Tomohiko Jimbo^{1}, Takashi Ozaki^{1}, Norikazu Ohta^{1} and Kanae Hamaguchi^{1}^{1}The 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.

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

### 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 $\beta$, which is the rotation angle around the Y-axis of the wing coordinate system.

The body force $f\in\mathbb{R}^{3}$ and torque ${\tau}\in\mathbb{R}^{3}$ are given by

$\begin{bmatrix}f\\\tau\end{bmatrix}=Mf_{w}$ | (1) |

where ${f}_{w}=[f_{1},f_{2},f_{3},f_{4}]^{\top}$ is the vector of wing forces generated by applying flapping amplitude $V$,and

$M=\begin{bmatrix}M_{1}\\M_{2}\end{bmatrix}=\begin{bmatrix}{e}_{1}&{e}_{2}&{e}_{3}&{e}_{4}\\{p}_{1}\times{e}_{1}&{p}_{2}\times{e}_{2}&{p}_{3}\times{e}_{3}&{p}_{4}\times{e%}_{4}\end{bmatrix}\in\mathbb{R}^{6\times 4}$ |

is the mixing matrix.Here, for the wing $i$,

$\displaystyle e_{i}=\begin{bmatrix}\cos\gamma_{i}\sin\beta_{i}\\\sin\gamma_{i}\sin\beta_{i}\\\cos\beta_{i}\end{bmatrix},p_{i}=\begin{bmatrix}a_{i}\\b_{i}\\0\end{bmatrix}+\begin{bmatrix}\cos\gamma_{i}\cos\beta_{i}\\\sin\gamma_{i}\cos\beta_{i}\\-\sin\beta_{i}\end{bmatrix},$ | ||

$\displaystyle a_{1}=a_{4}=a,\quad a_{2}=a_{3}=-a,$ | ||

$\displaystyle b_{1}=b_{2}=b,\quad b_{3}=b_{4}=-b,$ | ||

$\displaystyle\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=-\beta,$ | ||

$\displaystyle\gamma_{1}=\gamma,\quad\gamma_{2}=\pi-\gamma,\quad\gamma_{3}=\pi+%\gamma,\quad\gamma_{4}=2\pi-\gamma,$ |

$a$ and $b$ are the lengths of the body along the X- and Y-axes, respectively,$\gamma$ is the rotation angle around the Z-axis of the body coordinate system, and$l$ 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 $f_{body}(=[f_{body,x},f_{body,y},f_{body,z}]^{\top})$ and torque $\tau_{body}$ are given by

$\begin{bmatrix}{f}_{body}\\{\tau}_{body}\end{bmatrix}=\biggl{(}{M}+\delta_{M}\biggr{)}\biggl{(}{f}_{w}+%\delta_{f_{w}}\biggr{)}=\begin{bmatrix}{f}-{f}_{o}\\{\tau}-{\tau}_{o}\end{bmatrix},$ | (2) |

wherethe offset force and torque are given by

$\begin{bmatrix}{f}_{o}\\{\tau}_{o}\end{bmatrix}\approx-{M}\delta_{f_{w}}-\delta_{M}{f}_{w}.$ | (3) |

They depend on $\delta_{f_{w}}\in\mathbb{R}^{4}$ and $\delta_{M}\in\mathbb{R}^{6\times 4}$.$\delta_{f_{w}}$ corresponds to the error of the wing force voltage modeling $f_{w}=h(V)$.$\delta_{M}$ is the misalignment with respect to $\beta$, $\gamma$, and $l$.

### II-C Flight Dynamics

Consider the rigid-body model of the robot shown in Fig. 2.Using the actual force and torque, ${f}_{body}$ and ${\tau}_{body}$ in (2),the dynamics are represented by

$\displaystyle m\dot{{v}}$ | $\displaystyle=$ | $\displaystyle{R}{f}_{body}-m{g},$ | (4) | ||

$\displaystyle{J}\dot{{\omega}}$ | $\displaystyle=$ | $\displaystyle{\tau}_{body}-\left({\omega}\times{J}{\omega}\right),$ | (5) |

where $v(=[v_{x},v_{y},v_{z}]^{\top})$ is the translational velocity of the body in the global coordinate system,$\omega(=[\omega_{B,x},\omega_{B,y},\omega_{B,z}]^{\top})$ is the angular velocity of the body in the body coordinate system,$m$ and ${J}\in\mathbb{R}^{3\times 3}$ are the mass and inertia of the body, respectively,${R}$ is the rotation matrix,${g}=[0,0,g]^{\top}$, $g$ is the gravitational acceleration.

The relationship between $\omega$ and the attitude $\eta=[\phi,\theta,\psi]^{\top}$ of the body is described by

${\omega}={G}\dot{{\eta}},$ | (6) |

where

${G}=\begin{bmatrix}1&0&-\sin\theta\\0&\cos\phi&\cos\theta\sin\phi\\0&-\sin\phi&\cos\theta\cos\phi\end{bmatrix}.$ |

### II-D Control-Oriented Model

The translational dynamics on the horizontal plane is derived from (4), assuming the hovering state, $v_{z}\approx 0$, and $\omega_{z}\approx 0$, to obtain

$\begin{bmatrix}\dot{v}_{x}^{B}\\\dot{v}_{y}^{B}\end{bmatrix}=\begin{bmatrix}\theta\\-\phi\end{bmatrix}g,$ | (7) |

where $v_{x}^{B}$ and $v_{y}^{B}$ are the translational velocities in the body coordinate system.

When approximating the rotation matrix $R$ using $\phi$ and $\theta$, the dynamics in the vertical direction, that is, along the Z-axis, is given by

$m\ddot{z}=f_{z}-f_{o,z}-mg,$ | (8) |

where $z$ is the altitude, that is, the vertical position of the body.Note that we assume $|\theta mg|\gg|f_{body,x}|$, $|\phi mg|\gg|f_{body,y}|$, and $|f_{body,z}|\gg|-\theta f_{body,x}+\phi f_{body,y}|$.

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:

$\displaystyle T\dot{f}_{z}$ | $\displaystyle=$ | $\displaystyle f_{d,z}-f_{z},$ | (9) | ||

$\displaystyle T\dot{{\tau}}$ | $\displaystyle=$ | $\displaystyle{\tau}_{d}-{\tau},$ | (10) |

where $T(>0)$ is a time constant and $f_{d,z}\in\mathbb{R}$ and ${\tau}_{d}\in\mathbb{R}^{3}$ 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 $f_{w}$ and $V$ occurs due to damage during use.Therefore, in this study, an adaptive controller is adopted.

### III-A Velocity Control

To track the velocities $v_{x}^{B}$ and $v_{y}^{B}$ on the horizontal plane to the target $v_{x,d}^{B}$ and $v_{y,d}^{B}$, respectively,considering (7),the corresponding target angles of the pitch $\theta_{d}$ and roll $\phi_{d}$ are set to

$\displaystyle\theta_{d}$ | $\displaystyle=$ | $\displaystyle-h_{x}\left(v_{x}^{B}-v_{x,d}^{B}\right)/g,$ | ||

$\displaystyle\phi_{d}$ | $\displaystyle=$ | $\displaystyle h_{y}\left(v_{y}^{B}-v_{y,d}^{B}\right)/g,$ |

where $h_{x}$ and $h_{y}$ are positive constants.

### III-B Attitude Control

To track the attitude ${\eta}$ to the target ${\eta}_{d}(=[\phi_{d},\theta_{d},\psi_{d}]^{\top})$,from (6),the target angular velocity of the robot $\omega$ and higher-order derivatives are designed as follows:

$\displaystyle{\omega}_{d}$ | $\displaystyle=$ | $\displaystyle-G{K}_{\eta}{e}_{\eta},$ | (11) | ||

$\displaystyle\dot{{\omega}}_{d}$ | $\displaystyle\approx$ | $\displaystyle-G{K}_{\eta}\dot{{\eta}},$ | |||

$\displaystyle\ddot{{\omega}}_{d}$ | $\displaystyle\approx$ | $\displaystyle-G{K}_{\eta}\ddot{{\eta}},$ |

where ${e}_{\eta}={\eta}-{\eta}_{d}$,${K}_{\eta}(={\rm diag}(k_{\eta 1},k_{\eta 2},k_{\eta 3}))$ is a positive diagonal matrix.

To track the angular velocity ${\omega}$ to the target ${\omega}_{d}$,from (5) and (10),the demanded torque ${\tau}_{d}$ and the adaptive law to estimate the offset torque $\hat{{\tau}}_{o}$ are designed as

$\displaystyle{\tau}_{d}$ | $\displaystyle=$ | $\displaystyle-{K}_{\omega}{s}_{\omega}+J\dot{{\omega}}_{r}+{F}+T\left(J\ddot{{%\omega}}_{r}+\dot{{F}}\right)+\hat{{\tau}}_{o},$ | (12) | ||

$\displaystyle\dot{\hat{{\tau}}}_{o}$ | $\displaystyle=$ | $\displaystyle-{\Gamma}_{\omega}{s}_{\omega},$ | (13) |

where $F(=\omega\times J\omega)$ is the centrifugal Coriolis force,

$\displaystyle{s}_{\omega}$ | $\displaystyle=$ | $\displaystyle\dot{{e}}_{\omega}+{\Lambda}_{\omega}{e}_{\omega}=\dot{{\omega}}-%\dot{{\omega}}_{r},$ | (14) | ||

$\displaystyle{e}_{\omega}$ | $\displaystyle=$ | $\displaystyle{\omega}-{\omega}_{d},$ | |||

$\displaystyle\dot{{\omega}}_{r}$ | $\displaystyle=$ | $\displaystyle\dot{{\omega}}_{d}-{\Lambda}_{\omega}({\omega}-{\omega}_{d}),$ |

${\Lambda}_{\omega}(={\rm diag}(\lambda_{\omega 1},\lambda_{\omega 2},\lambda_{%\omega 3}))$,${K}_{\omega}(={\rm diag}(k_{\omega 1},k_{\omega 2},k_{\omega 3}))$, and${\Gamma}_{\omega}(={\rm diag}(\gamma_{\omega 1},\gamma_{\omega 2},\gamma_{%\omega 3}))$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 $s_{\omega}$ satisfies $s_{\omega}\rightarrow 0$ as $t\rightarrow\infty$.

###### Proof.

See Appendix A for the proof.∎

From Theorem 1, considering (14), $\omega$ asymptotically converges to $\omega_{d}$ as $t\rightarrow\infty$.Similarly, from (6) and (11), $\eta$ asymptotically converges to $\eta_{d}$ as $t\rightarrow\infty$.

### III-C Vertical Control

To control the flapping-wing robot in the vertical direction,from (8) and (9),the demanded force $f_{d,z}$ and the adaptive law to estimate the offset force $\hat{f}_{o,z}$ are designed as

$\displaystyle\frac{f_{d,z}}{m}$ | $\displaystyle=$ | $\displaystyle-k_{z}s_{z}+\ddot{z}_{r}+T\dddot{z}_{r}+g+\frac{\hat{f}_{o,z}}{m},$ | (15) | ||

$\displaystyle\dot{\hat{f}}_{o,z}$ | $\displaystyle=$ | $\displaystyle-\frac{\gamma_{z}}{m}s_{z},$ | (16) |

where, for the vertical velocity control to track $\dot{z}$ to the target $\dot{z}_{d}$,

$\displaystyle s_{z}$ | $\displaystyle=$ | $\displaystyle\ddot{e}_{z}+\lambda_{z}\dot{e}_{z}=\ddot{z}-\ddot{z}_{r},$ | (17) | ||

$\displaystyle\ddot{z}_{r}$ | $\displaystyle=$ | $\displaystyle-\lambda_{z}(\dot{z}-\dot{z}_{d}),$ |

and for the vertical position (altitude) control to track $z$ to the target $z_{d}$,

$\displaystyle s_{z}$ | $\displaystyle=$ | $\displaystyle\ddot{e}_{z}+2\lambda_{z}\dot{e}_{z}+\lambda_{z}^{2}e_{z}=\ddot{z%}-\ddot{z}_{r},$ | (18) | ||

$\displaystyle\ddot{z}_{r}$ | $\displaystyle=$ | $\displaystyle-2\lambda_{z}\dot{z}-\lambda_{z}^{2}(z-z_{d}).$ |

Here, $e_{z}=z-z_{d}$.$\lambda_{z}$, $k_{z}$, and $\gamma_{z}$ 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 $s_{z}$ satisfies $s_{z}\rightarrow 0$ as $t\rightarrow\infty$.

###### Proof.

See Appendix B for the proof.∎

From Theorem 2, considering (18), $z$ asymptotically converges to $z_{d}$ as $t\rightarrow\infty$.

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

From the demanded torque $\tau_{d}$ and force $f_{d}$ in (12) and (15) as well as the mixing matrix in (1), the lift-force demand is given by

${f}_{w,d}=\begin{bmatrix}{M}_{1}(3,:)\\{M}_{2}\end{bmatrix}^{-1}\begin{bmatrix}{f}_{d,z}\\\boldsymbol{\tau}_{d}\end{bmatrix}.$ | (19) |

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

$f_{w}=h(V).$ | (20) |

As a result, the required flapping amplitude $V_{d}\in\mathbb{R}^{4}$ is derived from (19) and (20) as $V_{d}=h^{-1}(f_{w,d})$.Then, $V_{d}$ 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, $J$ 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].

Name | Symbol | Value | Unit |
---|---|---|---|

Boby mass | $m$ | $2.0*10^{-3}$ | $\rm kg$ |

Body intertia | $J_{1}$ | $1.50*10^{-7}$ | $\rm kg\cdot m^{2}$ |

$(J={\rm diag}(J_{1},J_{2},J_{3}))$ | $J_{2}$ | $1.35*10^{-7}$ | $\rm kg\cdot m^{2}$ |

$J_{3}$ | $2.21*10^{-7}$ | $\rm kg\cdot m^{2}$ | |

Body lengths | $a$ | $20.0*10^{-3}$ | $\rm m$ |

$b$ | $5.0*10^{-3}$ | $\rm m$ | |

Wing angles | $\beta$ | $20*\pi/180$ | $\rm rad$ |

$\gamma$ | $60*\pi/180$ | $\rm rad$ | |

Length between lift center | |||

and wing root | $l$ | $40.0*10^{-3}$ | $\rm m$ |

Time constant of first-order lag | $T$ | $0.013$ | sec |

Gravitational acceleration | $g$ | $9.81$ | $\rm kg\cdot m/sec^{2}$ |

Symbol | Value | Unit | |
---|---|---|---|

Attitude | $h_{x}$ | $1/0.5$ | $\rm 1/sec$ |

$h_{y}$ | $1/0.5$ | $\rm 1/sec$ | |

$k_{\eta 1}$ | $1/0.1$ | $\rm 1/sec$ | |

$k_{\eta 2}$ | $1/0.1$ | $\rm 1/sec$ | |

$k_{\eta 3}$ | $1/0.1$ | $\rm 1/sec$ | |

$\lambda_{\omega 1}$ | $1/0.1$ | $\rm 1/sec$ | |

$\lambda_{\omega 2}$ | $1/0.1$ | $\rm 1/sec$ | |

$\lambda_{\omega 3}$ | $1/0.1$ | $\rm 1/sec$ | |

$k_{\omega 1}$ | $9.50*10^{-8}$ | - | |

$k_{\omega 2}$ | $8.55*10^{-8}$ | - | |

$k_{\omega 3}$ | $1.40*10^{-7}$ | - | |

$\gamma_{\omega 1}$ | $7.70*10^{-6}$ | - | |

$\gamma_{\omega 2}$ | $6.93*10^{-6}$ | - | |

$\gamma_{\omega 3}$ | $1.13*10^{-5}$ | - | |

$\hat{\tau}_{o,x}(t=0)$ | $0$ | ${\rm Nm}$ | |

$\hat{\tau}_{o,y}(t=0)$ | $0$ | ${\rm Nm}$ | |

$\hat{\tau}_{o,z}(t=0)$ | $0$ | ${\rm Nm}$ | |

Altitude | $\lambda_{z}$ | $1/0.5$ | $\rm 1/sec$ |

$k_{z}$ | $6.34*10^{-1}$ | - | |

$\gamma_{z}$ | $2.05*10^{-4}$ | - | |

$\hat{f}_{o,z}(t=0)$ | $0$ | ${\rm N}$ |

Symbol | Value | Unit |
---|---|---|

$\delta\beta$ | ${10}*\pi/180$ | $\rm rad$ |

$\delta\gamma$ | ${10}*\pi/180$ | $\rm rad$ |

$\delta l$ | $5*10^{-3}$ | $\rm m$ |

${\delta f}_{w}$ | $mg/4/{3}*[0,-1,0,0]^{\top}$ | $\rm N$ |

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 $\delta\beta$ and $\delta\gamma$ that occur during manufacturing were both set to 10 degrees.The error $\delta l$ in the distance $l$ 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 ${\delta f}_{w}$ in the lift-force model of (20).In this case, only the second wing has a modeling error of one-third of $mg/4$, 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 $\hat{\tau}_{o}$ and force $\hat{f}_{o,z}$, estimated previously in Case 2, were used as the initial values of the parameter adaptation laws in equations (13) and (16).

### IV-B Comparison Results

First, we performed a numerical experiment without the offsets in Table III.Here, a target value of $\boldsymbol{r}=[v_{x,d}^{B},v_{y,d}^{B},v_{z,d}^{B},\psi_{d}]^{\top}=[0.5,0.5,%0.5,1]^{\top}$ 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 $1$ 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 $v_{z}$ 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 $v_{x}$ and $v_{z}$ occurred for the LQI controller.$v_{y}$ had an overshoot.In contrast, for the proposed controller, the response of $v_{z}$ was the same as that in Fig. 6.However, a reverse response of $v_{x}$ occurred, and $v_{y}$ 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.

## 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

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 $3$ 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 $\psi_{d}=0$, $z_{d}=0.05$, and $v_{x,d}^{B}=v_{y,d}^{B}=0$. As $v_{x,d}^{B}=v_{y,d}^{B}=0$, the target values in the absolute coordinate system are also $v_{x,d}=v_{y,d}=0$. Considering the weak yaw torque, the target angular velocity $\boldsymbol{\omega}_{d}$ of the attitude control is derived from the target roll angle $\phi_{d}$, target pitch angle $\theta_{d}$, and current yaw angle $\psi$.

The yaw angle $\psi$ shown in Fig. 10 gradually deviates from the target value $\psi_{d}=0$ 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 $z_{d}=0.05$ m.Furthermore, the translational velocity in Fig.10 is mostly controlled around the target value $v_{x,d}=v_{y,d}=0$.The oscillation is caused by ignoring the infinitesimal forces $f_{body,x}$ and $f_{body,y}$.

## 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:

$V_{\omega}=\frac{1}{2}s_{\omega}^{\top}(TJ)s_{\omega}+\frac{1}{2}\tilde{\tau}_%{o}^{\top}\Gamma_{\omega}^{-1}\tilde{\tau}_{o}.$ | (21) |

where $\tilde{\tau}_{o}=\hat{\tau}_{o}-\tau_{o}$.Note that the dynamics of $s_{\omega}$ is derived from (5), (10) and (14) with (12):

$TJ\dot{s}_{\omega}+(J+K_{\omega})s_{\omega}=\tilde{\tau}_{o}$ | (22) |

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

$\dot{V}_{\omega}=s_{\omega}^{\top}(TJ\dot{s}_{\omega})+\tilde{\tau}_{o}^{\top}%\Gamma_{\omega}^{-1}\dot{\hat{\tau}}_{o}=-s_{\omega}^{\top}(J+K_{\omega})s_{%\omega}\leq 0.$ |

As a result,the attitude control system is theoretically, globally, and asymptotically stable according to the invariant set theorem to satisfy $s_{\omega}\rightarrow 0$ as $t\rightarrow\infty$.

## APPENDIX B: Proof of Theorem 2

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

$V_{z}=\frac{T}{2}{s_{z}}^{2}+\frac{1}{2\gamma_{z}}{\tilde{f}_{o,z}}^{2}$ | (23) |

where $\tilde{f}_{o,z}=\hat{f}_{o,z}-f_{o,z}$.Note that the dynamics of $s_{z}$ is derived from (8), (9), and (17) or (18) with (15):

$T\dot{s}_{z}+(1+k_{z})s_{z}=\frac{\tilde{f}_{o,z}}{m}$ | (24) |

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

$\dot{V}_{z}=Ts_{z}\dot{s}_{z}+\frac{1}{\gamma_{z}}\tilde{f}_{o,z}\dot{\hat{f}}%_{o,z}=-(1+k_{z})s_{z}^{2}\leq 0.$ |

As a result,the vertical control system is theoretically, globally, and asymptotically stable according to the invariant set theorem to satisfy $s_{z}\rightarrow 0$ as $t\rightarrow\infty$.

## References

- [1]M.H. Dickinson, F.-O. Lehmann, and S.P. Sane, “Wing rotation and the aerodynamic basis of insect flight,”
*Science*, vol. 284, pp. 1954–1960, 1999. - [2]C.T. Orlowski and A.R. Girard, “Dynamics, stability, and control analyses of flapping wing micro-air vehicles,”
*Progress in Aerospace Sciences*, vol.51, pp. 18–30, 2012. - [3]E.F. Helbling and R.J. Wood, “A review of propulsion, power, and control architectures for insect-scale flapping-wing vehicles,”
*Applied Mechanics Reviews*, vol.70, no.1, p. 010801, 2018. - [4]H.V. Phan and H.C. Park, “Insect-inspired, tailless, hover-capable flapping-wing robots: Recent progress, challenges, and future directions,”
*Progress in Aerospace Sciences*, vol. 111, p. 100573, 2019. - [5]G.C. H.E. deCroon, K.M.E. deClercq, R.Ruijsink, B.Remes, and C.deWagter, “Design, aerodynamics, and vision-based control of the DelFly,”
*International Journal of Micro Air Vehicles*, vol.1, no.2, pp. 71–97, 2009. - [6]M.Keennon, K.Klingebiel, H.Won, and A.Andriukov, “Development of the nano hummingbird: A tailless flapping wing micro air vehicle,”
*Proceedings on 50th AIAA Aerosp. Sci. Meeting Including New Horizons Forum Aerosp.*, pp. 1–24, 2012. - [7]Q.-V. Nguyen, W.L. Chan, and D.Marco, “An insect-inspired flapping wing micro air vehicle with double wing clap-fling effects and capability of sustained hovering,”
*Proceedings of the SPIE*, pp. 136––146, 2015. - [8]M.Karásek, F.T. Muijres, C.D. Wagter, B.D.W. Remes, and G.C. H.E. deCroon, “A tailless aerial robotic flapper reveals that flies use torque coupling in rapid banked turns,”
*Science*, pp. 1089––1094, 2018. - [9]H.V. Phan, S.Aurecianus, T.Kang, and H.C. Park, “Kubeetle-s: An insect-like, tailless, hover-capable robot that can fly with a low-torque control mechanism,”
*International Journal of Micro Air Vehicles*, pp. 1089––1094, 2019. - [10]R.J. Wood, “The first takeoff of a biologically inspired at-scale robotic insect,”
*IEEE Transactions on Robotics*, vol.24, no.2, pp. 341–347, 2008. - [11]K.Y. Ma, P.Chirarattananon, S.B. Fuller, and R.J. Wood, “Controlled flight of a biologically inspired, insect-scale robot,”
*Science*, vol. 340, no. 6132, pp. 603–607, May 2013. - [12]T.Ozaki and K.Hamaguchi, “Bioinspired flapping-wing robot with direct-driven piezoelectric actuation and its takeoff demonstration,”
*IEEE Robotics and Automation Letters*, vol.3, no.4, pp. 4217–4224, 2018. - [13]Y.Chen, H.Zhao, P.C. JieMao, E.F. Helbling, N.seung PatrickHyun, D.R. Clarke, and R.J. Wood, “Controlled flight of a microrobot powered by soft artificial muscles,”
*Nature*, vol. 575, pp. 324––329, 2019. - [14]E.F. Helbling, S.B. Fuller, and R.J. Wood, “Pitch and yaw control of a robotic insect using an onboard magnetometer,”
*Proceedings on IEEE International Conference on Robotics and Automation (ICRA)*, pp. 5516–5522, 2014. - [15]Z.E. Teoh, “Design of hybrid passive and active mechanisms for control of insect-scale flapping-wing robots,”
*Ph.D. dissertation*, 2015. - [16]S.B. Fuller, “Four wings: An insect-sized aerial robot with steering ability and payload capacity for autonomy,”
*IEEE Robotics and Automation Letters*, vol.4, no.2, pp. 570–577, 2019. - [17]R.Steinmeyer, N.seung P.Hyun, E.F. Helbling, and R.J. Wood, “Yaw torque authority for a flapping-wing micro-aerial vehicle,”
*Proceedings on IEEE International Conference on Robotics and Automation (ICRA)*, pp. 2481–2487, 2019. - [18]Y.M. Chukewad and S.Fuller, “Yaw control of a hovering flapping-wing aerial vehicle with a passive wing hinge,”
*IEEE Robotics and Automation Letters*, vol.6, no.2, pp. 1864–1871, 2021. - [19]T.Ozaki and K.Hamaguchi, “Performance of direct-driven flapping-wing actuator with piezoelectric single-crystal PIN-PMN-PT,”
*Journal of Micromechanics and Microengineering*, vol.28, 2018. - [20]T.Ozaki and K.Hamaguchi, “Electro-aero-mechanical model of piezoelectric direct-driven flapping-wing actuator,”
*Applied Sciences*, vol.8, no.9, p. 1699, 2018. - [21]T.Jimbo, T.Ozaki, Y.Amano, and K.Fujimoto, “Flight control of flapping-wing robot with three paired direct-driven piezoelectric actuators,”
*Proceedings of 21st IFAC World Congress*, p. 936, 2020. - [22]T.Ozaki, N.Ohta, T.Jimbo, and K.Hamaguchi, “Takeoff of a 2.1 g fully untethered tailless flapping-wing micro aerial vehicle with integrated battery,”
*IEEE Robotics and Automation Letters*, vol.8, no.6, pp. 3574–3580, 2023.