#
Simulation and Optimization of Control of Selected Phases of Gyroplane Flight

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## Abstract

**:**

## 1. Introduction

- longitudinal and lateral tilting of the rotor shaft to ensure a pitch and roll control;
- deflections of the rudder to ensure a yaw control.

- a pre-rotator that drives the rotor to initiate its rotation,
- a changeable collective pitch of rotor blades that may be used for torque reduction in pre-rotation and is necessary to conduct so-called “jump takeoff.”

- To develop a computational methodology for the simulation of gyroplane flight, especially directed towards high-fidelity simulations of takeoff and ascent.
- To develop a numerical optimization methodology that would be applicable to solving problems concerning the optimization of rotorcraft flight control procedures.
- To optimize flight control procedures (i.e., functions describing time-varying settings of the flight control devices) during the classic takeoff and jump takeoff of the gyroplane, so as to achieve measurable improvement in the aircraft performance of these flight modes.

- The rotorcraft flight control optimization is based on advanced, gradient-based optimization methods coupled with advanced CFD methods used directly during the rotorcraft flight simulation for the current determination of aerodynamic loads acting on the aircraft.
- The developed methodology is applied to optimize the gyroplane flight control procedures (while most of the applications cited in the literature are focused on the optimization of helicopter flight control).

## 2. Research Methodology

^{6}.

^{6}÷ 5.7 × 10

^{6}. For the propeller blade airfoils, these ranges were: 0.2 ÷ 0.93 for Mach number and 1 × 10

^{6}÷ 6.5 × 10

^{6}for Reynolds number.

_{0}, t

_{1}, …, t

_{M}) were knots defining the function domain while (f

_{0}, f

_{1}, …, f

_{M}) were function values in subsequent knots. In the presented approach both the function knots {t

_{i}}

_{i=0,1,…,M}and values {f

_{i}}

_{i=0,1,…,M}were expressed as linear functions of a few unknown real numbers—the design parameters. In practice, when defining a given optimization problem, small values of M are preferred so as to reduce the number of independent design parameters, and thus the dimension of the problem. On the other hand, in every case the number M was selected so as to ensure sufficient degree of freedom of flight control procedures. The objective function, considered as a function of unknown design parameters, was defined as the altitude that would be achieved by the gyroplane after reaching the assumed distance from the takeoff point.

## 3. Optimization of Gyroplane Takeoff Control Procedures

- total mass of the gyroplane: 600 kg;
- maximum static thrust of the propeller: 2943 N (during takeoff, when the forward velocity of the vehicle was growing, the thrust of the propeller was changing, which was a result of modeling the propeller effects through VBM methodology).

#### 3.1. Classic Takeoff of the Gyroplane

_{R}), which was considered the only flight control parameter. The angle of collective pitch of rotor blades (θ

_{0}), unchangeable during the flight, has been assumed as an additional unknown parameter. Based on these assumptions, the flight control parameters ϕ

_{R}(t) and θ

_{0}(t) were assumed to have the form defined by Equation (1) as piecewise linear functions Ψ

_{M}(t). Parameters M, {t

_{i}}

_{i=0,1,…,M}and {f

_{i}}

_{i=0,1,…,M}of these functions were related to the design parameters D

_{1}, F

_{1}and F

_{2}presented in Figure 5, according to the following dependencies:

_{R}(t) = Ψ

_{2}(t), for: t

_{0}= 0, t

_{1}= D

_{1}, t

_{2}= +∞, f

_{0}= 0, f

_{1}= F

_{1}, f

_{2}= F

_{1}

_{0}(t) = Ψ

_{1}(t), for: t

_{0}= 0, t

_{1}= +∞, f

_{0}= F

_{2}, f

_{1}= F

_{2}.

_{1}, F

_{1}and F

_{2}maximizing the following objective function Φ:

_{1}, F

_{1}, F

_{2}) = H(X = 200 m),

_{1}> 1,

_{2}≤ θ

_{0max},

_{1}/D

_{1}| ≤ λ

_{1},

_{1}≤ ϕ

_{Rmax},

_{1}—limit of angular speed of change of ϕ

_{R}, ϕ

_{Rmax}—maximum of rotor pitch and θ

_{0max}—maximum of blade collective pitch.

#### 3.2. Jump Takeoff of the Gyroplane

_{R}) and the angle of collective pitch of the rotor blades (θ

_{0}). Based on these assumptions, the time-varying flight control parameters ϕ

_{R}(t) and θ

_{0}(t) were assumed to have the form defined by Equation (1) as piecewise linear functions Ψ

_{M}(t). Parameters M, {t

_{i}}

_{i=0,1,…,M}and {f

_{i}}

_{i=0,1,…,M}of these functions were related with the presented in Figure 14 design parameters D

_{1}, D

_{2}, D

_{3}, D

_{4}, F

_{1}, F

_{2}and F

_{3}according to the following dependencies:

_{R}(t) = Ψ

_{3}(t), for: t

_{0}= 0, t

_{1}= D

_{1}, t

_{2}= D

_{1}+ D

_{2}, t

_{3}= +∞, f

_{0}= 0, f

_{1}= 0, f

_{2}= F

_{1}, f

_{3}= F

_{1}

_{0}(t) = Ψ

_{4}(t), for: t

_{0}= 0, t

_{1}= 1, t

_{2}= 1 + D

_{3}, t

_{3}= 1 + D

_{3}+ D

_{4}, t

_{4}= +∞, f

_{0}= 0, f

_{1}= F

_{2}, f

_{2}= F

_{2}, f

_{3}= F

_{3}, f

_{4}= F

_{3}.

_{1}, D

_{2}, D

_{3}, D

_{4}, F

_{1}, F

_{2}and F

_{3}maximizing the following objective function Φ:

_{1}, D

_{2}, D

_{3}, D

_{4}, F

_{1}, F

_{2}, F

_{3}) = H(X = 100 m),

_{1}≥ 0, D

_{2}≥ 1, D

_{3}≥ 0, D

_{4}≥ 1,

_{2}≤ θ

_{0max},

_{1}/D

_{2}| ≤ λ

_{1},

_{2}− F

_{3})/D

_{4}| ≤ λ

_{2},

_{1}≤ ϕ

_{Rmax},

_{1}, λ

_{2}—limits of angular speed of changes of ϕ

_{R}and θ

_{0}respectively, ϕ

_{Rmax}—maximum of rotor pitch and θ

_{0max}maximum of blade collective pitch. The defined optimization problem has been solved by application of the BFGS algorithm. At every step of the iterative process of the optimization, the gradient of the objective function Equation (11) was determined using the one-sided finite-difference approximation. This required conducting at least N + 1 (where N = 7) independent simulations of gyroplane jump takeoff for different sets of values of unknown design parameters D

_{1}, D

_{2}, D

_{3}, D

_{4}, F

_{1}, F

_{2}, and F

_{3}. In addition, to solve the auxiliary one-dimensional problem of finding the optimal movement in the newfound search direction, eight additional gyroplane jump takeoff simulations were performed at each optimization step.

_{2}and F

_{3}. This means that these two parameters might be replaced by only one in the assumed parametric model of flight control strategy (shown in Figure 14) and the phase of decreasing of the rotor pitch, which we assume might be omitted in this model. Figure 18 compares gyroplane flight trajectories during the jump takeoff for two flight control strategies: baseline and optimized. It may be concluded that the optimized trajectory is growing monotonically while the baseline trajectory has a local minimum. Additionally, for the optimized flight control strategy the result of Equation (11) is higher by approximately 5 m than for the baseline strategy. As shown in Figure 19, during the jump takeoff and initial stage of ascent, the flight velocity (V) of the gyroplane controlled by optimized procedure, after 8 s the flight stabilized itself at a level of approx. 17 m/s. In this case, to increase the flight velocity, after the jump takeoff and ascent the backward pitch of the rotor should be reduced. For the baseline flight control procedure, the flight velocity was still growing and it reached V ≈ 25 m/s at t = 25 s. Figure 20, Figure 21 and Figure 22 present snapshots of flow field (velocity magnitude contours) around the gyroplane taken during the jump takeoff at t = 0.5, 1.5, 10 s (time elapsed from the beginning of the takeoff), for two compared flight control strategies: baseline and optimized.

## 4. Discussion

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Cross-section of computational mesh around the gyroplane in two different stages of flight

**Left**: a ground pre-rotation,

**right**: a forward flight a few meters above the ground.

**Figure 5.**Parametric model of the gyroplane flight control procedure utilized in the optimization of the classic takeoff of the gyroplane.

**Figure 7.**Values of the maximized objective Φ in sequential iterative steps of the process of optimization of the classic takeoff control procedure of the gyroplane.

**Figure 8.**Comparison of the baseline (

**left**) and optimized (

**right**) classic takeoff control procedures, showing the pitch angle of main rotor (ϕ

_{R}), collective pitch of rotor blades (θ

_{0}) and collective pitch of propeller blades (θ

_{P}) as functions of time (t).

**Figure 9.**Aircraft trajectories obtained for the baseline and optimized procedures of flight control during the classic takeoff of the gyroplane.

**Figure 10.**Aircraft flight velocity (V) vs. time (t) during the classic takeoff, for the baseline and optimized procedures of gyroplane flight control.

**Figure 11.**Comparison of velocity magnitude contours around the gyroplane during the classic takeoff, for two configurations related to baseline (

**left**) and optimized (

**right**) flight control procedures. The initial time (t = 0 s) of the aircraft run on a runway.

**Figure 12.**Comparison of velocity magnitude contours around the gyroplane during the classic takeoff, for two configurations related to baseline (

**left**) and optimized (

**right**) flight control procedures. Time elapsed from the beginning of the aircraft run: t = 10 s.

**Figure 13.**Comparison of velocity-magnitude contours around the gyroplane during the classic takeoff, for two configurations related to baseline (

**left**) and optimized (

**right**) flight control procedures. Time elapsed from the beginning of the aircraft run: t = 17.5 s.

**Figure 14.**Parametric model of the gyroplane flight control procedure utilized in the optimization of jump takeoff of the gyroplane.

**Figure 15.**Definition of the objective (Φ) for the optimization of jump takeoff control strategy on the example of a gyroplane trajectory controlled by the baseline procedure.

**Figure 16.**Values of the maximized objective Φ in sequential iterative steps of the process of optimization of the jump takeoff control procedure of the gyroplane.

**Figure 17.**Comparison of the baseline (

**left**) and optimized (

**right**) jump takeoff control procedures, showing the pitch angle of main rotor (ϕ

_{R}), collective pitch of rotor blades (θ

_{0}) and collective pitch of propeller blades (θ

_{P}) as functions of time (t).

**Figure 18.**Aircraft trajectories obtained for the baseline and optimized procedures of flight control during the jump takeoff of the gyroplane.

**Figure 19.**Aircraft flight velocity (V) vs. time (t) during the jump takeoff, for the baseline and optimized procedures of gyroplane flight control.

**Figure 20.**Comparison of velocity magnitude contours around the gyroplane during the jump takeoff, for two configurations related to baseline (

**left**) and optimized (

**right**) flight control procedures. The initial time moment (t = 0.5 s) of the jump takeoff.

**Figure 21.**Comparison of velocity magnitude contours around the gyroplane during the jump takeoff, for two configurations related to baseline (

**left**) and optimized (

**right**) flight control procedures. Time elapsed from the beginning of the jump takeoff: t = 1.5 s.

**Figure 22.**Comparison of velocity magnitude contours around the gyroplane during the jump takeoff, for two configurations related to baseline (

**left**) and optimized (

**right**) flight control procedures. Time elapsed from the beginning of the jump takeoff: t = 10 s.

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**MDPI and ACS Style**

Stalewski, W.
Simulation and Optimization of Control of Selected Phases of Gyroplane Flight

. *Computation* **2018**, *6*, 16.
https://doi.org/10.3390/computation6010016

**AMA Style**

Stalewski W.
Simulation and Optimization of Control of Selected Phases of Gyroplane Flight

. *Computation*. 2018; 6(1):16.
https://doi.org/10.3390/computation6010016

**Chicago/Turabian Style**

Stalewski, Wienczyslaw.
2018. "Simulation and Optimization of Control of Selected Phases of Gyroplane Flight

" *Computation* 6, no. 1: 16.
https://doi.org/10.3390/computation6010016