# Simplified Model for Forward-Flight Transitions of a Bio-Inspired Unmanned Aerial Vehicle

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## Abstract

**:**

## 1. Introduction

## 2. Flight Model

#### 2.1. Non-Dimensional Newton–Euler Equations

#### 2.2. Aerodynamic Characterization

## 3. Numerical Results and Experimental Validation

#### 3.1. Numerical Results

#### 3.2. Experimental Validation

## 4. Analytical Perturbation Solution and Its Comparison with the Numerical Solution

#### 4.1. Scales Analysis

- Small flight angles: $\gamma \sim \theta \sim \alpha \sim {\delta}_{t}\sim \u03f5$. Reduced angles of attack are required by the present aerodynamic formulation, as well as small deflections of the tail. All of them are assumed to be in the same order of magnitude as the flapping amplitude $\u03f5$. Small $\gamma $ and $\theta $ mean that aggressive climbing trajectories are not considered. Moreover, small flapping amplitude implies a limited thrust, which is proportional to ${\u03f5}^{2}$ (see below), so reaching large flight path angles would be possible only with high reduced frequencies, which are difficult to obtain in bird-scale ornithopters.
- Thrust coefficient: ${C}_{T}^{*}\sim {\u03f5}^{2}$. Clearly from the Formulation (9), with products of the aerodynamic angles and the flapping amplitude.
- Drag coefficients: ${C}_{D}^{*}\sim {C}_{Dt}^{*}\sim L{i}^{*}\sim {\u03f5}^{2}$. This is clear for the induced drag, as it is proportional to the square of the lift coefficient. On the other hand, friction drag cannot be larger than thrust to maintain flight.
- Non-dimensional inertia and reduced frequency: ${\mathcal{M}}^{2}\chi \sim \mathcal{M}{k}_{0}\sim 1$. Given in Table 1. Even though the frequency is a control variable, we can consider both terms of order unity in the typical range of bird-like ornithopters.
- All the remaining non-dimensional parameters either do not affect the structure of the solution or are order unity.

#### 4.2. Perturbation Solution

#### 4.3. Comparison between Analytical and Numerical Results

- For the large-time asymptotic values of the velocity and the pitch angle, we consider terms up to order ${\u03f5}^{2}$, while for the angle of attack, the approximation is even better, up to order ${\u03f5}^{3}$.
- Transient terms are computed up to order $\u03f5$ for the velocity and the pitch angle, and to order ${\u03f5}^{2}$ for the angle of attack.
- Harmonic oscillations with the flapping frequencies are considered up to order ${\u03f5}^{2}$ for the three variables, including asymptotic and transient values.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Aerodynamic Coefficients in Equations (9) and (10)

## Appendix B. Higher Order Perturbation Terms

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**Figure 1.**Scheme of forces, axis, and control and state variables. All vectors are on the plane ($X,Z$) of the longitudinal flight.

**Figure 2.**Transient phase of non-dimensional velocity U for three different frequencies. (

**a**) Normalized velocity vs. non-dimensional time; (

**b**) normalized velocity vs. time in seconds.

**Figure 6.**Comparison between theoretical and numerical final state at 5Hz. (

**a**) Airspeed; (

**b**) pitch angle.

**Figure 7.**Comparison between theoretical and numerical transient phase at 5Hz. (

**a**) Airspeed evolution; (

**b**) pitch angle evolution.

f | $\mathcal{M}{\mathit{k}}_{0}$ | $\mathcal{M}$ | ${\mathcal{M}}^{2}\mathit{\chi}$ | $\mathbf{\Lambda}$ | ${\mathit{l}}_{\mathit{w}}$ | ${\mathit{l}}_{\mathit{t}}\mathbf{\Lambda}$ | ${\mathit{h}}_{\mathit{w}}$ | ${\mathbf{Li}}^{*}$ | |
---|---|---|---|---|---|---|---|---|---|

2 Hz | $0.79$ | $2.54$ | $2.12$ | $0.25$ | $0.55$ | $-1.16$ | $0.38$ | $0.0048$ | $5.14$ |

5 Hz | $1.98$ | $2.54$ | $2.12$ | $0.25$ | $0.55$ | $-1.16$ | $0.38$ | $0.0048$ | $5.14$ |

7 Hz | $2.77$ | $2.54$ | $2.12$ | $0.25$ | $0.55$ | $-1.16$ | $0.38$ | $0.0048$ | $5.14$ |

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**MDPI and ACS Style**

Sanchez-Laulhe, E.; Fernandez-Feria, R.; Ollero, A.
Simplified Model for Forward-Flight Transitions of a Bio-Inspired Unmanned Aerial Vehicle. *Aerospace* **2022**, *9*, 617.
https://doi.org/10.3390/aerospace9100617

**AMA Style**

Sanchez-Laulhe E, Fernandez-Feria R, Ollero A.
Simplified Model for Forward-Flight Transitions of a Bio-Inspired Unmanned Aerial Vehicle. *Aerospace*. 2022; 9(10):617.
https://doi.org/10.3390/aerospace9100617

**Chicago/Turabian Style**

Sanchez-Laulhe, Ernesto, Ramon Fernandez-Feria, and Anibal Ollero.
2022. "Simplified Model for Forward-Flight Transitions of a Bio-Inspired Unmanned Aerial Vehicle" *Aerospace* 9, no. 10: 617.
https://doi.org/10.3390/aerospace9100617