Research ArticleResearch on Innovative Trim Method for Tiltrotor Aircraft Take-Off Based on Genetic Algorithm

Xueyun Wang ,1,2 Jiyang Chen,1 Qian Zhang ,2,3 Jingjuan Zhang,1,2 and Hao Cong1,2

1School of Instrument Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China2Hefei Innovation Research Institute, Beihang University, Anhui 230012, China3School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Qian Zhang; zhangqian528@buaa.edu.cn

Received 3 June 2020; Revised 26 August 2020; Accepted 30 November 2020; Published 15 December 2020

Academic Editor: Antonio Fernández-Caballero

Copyright © 2020 Xueyun Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Tiltrotor aircraft possesses redundant actuators in take-off phase and its flight control is more complicated than ordinary aircraftbecause the structural and dynamic characteristics keep changing due to tilting rotors. One of the fundamental bases for flightcontrol is trim, which provides steady flight states under various conditions and then constructs the reference trajectory.Tiltrotor aircraft trim models are described by multivariate nonlinear equations whose initial values are difficult to determineand bad initials could lead to incorrect solution for flight control. Therefore, an innovative trim method is proposed to solve thisissue. Firstly, genetic algorithm (GA), which possesses strong capability in searching global optimum, is adopted to identify acoarse solution. Secondly, the coarse solution is further refined by the Levenberg-Marquardt (LM) method for precise localoptimum. The innovative trim method combines the advantages of these two algorithms and is applied to a tiltrotor aircraft’sflight control in the transition process of incline take-off. The limitation of trajectory is discussed, and tilt corridor isconstructed. Finally, the incline take-off simulations are conducted and the effectiveness of the proposed trim method is verifiedthrough good match with the designed reference trajectory.

1. Introduction

Tiltrotor aircraft which is capable of taking-off/landing verti-cally and cruise at high speed simultaneously combines theadvantages of rotorcraft and fixed-wing aircraft. These char-acteristics make tiltrotor aircraft a research hotspot for recentyears [1–3]. Tilt rotors provide the aircraft with the ability ofvector tension, which not only expands its operating condi-tions but also entangles its flight control [4, 5]. In the processof rotor tilting, it is very important yet very difficult to imple-ment the stability control of aircraft attitudes. The adjust-ment of the rotor tilt angle during take-off or landingcauses significant changes in the structural and mechanicalproperties of the aircraft, eventually resulting in obviouschanges in flight control parameters. Under such circum-stances, accidents are more likely to happen during take-offor landing [6–10].

To achieve attitude stabilization, trim is needed, which isapplied to determine the steady flight states in the design of

the flight control algorithms. The equilibrium states of theaircraft play an important role in aircraft attitude responsecalculation, stability analysis, and flight control law design[11]. After trim then come the flight simulation and param-eter adjustment for better flight performance [12]. However,since the tiltrotor aircraft has more actuators (control planesand propellers) than ordinary fixed-wing aircraft or rotor-craft, the aerodynamic of tiltrotor aircraft is more complex,which leads to the trim of the tiltrotor aircraft being morecomplicated and difficult than that of ordinary aircrafts[13]. Especially in the stage of rotor tilting when take-off orlanding, the aircraft is controlled through both the directforce provided by the rotating propellers and the aerody-namic force derived from control planes. The actuatorredundancy and force complexity lead to difficulty in attitudestabilization, which in turn demands more for trim. Tradi-tional trim methods designed for ordinary aircrafts revealthe disadvantages such as lower efficiency and poor robust-ness. Therefore, a more efficient, robust, and accurate trim

HindawiJournal of SensorsVolume 2020, Article ID 8876867, 13 pageshttps://doi.org/10.1155/2020/8876867

method is needed in the design of tiltrotor aircraft fightcontrol.

In the process of trim, it is required to solve the multivar-iable time-varying nonlinear equations whose initial valuesare difficult to determine, and the globally optimal solutionsare usually not unique. Several traditional methods such asthe Newton method, gradient descent method, andLevenberg-Marquardt (LM) method have been used in trim[14–16], but they are so dependent on the initial values thattiny difference may lead to the locally optimal solution ratherthan the globally optimal solution. Besides, inaccurate initialvalues increase the number of iterations or even cause diver-gence of the solution. Genetic algorithm (GA) is capable ofsearching the optimal solution effectively in global scopeand achieving a robust convergent in wide range, which issuitable for solving complex nonlinear optimization prob-lems [17, 18]. However, GA’s ability of local search for opti-mum is weak, so the premier results derived from GA shouldbe further promoted. They can be applied as the initial valuesfor the traditional nonlinear programming algorithms forbetter local search performance. Combined with their respec-

tive advantages, the globally optimal solution for nonlinearproblems can be obtained in a fast, robust, and accurate way.

In this paper, an innovative trim method for tiltrotor air-craft is proposed. Firstly, GA is adopted in the primary trimprocedure to achieve a coarse solution in global scale, andsecondly, the coarse solution is assigned as the initial valueof the LM method for more precise local solution. In thisway, GA’s solution guarantees the optimality of the solutionin global scale while LM’s solution insures the precision ofthe solution in local scale. Hence, the nonlinear trim equa-tions can be solved efficiently, robustly, and precisely. Finally,the proposed innovative trim method is verified throughflight simulation of the tiltrotor aircraft under the referencetake-off trajectory.

2. Mechanical Model

The tiltrotor aircraft studied in this paper, JDW-1-1, is anelectric blended wing body (BWB) Unmanned Aerial Vehicledesigned and manufactured by our team as shown inFigure 1. JDW-1 has several emerging technologies that make

Figure 1: The tiltrotor aircraft studied in this paper.

Xm1

Om1

Ow

Om2

Xm2

Ob

Yb

Yw

L1

L2

Ym2

Ym1

Zm1

Zm2

𝛼2

𝛼1

Xn

Xb

m.g

h

n

l

Xw

Zn

Yn

Zw

Zb

On

𝜔3

𝜔1

𝜔2

Xm3

Ym3

Zm3

Om3

Figure 2: Coordinate systems of the tiltrotor aircraft.

2 Journal of Sensors

it an advanced and unique aircraft. First, the tiltrotors giveJDW-1 the ability to take off and land without a runway,the flexibility being improved. Second, the blended wingbody (BWB) configuration provides good stealth and relativehigh lift-to-drag ratio at the same time. Finally, JDW-1 ispurely electric powered without any combustion engines, sothe flight control efficiency is higher and more environmen-tally favorable.

JDW-1 has two main propellers which can be tilted con-tinuously by steering engines on both tips of the wings and asupplementary shrouded propeller at the tail for pitch stabi-lization when taking off or landing. The supplementaryshrouded propeller can also be tilted along the lateral axisof the aircraft, but the range of tilt angle is only from −π/6to π/6. Besides, it has two flaps for attitude control in levelflight. Based on this structure, the mechanical model isdetailed in the following aspects.

2.1. Coordinate System. The aerodynamic forces andmoments are defined in airflow frame while the forces andmoments of rotors are computed in rotor frame. Besidesbody frame and navigation frame also need to be set up inthe mechanical model of six degree of freedom (DOF). Aseries of coordinate systems (CSs) are presented in Figure 2.The five CSs are the navigation CS On − XnYnZn, body CSOb − XbYbZb, airflow CS Ow − XwYwZw, three rotor-fixedCS Omi − XmiYmiZmi ði = 1,2,3Þ, and centroid CS coincidingwith the body CS.

On − XnYnZn denotes the navigation frame which coin-cides with the geographic frame, north east and down (NED).

Ob − XbYbZb denotes the aircraft body frame. Ob is at thecenter of gravity, Xb is pointing to the front of the aircraft, Ybis pointing right, and Zb is defined by the right-hand rule.The rotation from NED to the body CS is defined by Eulerangles.

Ow − XwYwZw denotes the air speed direction. Ow is atthe center of gravity, Xw is pointing at the direction of air-speed Va, Zw is pointing down of the aircraft, and Yb isdefined by the right-hand rule. Rotation from the body CSto the airflow CS is defined by the angle of attack α andsideslip β.

Omi − XmiYmiZmi ði = 1,2,3Þ are the rotor coordinates sys-tem which are fixed with three rotor engines, respectively,and tilt with them as shown below.

2.2. Dynamic Equations. The tiltrotor aircraft is assumed tobe symmetrical relative to the plane XbObZb of the body CSwhile ignoring the centroid changes caused by the rotor tilt-ing. According to Newton’s second law, F being the externalforce acting on the airplane’s center of mass and m being themass of the aircraft, their relationship can be shown asfollows:

F = ddt

mVð Þ =mδVδt

+ ω ×V� �

, ð1Þ

where δV/δt = _ui + _vj + _wk, where V is the aircraft speed inthe body CS and u, v,w are the projection of V in the body

CS. ω = pi + qj + rk are the projections of the angular velocityof the body in the body CS. The component form is asfollows:

Fx =m _u + qw − rvð Þ,Fy =m _v + ru − pwð Þ,Fz =m _w + pv − quð Þ:

8>><>>: ð2Þ

According to Euler’s equation of rotation M = I _ω + ω ×Iω = I _ω +ΩIω, the external moment M is derived in acomponent form as follows:

Mx = Ixx _p − Iyy − Izz� �

qr − Ixz _r + pqð Þ,My = Iyy _q − Izz − Ixxð Þpr − Ixz p2 − r2

� �,

Mz = Izz _r − Ixx − Iyy� �

pq + Ixz qr − _pð Þ:

8>><>>: ð3Þ

Equations (2) and (3) are the six degree of freedomdynamic equations for tiltrotor aircraft.

2.3. Rotor Model. The tension and moments of the rotorsexpressed in body CS can be converted from rotor-fixed CSaccording to the following equations:

Trx

Try

Trz

2664

3775 =

cos −αrð Þ0

−sin −αrð Þ

010

sin −αrð Þ0

cos −αrð Þ

2664

3775

00Tr

2664

3775,

Mrx

Mry

Mrz

2664

3775 =

0 −zr yr

zr 0 −xr−yr xr 0

2664

3775

Trx

Try

Trz

2664

3775,

ð4Þ

where Tr is the tension provided by the rotor and αr is the tiltangle with the rotor pointing vertically being π/2 andhorizontally being 0. xr , yr , and zr are the distances betweenthe rotor and body center.

Slipstream free zone

Slipstream zone

Figure 3: Zone division on the wing.

3Journal of Sensors

2.4. Wing Model. The aerodynamic forces and momentsacting on the tiltrotor aircraft, such as lift force Lw, resistanceDw, lateral force Cw, rolling moment LA, pitching momentMA, and yaw moment NA in the airflow CS are calculated inthe airflowCS and then converted into the body CS. Since partof the wing is under themain propeller when it is pointing ver-tically, the propeller’s wake will aerodynamically interfere withthe wing [19, 20]. Therefore, it is necessary to divide the wingarea into slipstream zone and slipstream free zone to analyzethe aerodynamic forces and moments, respectively. As shownin Figure 3, the slipstream zone refers to the area directlyaffected by the propeller’s wake, while the free zone refers tothe area unaffected. According to References [21, 22], the areaof slipstream zone is the largest when the tiltrotor aircraft isvertically taking off and landing. When the rotor tilt angle isless than π/6, the area of the slipstream zone is 0. In practice,the slipstream area can be calculated approximately accordingto the following formula:

Swsr = Smax sin aπ

2 − αr� �� �

+ cos bπ

2 − αr� �� �h i umax − u

umax, αr >

π

6 ,

ð5Þ

where Smax = 2ηsrRrcW is the maximum area of slipstreamzone, namely, the area swept by the rotor’s radius on the wingwhen the aircraft is hovering; ηsr is the slipstream correctionfactor; and umax is the maximum x component speed in bodyCS. Parameters a and b yield the following constraints and canbe obtained numerically: a = 1:386 and b = 3:114.

sin aπ

2� �

+ cos bπ

2� �

= 1,

sin aπ

3� �

+ cos bπ

3� �

= 0:

8><>: ð6Þ

Assume the right wing area of the aircraft is Swr while thearea of slipstream-free zone is Swf r = Swr − Swsr. The coordi-nates of the aerodynamic pressure centers of the slipstreamzone and slipstream free zone on the right wing are ðxwsr ,ywsr , zwsrÞ and ðxwf r , ywf r , zwf rÞ, respectively. Then, thevelocity of the slipstream zone and slipstream free zone at thispoint can be expressed as follows:

Vwsr = Va +Ωwsr ⋅ ω,

Vwf r =Va +Ωwf r ⋅ ω +uif s

0wif s

2664

3775, ð7Þ

where the skew symmetric matrix

Ω =0 z −y

z 0 x

y −x 0

2664

3775 ð8Þ

is defined and uif s andwif s are the decomposition of the inter-

ference of the propeller wake to the wing in the body CS. Then,the angle of attack, sideslip of slipstream zone, and slipstreamfree zone of the wing can be calculated as follows:

αwsr = sin−1 wwsrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2wsr +w2

wsr

p !

, βwsr = sin−1 vwsrVwsrj j

� �,

αwf r = sin−1wwf rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2wf r +w2wf r

q0B@

1CA, βwf r = sin−1

vwf r

Vwf r

!

:

ð9Þ

Then, the force and moment generated by the right wingthat determine the attitude of the aircraft are given by

Lwsr =12 ρV

2wsrSwsrClwsr ,

Dwsr =12 ρV

2wsrSwsrCdwsr ,

Mwsr =12 ρV

2wsrSwsrCmwsrlwr ,

Lwf r =12 ρV

2wf rSwf rClwf r ,

Dwrf =12 ρV

2wf rSwf rCdwf r ,

Mwf r =12 ρV

2wf rSwf rCmwf rlwr ,

ð10Þ

where lwr is the mean geometric chord length of the rightwing. In addition, since lateral force Cw, rolling momentLA, and yaw moment NA are less affected by the wake, a

Start

Initial population

Fitness evaluation

Selection

Crossover

Mutation

Fitness evaluation

Satisfy

OverY

Ntermination?

Figure 4: Flow chart of genetic algorithm.

4 Journal of Sensors

formula without interference of the wake can be expressedas follows:

Cw = 12 ρV

2wSwCC ,

LA =12 ρV

2wSwClb,

NA =12 ρV

2wSwCnb:

ð11Þ

Finally, the determined angle of attack and the side slipangle and the force and moment generated by the left andright wings are converted into the body CS.

2.5. Gravity Model. Convert the aircraft gravity into the bodyCS through the following equations:

Tgx

Tgy

Tgz

2664

3775 =

−sin θ

sin φ cos θcos φ cos θ

2664

3775mg: ð12Þ

3. Trim Procedure

3.1. Genetic Algorithm. Genetic algorithm is a random searchalgorithm that simulates natural selection and genetic behav-ior in living nature. The procedure of the genetic algorithm is

0Generation

Best, worst, and mean scores

20 40 60 80 100 1200

0.5

1

1.5

2×104

0Individuals

Selection function

20 140 1800

2

6

4

8

10

40 160 20060 80 100 120

Generation

Aver

gae d

istan

ceN

umbe

r of c

hild

ren

50 100 150 200 250 3000

0.5

1

1.5

2

00

Generation

Fitn

ess v

alue

Best fitnessMean fitness

50 100 150 200 250 3000

200

400

600

800 Best: 1.45942e‑06 Mean: 0.000277033Average distance between individuals

Figure 5: Result of GA when the tilt angle of main rotors is 50 degrees.

Table 1: Trim results with different size of population.

PopulationX

Minimum fitness score Mean fitness scoreTm (N) Tn (N) φ (degree) Va (m/s) δt (degree) δd (degree)

60 -92.112 5.525 0.05521 29.273 1.3932 3.5531 6989.4 7009.1

100 -29.868 -0.2532 0.03731 32.426 0.3043 0.1151 2:06 × 10-5 0.0615

200 -29.861 -0.2384 0.03691 32.218 0.3039 0.1140 1:52 × 10-6 8:22 × 10-5

500 -29.861 -0.2384 0.03691 32.218 0.3039 0.1140 2:72 × 10-6 1:92 × 10-6

5Journal of Sensors

shown in Figure 4. The parameters to be solved are encodedto represent as chromosomes. One unsolved parameter cor-responds to one chromosome, and all chromosomes consti-tute an individual, namely, a full state, which is a whole setof solution to equations. At the beginning, several individualsare randomly generated within the boundaries as the initialpopulation. Then, selection, crossover, mutation, and otheroperations are carried out by an iterative method to exchangechromosome information. Better individuals are screenedaccording to the fitness of the chromosomes. Finally, thechromosomes that meet the optimization target aregenerated.

3.2. Trim States. We demonstrate the trim results by takingthe flight states when its rotor tilt angle is 50° as an example,since both vertical and the horizontal states are involved,making it a representative case. The trim states to be deter-mined include the synchronous tilt angles of the main rotorsαm = αr = αl, differential tilt angles of the main rotors βm =−βr = βl, tensions of both main propellers Tl and Tr , tensionand the tilt angle of the back supplementary shrouded pro-peller Tb and αb, translational angle δt and differential angleδd of the aileron, and airspeed Va and three aircraft attitudes,namely, pitch, roll, and yaw. The vector with a total of 12states is as follows:

Generation

Best fitness

Best: 1.86697e‑07

Mean fitness

Mean: 5.13272e‑05

50 100 150 200 250 3000

1000

2000

3000

0

Average distance between individuals

Fitn

ess v

alue

Generation50 100 150 200 250 300

00.5

11.5

22.5

0

Aver

age d

istan

ce

(a)

Best fitnessMean fitness

Generation

Best: 2.38426e‑07 Mean: 0.347143

20 40 80 100 140 1600

500

1000

1500

2000

0 60 120

Fitn

ess v

alue

Average distance between individuals

Generation40 60 80 100 140 160

0

1

2

3

20 120

Aver

age d

istan

ce

(b)

Figure 6: Convergence result for population size of 100 (a) and 500 (b).

Generation

Best fitness

Best: 9.48124e‑07

Mean fitness

Mean: 2.45609e‑05

50 100 150 200 2500

1000

2000

3000

4000

0

Average distance between individuals

Fitn

ess v

alue

Generation50 100 150 200 250

0

1

3

2

4

0

Aver

age d

istan

ce

(a)

Best fitnessMean fitness

Generation

Best: 4.66178e‑07 Mean: 1.79232e‑05

50 150 250

0

500

1000

1500

0 100 200

Fitn

ess v

alue

Average distance between individuals

Generation50 100 150 250

0

0.5

11.5

2

0 200

Aver

age d

istan

ce

(b)

Figure 7: Convergence result for crossover probability of 0.2 (a) and 0.8 (b).

6 Journal of Sensors

X = αm βm Tl Tr Tb αb δt δd Va θ φ ψ½ �:ð13Þ

As the aircraft does not achieve enough airspeed whenthe tilt angle of main rotor is 50 degrees, the control efficiencyof the supplementary shrouded propeller is relatively low.Therefore, it is assumed that its tension Tb and its inclinationangle αb are 0. For providing an appropriate angle of attack,assume that the aircraft’s pitch is fixed at this time and theyaw is 0. For convenience, the tension of the two main rotorsis expressed as the translational tension of main propellersTm = ðTr + TlÞ/2 and their differential tension Tn = ðTr – TlÞ/2. For instance, if both the tilt angles of main rotors are50 degrees, the translational tilt angle αm = 50° and the differ-ential tilt angle αn = 0. The parameters to be trimmed can beexpressed as follows:

X = Tm Tn φ Va δt δd½ �: ð14Þ

As the states are the solution to the equations in the for-mat of float real number, the encoding and decoding for thetrim GA are very simple that the chromosomes are the statesdirectly.

3.3. Trim Fitness Function. According to the dynamic equa-tions, the forces and moments generated by each rotor whenthe tilt angle of main rotor is 50 degree are expressed in thebody CS.

Tex

Tey

Tez

2664

3775 =

−Tr cos 50° − Tl cos 50°

0Tr sin 50° + Tl sin 50°

2664

3775,

Mex

Mey

Mez

2664

3775 =

yrTr sin 50° + ylTl sin 50°

− zrTr + zlTlð Þ cos 50° − xrTr + xlTlð Þ sin 50°

yrTr cos 50° + ylTl cos 50°

2664

3775:

ð15Þ

Aerodynamic forces and moments can be expressed inthe body CS as follows:

Twx

Twy

Twz

2664

3775 =

cos αw 0 −sin αw

0 1 0sin αw 0 cos αw

2664

3775

cos βw −sin βw 0sin βw cos βw 0

0 0 1

2664

3775

−Dw

Cw

−Lw

2664

3775,

Mwx

Mwy

Mwz

2664

3775 =

cos αw 0 −sin αw

0 1 0sin αw 0 cos αw

2664

3775

cos βw −sin βw 0sin βw cos βw 0

0 0 1

2664

3775

LA

MA

NA

2664

3775,

ð16Þ

where the angle of attack αw = 2° and sideslip angle βw = 0.

Combined force and moment of the aircraft can be intro-duced as follows:

Tx = Tex + Twx + Tgx ,Ty = Tey + Twy + Tgy ,Tz = Tez + Twz + Tgz ,Mx =Mex +Mwx ,My =Mey +Mwy ,Mz =Mez +Mwz:

8>>>>>>>>>>><>>>>>>>>>>>:

ð17Þ

When the resultant forces and moments are zeros, theaircraft is in trim state. Therefore, the fitness function is setto be the sum of the squares of all the resultant forces andthe moments as follows:

FitnessFcn = T2x + T2

y + T2z +M2

x +M2y +M2

z : ð18Þ

3.4. Trim Optimization Settings and Results. As the GAparameters have impact on the optimization result, they areadjusted with an experimental design method [23] and somecomparisons are made here.

The GA solver of optimization toolbox in Matlab isadopted to realize the trim GA optimization. The fitnessfunction with 6 states is established as a previous chapterdescribes. According to the specific situation of the tiltrotoraircraft, the lower and upper boundaries are as follows:

Lower boundaries= −500N 500N −10 degree 0m/s −10 degree −10 degree½ �,

Upper boundaries= 0N −500N 10 degree 50m/s 10 degree 10 degree½ �:

ð19Þ

Since fixed pitch propellers are used in the JDW-1-1, onlyone-direction tensions can be provided, so the translationaltension of main propellers Tm is between 0N and -500N inthe rotor coordinate system while the differential tension isbetween 500N and -500N. The roll, translational angle δt ,and differential angle δd of the aileron are all constrainedbetween -10 degree and 10 degree for flight safety.

The population size is set to 200, and the maximum limitof evolutionary algebra is 400. The crossover probability is0.6, and the elite number is 50. The elite preservation strategyis that elites are guaranteed to survive to the next generation.

In order to remove the effect of the spread of the FitnessFunction raw scores, the rank of the scores is adopted for fit-ness scaling rather than the score itself.

The mutation probability is not just a simple fixed ratiobut a mutation function provided by Matlab. For the muta-tion function, adaptive feasible mode is selected, which ran-domly generates directions that are adaptive with respect tothe last successful or unsuccessful generation. A step lengthis chosen along each direction so that boundaries aresatisfied.

7Journal of Sensors

00

5 10 15Airspeed (m/s)

Roto

r tilt

angl

e (de

gree

)

Maximum rotor tilt angleMinimum rotor tilt angleForce balance position

20 25 30 35

10

20

30

40

50

60

70

80

90

Figure 9: Tilt corridor of the tiltrotor aircraft.

0Generation

Best, worst, and mean scores

100 200 300 4000

1

2

3

4 ×106

0Individuals

Selection function

20 140 1800

2

6

4

8

10

40 160 20060 80 100 120

Num

ber o

f chi

ldre

n

0Generation

Fitn

ess v

alue

Best fitnessMean fitness

100 200 300 400

0

2

4

6

8×104 Best: 67.6912 Mean: 43929.8

Generation

Aver

age d

istan

ce

50 100 150 200 250

0

20

40

60

80

100

0 400350300

Average distance between individuals

Figure 8: Results for mutation probability of 0.05.

8 Journal of Sensors

The crossover function of scattered is chosen, in which arandomly generated binary vector determines how the cross-over is conducted. The crossover algorithm selects the geneswhere the binary vector is a 1 from the first parent and thegenes where the vector is a 0 from the second parent andcombines the genes to form the child. For example,

Parent 1 = 0:1 0:2 0:3 0:4 0:5 0:6½ �,Parent 2 = 1 2 3 4 5 6½ �:

ð20Þ

If the randomly generated binary vector is rb =1 1 0 1 0 0½ �, then, the child is child =

0:1 0:2 3 0:4 5 6½ �. It should be noted that individ-uals are randomly chosen for crossover.

For the selection function, the mode of stochastic uni-form is chosen, which lays out a line in which each parentcorresponds to a section of the line of length proportionalto its expectation. The algorithm moves along the line insteps of equal size, one step for each parent. At each step,the algorithm allocates a parent from the section it landson. The first step is a uniform random number less thanthe step size.

The GA trim results are shown in Figure 5.According to the GA results, its fitness value continu-

ously drops down and reaches nearly zeros after around

0 5 10 15 20 25 30 35

Airspeed (m/s)

−100

0

100

200

300

400

500Fo

rwar

d fo

rce r

ange

(N)

0 5 10 15 20 25 30 35

Airspeed (m/s)

−600

−400

−200

0

200

400

600

Ver

tical

forc

e ran

ge (N

)

Figure 10: External force range of the tiltrotor aircraft.

0 100 200 300 400 500 600

Forward distance (m)

0

10

20

30

40

Hei

ght (

m)

0 5 10 15 20 25 30 35Time (s)

0100200300400500600

Forw

ard

dista

nce (

m)

0 5 10 15 20 25 30 35Time (s)

0

10

20

30

40

Hei

ght (

m)

0 5 10 15 20 25 30 35Time (s)

1030507090

0 5 10 15 20 25 30 35Time (s)

02468

10

0 5 10 15 20 25 30 35Time (s)

012345

Roto

r tilt

angl

e (°)

Pitc

h an

gle (

°)A

ngle

of a

ttack

(°)

Figure 11: Designed aircraft take-off trajectory and attitude.

9Journal of Sensors

150 generations. Although the maximum limit of evolution-ary algebra is 400, the deviation of mean fitness between twogenerations is less than 1 × 10−5, when the generation reaches262, so GA terminates and provides final optimizationresults.

The best individual, namely, the globally optimal solutionof trim state, is as follows:

XTGA =

Tm

Tn

φ

Va

δt

δd

2666666666664

3777777777775=

−29:861N−0:2384N0:0369 deg32:218m/s0:3039 deg0:1140 deg

2666666666664

3777777777775: ð21Þ

From Figure 5, the average distance between individualsreaches nearly 0 after 150 generations, indicating that theGA algorithm achieves good convergence. The worst andmean scores drop quickly in the first 60 generations, showingthat the GA algorithm converges rapidly. The last subfigureshows the distribution of selection function, namely, thenumber of children of each individual for the second last gen-erations. The children are generated mainly by the first 50elites. In summary, the GA trim results achieve good conver-gence and succeed in providing a globally optimal solution.

Some comparisons are made with different GAparameters.

(1) Size of population: the size of population does notaffect the optimization result much once it is morethan 100. However, less than 100 populations doachieve a poor result. The trim results when the pop-ulation is 60, 100, 200, and 500 are summarized inTable 1. Other parameters are fixed as previouslystated. It is obvious that the differences of the finalsolution among the population of 100, 200, and 500are negligible. However, the convergence speed islower for 100 population case, as Figure 6 shows,compared with the first subfigure of Figure 5. Besides,the generation reaches 340, which also indicatespoorer convergence performance for the case of 100

0 5 10 15 20 25 30 35

Time (s)

0

20

40

60

80

100

Forw

ard

forc

e (N

)

0 5 10 15 20 25 30 35Time (s)

−40

−30

−20

−10

0

10

20

Vert

ical

forc

e (N

)

0 5 10 15 20 25 30

Airspeed (m/s)

0

20

40

60

80

100

Forw

ard

forc

e (N

)

0 5 10 15 20 25 30

Airspeed (m/s)

−40

−30

−20

−10

0

10

20

Vert

ical

forc

e (N

)

Figure 12: Required force for the aircraft to take off.

Table 2: Trim results of several typical moments during take-off.

Am(°)

Tm (N) Tn (N)φ

(degree)Va (m/s)

δt(degree)

δd(degree)

15 -276.904 -0.06976 0.003355 11.56308 17.85581 -0.08725

20 -219.458 -0.14263 0.010518 16.69597 5.707258 -0.03233

30 -152.655 -0.13814 0.016089 20.82226 0.941265 0.011532

40 -115.473 -0.12574 0.019347 22.80624 -0.66647 -0.00603

50 -95.2876 -0.11508 0.023792 24.38439 -1.17095 0.075036

60 -83.6407 -0.11532 0.024298 26.18835 -1.03244 0.149144

70 -75.7513 -0.13062 0.027013 28.15163 -0.70624 0.210545

75 -72.3484 -0.13696 0.029101 29.17739 -0.51581 0.230538

80 -66.8869 -0.15094 0.03261 31.04621 -0.05611 0.233952

85 -52.493 -0.16149 0.035012 32.45665 0.194407 0.217035

90 -26.6569 -0.16494 0.036658 32.94414 0.156521 0.216939

10 Journal of Sensors

populations. Hence, for better accuracy and conver-gence considerations, the population size of 200 isselected in our research.

(2) Crossover probability: the crossover probabilities of0.2 and 0.8 are compared with the case of 0.6. Thefinal optimization results are almost the same, yetthe convergence performance shown in Figure 7 indi-cates that 0.6 is the best in the three cases.

(3) Mutation function: mutation function has a greatimpact on the optimal solution. When the mutationprobability is fixed to 0.05 with other settings beingthe same, the optimization results got much worse,which can be seen from Figure 8. Not only the fitnessvalue keeps high, but also the convergence is poor.The final solution is not reliable. Different mutationprobabilities have been testified, and the results arenot satisfied. Therefore, the adaptive feasible muta-tion function is adopted in our research.

The premier results obtained by the GA are adopted asthe initial values of the LMmethod to refine the trim solutionin local scale, and the final trim results are obtained as fol-lows:

XT =

Tm

Tn

φ

Va

δt

δd

2666666666664

3777777777775=

−29:852N−0:2231N0:0365 deg32:212m/s0:3038 deg0:1122 deg

2666666666664

3777777777775: ð22Þ

3.5. Tilt Corridor Design for Tiltrotor Aircraft Take-Off. Basedon the innovative trim method proposed previously, theequilibrium flight states, for instance, airspeed, tensions ofthe rotors, and angles of flaps, can be determined under dif-ferent tilting angles on a condition that proper actions aretaken by the flight controller. However, the aircraft cannotalways be in a “trimmed state” but also the “transition state”.Therefore, the limitations of the flight states should also bedetermined in order to keep the tiltrotor aircraft under con-trol. For example, when the tiltrotor aircraft is changing thetilt angle, if the airspeed is too low but the tilt angle too small,namely, the rotor tension pointing horizontally too much,the altitude of the aircraft will decrease rapidly, and evenworse, the stall will occur since the lift is not enough; Onthe other hand, due to the limitation of rotor performanceand aircraft aerodynamic characteristics, there are maxi-mums for the airspeed and external forces applied to theaircraft.

In order to provide limitations for the transition statesand the reference for the design of the flight trajectory, it is

necessary to specify the relationship between the airspeedand the rotor tilt angle, which creates a rotor tilt angle-airspeed envelop called the tilt corridor.

According to the dynamic model of the tiltrotor aircraft,the maximum tilt angle is constrained by the z-axis force inthe body CS, and the minimum tilt angle is determined bythe x-axis force in the body CS. The aerodynamic force Fwand the rotor tension T are related to the airspeed. Whenthe maximum of the rotor tension is reached, it yields tothe following:

2Tm max cos Am max +mg + Tb max + Fwz ⩽ 0,T sin Am min − Fwx ⩾ 0,Fx max = −2Tm max sin Am max + Fwx ,Fz max = 2Tm max cos Am min +mg + Fwz ,

8>>>>><>>>>>:

ð23Þ

where Tm max is the maximum main rotor tension, Tb max isthe maximum back rotor tension; theAm max is the maximumtilt angle; Fwz and Fwx are the aerodynamic force along the z-axis and x-axis in the body CS, respectively; and Fz max andFx max are the maximum of the aerodynamic force along the z-axis and x-axis in the body CS, respectively.

As shown in Figure 9, the area enclosed by the dotted lineand the long broken line is the feasible range of the rotor tiltangle at a specific airspeed. The solid line with circles is theequilibrium state of the aircraft calculated by the innovativetrim method, in which the aircraft maintains force balance.When the tiltrotor aircraft state is within the envelope ofmaximum tilt angle curve and the balance curve, the aircraftis accelerating. On the contrary, in another area, the aircraftis decelerating. These constitute the transition states of thetiltrotor.

Furthermore, the external force limitation must also beconsidered when designing the flight path of take-off, so therange of the external forces along the x-axis (forward) andz-axis (vertical) of the tiltrotor aircraft according to airspeedare also derived and shown in Figure 10.

4. Take-Off Simulation and Verification

In this paper, an incline flight trajectory in which both heightand airspeed of tiltrotor aircraft increase simultaneously inthe take-off phase is designed based on the proposed trimmethod and the tilt corridor. The external forces, tiltingangles, angle of attack, etc. with respect to time or airspeedare depicted in Figures 11 and 12.

Several typical moments during take-off are selected to fitinto the trim equations in Chapter 2, in which the values ofTx and Tz in Equation (17) are external forces in perpendic-ular directions. It can be verified from the comparisonbetween Figure 10 and Figure 12 that the required externalforces for tiltrotor aircraft to take off according to thedesigned trajectory are all within the allowable range. Thefollowing results shown in Table 2 are obtained when thetiltrotor aircraft is in take-off phase.

The whole process of take-off is simulated afterwards.The trim results detailed above are applied as the target state

11Journal of Sensors

of the flight control during the take-off. The flight simulationresults and the trim values are compared to be in good agree-ment (Figure 13), so the effectiveness of the innovative trimmethod is verified through flight simulation.

Besides, the simulated flight trajectory of the tiltrotor air-craft is basically consistent with the designed flight trajectory(Figure 14).

5. Conclusion

An innovative trim method for tiltrotor aircraft take-offbased on a genetic algorithm is proposed in this paper.

Firstly, the genetic algorithm, which possesses strong capa-bility in searching global optimum, is adopted to identify acoarse solution. Secondly, the coarse solution of the trim isfurther refined by the Levenberg-Marquardt method forprecise local optimum. In addition, the innovative trimmethod is applied to a tiltrotor aircraft’s flight control inthe transition process of incline take-off. The limitationof trajectory is discussed, and the tilt corridor is con-structed. Finally, the incline take-off simulations are con-ducted, and the effectiveness of the proposed trimmethod is verified through good match with the designedreference trajectory.

0 100 200 300 400 500 600Forward distance (m)

0

10

20

30

40

Hei

ght (

m)

Design trajectorySimulation trajectory

0 100 200 300 400 500 600

Forward distance (m)

−0.2−0.1

00.10.20.30.4

Erro

r of h

eigh

t (m

)

Figure 14: Comparison of design trajectory and simulation.

0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

80

90

Trimming pointsSimulation of take-off

Airspeed (m/s)

Figure 13: Comparison of trim results and simulation.

12 Journal of Sensors

Data Availability

The airplane design data used to support the findings of thisstudy have not been made available because the projectinvolves intellectual property rights.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

The research is funded by the Beijing Natural Science Foun-dation (4204103) and Aeronautical Science Foundation ofChina (2019ZC051009).

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