Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (2024)

Tomohiko Jimbo1, Takashi Ozaki1, Norikazu Ohta1 and Kanae Hamaguchi11The authors are with Toyota Central R&D Labs., inc., 480-1192 Aichi, Japan t-jmb@mosk.tytlabs.co.jp

Abstract

Insect-scale micro-aerial vehicles, especially, lightweight, flapping-wing robots, are becoming increasingly important for safe motion sensing in spatially constrained environments such as living spaces. However, yaw control using flapping wings is fundamentally more difficult than using rotating wings. In this study, an insect-scale, tailless robot with four paired tilted flapping wings (weighing 1.52 g) to enable yaw control was fabricated. It benefits from the simplicity of a directly driven wing actuator with no transmission and a lift control signal; however, it still has an offset in the lift force. Therefore, an adaptive controller was designed to alleviate the offset. Numerical experiments confirm that the proposed controller outperforms the linear quadratic integral controller. Finally, in a tethered and controlled demonstration flight, the yaw drift was suppressed by the wing-tilting arrangement and the proposed controller. The simple structure drive system demonstrates the potential for future controlled flights of battery-powered, tailless, flapping-wing robots weighing less than 10 grams.

I INTRODUCTION

Unmanned aerial vehicles (UAVs) are beginning to be used for extensive monitoring and inspection of farms, roads, buildings, and bridges. However, because they generate their lift using rotating propellers, their movement in spatially constrained environments, such as living spaces, poses safety risks in the event of collisions. Therefore, micro-aerial vehicles, especially , lightweight, flapping-wing robots inspired by insects and birds, are being explored. Various studies on aerodynamics, kinematics, and control have reported interesting features, such as high maneuverability and efficiency with flapping wings [1, 2, 3, 4]. This study focuses on a hover-capable, tailless flapping-wing robot.

Various insect-inspired robots equipped with electromagnetic motors have been proposed [5, 6, 7, 8, 9]. In particular, a 28.2-g untethered battery-powered robot that can move freely in all directions and hover was demonstrated for the first time [8]. However, robots driven by electromagnetic motors require gears and transmission mechanisms to convert the rotating motion of the motor into high-speed reciprocating motion, and they are heavy, weighing 10 g or more.Extremely insect-scale robots weighing less than 1 g have been proposed [10, 11, 12, 13]. Their translational wing motion is generated directly by piezoelectric or dielectric elastomer actuators; gears are not required, but transmission may be necessary. However, in flight experiments, they are not battery-powered but are externally driven by wires (tethered flight). Therefore, controlled flights of battery-powered insect-scale robots weighing less than 10 g are still a challenge.

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (1)

Yaw control keeps on-board sensors pointing in the required direction and simplifies the horizontal controller design. However, yaw control using flapping wings is fundamentally more difficult than using rotating wings. Yaw torque generation using flapping wings is complex, as shown by many yaw control studies [14, 15, 16, 17, 18]. As demonstrated in [17], the speed ratio of the upstroke and downstroke of the piezoelectric actuator can be changed by adding a second-harmonic frequency to the fundamental frequency, generating yaw torque. This has been confirmed in experiments under conditions where the aircraft is fixed in the vertical direction. In [18], a yaw-controlled flight was demonstrated by a flapping-wing robot with two piezoelectric actuators, two wings, and a transmission. Here, the effect of phase shift in the second harmonic on the yaw torque was investigated and found to highly depend on manufacturing errors and other parameter changes. Therefore, the actuator drive system of the robot wing should be simple for yaw control. Furthermore, as the control signals for flapping wings become more complex, the circuits that generate the signals become more complex. For untethered, battery-powered flights, the weight and power consumption of the circuit must be minimized, and the robot should be able to operate with simple control signals.

We previously investigated a direct-driven flapping wing using a piezoelectric unimorph actuator [19, 20]. It is simple because it has no displacement-enhancing structure (transmission-free structure),and the wings are driven by simply changing the voltage amplitude at the resonance frequency.Furthermore, to suppress the coupling effect between the wings and the body [12], we fabricated a robot with three paired-wing actuators [21]. However, it was unable to generate sufficient yaw torque, thus the yaw angle drifted. Therefore, we propose an insect-scale flapping-wing robot weighing 1.52 g (Fig. 1), which generates yaw torque using a tilting arrangement of four paired-wing actuators. The simpler structure of the system results in er manufacturing errors; however, it still has an offset in the lift force. In addition, it is difficult to accurately measure the instantaneous lift forces generated during the flapping motion. Therefore, we modeled the offset force and torque acting on the body and designed an adaptive control system. The major contributions of this study are summarized below:

  • A novel flapping-wing robot with four paired tilted wings is fabricated to enable yaw control. The wings are driven by piezoelectric direct drive actuators; there is no transmission, and the lift force can be controlled by simply changing the voltage amplitude.

  • An adaptive controller is designed to account for the offset in the lift force.

  • We compare the performance of our controller with that of a linear quadratic integral (LQI) controller using numerical experiments and confirm the adaptability of our controller in the presence of an unknown lift offset.

  • We confirm that the yaw drift of the insect-scale flapping-wing robot can be suppressed through a controlled flight experiment.

The remainder of this paper is organized as follows:In Section 2, the control-oriented model of the flapping-wing robot with four paired-wing actuators and the uncertainties of the lift force and torque are derived. An adaptive controller is proposed, and voltage amplitudes are calculated in Section 3. The performance of the LQI controller and our controller is compared for different lift force offsets through numerical experiments in Section 4. Then, an experimental evaluation of the proposed method is performed in Section 5. Finally, Section 6 concludes the paper.

II MODELING

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (2)

II-A Tilted Wing Arrangement

Fig. 2 shows the wing arrangement of the newly developed four-paired-wing robot.The wing arrangement is different from that of the three-paired-wing robot developed in our previous study [21].Furthermore, to generate yaw torque, the four paired wings are tilted slightly by an angle β𝛽\betaitalic_β, which is the rotation angle around the Y-axis of the wing coordinate system.

The body force f3𝑓superscript3f\in\mathbb{R}^{3}italic_f ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and torque τ3𝜏superscript3{\tau}\in\mathbb{R}^{3}italic_τ ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT are given by

[fτ]=Mfwmatrix𝑓𝜏𝑀subscript𝑓𝑤\begin{bmatrix}f\\\tau\end{bmatrix}=Mf_{w}[ start_ARG start_ROW start_CELL italic_f end_CELL end_ROW start_ROW start_CELL italic_τ end_CELL end_ROW end_ARG ] = italic_M italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT(1)

where fw=[f1,f2,f3,f4]subscript𝑓𝑤superscriptsubscript𝑓1subscript𝑓2subscript𝑓3subscript𝑓4top{f}_{w}=[f_{1},f_{2},f_{3},f_{4}]^{\top}italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = [ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT is the vector of wing forces generated by applying flapping amplitude V𝑉Vitalic_V,and

M=[M1M2]=[e1e2e3e4p1×e1p2×e2p3×e3p4×e4]6×4𝑀matrixsubscript𝑀1subscript𝑀2matrixsubscript𝑒1subscript𝑒2subscript𝑒3subscript𝑒4subscript𝑝1subscript𝑒1subscript𝑝2subscript𝑒2subscript𝑝3subscript𝑒3subscript𝑝4subscript𝑒4superscript64M=\begin{bmatrix}M_{1}\\M_{2}\end{bmatrix}=\begin{bmatrix}{e}_{1}&{e}_{2}&{e}_{3}&{e}_{4}\\{p}_{1}\times{e}_{1}&{p}_{2}\times{e}_{2}&{p}_{3}\times{e}_{3}&{p}_{4}\times{e%}_{4}\end{bmatrix}\in\mathbb{R}^{6\times 4}italic_M = [ start_ARG start_ROW start_CELL italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] = [ start_ARG start_ROW start_CELL italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL italic_e start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT × italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT × italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT × italic_e start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ∈ blackboard_R start_POSTSUPERSCRIPT 6 × 4 end_POSTSUPERSCRIPT

is the mixing matrix.Here, for the wing i𝑖iitalic_i,

ei=[cosγisinβisinγisinβicosβi],pi=[aibi0]+[cosγicosβisinγicosβisinβi],formulae-sequencesubscript𝑒𝑖matrixsubscript𝛾𝑖subscript𝛽𝑖subscript𝛾𝑖subscript𝛽𝑖subscript𝛽𝑖subscript𝑝𝑖matrixsubscript𝑎𝑖subscript𝑏𝑖0matrixsubscript𝛾𝑖subscript𝛽𝑖subscript𝛾𝑖subscript𝛽𝑖subscript𝛽𝑖\displaystyle e_{i}=\begin{bmatrix}\cos\gamma_{i}\sin\beta_{i}\\\sin\gamma_{i}\sin\beta_{i}\\\cos\beta_{i}\end{bmatrix},p_{i}=\begin{bmatrix}a_{i}\\b_{i}\\0\end{bmatrix}+\begin{bmatrix}\cos\gamma_{i}\cos\beta_{i}\\\sin\gamma_{i}\cos\beta_{i}\\-\sin\beta_{i}\end{bmatrix},italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL roman_cos italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_sin italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL roman_sin italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_sin italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL roman_cos italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ] + [ start_ARG start_ROW start_CELL roman_cos italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_cos italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL roman_sin italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_cos italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL - roman_sin italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ,
a1=a4=a,a2=a3=a,formulae-sequencesubscript𝑎1subscript𝑎4𝑎subscript𝑎2subscript𝑎3𝑎\displaystyle a_{1}=a_{4}=a,\quad a_{2}=a_{3}=-a,italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_a , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - italic_a ,
b1=b2=b,b3=b4=b,formulae-sequencesubscript𝑏1subscript𝑏2𝑏subscript𝑏3subscript𝑏4𝑏\displaystyle b_{1}=b_{2}=b,\quad b_{3}=b_{4}=-b,italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_b , italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = - italic_b ,
β1=β2=β3=β4=β,subscript𝛽1subscript𝛽2subscript𝛽3subscript𝛽4𝛽\displaystyle\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=-\beta,italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = - italic_β ,
γ1=γ,γ2=πγ,γ3=π+γ,γ4=2πγ,formulae-sequencesubscript𝛾1𝛾formulae-sequencesubscript𝛾2𝜋𝛾formulae-sequencesubscript𝛾3𝜋𝛾subscript𝛾42𝜋𝛾\displaystyle\gamma_{1}=\gamma,\quad\gamma_{2}=\pi-\gamma,\quad\gamma_{3}=\pi+%\gamma,\quad\gamma_{4}=2\pi-\gamma,italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_γ , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_π - italic_γ , italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_π + italic_γ , italic_γ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 2 italic_π - italic_γ ,

a𝑎aitalic_a and b𝑏bitalic_b are the lengths of the body along the X- and Y-axes, respectively,γ𝛾\gammaitalic_γ is the rotation angle around the Z-axis of the body coordinate system, andl𝑙litalic_l is the distance between the mounting position of each wing and lift-force center.

II-B Unknown Offsets

The uncertainties of the body force and torque in (1) are expressed as offsets. They are due to unmodeled aerodynamic effects, manufacturing errors, and variations over time in system properties.The actual body force fbody(=[fbody,x,fbody,y,fbody,z])annotatedsubscript𝑓𝑏𝑜𝑑𝑦absentsuperscriptsubscript𝑓𝑏𝑜𝑑𝑦𝑥subscript𝑓𝑏𝑜𝑑𝑦𝑦subscript𝑓𝑏𝑜𝑑𝑦𝑧topf_{body}(=[f_{body,x},f_{body,y},f_{body,z}]^{\top})italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT ( = [ italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_x end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_y end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_z end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) and torque τbodysubscript𝜏𝑏𝑜𝑑𝑦\tau_{body}italic_τ start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT are given by

[fbodyτbody]=(M+δM)(fw+δfw)=[ffoττo],matrixsubscript𝑓𝑏𝑜𝑑𝑦subscript𝜏𝑏𝑜𝑑𝑦𝑀subscript𝛿𝑀subscript𝑓𝑤subscript𝛿subscript𝑓𝑤matrix𝑓subscript𝑓𝑜𝜏subscript𝜏𝑜\begin{bmatrix}{f}_{body}\\{\tau}_{body}\end{bmatrix}=\biggl{(}{M}+\delta_{M}\biggr{)}\biggl{(}{f}_{w}+%\delta_{f_{w}}\biggr{)}=\begin{bmatrix}{f}-{f}_{o}\\{\tau}-{\tau}_{o}\end{bmatrix},[ start_ARG start_ROW start_CELL italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_τ start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] = ( italic_M + italic_δ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = [ start_ARG start_ROW start_CELL italic_f - italic_f start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_τ - italic_τ start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ,(2)

wherethe offset force and torque are given by

[foτo]MδfwδMfw.matrixsubscript𝑓𝑜subscript𝜏𝑜𝑀subscript𝛿subscript𝑓𝑤subscript𝛿𝑀subscript𝑓𝑤\begin{bmatrix}{f}_{o}\\{\tau}_{o}\end{bmatrix}\approx-{M}\delta_{f_{w}}-\delta_{M}{f}_{w}.[ start_ARG start_ROW start_CELL italic_f start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_τ start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ≈ - italic_M italic_δ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT .(3)

They depend on δfw4subscript𝛿subscript𝑓𝑤superscript4\delta_{f_{w}}\in\mathbb{R}^{4}italic_δ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT and δM6×4subscript𝛿𝑀superscript64\delta_{M}\in\mathbb{R}^{6\times 4}italic_δ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 6 × 4 end_POSTSUPERSCRIPT.δfwsubscript𝛿subscript𝑓𝑤\delta_{f_{w}}italic_δ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT corresponds to the error of the wing force voltage modeling fw=h(V)subscript𝑓𝑤𝑉f_{w}=h(V)italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_h ( italic_V ).δMsubscript𝛿𝑀\delta_{M}italic_δ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT is the misalignment with respect to β𝛽\betaitalic_β, γ𝛾\gammaitalic_γ, and l𝑙litalic_l.

II-C Flight Dynamics

Consider the rigid-body model of the robot shown in Fig. 2.Using the actual force and torque, fbodysubscript𝑓𝑏𝑜𝑑𝑦{f}_{body}italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT and τbodysubscript𝜏𝑏𝑜𝑑𝑦{\tau}_{body}italic_τ start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT in (2),the dynamics are represented by

mv˙𝑚˙𝑣\displaystyle m\dot{{v}}italic_m over˙ start_ARG italic_v end_ARG=\displaystyle==Rfbodymg,𝑅subscript𝑓𝑏𝑜𝑑𝑦𝑚𝑔\displaystyle{R}{f}_{body}-m{g},italic_R italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT - italic_m italic_g ,(4)
Jω˙𝐽˙𝜔\displaystyle{J}\dot{{\omega}}italic_J over˙ start_ARG italic_ω end_ARG=\displaystyle==τbody(ω×Jω),subscript𝜏𝑏𝑜𝑑𝑦𝜔𝐽𝜔\displaystyle{\tau}_{body}-\left({\omega}\times{J}{\omega}\right),italic_τ start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y end_POSTSUBSCRIPT - ( italic_ω × italic_J italic_ω ) ,(5)

where v(=[vx,vy,vz])annotated𝑣absentsuperscriptsubscript𝑣𝑥subscript𝑣𝑦subscript𝑣𝑧topv(=[v_{x},v_{y},v_{z}]^{\top})italic_v ( = [ italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) is the translational velocity of the body in the global coordinate system,ω(=[ωB,x,ωB,y,ωB,z])annotated𝜔absentsuperscriptsubscript𝜔𝐵𝑥subscript𝜔𝐵𝑦subscript𝜔𝐵𝑧top\omega(=[\omega_{B,x},\omega_{B,y},\omega_{B,z}]^{\top})italic_ω ( = [ italic_ω start_POSTSUBSCRIPT italic_B , italic_x end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT italic_B , italic_y end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT italic_B , italic_z end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) is the angular velocity of the body in the body coordinate system,m𝑚mitalic_m and J3×3𝐽superscript33{J}\in\mathbb{R}^{3\times 3}italic_J ∈ blackboard_R start_POSTSUPERSCRIPT 3 × 3 end_POSTSUPERSCRIPT are the mass and inertia of the body, respectively,R𝑅{R}italic_R is the rotation matrix,g=[0,0,g]𝑔superscript00𝑔top{g}=[0,0,g]^{\top}italic_g = [ 0 , 0 , italic_g ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT, g𝑔gitalic_g is the gravitational acceleration.

The relationship between ω𝜔\omegaitalic_ω and the attitude η=[ϕ,θ,ψ]𝜂superscriptitalic-ϕ𝜃𝜓top\eta=[\phi,\theta,\psi]^{\top}italic_η = [ italic_ϕ , italic_θ , italic_ψ ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT of the body is described by

ω=Gη˙,𝜔𝐺˙𝜂{\omega}={G}\dot{{\eta}},italic_ω = italic_G over˙ start_ARG italic_η end_ARG ,(6)

where

G=[10sinθ0cosϕcosθsinϕ0sinϕcosθcosϕ].𝐺matrix10𝜃0italic-ϕ𝜃italic-ϕ0italic-ϕ𝜃italic-ϕ{G}=\begin{bmatrix}1&0&-\sin\theta\\0&\cos\phi&\cos\theta\sin\phi\\0&-\sin\phi&\cos\theta\cos\phi\end{bmatrix}.italic_G = [ start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL - roman_sin italic_θ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL roman_cos italic_ϕ end_CELL start_CELL roman_cos italic_θ roman_sin italic_ϕ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - roman_sin italic_ϕ end_CELL start_CELL roman_cos italic_θ roman_cos italic_ϕ end_CELL end_ROW end_ARG ] .

II-D Control-Oriented Model

The translational dynamics on the horizontal plane is derived from (4), assuming the hovering state, vz0subscript𝑣𝑧0v_{z}\approx 0italic_v start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≈ 0, and ωz0subscript𝜔𝑧0\omega_{z}\approx 0italic_ω start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≈ 0, to obtain

[v˙xBv˙yB]=[θϕ]g,matrixsuperscriptsubscript˙𝑣𝑥𝐵superscriptsubscript˙𝑣𝑦𝐵matrix𝜃italic-ϕ𝑔\begin{bmatrix}\dot{v}_{x}^{B}\\\dot{v}_{y}^{B}\end{bmatrix}=\begin{bmatrix}\theta\\-\phi\end{bmatrix}g,[ start_ARG start_ROW start_CELL over˙ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] = [ start_ARG start_ROW start_CELL italic_θ end_CELL end_ROW start_ROW start_CELL - italic_ϕ end_CELL end_ROW end_ARG ] italic_g ,(7)

where vxBsuperscriptsubscript𝑣𝑥𝐵v_{x}^{B}italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT and vyBsuperscriptsubscript𝑣𝑦𝐵v_{y}^{B}italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT are the translational velocities in the body coordinate system.

When approximating the rotation matrix R𝑅Ritalic_R using ϕitalic-ϕ\phiitalic_ϕ and θ𝜃\thetaitalic_θ, the dynamics in the vertical direction, that is, along the Z-axis, is given by

mz¨=fzfo,zmg,𝑚¨𝑧subscript𝑓𝑧subscript𝑓𝑜𝑧𝑚𝑔m\ddot{z}=f_{z}-f_{o,z}-mg,italic_m over¨ start_ARG italic_z end_ARG = italic_f start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT - italic_m italic_g ,(8)

where z𝑧zitalic_z is the altitude, that is, the vertical position of the body.Note that we assume |θmg||fbody,x|much-greater-than𝜃𝑚𝑔subscript𝑓𝑏𝑜𝑑𝑦𝑥|\theta mg|\gg|f_{body,x}|| italic_θ italic_m italic_g | ≫ | italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_x end_POSTSUBSCRIPT |, |ϕmg||fbody,y|much-greater-thanitalic-ϕ𝑚𝑔subscript𝑓𝑏𝑜𝑑𝑦𝑦|\phi mg|\gg|f_{body,y}|| italic_ϕ italic_m italic_g | ≫ | italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_y end_POSTSUBSCRIPT |, and |fbody,z||θfbody,x+ϕfbody,y|much-greater-thansubscript𝑓𝑏𝑜𝑑𝑦𝑧𝜃subscript𝑓𝑏𝑜𝑑𝑦𝑥italic-ϕsubscript𝑓𝑏𝑜𝑑𝑦𝑦|f_{body,z}|\gg|-\theta f_{body,x}+\phi f_{body,y}|| italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_z end_POSTSUBSCRIPT | ≫ | - italic_θ italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_x end_POSTSUBSCRIPT + italic_ϕ italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_y end_POSTSUBSCRIPT |.

For the rotational dynamics, (2) and (5) are directly used to design the flight controller.

The transient response of the wing force to the flapping amplitude exhibits a lag in relation to the dynamics of the wing position.The lag can be approximated by the following first-order lag system:

Tf˙z𝑇subscript˙𝑓𝑧\displaystyle T\dot{f}_{z}italic_T over˙ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT=\displaystyle==fd,zfz,subscript𝑓𝑑𝑧subscript𝑓𝑧\displaystyle f_{d,z}-f_{z},italic_f start_POSTSUBSCRIPT italic_d , italic_z end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ,(9)
Tτ˙𝑇˙𝜏\displaystyle T\dot{{\tau}}italic_T over˙ start_ARG italic_τ end_ARG=\displaystyle==τdτ,subscript𝜏𝑑𝜏\displaystyle{\tau}_{d}-{\tau},italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT - italic_τ ,(10)

where T(>0)annotated𝑇absent0T(>0)italic_T ( > 0 ) is a time constant and fd,zsubscript𝑓𝑑𝑧f_{d,z}\in\mathbb{R}italic_f start_POSTSUBSCRIPT italic_d , italic_z end_POSTSUBSCRIPT ∈ blackboard_R and τd3subscript𝜏𝑑superscript3{\tau}_{d}\in\mathbb{R}^{3}italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT are the demanded force and torque, respectively. Note that the lift force oscillates depending on the reciprocating motion of the wing.In this study, the oscillation component is not considered.

III FLIGHT CONTROLLER

The offset force and torque in (3) are unknown parameters.They are uncertain and cannot be measured.Furthermore, a change in the relationship between fwsubscript𝑓𝑤f_{w}italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT and V𝑉Vitalic_V occurs due to damage during use.Therefore, in this study, an adaptive controller is adopted.

III-A Velocity Control

To track the velocities vxBsuperscriptsubscript𝑣𝑥𝐵v_{x}^{B}italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT and vyBsuperscriptsubscript𝑣𝑦𝐵v_{y}^{B}italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT on the horizontal plane to the target vx,dBsuperscriptsubscript𝑣𝑥𝑑𝐵v_{x,d}^{B}italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT and vy,dBsuperscriptsubscript𝑣𝑦𝑑𝐵v_{y,d}^{B}italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, respectively,considering (7),the corresponding target angles of the pitch θdsubscript𝜃𝑑\theta_{d}italic_θ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT and roll ϕdsubscriptitalic-ϕ𝑑\phi_{d}italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT are set to

θdsubscript𝜃𝑑\displaystyle\theta_{d}italic_θ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT=\displaystyle==hx(vxBvx,dB)/g,subscript𝑥superscriptsubscript𝑣𝑥𝐵superscriptsubscript𝑣𝑥𝑑𝐵𝑔\displaystyle-h_{x}\left(v_{x}^{B}-v_{x,d}^{B}\right)/g,- italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) / italic_g ,
ϕdsubscriptitalic-ϕ𝑑\displaystyle\phi_{d}italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT=\displaystyle==hy(vyBvy,dB)/g,subscript𝑦superscriptsubscript𝑣𝑦𝐵superscriptsubscript𝑣𝑦𝑑𝐵𝑔\displaystyle h_{y}\left(v_{y}^{B}-v_{y,d}^{B}\right)/g,italic_h start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) / italic_g ,

where hxsubscript𝑥h_{x}italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and hysubscript𝑦h_{y}italic_h start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT are positive constants.

III-B Attitude Control

To track the attitude η𝜂{\eta}italic_η to the target ηd(=[ϕd,θd,ψd])annotatedsubscript𝜂𝑑absentsuperscriptsubscriptitalic-ϕ𝑑subscript𝜃𝑑subscript𝜓𝑑top{\eta}_{d}(=[\phi_{d},\theta_{d},\psi_{d}]^{\top})italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( = [ italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT , italic_ψ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ),from (6),the target angular velocity of the robot ω𝜔\omegaitalic_ω and higher-order derivatives are designed as follows:

ωdsubscript𝜔𝑑\displaystyle{\omega}_{d}italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT=\displaystyle==GKηeη,𝐺subscript𝐾𝜂subscript𝑒𝜂\displaystyle-G{K}_{\eta}{e}_{\eta},- italic_G italic_K start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ,(11)
ω˙dsubscript˙𝜔𝑑\displaystyle\dot{{\omega}}_{d}over˙ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT\displaystyle\approxGKηη˙,𝐺subscript𝐾𝜂˙𝜂\displaystyle-G{K}_{\eta}\dot{{\eta}},- italic_G italic_K start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT over˙ start_ARG italic_η end_ARG ,
ω¨dsubscript¨𝜔𝑑\displaystyle\ddot{{\omega}}_{d}over¨ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT\displaystyle\approxGKηη¨,𝐺subscript𝐾𝜂¨𝜂\displaystyle-G{K}_{\eta}\ddot{{\eta}},- italic_G italic_K start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT over¨ start_ARG italic_η end_ARG ,

where eη=ηηdsubscript𝑒𝜂𝜂subscript𝜂𝑑{e}_{\eta}={\eta}-{\eta}_{d}italic_e start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT = italic_η - italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT,Kη(=diag(kη1,kη2,kη3))annotatedsubscript𝐾𝜂absentdiagsubscript𝑘𝜂1subscript𝑘𝜂2subscript𝑘𝜂3{K}_{\eta}(={\rm diag}(k_{\eta 1},k_{\eta 2},k_{\eta 3}))italic_K start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ( = roman_diag ( italic_k start_POSTSUBSCRIPT italic_η 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT italic_η 2 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT italic_η 3 end_POSTSUBSCRIPT ) ) is a positive diagonal matrix.

To track the angular velocity ω𝜔{\omega}italic_ω to the target ωdsubscript𝜔𝑑{\omega}_{d}italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT,from (5) and (10),the demanded torque τdsubscript𝜏𝑑{\tau}_{d}italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT and the adaptive law to estimate the offset torque τ^osubscript^𝜏𝑜\hat{{\tau}}_{o}over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT are designed as

τdsubscript𝜏𝑑\displaystyle{\tau}_{d}italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT=\displaystyle==Kωsω+Jω˙r+F+T(Jω¨r+F˙)+τ^o,subscript𝐾𝜔subscript𝑠𝜔𝐽subscript˙𝜔𝑟𝐹𝑇𝐽subscript¨𝜔𝑟˙𝐹subscript^𝜏𝑜\displaystyle-{K}_{\omega}{s}_{\omega}+J\dot{{\omega}}_{r}+{F}+T\left(J\ddot{{%\omega}}_{r}+\dot{{F}}\right)+\hat{{\tau}}_{o},- italic_K start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT + italic_J over˙ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_F + italic_T ( italic_J over¨ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + over˙ start_ARG italic_F end_ARG ) + over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ,(12)
τ^˙osubscript˙^𝜏𝑜\displaystyle\dot{\hat{{\tau}}}_{o}over˙ start_ARG over^ start_ARG italic_τ end_ARG end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT=\displaystyle==Γωsω,subscriptΓ𝜔subscript𝑠𝜔\displaystyle-{\Gamma}_{\omega}{s}_{\omega},- roman_Γ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ,(13)

where F(=ω×Jω)annotated𝐹absent𝜔𝐽𝜔F(=\omega\times J\omega)italic_F ( = italic_ω × italic_J italic_ω ) is the centrifugal Coriolis force,

sωsubscript𝑠𝜔\displaystyle{s}_{\omega}italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT=\displaystyle==e˙ω+Λωeω=ω˙ω˙r,subscript˙𝑒𝜔subscriptΛ𝜔subscript𝑒𝜔˙𝜔subscript˙𝜔𝑟\displaystyle\dot{{e}}_{\omega}+{\Lambda}_{\omega}{e}_{\omega}=\dot{{\omega}}-%\dot{{\omega}}_{r},over˙ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT + roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT = over˙ start_ARG italic_ω end_ARG - over˙ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ,(14)
eωsubscript𝑒𝜔\displaystyle{e}_{\omega}italic_e start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT=\displaystyle==ωωd,𝜔subscript𝜔𝑑\displaystyle{\omega}-{\omega}_{d},italic_ω - italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ,
ω˙rsubscript˙𝜔𝑟\displaystyle\dot{{\omega}}_{r}over˙ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT=\displaystyle==ω˙dΛω(ωωd),subscript˙𝜔𝑑subscriptΛ𝜔𝜔subscript𝜔𝑑\displaystyle\dot{{\omega}}_{d}-{\Lambda}_{\omega}({\omega}-{\omega}_{d}),over˙ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT - roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_ω - italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ,

Λω(=diag(λω1,λω2,λω3))annotatedsubscriptΛ𝜔absentdiagsubscript𝜆𝜔1subscript𝜆𝜔2subscript𝜆𝜔3{\Lambda}_{\omega}(={\rm diag}(\lambda_{\omega 1},\lambda_{\omega 2},\lambda_{%\omega 3}))roman_Λ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( = roman_diag ( italic_λ start_POSTSUBSCRIPT italic_ω 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT italic_ω 2 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT italic_ω 3 end_POSTSUBSCRIPT ) ),Kω(=diag(kω1,kω2,kω3))annotatedsubscript𝐾𝜔absentdiagsubscript𝑘𝜔1subscript𝑘𝜔2subscript𝑘𝜔3{K}_{\omega}(={\rm diag}(k_{\omega 1},k_{\omega 2},k_{\omega 3}))italic_K start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( = roman_diag ( italic_k start_POSTSUBSCRIPT italic_ω 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT italic_ω 2 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT italic_ω 3 end_POSTSUBSCRIPT ) ), andΓω(=diag(γω1,γω2,γω3))annotatedsubscriptΓ𝜔absentdiagsubscript𝛾𝜔1subscript𝛾𝜔2subscript𝛾𝜔3{\Gamma}_{\omega}(={\rm diag}(\gamma_{\omega 1},\gamma_{\omega 2},\gamma_{%\omega 3}))roman_Γ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( = roman_diag ( italic_γ start_POSTSUBSCRIPT italic_ω 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT italic_ω 2 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT italic_ω 3 end_POSTSUBSCRIPT ) )are positive diagonal matrices.

Theorem 1.

On applying the control input of (12) and the adaptive law of (13) with (14) to (5) and (10), the sliding variable sωsubscript𝑠𝜔s_{\omega}italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT satisfies sω0subscript𝑠𝜔0s_{\omega}\rightarrow 0italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT → 0 as t𝑡t\rightarrow\inftyitalic_t → ∞.

Proof.

See Appendix A for the proof.∎

From Theorem 1, considering (14), ω𝜔\omegaitalic_ω asymptotically converges to ωdsubscript𝜔𝑑\omega_{d}italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT as t𝑡t\rightarrow\inftyitalic_t → ∞.Similarly, from (6) and (11), η𝜂\etaitalic_η asymptotically converges to ηdsubscript𝜂𝑑\eta_{d}italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT as t𝑡t\rightarrow\inftyitalic_t → ∞.

III-C Vertical Control

To control the flapping-wing robot in the vertical direction,from (8) and (9),the demanded force fd,zsubscript𝑓𝑑𝑧f_{d,z}italic_f start_POSTSUBSCRIPT italic_d , italic_z end_POSTSUBSCRIPT and the adaptive law to estimate the offset force f^o,zsubscript^𝑓𝑜𝑧\hat{f}_{o,z}over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT are designed as

fd,zmsubscript𝑓𝑑𝑧𝑚\displaystyle\frac{f_{d,z}}{m}divide start_ARG italic_f start_POSTSUBSCRIPT italic_d , italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG=\displaystyle==kzsz+z¨r+Tz˙˙˙r+g+f^o,zm,subscript𝑘𝑧subscript𝑠𝑧subscript¨𝑧𝑟𝑇subscript˙˙˙𝑧𝑟𝑔subscript^𝑓𝑜𝑧𝑚\displaystyle-k_{z}s_{z}+\ddot{z}_{r}+T\dddot{z}_{r}+g+\frac{\hat{f}_{o,z}}{m},- italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + over¨ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_T over˙˙˙ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_g + divide start_ARG over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG ,(15)
f^˙o,zsubscript˙^𝑓𝑜𝑧\displaystyle\dot{\hat{f}}_{o,z}over˙ start_ARG over^ start_ARG italic_f end_ARG end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT=\displaystyle==γzmsz,subscript𝛾𝑧𝑚subscript𝑠𝑧\displaystyle-\frac{\gamma_{z}}{m}s_{z},- divide start_ARG italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ,(16)

where, for the vertical velocity control to track z˙˙𝑧\dot{z}over˙ start_ARG italic_z end_ARG to the target z˙dsubscript˙𝑧𝑑\dot{z}_{d}over˙ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT,

szsubscript𝑠𝑧\displaystyle s_{z}italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT=\displaystyle==e¨z+λze˙z=z¨z¨r,subscript¨𝑒𝑧subscript𝜆𝑧subscript˙𝑒𝑧¨𝑧subscript¨𝑧𝑟\displaystyle\ddot{e}_{z}+\lambda_{z}\dot{e}_{z}=\ddot{z}-\ddot{z}_{r},over¨ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT over˙ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = over¨ start_ARG italic_z end_ARG - over¨ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ,(17)
z¨rsubscript¨𝑧𝑟\displaystyle\ddot{z}_{r}over¨ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT=\displaystyle==λz(z˙z˙d),subscript𝜆𝑧˙𝑧subscript˙𝑧𝑑\displaystyle-\lambda_{z}(\dot{z}-\dot{z}_{d}),- italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( over˙ start_ARG italic_z end_ARG - over˙ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ,

and for the vertical position (altitude) control to track z𝑧zitalic_z to the target zdsubscript𝑧𝑑z_{d}italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT,

szsubscript𝑠𝑧\displaystyle s_{z}italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT=\displaystyle==e¨z+2λze˙z+λz2ez=z¨z¨r,subscript¨𝑒𝑧2subscript𝜆𝑧subscript˙𝑒𝑧superscriptsubscript𝜆𝑧2subscript𝑒𝑧¨𝑧subscript¨𝑧𝑟\displaystyle\ddot{e}_{z}+2\lambda_{z}\dot{e}_{z}+\lambda_{z}^{2}e_{z}=\ddot{z%}-\ddot{z}_{r},over¨ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + 2 italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT over˙ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = over¨ start_ARG italic_z end_ARG - over¨ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ,(18)
z¨rsubscript¨𝑧𝑟\displaystyle\ddot{z}_{r}over¨ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT=\displaystyle==2λzz˙λz2(zzd).2subscript𝜆𝑧˙𝑧superscriptsubscript𝜆𝑧2𝑧subscript𝑧𝑑\displaystyle-2\lambda_{z}\dot{z}-\lambda_{z}^{2}(z-z_{d}).- 2 italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT over˙ start_ARG italic_z end_ARG - italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_z - italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) .

Here, ez=zzdsubscript𝑒𝑧𝑧subscript𝑧𝑑e_{z}=z-z_{d}italic_e start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = italic_z - italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT.λzsubscript𝜆𝑧\lambda_{z}italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, kzsubscript𝑘𝑧k_{z}italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, and γzsubscript𝛾𝑧\gamma_{z}italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT are positive constants.

Theorem 2.

On applying the control input of (15) and the adaptive law of (16) with (17) or (18) to (8) and (9), the sliding variable szsubscript𝑠𝑧s_{z}italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT satisfies sz0subscript𝑠𝑧0s_{z}\rightarrow 0italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT → 0 as t𝑡t\rightarrow\inftyitalic_t → ∞.

Proof.

See Appendix B for the proof.∎

From Theorem 2, considering (18), z𝑧zitalic_z asymptotically converges to zdsubscript𝑧𝑑z_{d}italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT as t𝑡t\rightarrow\inftyitalic_t → ∞.

III-D Lift-Force Demand and Flapping-Amplitude Control

From the demanded torque τdsubscript𝜏𝑑\tau_{d}italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT and force fdsubscript𝑓𝑑f_{d}italic_f start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT in (12) and (15) as well as the mixing matrix in (1), the lift-force demand is given by

fw,d=[M1(3,:)M2]1[fd,z𝝉d].subscript𝑓𝑤𝑑superscriptmatrixsubscript𝑀13:subscript𝑀21matrixsubscript𝑓𝑑𝑧subscript𝝉𝑑{f}_{w,d}=\begin{bmatrix}{M}_{1}(3,:)\\{M}_{2}\end{bmatrix}^{-1}\begin{bmatrix}{f}_{d,z}\\\boldsymbol{\tau}_{d}\end{bmatrix}.italic_f start_POSTSUBSCRIPT italic_w , italic_d end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 3 , : ) end_CELL end_ROW start_ROW start_CELL italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT [ start_ARG start_ROW start_CELL italic_f start_POSTSUBSCRIPT italic_d , italic_z end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .(19)

The flapping-wing actuator used in this study is driven by simply changing the flapping amplitude [19].The relationship between the wing force fwsubscript𝑓𝑤f_{w}italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT and the flapping amplitude V𝑉Vitalic_V is modeled around the voltage at which the robot supports its own weight as follows [21]:

fw=h(V).subscript𝑓𝑤𝑉f_{w}=h(V).italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_h ( italic_V ) .(20)

As a result, the required flapping amplitude Vd4subscript𝑉𝑑superscript4V_{d}\in\mathbb{R}^{4}italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT is derived from (19) and (20) as Vd=h1(fw,d)subscript𝑉𝑑superscript1subscript𝑓𝑤𝑑V_{d}=h^{-1}(f_{w,d})italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_f start_POSTSUBSCRIPT italic_w , italic_d end_POSTSUBSCRIPT ).Then, Vdsubscript𝑉𝑑V_{d}italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT is multiplied by a sinusoidal wave and then applied to the robot. Note that this can be performed using a digital circuit,such as a pulse-width modulator or similar modulator,which is much simpler than a circuit that generates complex analog waveforms [22].

IV SIMULATION

The effectiveness of the proposed controller was validated through numerical simulations under unknown offsets.The proposed controller was compared with a conventional LQI controller derived from the linearization of (4) and (5), including the first-order lag of the wing forces.

IV-A Simulation Setup

Tables III and III show the model parameters of the robot and the control parameters in some numerical simulations, respectively.Here, the model parameters are not correct. Notably, J𝐽Jitalic_J is the value of a three-paired-wing robot in [21].The weight is set at 2.0 g considering the untethered flapping-wing robot with an integrated battery [22].

NameSymbolValueUnit
Boby massm𝑚mitalic_m2.01032.0superscript1032.0*10^{-3}2.0 ∗ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPTkgkg\rm kgroman_kg
Body intertiaJ1subscript𝐽1J_{1}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT1.501071.50superscript1071.50*10^{-7}1.50 ∗ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPTkgm2kgsuperscriptm2\rm kg\cdot m^{2}roman_kg ⋅ roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
(J=diag(J1,J2,J3))𝐽diagsubscript𝐽1subscript𝐽2subscript𝐽3(J={\rm diag}(J_{1},J_{2},J_{3}))( italic_J = roman_diag ( italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) )J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT1.351071.35superscript1071.35*10^{-7}1.35 ∗ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPTkgm2kgsuperscriptm2\rm kg\cdot m^{2}roman_kg ⋅ roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
J3subscript𝐽3J_{3}italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT2.211072.21superscript1072.21*10^{-7}2.21 ∗ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPTkgm2kgsuperscriptm2\rm kg\cdot m^{2}roman_kg ⋅ roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
Body lengthsa𝑎aitalic_a20.010320.0superscript10320.0*10^{-3}20.0 ∗ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPTmm\rm mroman_m
b𝑏bitalic_b5.01035.0superscript1035.0*10^{-3}5.0 ∗ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPTmm\rm mroman_m
Wing anglesβ𝛽\betaitalic_β20π/18020𝜋18020*\pi/18020 ∗ italic_π / 180radrad\rm radroman_rad
γ𝛾\gammaitalic_γ60π/18060𝜋18060*\pi/18060 ∗ italic_π / 180radrad\rm radroman_rad
Length between lift center
and wing rootl𝑙litalic_l40.010340.0superscript10340.0*10^{-3}40.0 ∗ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPTmm\rm mroman_m
Time constant of first-order lagT𝑇Titalic_T0.0130.0130.0130.013sec
Gravitational accelerationg𝑔gitalic_g9.819.819.819.81kgm/sec2kgmsuperscriptsec2\rm kg\cdot m/sec^{2}roman_kg ⋅ roman_m / roman_sec start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
SymbolValueUnit
Attitudehxsubscript𝑥h_{x}italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT1/0.510.51/0.51 / 0.51/sec1sec\rm 1/sec1 / roman_sec
hysubscript𝑦h_{y}italic_h start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT1/0.510.51/0.51 / 0.51/sec1sec\rm 1/sec1 / roman_sec
kη1subscript𝑘𝜂1k_{\eta 1}italic_k start_POSTSUBSCRIPT italic_η 1 end_POSTSUBSCRIPT1/0.110.11/0.11 / 0.11/sec1sec\rm 1/sec1 / roman_sec
kη2subscript𝑘𝜂2k_{\eta 2}italic_k start_POSTSUBSCRIPT italic_η 2 end_POSTSUBSCRIPT1/0.110.11/0.11 / 0.11/sec1sec\rm 1/sec1 / roman_sec
kη3subscript𝑘𝜂3k_{\eta 3}italic_k start_POSTSUBSCRIPT italic_η 3 end_POSTSUBSCRIPT1/0.110.11/0.11 / 0.11/sec1sec\rm 1/sec1 / roman_sec
λω1subscript𝜆𝜔1\lambda_{\omega 1}italic_λ start_POSTSUBSCRIPT italic_ω 1 end_POSTSUBSCRIPT1/0.110.11/0.11 / 0.11/sec1sec\rm 1/sec1 / roman_sec
λω2subscript𝜆𝜔2\lambda_{\omega 2}italic_λ start_POSTSUBSCRIPT italic_ω 2 end_POSTSUBSCRIPT1/0.110.11/0.11 / 0.11/sec1sec\rm 1/sec1 / roman_sec
λω3subscript𝜆𝜔3\lambda_{\omega 3}italic_λ start_POSTSUBSCRIPT italic_ω 3 end_POSTSUBSCRIPT1/0.110.11/0.11 / 0.11/sec1sec\rm 1/sec1 / roman_sec
kω1subscript𝑘𝜔1k_{\omega 1}italic_k start_POSTSUBSCRIPT italic_ω 1 end_POSTSUBSCRIPT9.501089.50superscript1089.50*10^{-8}9.50 ∗ 10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT-
kω2subscript𝑘𝜔2k_{\omega 2}italic_k start_POSTSUBSCRIPT italic_ω 2 end_POSTSUBSCRIPT8.551088.55superscript1088.55*10^{-8}8.55 ∗ 10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT-
kω3subscript𝑘𝜔3k_{\omega 3}italic_k start_POSTSUBSCRIPT italic_ω 3 end_POSTSUBSCRIPT1.401071.40superscript1071.40*10^{-7}1.40 ∗ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT-
γω1subscript𝛾𝜔1\gamma_{\omega 1}italic_γ start_POSTSUBSCRIPT italic_ω 1 end_POSTSUBSCRIPT7.701067.70superscript1067.70*10^{-6}7.70 ∗ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT-
γω2subscript𝛾𝜔2\gamma_{\omega 2}italic_γ start_POSTSUBSCRIPT italic_ω 2 end_POSTSUBSCRIPT6.931066.93superscript1066.93*10^{-6}6.93 ∗ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT-
γω3subscript𝛾𝜔3\gamma_{\omega 3}italic_γ start_POSTSUBSCRIPT italic_ω 3 end_POSTSUBSCRIPT1.131051.13superscript1051.13*10^{-5}1.13 ∗ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT-
τ^o,x(t=0)subscript^𝜏𝑜𝑥𝑡0\hat{\tau}_{o,x}(t=0)over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o , italic_x end_POSTSUBSCRIPT ( italic_t = 0 )00NmNm{\rm Nm}roman_Nm
τ^o,y(t=0)subscript^𝜏𝑜𝑦𝑡0\hat{\tau}_{o,y}(t=0)over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o , italic_y end_POSTSUBSCRIPT ( italic_t = 0 )00NmNm{\rm Nm}roman_Nm
τ^o,z(t=0)subscript^𝜏𝑜𝑧𝑡0\hat{\tau}_{o,z}(t=0)over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT ( italic_t = 0 )00NmNm{\rm Nm}roman_Nm
Altitudeλzsubscript𝜆𝑧\lambda_{z}italic_λ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT1/0.510.51/0.51 / 0.51/sec1sec\rm 1/sec1 / roman_sec
kzsubscript𝑘𝑧k_{z}italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT6.341016.34superscript1016.34*10^{-1}6.34 ∗ 10 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-
γzsubscript𝛾𝑧\gamma_{z}italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT2.051042.05superscript1042.05*10^{-4}2.05 ∗ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT-
f^o,z(t=0)subscript^𝑓𝑜𝑧𝑡0\hat{f}_{o,z}(t=0)over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT ( italic_t = 0 )00NN{\rm N}roman_N
SymbolValueUnit
δβ𝛿𝛽\delta\betaitalic_δ italic_β10π/18010𝜋180{10}*\pi/18010 ∗ italic_π / 180radrad\rm radroman_rad
δγ𝛿𝛾\delta\gammaitalic_δ italic_γ10π/18010𝜋180{10}*\pi/18010 ∗ italic_π / 180radrad\rm radroman_rad
δl𝛿𝑙\delta litalic_δ italic_l51035superscript1035*10^{-3}5 ∗ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPTmm\rm mroman_m
δfw𝛿subscript𝑓𝑤{\delta f}_{w}italic_δ italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPTmg/4/3[0,1,0,0]𝑚𝑔43superscript0100topmg/4/{3}*[0,-1,0,0]^{\top}italic_m italic_g / 4 / 3 ∗ [ 0 , - 1 , 0 , 0 ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPTNN\rm Nroman_N

In the simulations, we consider some case studies with lift force offsets, as shown in Table III.

Case 1. Offset due to manufacturing errors


The misalignments δβ𝛿𝛽\delta\betaitalic_δ italic_β and δγ𝛿𝛾\delta\gammaitalic_δ italic_γ that occur during manufacturing were both set to 10 degrees.The error δl𝛿𝑙\delta litalic_δ italic_l in the distance l𝑙litalic_l from the mounting position of each wing to the center of thrust was set to 5 mm for each wing.

Case 2. Offset due to modeling error


There exists an error δfw𝛿subscript𝑓𝑤{\delta f}_{w}italic_δ italic_f start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT in the lift-force model of (20).In this case, only the second wing has a modeling error of one-third of mg/4𝑚𝑔4mg/4italic_m italic_g / 4, which is necessary when four paired wings share the weight.

Case 3. After adaptating for modeling error


The numerical experiment for Case 2 was performed again. However, the offset torque τ^osubscript^𝜏𝑜\hat{\tau}_{o}over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT and force f^o,zsubscript^𝑓𝑜𝑧\hat{f}_{o,z}over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT, estimated previously in Case 2, were used as the initial values of the parameter adaptation laws in equations (13) and (16).

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (3)

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (4)

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (5)

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (6)

IV-B Comparison Results

First, we performed a numerical experiment without the offsets in Table III.Here, a target value of 𝒓=[vx,dB,vy,dB,vz,dB,ψd]=[0.5,0.5,0.5,1]𝒓superscriptsuperscriptsubscript𝑣𝑥𝑑𝐵superscriptsubscript𝑣𝑦𝑑𝐵superscriptsubscript𝑣𝑧𝑑𝐵subscript𝜓𝑑topsuperscript0.50.50.51top\boldsymbol{r}=[v_{x,d}^{B},v_{y,d}^{B},v_{z,d}^{B},\psi_{d}]^{\top}=[0.5,0.5,%0.5,1]^{\top}bold_italic_r = [ italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT italic_z , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_ψ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = [ 0.5 , 0.5 , 0.5 , 1 ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT was set.Note that the control parameters were adjusted to obtain similar closed-loop responses from both controllers.As shown in Fig. 6,the velocity and attitude controlled by each controller converged to the targets at about 1111 s without large overshoots.

Second, Fig. 6 shows the result for Case 1 with the misalignments shown in Table III.The control performance of the LQI controller was worse than that without the offset shown in Fig. 6.The reverse response of vzsubscript𝑣𝑧v_{z}italic_v start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT occurred for the LQI controller.In contrast, the control performance of the proposed controller was almost the same as that without the offset shown in Fig. 6.

Third, Fig. 6 shows the result for Case 2 with the wing force voltage modeling error shown in Table III.The control performance of the LQI controller was worse than that without the offset shown in Fig. 6.The reverse responses of vxsubscript𝑣𝑥v_{x}italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and vzsubscript𝑣𝑧v_{z}italic_v start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT occurred for the LQI controller.vysubscript𝑣𝑦v_{y}italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT had an overshoot.In contrast, for the proposed controller, the response of vzsubscript𝑣𝑧v_{z}italic_v start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT was the same as that in Fig. 6.However, a reverse response of vxsubscript𝑣𝑥v_{x}italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT occurred, and vysubscript𝑣𝑦v_{y}italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT had a larger overshoot than that of the LQI controller.

Finally, Fig. 6 shows the result for Case 3 using the proposed controller with the parameters adapted for modeling errors.The response of the proposed controller was neither a reverse response nor an overshoot.The result was clearly improved compared to that shown in Fig. 6.

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (7)

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (8)

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (9)

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (10)

V EXPERIMENT

The effectiveness of the proposed controller was demonstrated through a flight experiment.The weight, which is one of the model parameters, was set to 1.52 g for the flapping-wing robot, as shown in Fig. 1.

V-A Experimental Setup

Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (11)

Our experimental system is depicted in Fig. 10.This system tracks markers attached to the body using the OptiTrack Prime 17 W motion capture system and a computer (Intel Core i9-9900K, 8-core 3.6 GHz, 64 GB of RAM), which calculates the position and orientation of the robot.These values are then sent to a control computer (Intel Core i7-7700K, 4-core 4.2 GHz, 32 GB of RAM) within 3333 ms.

The proposed controller calculates the flapping amplitude, which is then multiplied by a sinusoidal wave at 115 Hz, generated by a function generator (Precision 4050B, B&K), using a multiplier (AD633, Analog Devices). The amplitude is further amplified 30 times using an amplifier (HJPZ-0.3P×3, Matsusada Precision) and then applied to the robot through enameled wires. Note that the same flapping amplitude is applied to paired wings to generate an equal force.The control circuitry, along with the battery, can be implemented in digital circuitry when mounted in the body [22].

V-B Results

The results of a flight experiment with the four tilted paired-wing robot using the proposed controller are shown in Figs. 10, 10, and 10. In Fig. 10, the result of the three paired-wing robot in [21] is also shown.Fig. 11 shows the sequential shots taken during the controlled flight.Here, the target values are set to ψd=0subscript𝜓𝑑0\psi_{d}=0italic_ψ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0, zd=0.05subscript𝑧𝑑0.05z_{d}=0.05italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0.05, and vx,dB=vy,dB=0superscriptsubscript𝑣𝑥𝑑𝐵superscriptsubscript𝑣𝑦𝑑𝐵0v_{x,d}^{B}=v_{y,d}^{B}=0italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = 0. As vx,dB=vy,dB=0superscriptsubscript𝑣𝑥𝑑𝐵superscriptsubscript𝑣𝑦𝑑𝐵0v_{x,d}^{B}=v_{y,d}^{B}=0italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = 0, the target values in the absolute coordinate system are also vx,d=vy,d=0subscript𝑣𝑥𝑑subscript𝑣𝑦𝑑0v_{x,d}=v_{y,d}=0italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT = 0. Considering the weak yaw torque, the target angular velocity 𝝎dsubscript𝝎𝑑\boldsymbol{\omega}_{d}bold_italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT of the attitude control is derived from the target roll angle ϕdsubscriptitalic-ϕ𝑑\phi_{d}italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT, target pitch angle θdsubscript𝜃𝑑\theta_{d}italic_θ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT, and current yaw angle ψ𝜓\psiitalic_ψ.

The yaw angle ψ𝜓\psiitalic_ψ shown in Fig. 10 gradually deviates from the target value ψd=0subscript𝜓𝑑0\psi_{d}=0italic_ψ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0 but converges to a final value.The amount of the yaw drift is suppressed compared to the yaw angle of the three-paired-wing robot.On the other hand, because controlling the yaw angle reduces the lift force in the vertical direction,the altitude in Fig. 10 oscillates,but it can be controlled to tend to the target value of zd=0.05subscript𝑧𝑑0.05z_{d}=0.05italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 0.05 m.Furthermore, the translational velocity in Fig.10 is mostly controlled around the target value vx,d=vy,d=0subscript𝑣𝑥𝑑subscript𝑣𝑦𝑑0v_{x,d}=v_{y,d}=0italic_v start_POSTSUBSCRIPT italic_x , italic_d end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_y , italic_d end_POSTSUBSCRIPT = 0.The oscillation is caused by ignoring the infinitesimal forces fbody,xsubscript𝑓𝑏𝑜𝑑𝑦𝑥f_{body,x}italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_x end_POSTSUBSCRIPT and fbody,ysubscript𝑓𝑏𝑜𝑑𝑦𝑦f_{body,y}italic_f start_POSTSUBSCRIPT italic_b italic_o italic_d italic_y , italic_y end_POSTSUBSCRIPT.

VI CONCLUSION

In this study, a flapping-wing robot with four paired tilted wings to enable yaw control was fabricated. The wings are directly driven by piezoelectric actuators without transmission, and lift control is achieved simply by changing the voltage amplitude.However, it incurred an offset in the lift force; therefore, we designed an adaptive controller to alleviate the offset problem. Numerical experiments confirm that the proposed controller shows improved control performance compared to the LQI controller by adapting to unknown lift offsets. Finally, a tethered, controlled flight was performed, and the yaw drift was suppressed by the wing tilting arrangement and the proposed controller.

We reported the first takeoff of the lightest battery-powered, tailless, flapping-wing robot (2.1 g insect scale) in [22]. The robot was equipped with a low-power circuit specifically for digital duty ratio control and a sensor unit in addition to a LiPo battery; however, it was not a controlled flight. In the future, we will develop a hover-capable, battery-powered, insect-scale, tailless, flapping-wing robot weighing less than 10 g by adopting the wing tilt arrangement and the proposed controller into the system developed in [22]. Furthermore, to quickly control the yaw angle in experiments rather than in simulations, it is essential to improve yaw torque generation.

APPENDIX A: Proof of Theorem 1

Consider the following Lyapunov function candidate for the attitude control system:

Vω=12sω(TJ)sω+12τ~oΓω1τ~o.subscript𝑉𝜔12superscriptsubscript𝑠𝜔top𝑇𝐽subscript𝑠𝜔12superscriptsubscript~𝜏𝑜topsuperscriptsubscriptΓ𝜔1subscript~𝜏𝑜V_{\omega}=\frac{1}{2}s_{\omega}^{\top}(TJ)s_{\omega}+\frac{1}{2}\tilde{\tau}_%{o}^{\top}\Gamma_{\omega}^{-1}\tilde{\tau}_{o}.italic_V start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( italic_T italic_J ) italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG over~ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT .(21)

where τ~o=τ^oτosubscript~𝜏𝑜subscript^𝜏𝑜subscript𝜏𝑜\tilde{\tau}_{o}=\hat{\tau}_{o}-\tau_{o}over~ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT = over^ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT - italic_τ start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT.Note that the dynamics of sωsubscript𝑠𝜔s_{\omega}italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT is derived from (5), (10) and (14) with (12):

TJs˙ω+(J+Kω)sω=τ~o𝑇𝐽subscript˙𝑠𝜔𝐽subscript𝐾𝜔subscript𝑠𝜔subscript~𝜏𝑜TJ\dot{s}_{\omega}+(J+K_{\omega})s_{\omega}=\tilde{\tau}_{o}italic_T italic_J over˙ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT + ( italic_J + italic_K start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ) italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT = over~ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT(22)

The time derivative of (21) becomes a negative definite considering (13) and (22):

V˙ω=sω(TJs˙ω)+τ~oΓω1τ^˙o=sω(J+Kω)sω0.subscript˙𝑉𝜔superscriptsubscript𝑠𝜔top𝑇𝐽subscript˙𝑠𝜔superscriptsubscript~𝜏𝑜topsuperscriptsubscriptΓ𝜔1subscript˙^𝜏𝑜superscriptsubscript𝑠𝜔top𝐽subscript𝐾𝜔subscript𝑠𝜔0\dot{V}_{\omega}=s_{\omega}^{\top}(TJ\dot{s}_{\omega})+\tilde{\tau}_{o}^{\top}%\Gamma_{\omega}^{-1}\dot{\hat{\tau}}_{o}=-s_{\omega}^{\top}(J+K_{\omega})s_{%\omega}\leq 0.over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( italic_T italic_J over˙ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ) + over~ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over˙ start_ARG over^ start_ARG italic_τ end_ARG end_ARG start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT = - italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( italic_J + italic_K start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ) italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ≤ 0 .

As a result,the attitude control system is theoretically, globally, and asymptotically stable according to the invariant set theorem to satisfy sω0subscript𝑠𝜔0s_{\omega}\rightarrow 0italic_s start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT → 0 as t𝑡t\rightarrow\inftyitalic_t → ∞.

APPENDIX B: Proof of Theorem 2

Similar to Appendix A, consider the following Lyapunov function candidate for the vertical control system:

Vz=T2sz2+12γzf~o,z2subscript𝑉𝑧𝑇2superscriptsubscript𝑠𝑧212subscript𝛾𝑧superscriptsubscript~𝑓𝑜𝑧2V_{z}=\frac{T}{2}{s_{z}}^{2}+\frac{1}{2\gamma_{z}}{\tilde{f}_{o,z}}^{2}italic_V start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = divide start_ARG italic_T end_ARG start_ARG 2 end_ARG italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT(23)

where f~o,z=f^o,zfo,zsubscript~𝑓𝑜𝑧subscript^𝑓𝑜𝑧subscript𝑓𝑜𝑧\tilde{f}_{o,z}=\hat{f}_{o,z}-f_{o,z}over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT = over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT.Note that the dynamics of szsubscript𝑠𝑧s_{z}italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT is derived from (8), (9), and (17) or (18) with (15):

Ts˙z+(1+kz)sz=f~o,zm𝑇subscript˙𝑠𝑧1subscript𝑘𝑧subscript𝑠𝑧subscript~𝑓𝑜𝑧𝑚T\dot{s}_{z}+(1+k_{z})s_{z}=\frac{\tilde{f}_{o,z}}{m}italic_T over˙ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + ( 1 + italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = divide start_ARG over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG(24)

The time derivative of (23) becomes a negative definite considering (16) and (24):

V˙z=Tszs˙z+1γzf~o,zf^˙o,z=(1+kz)sz20.subscript˙𝑉𝑧𝑇subscript𝑠𝑧subscript˙𝑠𝑧1subscript𝛾𝑧subscript~𝑓𝑜𝑧subscript˙^𝑓𝑜𝑧1subscript𝑘𝑧superscriptsubscript𝑠𝑧20\dot{V}_{z}=Ts_{z}\dot{s}_{z}+\frac{1}{\gamma_{z}}\tilde{f}_{o,z}\dot{\hat{f}}%_{o,z}=-(1+k_{z})s_{z}^{2}\leq 0.over˙ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = italic_T italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT over˙ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT over˙ start_ARG over^ start_ARG italic_f end_ARG end_ARG start_POSTSUBSCRIPT italic_o , italic_z end_POSTSUBSCRIPT = - ( 1 + italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 0 .

As a result,the vertical control system is theoretically, globally, and asymptotically stable according to the invariant set theorem to satisfy sz0subscript𝑠𝑧0s_{z}\rightarrow 0italic_s start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT → 0 as t𝑡t\rightarrow\inftyitalic_t → ∞.

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Insect-Scale Tailless Robot with Flapping Wings: A Simple Structure and Drive for Yaw Control (2024)

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