for | \frac{ω}{ω_{p}} | < 1 Ψ_{N} (\frac{ω}{ω_{p}}) = \cosh (N \cosh^{- 1} (\frac{ω}{ω_{p}})) : for | \frac{ω}{ω_{p}} | > 1

Here, $Ψ_{N}$ is a Chebyshev polynomial of the order N, ω is the wing flapping frequency, and $λ = \sqrt{\frac{H_{0}^{2}}{H_{p}^{2}} - 1}$ is the ripple factor. The order of the filter N is defined as

N \geq N_{Chebyshev} = \frac{\cosh^{- 1} \sqrt{\frac{(\frac{H_{0}^{2}}{H_{s}^{2}}) - 1}{(\frac{H_{0}^{2}}{H_{p}^{2}}) - 1}}}{\cosh^{- 1} (\frac{ω_{s}}{ω_{p}})}

(6)

As shown in Figure 4(b), H₀ denotes maximum gain of the Low Pass Filter (LPF), H_p denotes minimum allowed gain in the passband, and H_s indicates maximum allowed gain in the stopband. The passband ends at ω_p, and the stop band starts at ω_s. The frequency region between ω_p and ω_s is the transition band of the LPF. The pole locations are found at

p_{k} = - \sinh (φ_{2}) \sin θ_{k} + \cosh (φ_{2}) \cos θ_{k} : k = 1, 2, 3 \dots . N

(7)

Here, φ_{2} = \frac{1}{N} \sinh^{- 1} (\frac{1}{λ}) and θ_{k} = \frac{(2 k - 1)}{2 N} π

This yields the transfer function of the filter as

H (s) = \frac{\frac{H_{0}}{c_{N} . λ}}{Π_{k = 1}^{N} (s - p_{k})}

(8)

Here, $c_{N} = 2^{(N - 1)}$ . We have adopted Sallen–Key topology²⁶ and considered opamp-based design for the filter circuit.²⁷ Figure 4(c) shows a circuit diagram of a third-order active low-pass filter using Sallen–Key topology. Transfer function of a third-order Sallen–Key low-pass filter, which is shown in Figure 4(c), is

H (s) = \underset{2^{nd} orderfilter}{\underset{︸}{\frac{\frac{1}{R_{1} R_{2} C_{1} C_{2}}}{(s^{2} + (\frac{1}{R_{1} C_{1}} + \frac{1}{R_{2} C_{1}}) s + \frac{1}{R_{1} R_{2} C_{1} C_{2}})}}} \times \underset{1^{st} orderfilter}{\underset{︸}{\frac{\frac{1}{R_{3} C_{3}}}{(s + \frac{1}{R_{3} C_{3}})}}}

(9)

While using the LPF circuit, shown in Figure 4(c), to drive a piezoelectric actuator in flapping-wing MAV application, the capacitor C₃ is replaced by the capacitance of the piezoelectric actuator. For a series electrical connection of a piezoelectric bimorph actuator, capacitance C₃ consists of a series combination of two $2 C_{V}^{s_L}$ capacitance, which corresponds to each piezoelectric patch. Similarly, for a parallel electrical connection of a piezoelectric bimorph actuator, capacitance C₃ consists of a parallel combination of two $\frac{C_{V}^{p_L}}{2}$ capacitance, which corresponds to each piezoelectric patch. To design the driver, shown in Figure 4(c), first we decide an acceptable gains and the frequency range of operation of the driver for a flapping-wing MAV. This provides the order of the filter equation (6), the pole locations equation (7) and the all pole transfer function equation (9). Finalizing the geometry of the piezoelectric actuator by satisfying the mechanical properties for flapping-wing MAV, the corresponding capacitance is assigned to C₃, which is shown in Figure 4(c). Then the values of other circuit elements of Figure 4(c) are calculated from the transfer function. Then the value of resistor R₃ is decided by considering a pole of the transfer function. In an alternative approach, first resistor R₃ is set without considering the capacitance of a piezo-actuator. Then the value of capacitor C₃ is calculated. If the capacitance of the piezoelectric actuator is not equal to C₃, a compensation capacitor is connected externally either in series or in parallel with the piezoelectric actuator. This must ensure that the combination of the compensation capacitor and the piezoelectric capacitor equals C₃. In Figure 5(a), capacitor $C_{3}^{s}$ represents capacitance of the piezoelectric actuator, and capacitor $C_{C}^{s}$ is a compensating capacitor connected in series. The combination equals C₃ and must satisfy the pre-assigned resistance R₃. In Figure 5(b), capacitor $C_{3}^{p}$ represents capacitance of the piezoelectric actuator, and capacitor $C_{C}^{p}$ is a compensating capacitor connected in parallel connection. Here also the combination equals C₃ and must satisfy the pre-assigned resistance R₃. Equations (3) and (4) show that if we increase the capacitance of the actuator to satisfy the transfer function equation (9), the mass of the actuator also increases, which is a crucial parameter in flapping-wing MAV design. Therefore, the design approach, illustrated in this section, provides a systematic method to design the driver of a piezoelectric actuator for flapping-wing MAV applications.

Figure 5.

Third-order Chebyshev active low-pass filter with compensation capacitor in (a) series electrical connection and (b) parallel electrical connection.

Results and discussion

We have analytically discussed the design of an active low-pass Chebyshev filter for driving piezoelectric bending actuators in flapping-wing MAV applications. Average wing flapping frequency for dragonfly is 27 Hz,¹⁵ hawkmoth is 25 Hz,¹⁴ and butterfly is less than 20 Hz.¹³ Therefore, we have considered a flapping frequency range 30–50 Hz for designing a driver of a piezoelectric actuator. We have selected PZT 3203HD piezoelectric material and chosen a geometry of the actuator such that its first vibration mode exists beyond 50 Hz, which is the flapping frequency range. With this assumption, we have taken L = 50 mm, B_L=4 mm, γ = 4, t_p = 500 µm, and t_m = 500 µm. Piezoelectric strain constant of PZT 3203HD is $d_{31} = - 320 \times 10^{- 12}$ C/N. Relative dielectric constant of PZT 3203HD is $ɛ_{33}^{σ} = 3800$ . The nonlinear coupling coefficient $d_{331} = - 520 \times$ 10⁻¹⁸ m²·V⁻² and we neglect $ɛ_{333}^{σ}$ . The density of PZT 3203HD is $ρ_{p} = 7870$ kg·m⁻³, and the density of carbon epoxy is $ρ_{m} = 1500$ kg·m⁻³.²⁵ Considering such a piezoelectric flapping actuator, we find capacitance of the bimorph in series electrical connection is $C_{V}^{s_L}$ = 16 nF and in parallel electrical connection is $C_{V}^{p_L}$ = 64 nF. Now, to design a Chebyshev active low-pass filter as discussed in the previous section, we consider a maximum gain of the LPF $H_{0} = 1$ , minimum allowed gain in the passband H_p = 0.99H₀, maximum allowed gain in the stopband H_s = 0.01H₀, passband ending frequency ω_p = 2π.50 rev/s, and stopband starting frequency ω_s = 2π.300 rev/s. The frequency region between $ω_{p} = 2 π . 50$ rev/s to $ω_{s} = 2 π . 300$ rev/s is the transition band of the LPF. Solving equation (6), we find N = 2.9. The closest integer of 2.9 is 3. Therefore, the order of the filter is N = 3. Now, we get the pole locations at $p_{1} = - 0.503 + j 1.228, p_{2} = - 1.006$ and $p_{3} = - 0.503 - j 1.228$ by equation (7). This is an all-pole filter. Putting these three poles in equation (8), we get the transfer function of the filter as

H (s) = \frac{1.77}{(s + 1.006) (s^{2} + 1.006 s + 1.76)}

(10)

Now we compare the transfer function of equation (10) with the transfer function of equation (9). To find out the design of the second-order filter of equation (9), we put $R_{1} = R_{2} = 1$ , which gives $C_{1} = 1.998$ and $C_{2} = 0.2258$ . Considering ω_p = 2π.50 rev/s, and de-normalizing the values, we get $R_{1} = R_{2} = 10$ kΩ, $C_{1} = 633$ nF, and $C_{2} = 91$ nF. To find out the design of the first-order filter of equation (9), we consider the capacitance of the piezoelectric bimorph actuator. Taking the capacitance of the bimorph for series electrical connection, which is $C_{V}^{s_L}$ = 16 nF, we get $R_{3} = 197$ kΩ. Similarly, considering the capacitance of the bimorph for parallel electrical connection, which is $C_{V}^{p_L}$ = 64 nF, we get $R_{3} = 49$ kΩ. Now, if we want to reduce the value of R₃, which is shown in Figure 4(c), we incorporate compensation capacitors, which are shown in Figure 5(a) and (b). Considering $R_{3} = 10$ kΩ, if we connect the piezoelectric bimorph actuator in series electrical connection, we need a compensation capacitor $C_{C}^{p} = 299$ nF (Figure 5(b)); and for a parallel electrical connection, we need a compensation capacitor $C_{C}^{p} = 251$ nF (Figure 5(b)). Similarly, we can also connect a compensation capacitor $C_{C}^{s}$ in series connection (Figure 5(a)). We have designed a third-order active Chebyshev low-pass filter of Sallen–Key topology, which is shown in Figure 4(c), in OrCAD® version 16.6.²⁸ Considering a voltage V_HV = 70 V, $R_{1} = R_{2} = 10$ kΩ, $C_{1} = 633$ nF, $C_{2} = 91$ nF, $R_{3} = 10$ kΩ, $C_{3} = 315$ nF, and a flapping frequency 50 Hz, we have simulated the driver. The output voltage V, which is applied across the piezoelectric bimorph actuator, is plotted in Figure 6(a). The plot in Figure 6(a) shows an approximate sinusoidal voltage signal of 50 Hz across the actuator, which is desirable for a typical flapping-wing MAV. The bandwidth of this filter is plotted in Figure 6(b), which ensures a substantial rejection of the high-frequency components to produce a unipolar sinusoidal signal. In Figure 4(a) and (c), the two input switches are driven by pulses at flapping frequency. These pulses can be generated by radio-controlled communication, which provides facility to flap the wings of a MAV at any desired frequency.

Figure 6.

(a) Sinusoidal output driving voltage of the third-order Chebyshev active low-pass filter and (b) frequency bandwidth of the driver.

Conclusion

We have systematically presented the design of a piezoelectric driver for flapping-wing MAV applications. The driver uses a third-order, all-pole, Chebyshev active low-pass filter to produce a unipolar high-voltage sinusoidal signal for flapping-wing applications. The driver is analyzed and designed using Sallen–Key topology. The present design emphasizes on simple analog circuit and demonstrates lower number of circuit components and lower design complexity compared to the earlier drivers. The present driver is an inductor-less circuit, which facilitates fabricability of a single-driver chip for such applications. Reduction in the number of circuit components in the present driver can also reduce the cost of the driver. The suggested driver is compatible with the radio control communication system for adjusting the flapping frequency of the wings of a flapping-wing MAV.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work is supported by Indian Institute of Science, Bangalore.

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