# Training Sample Pattern Optimization Based on a Swarm Intelligence Algorithm for Tiltrotor Flight Dynamics Model Approximation

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## Abstract

**:**

## 1. Introduction

## 2. Approximation of Flight Dynamics Model

#### 2.1. Flight Dynamics Model of a Small-Scaled Tiltrotor Aircraft

#### 2.1.1. Rotor Aerodynamic Forces and Moments

_{b}, K, r

_{0}, and r

_{1}are the number of blades, azimuth stations, blade root cutout, and tip loss, respectively. Angles $\beta $ and $\psi $ are blade flapping and azimuth angles. Components dF

_{p}and dF

_{t}are the blade element perpendicular and tangential force elements, which can be represented by the element lift and drag as

_{p}and U

_{t}denote the velocity in-plane and normal components seen by the rotor. These components can be evaluated by the advance ratio and inflow ratio. The rotor inflow ratio (i.e., dimensionless induced velocity) is governed by the quasi-steady version of the Pitt–Peters’ dynamic in-flow model. The inflow ratio at each station of the rotor disk is expanded as the base and first harmonic term as follows:

_{nl}is related to the rotor wake angle $\alpha $, the hub total velocity V

_{T}, the advancing ratio $\mu $, and the resultant inflow ${\lambda}_{m}-{\mu}_{z}$ as follows:

#### 2.1.2. Airframe Forces and Moments

_{D}, C

_{L}, C

_{m}) and the lateral forces and moments coefficients (C

_{Y}, C

_{l}, C

_{n}) are functions of the fuselage states and controls, which can be written as

#### 2.1.3. Nonlinear Equations of Motion

#### 2.2. Aircraft Model Approximation Using a Neural Network

**C**is of proper size and time-invariant.

_{j}is the output of the j’s Gaussian basis function, ${c}_{:,j}\in {\mathbb{R}}^{17\times 1}$ and ${b}_{j}\in \mathbb{R}$ are the respective center coordinates and the width of the j’s basis function, and ${W}_{sNN}^{\mathrm{T}}\in {\mathbb{R}}^{3\times m}$ is the time-invariant network weight matrix. The network should be trained offline by the data covering the desired flight envelope derived from the flight dynamics model of the tiltrotor aircraft obtained from previous sections. It is worth noting that the training data should cover all of the valid ranges of each control deflection.

## 3. Optimization of the Neural Network Sample Distribution Pattern

#### 3.1. Problem Formularization and Methodology

**x**

^{(n)}and

**u**

^{(n)}are the sampled network inputs, and ${\dot{y}}^{(n)}$ is the corresponding output.

_{SR}and C

_{SL}, respectively):

_{IW}, C

_{SR}, and C

_{SL}are the respective inertia weight, self-recognition, and social learning factors, rand is the uniformly distributed random number between [0,1], and

**P**

_{ib}and

**P**

_{gb}are the optima encountered by the individual itself and by the flock in history (global optimum). Viewed as ordinary differential equations, the above model can be solved by several different numerical methods, which results in variants of schemes. A fourth-order Runge–Kutta method is adopted for its relatively high order of truncation error. Thus, the scheme with a step size of h can be represented as

_{IW}, C

_{SR}, and C

_{SL}of the optimizer are incorporated with the following time-varying form:

#### 3.2. Case Studies

#### 3.2.1. Approximation of a Nonlinear Function

#### 3.2.2. Parameter Analysis for Searching the Particle Number and Sample Number

#### 3.2.3. Time Domain Response Performance

## 4. Sample Distribution Pattern Optimization for the Flight Dynamics Model of a Tiltrotor Aircraft

#### 4.1. Results of the Early-Stage Conversion Mode

^{−4}. The global SE distribution of the optimum network is shown in Figure 19. The elite fitness values from 200 generations are shown in Figure 20, from which one can find rapid convergence within the first 10 generations.

#### 4.2. Results of the Late-Stage Conversion Mode

#### 4.3. Time-Domain Response Simulation

_{w}, f

_{q}, and control functions g

_{w}, g

_{q}are sampled from the full-state nonlinear tiltrotor model under its conversion mode of a 30 deg nacelle tilting angle. The forward velocity u = 59 m/s, which is considered to remain nearly constant during short-period oscillation. All of the lateral state and control variables are fixed at zero, as well as the redundant controls in the longitudinal channel. The simulation is started under a non-equilibrium flight condition. The control signal of a bipolar square wave of longitudinal cyclic control is imposed at simulation time t = 2 s, see Figure 27.

## 5. Discussion and Conclusions

- (1)
- For the function approximation problem, this work presented a novel methodology, that is, to use the metaheuristic optimization algorithm to search the optimum sample pattern of the training set, revealing its effectiveness and benefits. In the example case of approximating a highly-nonlinear function with two inputs, the RBF neural network trained by the optimum data set obtained by the DPSORK algorithm showed a significantly improved performance, with a network MSE decrease from 2 × 10
^{−3}to 3 × 10^{−5}(65 times lower) compared to the results of the random sample network. - (2)
- The DPSORK-optimized RBF neural network was applied to the flight dynamics model approximation of the pitching angular acceleration field of the tiltrotor aircraft. Examples of the early (30 deg nacelle tilting angle) and late stage (70 deg nacelle tilting angle) of its conversion mode were studied. The methodology presented in this work showed its generality and readiness of solving practical problems.
- (3)
- Although the overall error of the approximated network has been reduced by the presented algorithm, there are still certain deviations in each case (for example, the MSE of 8.7 × 10
^{−4}for control response at the late tilting stage). Since an RBF network is able to converge on a result with an arbitrarily small deviation with adequate neurons, the remaining problem would be to decrease the deviation under a neuron number as low as possible. Future work should study all of the parameters (including that of self-recognition and social learning factors) analytically and formulate the optimum of these parameters with respect to the studied cases.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wang, Y.; Li, A.; Yang, S.; Li, Q.; Ma, Z. A neural network based MRAC scheme with application to an autonomous nonlinear rotorcraft in the presence of input saturation. ISA Trans.
**2021**, 115, 1. [Google Scholar] [CrossRef] [PubMed] - Tang, D.; Tang, B.; Shen, W.; Zhu, K.; Quan, Q.; Deng, Z. On genetic algorithm and artificial neural network combined optimization for a Mars rotorcraft blade. Acta Astronaut.
**2023**, 203, 78–87. [Google Scholar] [CrossRef] - Tee, K.P.; Ge, S.S.; Tay, F.E.H. Adaptive Neural Network Control for Helicopters in Vertical Flight. IEEE Trans. Control Syst. Technol.
**2008**, 16, 753–762. [Google Scholar] - Razmi, H.; Afshinfar, S. Neural Network-Based Adaptive Sliding Mode Control Design for Position and Attitude Control of a Quadrotor UAV. Aerosp. Sci. Technol.
**2019**, 91, 12–27. [Google Scholar] [CrossRef] - Tavoosi, J. Hybrid Intelligent Adaptive Controller for Tiltrotor UAV. Int. J. Intell. Unmanned Syst.
**2021**, 9, 256–273. [Google Scholar] [CrossRef] - Yu, J.; Hesthaven, J.S. Flowfield Reconstruction Method Using Artificial Neural Network. AIAA J.
**2019**, 57, 482–498. [Google Scholar] [CrossRef] - Alhazmi, N.; Ghazi, Y.; Aldosari, M.; Tezaur, R.; Farhat, C. Training a Neural-Network-Based Surrogate Model for Aerodynamic Optimization Using a Gaussian Process. In Proceedings of the AIAA SciTech 2021 Forum, Virtual Event, 26 August 2021. [Google Scholar]
- Seo, J.; Kapania, R.K. Development of Deep Convolutional Neural Network for Structural Topology Optimization. AIAA J.
**2023**, 61, 1366–1379. [Google Scholar] [CrossRef] - Rostami, M.; Chung, J.; Park, H.U. Design optimization of multi-objective proportional–integral–derivative controllers for enhanced handling quality of a twin-engine, propeller-driven airplane. Adv. Mech. Eng.
**2020**, 12, 1687814020923178. [Google Scholar] [CrossRef] - Wang, L.; Diskin, B.; Biedron, R.T.; Nielsen, E.J.; Bauchau, O.A. Evaluation of high-fidelity multidisciplinary sensitivity-analysis framework for multipoint rotorcraft optimization. J. Aircr.
**2020**, 57, 830–842. [Google Scholar] [CrossRef] - An, J.Y.; Choi, Y.S.; Lee, I.R.; Lim, M.; Kim, C.J. Performance Analysis of a Conceptual Urban Air Mobility Configuration Using High-Fidelity Rotorcraft Flight Dynamic Model. Int. J. Aeronaut. Space Sci.
**2023**, 24, 1491–1508. [Google Scholar] [CrossRef] - Chen, R.; Yuan, Y.; Thomson, D. A review of mathematical modelling techniques for advanced rotorcraft configurations. Prog. Aerosp. Sci.
**2021**, 120, 100681. [Google Scholar] [CrossRef] - Rostami, M.; Bagherzadeh, S. Development and validation of an enhanced semi-empirical method for estimation of aerodynamic characteristics of light, propeller-driven airplanes. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2018**, 232, 638–648. [Google Scholar] [CrossRef] - Xue, J.; Shen, B. A Novel Swarm Intelligence Optimization Approach: Sparrow Search Algorithm. Syst. Sci. Control Eng.
**2020**, 8, 22–34. [Google Scholar] [CrossRef] - Zervoudakis, K.; Tsafarakis, S. A Mayfly Optimization Algorithm. Comput. Ind. Eng.
**2020**, 145, 106559. [Google Scholar] [CrossRef] - Alsattar, H.; Zaidan, A.A.; Zaidan, B.B. Novel Meta-Heuristic Bald Eagle Search Optimization Algorithm. Artif. Intell. Rev.
**2020**, 53, 2237–2264. [Google Scholar] [CrossRef] - Ye, Z.; Ma, L.; Chen, H. A hybrid rice optimization algorithm. In Proceedings of the 2016 11th International Conference on Computer Science & Education (ICCSE), Nagoya, Japan, 23–25 August 2016; pp. 169–174. [Google Scholar] [CrossRef]
- Shi, Y. Brain Storm Optimization Algorithm. In Proceedings of the International Conference in Swarm Intelligence, Chongqing, China, 12–15 June 2011; Springer: Berlin/Heidelberg, Germany, 2011; pp. 303–309. [Google Scholar]
- Zhang, Q.; Wang, R.; Yang, J.; Ding, K.; Li, Y.; Hu, J. Collective Decision Optimization Algorithm. Neurocomputing
**2017**, 221, 123–137. [Google Scholar] [CrossRef] - Moghdani, R.; Salimifard, K. Volleyball Premier League Algorithm. Appl. Soft Comput.
**2017**, 64, 161–185. [Google Scholar] [CrossRef] - Faramarzi, A.; Heidarinejad, M.; Stephens, B.; Mirjalili, S. Equilibrium Optimizer: A Novel Optimization Algorithm. Knowl. Based Syst.
**2020**, 191, 105190. [Google Scholar] [CrossRef] - Choudhury, S.; Khandelwal, N.; Kumar, A.; Shukla, A.; Jha, A.; Mohanty, M.; Dash, T. A Supervisory State of Charge and State of Power Management Control Strategy among Hybrid Energy Storage Systems through Thermal Exchange Optimization Technique. In Proceedings of the 2020 IEEE Calcutta Conference, Kolkata, India, 28–29 February 2020; IEEE: Piscataway, NJ, USA, 2020. [Google Scholar]
- Wen, J.; Song, Y.; Wang, H.; Han, D. Numerical Study on Tandem-Rotor Autorotation in Forward Flight. Aerospace
**2023**, 10, 15. [Google Scholar] [CrossRef] - Eberhart, R.; Kennedy, J. A New Optimizer Using Particle Swarm Theory. In Proceedings of the Mhs95 Sixth International Symposium on Micro Machine & Human Science, Nagoya, Japan, 6 August 2002; IEEE: Piscataway, NJ, USA, 2002. [Google Scholar]

**Figure 16.**Phase portraits of the optimized approximation, the random sample approximation, and the actual system.

**Figure 18.**Optimum sample network prediction of the $\dot{q}$-field with respect to airspeed and longitudinal cyclic pitch control in the early conversion stage.

**Figure 21.**Random sample network prediction of the $\dot{q}$-field with respect to airspeed and longitudinal cyclic pitch control in the early conversion stage.

**Figure 23.**Optimum sample network prediction of the $\dot{q}$-field with respect to elevator deflection and longitudinal cyclic pitch control in the early conversion stage.

**Figure 24.**Optimum sample network prediction of the $\dot{q}$-field with respect to airspeed and longitudinal cyclic pitch control in the late conversion stage.

**Figure 25.**Optimum sample network prediction of the $\dot{q}$-field with respect to elevator deflection and longitudinal cyclic pitch control in the late conversion stage.

**Figure 26.**Searching loci and scores of typical particles (with the global best particle). (

**a**) Particle No. 1. (

**b**) Particle No. 2. (

**c**) Particle No. 4 (global best). (

**d**) Particle No. 20.

Parameter | Value |
---|---|

Total number of sample points | 40 |

Total number of particles | 20 |

N_{max} | 200 |

C_{IW,i} | 0.9 |

C_{IW,f} | 0.5 |

C_{SR,i} | 2.5 |

C_{SR,f} | 0.5 |

C_{SL,i} | 0.5 |

C_{SL,f} | 2.5 |

Parameter | Value |
---|---|

Aircraft mass | 5896 Kg |

Rotor radius | 3.81 m |

Rotor solidity | 0.09 |

Rotor blade Lock Number | 3.8 |

Rotor speed | 565 rpm |

Airframe length | 12.8 m |

Airframe width | 2.9 m |

Wing span | 9.81 m |

Wing area | 15.71 m^{2} |

MAC | 1.6 m |

Horizontal tail area | 4.67 m^{2} |

Vertical tail area | 4.69 m^{2} |

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**MDPI and ACS Style**

Wen, J.; Song, Y.; Wang, H.; Han, D.; Yang, C.
Training Sample Pattern Optimization Based on a Swarm Intelligence Algorithm for Tiltrotor Flight Dynamics Model Approximation. *Aerospace* **2023**, *10*, 1006.
https://doi.org/10.3390/aerospace10121006

**AMA Style**

Wen J, Song Y, Wang H, Han D, Yang C.
Training Sample Pattern Optimization Based on a Swarm Intelligence Algorithm for Tiltrotor Flight Dynamics Model Approximation. *Aerospace*. 2023; 10(12):1006.
https://doi.org/10.3390/aerospace10121006

**Chicago/Turabian Style**

Wen, Jiayu, Yanguo Song, Huanjin Wang, Dong Han, and Changfa Yang.
2023. "Training Sample Pattern Optimization Based on a Swarm Intelligence Algorithm for Tiltrotor Flight Dynamics Model Approximation" *Aerospace* 10, no. 12: 1006.
https://doi.org/10.3390/aerospace10121006