genetic algorithm technique applied to a forced landing

24TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

1

Abstract

In this research a trajectory optimisation wasbeing undertaken using Genetic Algorithm tosearch for optimal landing manoeuvres for aforced landing of an airplane after enginefailure using a simplified analytical model forthe Beech Bonanza model E33A retractable-undercarriage aircraft. Vertical atmosphericdisturbances were simulated using anapproximated von Karman low altitude modelfor the atmospheric disturbance model based onthe MIL-F-8785C specifications. A forcedlanding manoeuvre analysis was carried out foran engine failure at 650 ft AGL for bank anglevarying from banking left at 45° to bankingright at 45° and with an aircraft’s speed varyingfrom 75.6 mph to 208 mph corresponding to 5%above airplane’s stall speed and airplane’smaximum speed respectively. The results showthe effects vertical disturbances have on thegeneral flight paths for three pre-selectedlanding locations.

1 IntroductionThe study of safe landing of aeroplanes is a veryimportant issue in the aviation field and isconsidered by pilots as the most demanding taskin every flight. Many accidents have occurredduring the landing phase of flights, some ofwhich were beyond the pilot’s control, somewere due to human error, while some couldhave been successful if only a more optimallanding manoeuvre was carried out.Unfortunately, it is the disastrous failed landingsthat became a statistics with the NationalTransportation Safety Board (NTSB) and those

who landed safely or with minor damagegenerally are not reported to Federal AviationAuthority (FAA) or NTSB. Hence, the statisticsgathered are skewed or biased towards failedattempts.

The research problem in this study has itsinterest in the search for the best landingtrajectory for a forced landing manoeuvre of anaircraft after an engine failure. Such situationcould occur after take-off [1] or during a levelflight at any altitude above ground level (AGL)[2]. When an engine failure occurs in an aircraftand no additional power is available, the pilotmust select a suitable location to land safelywith the limited amount of energy availablefrom the engine failure position. The generalrecommendation is to land straight ahead andRogers’ studies [1] confirmed the high rates ofusing this ingrained technique. However, hesuggests that, for forced landings from a higheraltitude, a turnback manoeuvre may be flownbecause higher altitude allows for more time inthe air. This research is also an extension ofRogers’ forced landing manoeuvre where thesearch for optimal landing paths begins after thepilot has selected a practical landing location onthe ground that is within range after an enginefailure. A study on the effects verticalatmospheric disturbances have on a forcedlanding manoeuvre was also carried out.

2 Problem Description – ForcedLandings

Landing an aircraft that has suffered an enginefailure during take-off is one of theclassifications of a forced landing and is the

GENETIC ALGORITHM APPLIED TO A FORCEDLANDING MANOEEUVREPeter Tong*, Cees Bil** George Galanis‡

*†RMIT University, ‡Defence science and Technology Organisation Australia

Keywords: genetic algorithm, forced landing, flight paths

Peter Tong*, Cees Bil** George Galanis‡

2

focus of this study. The generalrecommendation in the aviation literature forsuch a situation is to land straight ahead [3, 4].For example, FAA regulations recommend thatpilots land straight ahead and should neverattempt track reversals in an effort to land on thedeparture runway. The present researchproblem is an extension of a forced landingmanoeuvre on the Beech E33A Bonanza singleengine aircraft considered by Tong, Galanis andBil[5] based on Rogers[1]. The problemconsidered in this forced landing assumed anengine failure at 650 ft AGL after take-off. Itused the engine failure point as the referencepoint for all distances calculated. It wasassumed that the transition in speed occurredinstantaneously and the effects of landing gearretraction/extension were not considered.

A graphical interpretation of the forced landingafter an engine failure at an arbitrary altitude isshown in Fig. 1. This research took theapproach of an ensemble of probability oflanding within a specified tolerance from a pre-selected location and not as an optimal controlproblem of the deviation from the flighttrajectory during a forced landing manoeuvre.In other words, what are the chances of the pilotperforming the landing task with maximumprobability of landing on a pre-selected landingsite? The calculations performed in this studyused the general flight dynamics equations [6]and data based on the Beech Bonanza E33Aretractable aircraft characteristics obtained fromRogers as shown in Table 1. The data for initialtakeoff ground roll and distance to clear 50 ftobstacle are obtained from Rising Up AviationResources1.

1 Data available online athttp://www.risingup.com/planespecs/info/airplane117.shtml. 2003.

Table 1. Beech Bonanza Model 33A characteristics [1]Parameter Value

Gross Weight, lb 3300Wing Area, ft2 181L/Dmax 10.56Power, brake horsepower 285Propeller Constant speed 3-bladeVmax, mph 208Vcruise at 65%, mph 190Vstall(clean)

a power off, mph 72Vstall(dirty) power off, mph 61VL/Dmax

b, mph 122

Vγmax c at sea level, mph 91

VR/Cmax d at sea level, mph 112.5

R/C at sea level and 3300 lb, ft/min 1200Parabolic drag polar CD = 0.019 + 0.0917CL

2

Takeoff: Ground roll, ft 880Takeoff: Over 50 ft obstacle, ft 1225Landing: Ground roll, ft 625Landing: Over 50 ft obstacle, ft 1150a Gear and flaps retracted.b L/Dmax = maximum lift to drag ratio.c γ = glide angle.d R/Cmax = maximum rate of climb.

3 Atmospheric ModelThe simulation of atmospheric turbulence is ofconsiderable importance and is a criticalcomponent in any aircraft simulation and intrajectory optimisation development.Atmospheric disturbances are very random bynature and so are its magnitudes and thefrequencies of occurrence. They are affected,for example, by the geographical location, theweather and the time of the year.

For this study, the von Karman low altitudemodel and the medium altitude model foratmospheric disturbance model based on theMIL-F-8785C specifications were used tosimulate the atmospheric turbulence model [7].The atmospheric turbulence velocity for thisresearch was calculated using [8]’sapproximated transfer function from the vonKarman spectrum.

The vertical turbulence velocity terms werevectorially added to the aircraft’s velocity andwere assumed to have an instantaneous changeto the aircraft’s vertical velocity. The aircraftwas also assumed to be flying through a one-dimensional gust field where only the verticalvelocity changed and it was assumed that the

Engine failure point

Pre-selectedlanding locationLanding area

within range

Fig. 1. Forced Landing Area

3

Genetic Algorithm Technique Applied to a Forced Landing Manoeuvre

turbulence encountered was independent oftime. In other words, the turbulence profile wasfrozen or fixed in space and time.

The influence of wind velocities have on anaircraft could be roughly separated into 2 parts.The flight performance description of an aircraftdepends on the low frequency part of the windvector, the wind shear component. It is only thelow frequency part of the wind that influencesthe energy relation of the aircraft. The highfrequency wind components in the atmosphericturbulence and gust are considered highfrequency and have no effect on the aircrafttrajectory. Their effects are on the aircraft'sloading, the structural fatigue, the pilot'sworkload, passenger comfort, and on the flyingqualities of the aircraft. The eigenmotions ofthe aircraft, the phugoid and the short periodmotion are important frequencies for theseparation effects. If the frequency of the windperturbation is less than the phugoid frequency,the change of aircraft trajectory is directlyproportional to the wind angle of attack,meaning the low frequency directly changes theaircraft trajectory. If the range of frequency isabove the short period motion, the inertia of theaircraft avoids large change of trajectory. Inother words the flight path is only affected inphugoid mode where the pitch angle follows theflight path angle [9-11]. Using the airplane’sstall speed (Vs = 72 mph) and the well-knownphugoid period equation, the phugoid period

was found to be .sec152≈=

gVTPhugoid

π

100 atmospheric turbulence profiles weregenerated for reference speed of 15 kts at 650 ftAGL using the von Karman atmosphericturbulence model. An improvisation to theatmospheric turbulence model was made bytime averaging the von Karman atmosphericturbulence model to coincide with the airplane’sapproximated phugoid mode of 15 secs. This ineffect is one method of obtaining the lowfrequency velocity representations. Themaximum time averaged atmospheric

turbulence updraft is 1.48 ft/sec and themaximum downdraft is –1.67 ft/sec.

4 Genetic AlgorithmsDrawing parallels from natural selection,Holland[12] proposed the theory of GA in theearly 1970’s which imitates the evolutionaryprocesses in nature. Evolution can beconsidered as a form of an optimisation problemwhere only the fittest individuals will surviveand reproduce, also known as the “survival ofthe fittest”. GAs use crossovers – aprobabilistic mechanism for randomisingchromosomes, and mutations – a perturbationmechanism, as search mechanisms to generate asequence of populations. The most rudimentaryunit, the genes, which can take the form ofdifferent alleles, are combined to formchromosomes that control the “keys” to thesurvival of the individual in a competitiveenvironment. Evolution occurs when thechromosomes from two parents are combinedduring reproduction and a new gene pool isformed from combinations generated througheither crossover or mutation.

GAs perform parallel, non-comprehensivesearch in the hope of finding the globalmaximum, if not a very near optimal solution, tooptimisation problems. The procedure to solvea complex problem using GAs is to define thesearch space and to custom design a codingscheme for the solutions in the search spacetailored to the problem. This process is knownas genetic representation[13]. A fitness functionis then designed to evaluate the potentialsolutions, and the “better” ones are kept forsubsequent regenerations and the “inferior”solutions are discarded. The next generation ofsolutions are created by applying the crossoversand mutations genetic operators to evolvesolutions for further fitness evaluations. Theoptimisation process terminates when either anacceptable tolerance in results is obtained, orwhen it has processed a specific number ofgenerations, or when no improvement in fitnessvalue is encountered after a number of


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consecutive generations. The GA cycle isshown in Fig. 2. The three most importantfeatures are fitness function, the geneticencoding and the genetic operators.

GA is a relatively new optimisation methodcompared to the traditional gradient searchmethod, which has difficulties for discontinuousor non-smooth functions. In solvingoptimisation problems, GAs have the advantagethat no derivatives have to be found but theyhave only to utilize the governing equations inthe problem considered. The trade off in notusing gradient information is that it does notguarantee a minimum point but will locateresults that are very close to the optimalsolution. Other optimisation methods such asgradient method may be able to locate betteroptimal solutions but may suffer computationaltime. GA is not an alternative nor is it areplacement method to other traditionaloptimisation methods but it is a validcomplementary optimisation technique. Insimulation, obtaining acceptable results rapidlyis more valuable than spending an enormoustime searching for the optimal point and GA iscapable of doing so.

4.1 Real-Value GA OperatorsA real-value representation was used since ithelped to exploit the numerical properties of acandidate solution by exploiting the solutiongradients and information from the function’slandscape. The GA real-value chromosomes arerepresented by a vector xv = (x1, … xn), where n isthe chromosome length. The chromosomelength is equivalent to the number of variablesused to represent the domain. Each gene, (xk),

in the chromosome is bounded by an upper limit(xmax) and a lower limit (xmin) specific to thegene.

A brief description from Michalewicz[14] ispresented here for the different operators usedfor a real value encoding. The genetic operatorsused for real value encoding in this analysisconsist of three types of mutation operators andthree types of crossover operators:

i) Uniform mutation randomly mutates agene in the chromosome with uniformprobability distribution to any valuewithin the real-valued domain range.This operator is important in the earlyphases of the evolution process, as thesolutions are free to move within thesearch space.

ii) Boundary mutation mutates a gene toeither the lower boundary value or theupper boundary value for the real-valuedrange. This operator is very useful forGAs with constraints.

iii) Non-uniform mutation mutates a geneby a factor that is a function of thedifference in value between thatparticular gene and either of itsboundary value, and the generationnumber. This mutation probability willdecrease to 0 as the generation numberincreases. This type of mutation is usedfor local fine-tuning of genes where theoperator will initially search the spaceuniformly and very locally at latergenerations.

iv) Arithmetic crossover linearly combinesthe genes from two parents to producetwo children.

v) Simple crossover randomly selects apoint in a chromosome as a crossoverpoint which is very similar to thetraditional one-point crossover.

vi) Heuristic crossover uses the values ofthe objective function to determine thedirection of the search and it may or maynot produce an offspring. It isresponsible for local fine-tuning andsearch in the promising direction.

Elitism

Population(chromosomes)

Geneticoperation

Selection(mating pool)

Evaluation(fitness)

DecodedstringsParents

ReproductionManipulation

Mates

OffspringNew

generation

Fig. 2. Genetic Algorithm Cycle

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All of the six genetic operators described wererequired to explore the search space adequatelyand were used equally to prevent prematureconvergent without regard to fitness. Forexample, arithmetical crossover would tend todrive the population to the numerical center ofthe search space very quickly, regardless if ityields good fitness values, and boundarymutation would set the gene to either of itsboundary value. However, the use of otheroperators will prevent such problem. It isthrough the combination of these powerfulcrossover and mutation operators developed byMichalewicz that the search space can beexplored and good genetic material exploited.In order to randomise the use of the three typesof crossovers and mutations uniformly, thepopulations were randomly chosen for thedifferent types of crossovers and mutations.The three types of crossovers and mutationswere applied equally to all the randomisedpopulations in every generation to allow equaldistribution of genetic operators.

The GA in this analysis used tournamentselection, whereby two chromosomes wereselected from the population and compared atany one time to select the fitter chromosome forcrossover and mutation. This selection methodprevented fitness scaling where a few highly fitchromosomes may dominate the parentpopulation. It also used elitism where a certainnumber of the best chromosomes from theprevious generation are cloned to the presentgeneration.

4.2 Real-Value Control Parameters SelectionA common issue that arises in using GA is theproper selection of parameter settings [15-18].Using a large population of chromosomes willincrease computational time but it will preventpremature convergence to a local optimumwhile using a small population of chromosomeswill reduce computational time but it mayconverge prematurely to a sub optimal solution.In general, a high crossover rate increases therecombination of building structures but anexcessive crossover rate may lead to high-

performance structures being discarded fasterthan the selection process is able produceimprovement while a low crossover rate maystagnate the optimisation process. The selectionof the crossover rate also depends on thepopulation sizes where higher crossover rate forsmaller populations can prevent prematureconvergence while lower crossover rate be usedfor larger populations since they have largersearch space. A high mutation rate, which ismore important in later generations, resembles arandom search but it may help to reintroducelost structure while a low mutation rate maylead to a convergence to a local optimum.

Since the GAs method relies on stochasticprocesses and is optimisation objectivedependent, a control parameter test was carriedout to determine the best selection of populationsize, the number of generations, the crossoverrate and the mutation rate for the optimisationproblem considered in this study. Achromosome length (l) of 26 bits, correspondingto two variables, speed and bank angle, for eachof the 13 discrete altitude steps to ground levelfor an engine failure at 650 ft AGL was used. Agenetic control parameter selection process asshown in Table 2 was carried out for three setsof pre-selected touchdown locations. They werelocated at 0 ft laterally and –3100 ftlongitudinally, at 3000 ft laterally and 3000 ftlongitudinally, and at 500 ft longitudinally and200 ft laterally from the engine failure pointwith reference to the airplane’s headingdirection at that instance. The code forimplementation of the GA was developed inMatlab 5.3. The fitness value in this GAtrajectory optimisation was defined as thedistance between the touchdown location andthe pre-selected touchdown location.

Table 2. Real-Valued Control Parameters SelectionGA Control Parameters Range Step Size Cumulative runs

Coef. for non-uniformmutation (b)

2 – 8 2 4

Crossover rate 30% – 40% 10% 8Population size (N) 52 – 260 52 40Mutation rate 1% – 9% 2% 200Repetition for each test 100 1 20000

APeter Tong*, Cees Bil** George Galanis‡

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The GA control parameter selection was carriedout for an elitism of 10% of the population sizefor subsequent generations. A larger percentageof elitism used may stagnate the evolutionprocess while a smaller percentage of elitismused may slow down the convergence. Astopping criterion of 100 generations and arepetition of 100 GA runs were carried out foreach of the 200 combinations of parameters foreach test location to reduce the effect ofprobabilistic “noise”. For each run, the bestfitness value was recorded as measure for theGA’s performance. These values were averagedover 100 runs for every combination ofpopulation sizes (N), crossover rates,coefficients for non-uniform mutation (b) andmutation rates to provide a representativeperformance of a general GA run. Thecomputation cost is defined as a product of thepopulation size and the minimum number ofgenerations it takes to obtain the minimumfitness value for each trial limited to thestopping criterion of 100 generations used. Theaverage performance values for each set werenormalised with respect to those values fromruns with population size of 52 (2l) andmutation rate of 1%.

Calculations were carried out using altitudesteps of 50 ft for convenience instead ofconstant time interval steps. The reason for notusing constant time interval was the requirementto terminate the GA search at exactly 0 ft AGL.In this forced landing simulation, at every 50 ftdrop in altitude, the pilot continuously decideson the flying speed and bank angle. At eachaltitude step, the pilot may elect to turn bybanking continuously at ±45 deg flying withspeed varying continuously from 5% above stallspeed (75.6 mph) to maximum speed (208mph).

4.3 Results for Real-value ControlParameters SelectionSince similar trends in results were observed forall the three pre-selected touchdown locations,remarks will be made for the pre-selectedlocation of touching down at 0 ft laterally and –

3100 ft longitudinally. Larger population sizesusually provide a more accurate solution.However, for mutation rates of less than 5%,there is relatively little improvement in thefitness value for population size above 4l (N =104) as shown in Fig. 3 and fitter values wereobtained using a crossover rate of 30% asshown in Fig. 4 This minimal improvement infitness value is at the expense of greatercomputational cost, which in general is a linearincrease of computational effort for an increasein population size. While there is minimalimprovement in the best fitness for populationsgreater than 4l, the computational effortcontinues to increase as shown in Fig. 5.Hence, increasing the population size beyond 4ldid not appear to be worth the relatedcomputational cost. The results suggest that apopulation size of 4l (N = 104) is an appropriatecompromise for a best fitness value and areasonable computational effort which agreeswell with Williams and Crossley[18]. Acoefficient for non-uniform mutation of 4provided a fitter and more consistent best fitnessvalue for the different control parametercombinations tested. Very similar results forcomputational cost were observed for mutationrates ranging from 3% to 9%. Nevertheless, thefitness values for these mutation rates are not asfit as the fitness value for a mutation rate of 1%.The deterioration in fitness value may be due toexcessive increase of mutation in thechromosomes. Mutation rates have a significanteffect on the fitness performance but its effect isreduced as the population size increase beyond4l.

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0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5 GA Control Parameter Analysis - Best Fitness (0 ft.,-3100 ft.)

100 generations, coef for non-uniform mutation = 4

Normalized Population Size

Nor

mal

ized

Bes

t Ave

rage

d Fi

tnes

s

crossover rate = 30% mutation rate = 1%crossover rate = 30% mutation rate = 3%crossover rate = 30% mutation rate = 5%crossover rate = 30% mutation rate = 7%crossover rate = 30% mutation rate = 9%crossover rate = 40% mutation rate = 1%crossover rate = 40% mutation rate = 3%crossover rate = 40% mutation rate = 5%crossover rate = 40% mutation rate = 7%crossover rate = 40% mutation rate = 9%

Fig. 3. GA Control Parameter – Normalised Best Fitness

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140 GA Control Parameter Analysis - Best Fitness (0 ft.,-3100 ft.)



Touc

hdow

n D

ista

nce

from

Pre

-sel

ecte

d po

int (

ft)


Fig. 4. GA Control Parameter – Best Fitness

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 GA Control Parameter Analysis - Normalized Computation Cost (0 ft.,-3100 ft.)



Nor

mal

ized

Nor

mal

ized

Com

puta

tion

Cos

t


Fig. 5. GA Control Parameter Analysis – ComputationCost

Based on the GA parameter analysis carried out,a set of parameters were found suitable forfurther work on this particular trajectoryoptimisation for a forced landing manoeuvreupon engine failure. These are a population sizeof 104, a crossover rate of 30%, a mutation rateof 1%, a coefficient for non-uniform mutationof 4, using tournament selection and an elitismof 10% based on population.

GA convergence history is important to theunderstanding of its behaviour to ensuresufficient number of generations was run toobtain satisfactory results. Fig. 6 illustrates atypical convergence history for 100 generationsfor each of the 100 trials obtained from thecontrol parameter selection analysis. Nopremature convergence was observed and theselected generation of 100 has allowed asatisfactory development of the population.

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

4000

5000

6000

7000GA Optimisation Statistics

Generation Number

Fitn

ess

Valu

e

Population size = 104Crossover rate = 30% Mutation rate = 1%

Fig. 6. Evolution History for 100 Trials

5 Results for GA in Forced Landings withContinuous Speed and Continuous BankAngleA GA for forced landing with continuous speedand continuous bank angle was carried out forthree test locations; (0 ft, -3100 ft), (3000 ft,3000 ft) and at (500 ft, 200 ft), where the 1st

component represents the lateral distance andthe 2nd component represents the longitudinaldistance from the failure point. It used real-value representation and the control parameters


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determined in the previous section. In order totest the validity and the reliability of the real-value representation GA search proceduredeveloped, the results were compared to theresults obtained using the exhaustive searchmethod and similar results were obtained.

5.1 Results with Still Air ConditionsThe results for GA with continuous speed andcontinuous bank angle for touchdown distancesfrom the respective pre-selected landinglocations are shown in Table 3.

Table 3. Results for GA with Continuous Speed andContinuous Bank Angle

Pre-selectedLocation

GlobalMinimumDistance

Average MinimumDistance from 100

trials

Probability of Landing ≤Average Minimum

Distance from 100 trials(0 ft, -3100 ft) 0.0064 ft 0.4252 ft 91 %

(3000 ft, 3000 ft) 0.0005 ft 0.0312 ft 59 %(500 ft, 200 ft) 0.0022 ft 0.0426 ft 73 %

Based on the results obtained, the GA withcontinuous speed and continuous bank anglesearch method has successfully found suitablecombinations of aircraft speed and bank angle toland extremely close to the pre-selectedlocations with high probability for landingwithin the average minimum distance except forthe (3000 ft, 3000 ft) location. The lowerprobability in obtaining paths within the averageminimum distance for location (3000 ft, 3000 ft)is due to the nature of the solution landscape.For minimum global values with neighbourhoodvalues of very small differences, as illustratedby this particular test location, GA will have alower probability (59%) in locating paths thatare within the average minimum distance sinceGA concentrates in that solution space andnearby adjacent values maybe found instead. Aslightly higher mutation rate may assist inlocating the global minimum value better insuch a solution landscape but it is not necessarysince the average minimum distance from thepre-selected location is very small (0.0312 ft)for this particular location. The quality in theresults obtained for all the three pre-selectedlocations can be observed from the probabilityranging from 59% to 91% in landing less thanor equal to the average minimum distance fromthe 100 trials.

Two general flight paths for the pre-selectedlocation (0 ft, -3100 ft) exist since this locationis located along the airplane’s line of symmetryat failure point. The best flight paths found fromeach of the 100 runs and their average motionvariables for turning right are shown in Fig.7.The results show that in order to land at the pre-selected location with high probability, the pilotflies the following manoeuvre. Upon enginefailure point at 650 ft AGL, the pilot begins atight turn either by banking steeply right or leftat approximately 40° while flying at 76 mph.This is followed by increasing the airplane’sspeed towards 107 mph and widening of theturn radius by gradually reducing the bank angleto 23° as it descends to approximately 200 ftAGL. The pilot then continues flying atapproximately 106 mph while further reducingthe bank angle towards 0° until touchdown.

(a)

# RunsFinal Heading

50 %126° ≤ ψ ≤ 156°

50 %206° ≤ ψ ≤ 240°

0 100 20030040050060070

80

90

100

110

Average Flying Parameters for Forced Landing (0 ft,-3100 ft) Variable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

Height (ft)

Aver

age

Spee

d (m

ph)

0 100 200300400500600

-20

0

20

40

60

Aver

age

Bank

Ang

le (d

eg)

(b)Fig. 7. Optimal Forced Landing (0 ft, -3100 ft)Continuous Speed and Continuous Bank Angle at 650 ftAGL

The best flight paths found from each of the 100runs and their average motion variables for the

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pre-selected location (3000 ft, 3000 ft) areshown in Fig. 8. The general flying manoeuvreto land at the pre-selected location of (3000 ft,3000 ft) is, upon engine failure at 650 ft AGL,to begin with a left bank of approximately 9° at96 mph. This is followed by a right bank untilapproximately 15° at 500 ft AGL and a furtherright bank towards 22° while graduallydecreasing the airplane’s speed untilapproximately 87 mph at 250 ft AGL. The pilotthen increases the airplane’s speed toapproximately 91 mph along with a gradualdecrease in right bank angle towards 7° untiltouchdown. The landing manoeuvre for thispre-selected location resembles the 90° landingapproach in general aviation for flying from thebase leg to the final leg where the pilot begins toturn towards the pre-selected location when it isat 45° from the current position. The resultsalso show the double base leg flight paths whereat engine failure point the aeroplane turns rightand then followed by a left turn to touchdown.Although these are feasible flight paths, they aregenerally not recommended since they areharder to fly and will cause an increase in thepilot’s workload.

(a)

# RunsFinal Heading

2 %045° ≤ ψ < 090°

53 %090° ≤ ψ < 135°

42 %135° ≤ ψ < 180°

3 %340° ≤ ψ < 045°(double base leg)

0 100 20030040050060085

90

95

100

Average Flying Parameters for Forced Landing (3000 ft,3000 ft) Variable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

Height (ft)

Aver

age

Spee

d (m

ph)

0 100 200300400500600-20

0

20

40

Aver

age

Bank

Ang

le (d

eg)

(b)

# RunsFinal Heading

2 %045° ≤ ψ < 090°

53 %090° ≤ ψ < 135°

42 %135° ≤ ψ < 180°

3 %340° ≤ ψ < 045°(double base leg)

Fig. 8. Optimal Forced Landing (3000 ft, 3000 ft)Continuous Speed and Continuous Bank Angle at 650 ftAGL

The majority of the initial flight paths for thepre-selected location (500 ft, 200 ft) show that95% of the flight paths turn toward the pre-selected location while counter intuitive flightpaths of 5% turn away from the pre-selectedlocation. The counter intuitive manoeuvre isalso possible since the pre-selected location islocated near the airplane’s line of symmetry atengine failure point. The best flight paths foundfrom each of the 100 runs and their averagemotion variables for both right and left turns forthe pre-selected location (500 ft, 200 ft) areshown in Fig. 9-a. The general flyingmanoeuvre to land at the pre-selected locationof (500 ft, 200 ft) for the turn towards the pre-selected location (see Fig. 9-b) is, upon enginefailure, to fly at 100 mph, banking left atapproximately 7%, then gradually decreasingthe airplane’s speed to approximately 79 mphwhile increasing the right bank angle towards37° until 350 ft AGL. The pilot then maintainsthe flying speed at 79 mph while maintainingthe bank angle at 37° until 250 ft AGL. This isfollowed by gradually increasing the airplane’sspeed towards 96 mph and decreasing the bankangle towards 12° until touchdown. Thegeneral flying manoeuvre for the counterintuitive manoeuvre (see Fig. 9-c) is to fly at 96mph, banking left at approximately 5.5%,gradually decreasing the airplane’s speed toapproximately 77 mph while increasing the leftbank angle towards 37° until 550 ft AGL. Thepilot then maintains the flying speed at 75 mphwhile maintaining the left bank angle at 40°


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until 250 ft AGL. This is followed by graduallyincreasing the airplane’s speed towards 100mph and decreasing the left bank angle towards21° until touchdown. The landing manoeuvrefor both manoeuvres for this pre-selectedlocation resemble the 180° and 270° landingapproach in general aviation in flying from thebase leg to the final leg where the pilot begins toturn towards the pre-selected location when thepilot is abreast with the pre-selected landinglocation.

(a)

# RunsFinal Heading

5 %032° ≤ ψ ≤ 073°

95%222° ≤ ψ ≤ 288°

0 100200 300 400 500 600

80

100

120 Average Flying Parameters for Forced Landing (500 ft,200 ft) - Right Turn Variable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

Height (ft)

Aver

age

Spee

d (m

ph)

0 100200 300 400 500 600 -20

0

20

40

Aver

age

Bank

Ang

le (d

eg)

(b)

0 100200 300 400 500 600 70

80

90

100

Average Fly ing Parameters for Forced Landing (500 ft,200 ft) - Left TurnVariable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

Height (ft)

Aver

age

Spee

d (m

ph)

0 100200 300 400 500 600 -60

-40

-20

0

Aver

age

Bank

Ang

le (d

eg)

(c)Fig. 9. Optimal Forced Landing (500 ft, 200 ft)Continuous Speed and Continuous Bank Angle at 650 ftAGL

5.2 Results with Time Averaged VerticalAtmospheric Turbulence

The results for the GA with continuous speedand continuous bank angle, and with verticalatmospheric turbulence for touchdown distancefrom the respective pre-selected landinglocations are shown in Table 4.

The best landing flight paths from each of the100 atmospheric turbulence profiles as shown inFig. 10 – 12 trace a slightly wider landing flightpath envelope than in still air condition. Theaverage flying parameters with verticalatmospheric turbulence for all the three pre-selected locations are very similar to the still aircondition since the disturbance is mild.

The results show that the GA with continuousspeed and continuous bank angle method, andwith vertical atmospheric turbulence hassuccessfully found suitable combinations of theaeroplane’s speed and bank angle to land veryclose to the pre-selected locations. The GAwith vertical atmospheric turbulence found veryminute global minimum distance from each ofthe three pre-selected locations. Comparatively,the average touchdown distances from all thepre-selected locations with vertical atmosphericturbulence are farther than in still air condition.This is as expected since vertical atmosphericturbulence is a form of disturbances anduncertainties to the flying manoeuvre. A veryhigh probability (93%) of landing within theaverage minimum distance from the 10,000trials was obtained for the pre-selected location(0 ft, -3100 ft) but a relatively low probability(61%) was obtained for the pre-selected location(3000 ft, 3000 ft) because GA concentrated inthat solution space nearby where adjacentvalues were found instead.

Fig. 10-a shows the best forced landing flightpaths with vertical atmospheric turbulence fromeach of the 100 turbulence profiles runs for thepre-selected location (0 ft, -3100 ft). Twogeneral forced landing paths exist since the pre-selected location is located along the airplane’sline of symmetry at engine failure point. Theaeroplane’s average flying speed and bank angleat each 50 ft decrement in altitude, and theairplane’s final heading statistics to land at the


-Error! Main Document Only.

Table 4. Results for GA with Continuous Speed and Continuous Bank Angle, and Vertical Atmospheric TurbulencePre-selected

LocationGlobal

MinimumDistance

Average of the MinimumDistance from eachTurbulence Profile

Average MinimumDistance from10,000 trials

Probability of Landing ≤Average Minimum Distance

from 10,000 trials(0 ft, -3100 ft) 1.4847x10-4 ft 0.0123 ft 0.7244 ft 93 %

(3000 ft, 3000 ft) 9.2037x10-4 ft 0.0045 ft 0.0625 ft 61 %(500 ft, 200 ft) 1.4374x10-4 ft 0.0044 ft 0.0718 ft 70 %

pre-selected location are shown in Fig. 10-b.The results show that strong continuousdowndrafts during both the turn glidemanoeuvre and the straight glide manoeuvrewill have the most effect in an attempt to landclose to this pre-selected location. Downdraftsaffect the straight glide manoeuvre more thanthe turn glide manoeuvre in an attempt to landclose to the pre-selected location while updraftswill increase the probability of landing closer tothe pre-selected location.

(a)

# RunsFinal Heading

58 %122° ≤ ψ ≤ 154°

42 %201° ≤ ψ ≤ 237°

0 100200 300 400 500 600

80

100

120

Average Flying Parameters for Forced Landing (0 ft,-3100 ft) Variable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

w ith Vertical Atmospheric Turbulence

Height (ft)

Aver

age

Spee

d (m

ph)

0 100200 300 400 500 600

-20

0

20

40

Aver

age

Bank

Ang

le (d

eg)

(b)Fig. 10. Optimal Forced Landing (0 ft, -3100 ft)Continuous Speed and Continuous Bank Angle at 650 ftAGL with Vertical Atmospheric Turbulence

Fig. 11-a shows the best forced landing flightpaths with vertical atmospheric turbulence fromeach of the 100 turbulence profiles for the pre-

selected location (3000 ft, 3000 ft). Theaeroplane’s average flying speed and bank angleat each 50 ft decrement in altitude, and theairplane’s final heading statistics to land at thepre-selected location are shown in Fig. 11-b.The effects updrafts have on this landingmanoeuvre is a slightly wider flight path orlonger flight path since updrafts decrease thedescend rate and longer paths are required tobleed off the excess energy (altitude). Flyinglonger flight paths also have the effect oftouching down at the pre-selected location withfinal headings near the 180º heading relative tothe initial engine failure heading.

(a)

# RunsFinal Heading

6 %045° ≤ ψ < 090°

51 %090° ≤ ψ < 135°

42 %135° ≤ ψ < 180°

1 %ψ = 310°

(double base leg)

0 100 20030040050060085

90

95

100

Average Flying Parameters for Forced Landing (3000 ft,3000 ft) Variable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

w ith Vertical Atmospheric Turbulence

Height (ft)

Aver

age

Spee

d (m

ph)

0 100 200300400500600-20

0

20

40

Aver

age

Bank

Ang

le (d

eg)

(b)


12

Fig. 11. Optimal Forced Landing (3000 ft, 3000 ft)Continuous Speed and Continuous Bank Angle at 650 ftAGL with Vertical Atmospheric Turbulence

Fig. 12-a shows the best forced landing flightpaths with vertical atmospheric turbulence fromeach of the 100 turbulence profiles runs and theairplane’s final heading statistics for the pre-selected location (500 ft, 200 ft). Theaeroplane’s average flying speed and bank anglemanoeuvres at each 50 ft decrement in altitudeto land at the pre-selected location are shown inFig. 12-b,c. The random vertical disturbancesdo not seem to affect its performance in landingclose to this pre-selected location since verticaldisturbances do not affect the turn manoeuvresas much as the straight glide manoeuvres.

(a)

# RunsFinal Heading

3 %054° ≤ ψ ≤ 076°

97%220° ≤ ψ ≤ 288°

0 100200 300 400 500 600

80

100

120 Average Flying Parameters for Forced Landing (500 ft,200 ft) - Right TurnVariable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

with Vertical Atmospheric Turbulence

Height (ft)

Aver

age

Spee

d (m

ph)

0 100200 300 400 500 600 -20

0

20

40

Aver

age

Bank

Ang

le (d

eg)

(b)

0 100200 300 400 500 600

80

100

120 Average Flying Parameters for Forced Landing (500 ft,200 ft) - Left Turn Variable Speed & Variable Bank Angle for Engine Failure at 650 ft AGL

with Vertical Atmospheric Turbulence

Height (ft)

Aver

age

Spee

d (m

ph)

0 100200 300 400 500 600 -60

-40

-20

0

Aver

age

Bank

Ang

le (d

eg)

(c)

Fig. 12. Optimal Forced Landing (500 ft, 200 ft)Continuous Speed and Continuous Bank Angle at 650 ftAGL with Vertical Atmospheric Turbulence

6 DiscussionsFor the test location at (0 ft, -3100 ft), it isintuitive that that are 2 general paths since thepre-selected location lies along the airplane’sline of symmetry at engine failure. A doublebase leg landing can also be clearly seen for the(3000 ft, 3000 ft) location but this flyingmanoeuvre is not recommended because it ismore complicated to fly than the typical baseleg to final leg landing manoeuvre and it willalso increase the pilot’s workload. Statically,GA also found more flying paths for the typicalbase leg to final leg manoeuvre than for thedouble base leg landing manoeuvre. Lastly, forthe location (500 ft, 200 ft), it is possible to landat the pre-selected location by turning in theopposite direction instead of towards theintended location. This is possible because thepre-selected location is located at closeproximity to the airplane’s initial line ofsymmetry at engine failure. It is notrecommended to fly this manoeuvre to land atthis particular landing location becausestatistically, GA found more paths in flying themanoeuvre that continuously turn towards thepre-selected location. The lack of gradientinformation in GA is responsible for its inabilityto mathematically prove whether the resultsfound are optimum. The trade off in its abilityto search a large solution space is theperformance sacrifice as a true optimisationprocedure.

One of GA’s limitations is not being able tolocate all the possible trajectories and neithercan it guarantee a single best solution since it isa stochastic process based on randomness. Infact, different GA runs may produce differentoptimal results, possibly more than one uniquelanding path, perhaps causing uncertainty as towhich is the best landing path as can be seen forthe pre-selected landing locations tested. GAmay also generate results that are intuitive,those may provide more information that would

13


not have been thought of otherwise or resultsthat are counter intuitive as illustrated by thethree test locations.

7 ConclusionsThe results from various GA search haveconfirmed GA’s effectiveness to explore thesolution domain as well as its capability tosuccessfully identify the most promisingtrajectory paths to a forced landing manoeuvre.The results obtained trace a flight path envelopefor the most probable landing flight paths foreach of the pre-selected landing locationconsidered, touching down very close to theintended touchdown point on ground. Thevertical atmospheric turbulence has the effect ofwidening the landing path envelope for each ofthe three locations considered. This is expectedsince vertical turbulence velocity componentseffectively changes the vertical descend rateforcing a change in the forced landingmanoeuvre to land at the pre-selected landinglocations. The vertical disturbances have themost effect on straight glide manoeuvres and areless sensitive to the turning manoeuvres.Overall, the GA procedure developed clearlyidentified an ensemble of the most promisinglanding paths within the search domain.

8 References[1] Rogers, D.F., Possible "Impossible" Turn.

Journal of Aircraft, Vol. 32, No. 2, pp. 392-397,1995.

[2] Hoffren, J. and T. Raivio. Optimal ManeuveringAfter Engine Failure. AIAA Proceedings of theAtmospheric Flight Mechanics Conference,Denver, Colorado, AIAA Paper 2000-3992,2000.

[3] Bramson, A.F., Glide, flapless and otherabnormal landings. Make Better Landings.Martin Dunitz Ltd, London, 1982.

[4] Trevor Thom, C., The Forced Landing WithoutPower. Private Pilot Flying Training Guide forthe Student Pilot. Aviation Theory Centre, 1987.

[5] Tong, P., G. Galanis, and C. Bil, SensitivityAnalysis to a Forced Landing Manoeuvre.Journal of Aircraft, Vol. 40, No. 1, pp. 208-210,2003.

[6] Pamadi, B.N., Performance, Stability, Dynamics,and Control of Airplanes. AIAA, 1998.

[7] Anon, Flying Qualities of Piloted Airplanes,http://www.aa.washington.edu/courses/aa441/MIL-F-8785.pdf, Military Specifications, Editor.US Department of Defence, 1980.

[8] Newman, D.M. and K.C. Wong, Six Degree ofFreedom Flight Dynamic and PerformanceSimulation of a Remotely-Piloted Vehicle. TheUniversity of Sydney, Sydney. pp. 22-27, 1993.

[9] Hahn, K.-U., et al. Effect of Wind and WindVariation on Aircraft Flight - Paths.AGARDograph No. 301 Aircraft TrajectoriesComputation - Prediction - Control, 1990.

[10] Hahn, K.-U., et al. Wind Models for FlightSimulation. AGARDograph No. 301 AircraftTrajectories Computation - Prediction - Control,1990.

[11] Schanzer, G. Influence of Windshear, Downdraftand Turbulence on Flight Safety. AgardConference Proceedings No. 470 - Flight inAdverse Environment Conditions, 1989.

[12] Holland, J.H., Adaptation in Natural andArtificial Systems: An introduction Analysis withApplications to Biology, Control, and ArtificialIntelligence. The University of Michigan Press,1975.

[13] Goldberg, D.E., Genetic Algorithms in Search,Optimization, and Machine Learning. Addison-Wesley Publishing Company Inc., 1989.

[14] Michalewicz, Z., Genetic Algorithms + DataStructure = Evolution Programs. Springer-Verlag, 1996.

[15] Grefenstette, J.J., Optimization of ControlParameters for Genetic Algorithms. IEEETransactions on Systems, Man, and Cybernetics,Vol. SMC-16, No. 1, pp. 122-128, 1986.

[16] Srinivas, M., Genetic Algorithms: A Survey.Membership Magazine of the IEEE ComputerSociety, Vol., No. June, pp. 17-26, 1994.

[17] Goldberg, D.E., Optimal Initial Population Sizefor Binary-Coded Genetic Algorithms. TheClearing House for Genetic Algorithms, TheUniversity of Alabama, Tuscaloosa, Al,Alabama. pp. 1-14, 1985.

[18] Williams, E.A. and W.A. Crossley, Empirically-Derived Population Size and Mutation RateGuidelines for a Genetic Algorithm withUniform Crossover, Soft Computing inEngineering Design and Manufacturing, P.K.Chawdhry, R.Roy, and R.K. Pant, Editors.Springer-Verlag. pp. 163-172. 1998.

genetic algorithm technique applied to a forced landing

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