# Derivation and Flight Test Validation of Maximum Rate of Climb during Takeoff for Fixed-Wing UAV Driven by Propeller Engine

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of Maximum Rate of Climb Using Optimization Problems

#### 2.1. Equations of Motion during Climb

#### 2.2. Application of Optimization Problems to Climbing Airplane

_{1}, λ

_{2}. The thrust T is a function of the airspeed V, and the lift L and the drag D are also a function of the airspeed V and the angle of attack α.

_{1}, and λ

_{2}as variables, they are to be solved numerically. Numerical analysis was performed using MATLAB software (version: R2020b) [17].

#### 2.3. Formulation of Propeller Engine Thrust

#### 2.4. Target Airplane

#### 2.5. Numerical Result of Maximum Rate of Climb

## 3. Verification by Using 6-DOF Flight Simulation

#### 3.1. Simulation Condition

_{R}) was set to be equal to the stall speed (V

_{s1}) in the takeoff configuration obtained from Equation (14), as in reference [22]. The switching speed from the rotation phase to the climb phase (safety takeoff speed V

_{2}) was set to be 1.2 times the stall speed from reference [24], and these values are shown in Table 5. In addition, the simulation time was set to be 20 s by using the result of the takeoff of flight experiment in reference [4] in which the UAV used flew at a lower speed.

#### 3.2. Flight Control System for Model Airplane during Takeoff

_{R}), the airplane transits from the run phase to the rotation phase. The block diagram of the rotation control system is shown in Figure 6. In the lateral-directional control system, the ailerons are controlled to keep the airplane horizontal. In the longitudinal control system, the nose is raised to a designated angle to increase the lift until the airplane leaves the ground. The pitch angle during this phase is set to be 10 degrees with 2 degrees of margin to prevent the tail of the airplane from hitting the ground during the run.

_{2}), the airplane transits from the rotation phase to the climb phase. The block diagram of the climb control system is shown in Figure 7. During the climb phase, in the lateral-directional control system, the airplane climbs straight onto the runway, i.e., the lateral position of the airplane is maintained to be on the centerline of the runway by controlling the ailerons. The longitudinal control system aims at achieving a steady climb at the maximum rate of climb. During this steady climb, the airplane’s motion is balanced; if one of the airspeed, path angle, and angle of attack is determined, the remaining two values are uniquely determined. Therefore, the pitch angle is controlled with maximum thrust so that the airspeed matches the one (15.1 m/s) at which the maximum rate of climb can be achieved. When the airplane’s airspeed is greater than that of the optimal solution, the pitch angle is increased so as to convert kinetic energy to potential energy. On the other hand, if the airplane’s airspeed is below the optimal solution, the pitch angle is reduced so as to convert potential energy to kinetic energy. Here, the pitch angle of the optimal solution is added to the pitch angle command at the start of the climb phase. Furthermore, each control system uses PID controllers. The throttle is fixed at position where it generates maximum thrust during takeoff. However, the actual magnitude of the thrust in this case follows Equation (13). The airspeed used in the above flight control is measured by the ADS, and the attitude angle and position by the INS.

#### 3.3. Results of 6-DOF Flight Simulation

## 4. Flight Experiment

#### 4.1. Judgement Conditions

#### 4.2. Experimental Results

#### 4.3. Comparison of Flight Experiment and 6-DOF Flight Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

g | gravitational acceleration, m/s^{2} | c | chord length, m |

$\mathsf{\rho}$ | atmospheric density, kg/m^{3} | e | Oswald efficiency number |

$\mathrm{L}$ | lift, N | AR | aspect ratio |

$\mathrm{D}$ | drag, N | S | wing area, m^{2} |

$\mathrm{W}$ | weight, N | J | evaluation function |

$\mathsf{\alpha}$ | angle of attack, rad | H | Hamiltonian |

$\mathsf{\gamma}$ | path angle, rad | ${\mathsf{\lambda}}_{1},{\mathsf{\lambda}}_{2}$ | adjoint variable |

${\mathrm{C}}_{\mathrm{L}0}$ | coefficient of lift | V | airspeed, m/s |

${\mathrm{C}}_{\mathrm{L}\mathsf{\alpha}}$ | lift per unit AoA, 1/rad | ${\mathrm{V}}_{\mathrm{R}}$ | rotation speed, m/s |

${\mathrm{C}}_{\mathrm{D}0}$ | coefficient of drag | ${\mathrm{V}}_{2}$ | takeoff safety speed, m/s |

${\mathsf{\alpha}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}}$ | stall angle | T | thrust, N |

m | mass, kg | $\mathsf{\theta}$ | pitch angle, rad |

$\mathsf{\delta}$ | angle of aileron, elevator, steer | $\mathsf{\varphi}$ | roll angle, rad |

$\mathsf{\psi}$ | azimuth angle, rad | ||

Subscripts | |||

e | elevator | opt | optimal solution |

a | aileron | cmd | command |

$\mathrm{s}$ | steer |

## References

- Chauhan, S.S.; Martins, J.R. Tilt-Wing eVTOL Takeoff Trajectory Optimization. J. Aircr.
**2019**, 57, 93–112. [Google Scholar] [CrossRef] - Kaneko, S.; Martins, J.R. Simultaneous Optimization of Conceptual Design and Takeoff Trajectory of a Lift-Plus-Cruise UAV. In Proceedings of the 10th Autonomous VTOL Technical Meeting, Mesa, AZ, USA, 24–26 January 2023. [Google Scholar]
- Gao, J.; Zhang, Q.; Chen, J.; Wang, X. Take-Off Trajectory Optimization of Tilt-rotor Aircraft Based on Direct Allocation Method. IOP Conf. Ser. Mater. Sci. Eng.
**2022**, 768, 042004. [Google Scholar] [CrossRef] - Masazumi, U.; Tomohiro, K.; Sakurako, N.; Yusuke, M. Verification of Fully Autonomous Flight from Takeoff to Landing of a Low-Speed Model Airplane with Application to a Small Unmanned Supersonic Airplane. Trans. Jpn. Soc. Aeronaut. Space Sci. Aerosp. Technol. Jpn.
**2021**, 9, 667–675. [Google Scholar] - Ryuichi, I.; Seiji, T. Full Automatic Randing System and Long-Distance Flight of the Fixed-Wing UAV. In Proceedings of the 50th Automatic Control Union Conference, Yokohama, Japan, 24–25 November 2007. [Google Scholar]
- Meng, X.; Xu, Z.; Chang, M.; Bai, J. Performance Analysis and Flow Mechanism of Channel Wing Considering Propeller Slipstream. Chin. J. Aeronaut.
**2023**, 36, 165–184. [Google Scholar] [CrossRef] - Courtin, C.B. An Assessment of Electric STOL Aircraft. Master’s Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2019. [Google Scholar]
- Federal Aviation Administration. Instrument Procedures Handbook; Federal Aviation Administration: Washington, DC, USA, 2017; p. 33. [Google Scholar]
- Wing Commander; Kelly, L. Optimum Climb Technique for a Jet Propelled Aircraft; College of Aeronautics, Cranfield: Oxon, UK, 1952. [Google Scholar]
- Bryson, A.E.; Denham, W.F. A Steepest-Ascent Method for Solving Optimum Programming Problems. J. Appl. Mech.
**1962**, 29, 247–257. [Google Scholar] [CrossRef] - Ong, S.Y. A Model Comparison of a Supersonic Aircraft Minimum Time-to-Climb Problem. Master’s Thesis, Iowa State University of Science and Technology, Ames, IA, USA, 1986. [Google Scholar]
- Tomoyuki, Y.; Kanichiro, K. Minimum Time Climb for Airplane [Translated from Japanese]. In Proceedings of the 20th Airplane Symposium, Tokyo, Japan, 12 November 1982. [Google Scholar]
- Bryson, E., Jr.; Ho, Y.-C. Applied Optimal Control; Taylor & Francis Group: New York, NY, USA, 1975; pp. 8–9. [Google Scholar]
- Ostler, J.N.; Bowman, W.J.; Snyder, D.O.; McLain, T.W. Performance Flight Testing of Small Electric Powered Unmanned Aerial Vehicles. Int. J. Micro Air Veh.
**2009**, 1, 155–171. [Google Scholar] [CrossRef] - Masazumi, U.; Yuichi, T.; Kouhei, T.; Tomohiro, K. Design and Tests of Guidance and Control Systems for Autonomous Flight of a Low-Speed Model Airplane for Application to a Small-Scale Unmanned Supersonic Airplane. Trans. Jpn. Soc. Aeronaut. Space Sci. Aerosp. Technol. Jpn.
**2019**, 17, 220–226. [Google Scholar] - Kanichiro, K.; Akio, O.; Kenji, K. Introduction to Aircraft Dynamics; Translated from Japanese; The University of Tokyo Press: Tokyo, Japan, 2012. [Google Scholar]
- MATLAB. Available online: https://jp.mathworks.com/products/matlab.html?s_tid=hp_products_matlab (accessed on 9 March 2024).
- Rankine, W.J.M. On The Mechanical Principles of the Action of Propellers; Institution of Naval Architects: London UK, 1865; Volume 6, pp. 13–39. [Google Scholar]
- Glauert, H. The Elements of Aerofoil and Airscrew Theory, 2nd ed.; Cambridge at the University Press: Cambridge, UK, 2008; pp. 208–221. [Google Scholar]
- Performance Data. Available online: https://www.apcprop.com/technical-information/performance-data/ (accessed on 26 July 2023).
- Public Domain Aeronautical Software. Available online: https://www.pdas.com/index.html (accessed on 9 March 2024).
- Takahashi, K.; Ueba, M. Flight Experiment of Take-off Control System for Small-Scale Unmanned Supersonic Experimental Airplane. In Proceedings of the 54th Airplane Symposium, Toyama, Japan, 28 October 2016. [Google Scholar]
- Simulink. Available online: https://jp.mathworks.com/products/simulink.html (accessed on 9 March 2024).
- Aircraft Safety Division, Safety Department, Civil Aviation Bureau, Ministry of Land, Infrastructure, Transport and Tourism. Airworthiness Standards; Houbunnsyorin Publishing: Tokyo, Japan, 2012; p. 53. [Google Scholar]

Airplane Specification | Value |
---|---|

m | 6.0 kg |

c | $0.315$ m |

${\mathsf{\alpha}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}}$ | 10 deg. |

$\mathrm{S}$ | $0.649$ m^{2} |

$\mathrm{e}$ | $0.6$ |

$\mathrm{A}\mathrm{R}$ | 6.54 |

Longitudinal | Lateral | ||
---|---|---|---|

${\mathrm{C}}_{\mathrm{L}\mathsf{\alpha}}$ | 4.355 | ${{\mathrm{C}}_{\mathrm{y}}}_{\mathsf{\beta}}$ | −0.1623 |

${\mathrm{C}}_{\mathrm{L}0}$ | $0.176$ | ${{\mathrm{C}}_{\mathrm{y}}}_{\mathrm{p}}$ | −0.03861 |

${\mathrm{C}}_{\mathrm{D}0}$ | $0.0488$ | ${{\mathrm{C}}_{\mathrm{y}}}_{{\mathsf{\delta}}_{\mathrm{r}}}$ | 0.08116 |

${{\mathrm{C}}_{\mathrm{x}}}_{\mathrm{u}}$ | −0.3790 | ${{\mathrm{C}}_{\mathrm{y}}}_{\mathrm{r}}$ | 0.1431 |

${{\mathrm{C}}_{\mathrm{x}}}_{\mathsf{\alpha}}$ | 0.06371 | ${{\mathrm{C}}_{\mathrm{l}}}_{\mathsf{\beta}}$ | −0.05330 |

${{\mathrm{C}}_{\mathrm{z}}}_{\mathrm{u}}$ | 0 | ${{\mathrm{C}}_{\mathrm{l}}}_{{\mathsf{\delta}}_{\mathrm{a}}}$ | −0.4022 |

${{\mathrm{C}}_{\mathrm{z}}}_{\mathsf{\alpha}}$ | −4.355 | ${{\mathrm{C}}_{\mathrm{l}}}_{{\mathsf{\delta}}_{\mathrm{r}}}$ | 0.009653 |

${{\mathrm{C}}_{\mathrm{z}}}_{{\mathsf{\delta}}_{\mathrm{e}}}$ | −0.4324 | ${{\mathrm{C}}_{\mathrm{l}}}_{\mathrm{p}}$ | −0.7798 |

${{\mathrm{C}}_{\mathrm{z}}}_{\mathrm{q}}$ | −4.881 | ${{\mathrm{C}}_{\mathrm{l}}}_{\mathrm{r}}$ | 0.08940 |

${{\mathrm{C}}_{\mathrm{m}}}_{\mathrm{u}}$ | 0 | ${{\mathrm{C}}_{\mathrm{n}}}_{\mathsf{\beta}}$ | 0.04465 |

${{\mathrm{C}}_{\mathrm{m}}}_{\mathsf{\alpha}}$ | −1.359 | ${{\mathrm{C}}_{\mathrm{n}}}_{{\mathsf{\delta}}_{\mathrm{a}}}$ | 0 |

${{\mathrm{C}}_{\mathrm{m}}}_{{\mathsf{\delta}}_{\mathrm{e}}}$ | −1.220 | ${{\mathrm{C}}_{\mathrm{n}}}_{{\mathsf{\delta}}_{\mathrm{r}}}$ | −0.03578 |

${{\mathrm{C}}_{\mathrm{m}}}_{\mathrm{q}}$ | −13.78 | ${{\mathrm{C}}_{\mathrm{n}}}_{\mathrm{p}}$ | −0.01062 |

${{\mathrm{C}}_{\mathrm{m}}}_{\dot{\mathsf{\alpha}}}$ | −5.111 | ${{\mathrm{C}}_{\mathrm{n}}}_{\mathrm{r}}$ | −0.1052 |

Item | Value |
---|---|

$\mathsf{\rho}$ | 1.23 kg/m^{3} |

$\mathrm{g}$ | 9.81 m/s^{2} |

Flight Variable | Optimal Solution |
---|---|

$\mathrm{V}$ | 15.1 m/s |

$\mathsf{\alpha}$ | 5.4 deg. |

$\mathsf{\gamma}$ | 19.6 deg. |

$\mathsf{\theta}$ | 24.9 deg. |

$\mathrm{V}\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\gamma}$ | 5.1 m/s |

Flight Variable | Airspeed |
---|---|

${\mathrm{V}}_{\mathrm{R}}$ | 12.5 m/s |

${\mathrm{V}}_{2}$ | 15.1 m/s |

Flight Variable | Standard Deviation |
---|---|

Airspeed | 1.8 m/s |

Pitch angle | 0.5 deg. |

**Table 7.**Pitch angle at which the airplane is balanced when the airspeed value deviates due to the noise amount.

Flight Variable | Minimum | Nominal | Max |
---|---|---|---|

Airspeed discrepancy (true airspeed) | −1.8 m/s (13.3 m/s) | 0 m/s (15.1 m/s) | 1.8 m/s (16.9 m/s) |

Pitch angle | 29.0 deg. | 24.9 deg. | 21.0 deg. |

Flight Variable | Criteria |
---|---|

Airspeed | 3.6 m/s |

Pitch angle | 8.0 deg. |

Airspeed | Pitch Angle | Rate of Climb | |
---|---|---|---|

Optimal solution | 15.1 m/s | 24.9 deg. | 5.1 m/s |

Simulation | 15.1 m/s | 24.9 deg. | 4.9 m/s |

Experiment | 15.1 m/s | 22.0 deg. | 5.7 m/s |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Watanabe, K.; Shibata, T.; Ueba, M.
Derivation and Flight Test Validation of Maximum Rate of Climb during Takeoff for Fixed-Wing UAV Driven by Propeller Engine. *Aerospace* **2024**, *11*, 233.
https://doi.org/10.3390/aerospace11030233

**AMA Style**

Watanabe K, Shibata T, Ueba M.
Derivation and Flight Test Validation of Maximum Rate of Climb during Takeoff for Fixed-Wing UAV Driven by Propeller Engine. *Aerospace*. 2024; 11(3):233.
https://doi.org/10.3390/aerospace11030233

**Chicago/Turabian Style**

Watanabe, Katsumi, Takuma Shibata, and Masazumi Ueba.
2024. "Derivation and Flight Test Validation of Maximum Rate of Climb during Takeoff for Fixed-Wing UAV Driven by Propeller Engine" *Aerospace* 11, no. 3: 233.
https://doi.org/10.3390/aerospace11030233