Reachability Analysis of Landing Sites for Forced Landing of a UAS

J Intell Robot Syst (2014) 73:635–653DOI 10.1007/s10846-013-9920-9

Reachability Analysis of Landing Sites for ForcedLanding of a UAS

Matthew Coombes · Wen-Hua Chen · Peter Render

Received: 30 August 2013 / Accepted: 12 September 2013 / Published online: 13 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract This paper details a method to ascer-tain the reachability of known emergency landingsites for any fixed wing aircraft in a forced land-ing situation. With a knowledge of the aircraft’sstate and parameters, as well as a known windprofile, the area of maximum glide range can becalculated using aircraft equations of motion forgliding flight. A landing descent circuit techniqueused by human pilots carrying out forced landingscalled high key low key is employed to account forthe extra glide distance required for an approachand landing. By combining maximum glide rangeanalysis with the descent circuit, all the reachablelanding sites can be determined. X-Plane flightsimulator is used to demonstrate and validate thetechniques presented.

Keywords Reachability Analysis · Gliding ·Forced landings · Speed to fly · X-Plane

M. Coombes (B) · W.-H. Chen · P. RenderDepartment of Automotive and AeronauticalEngineering, Loughborough University,Loughborough, LE11 3TQ, UKe-mail: [email protected]

W.-H. Chene-mail: [email protected]

P. Rendere-mail: [email protected]

1 Introduction

Due to the recent miniaturisation of electronics,Unmanned Aerial Systems (UASs) are becomingsmall and cheap enough for the civilian market.A report by the US Department of Defense hasstated that 48 % of all UAS Class A failures couldbe attributed to communications and propulsionfailures [12]. It is issues such as these, that are thelargest stumbling blocks for aviation authoritieslike the Civil Aviation Authority (CAA), FederalAviation Authority (FAA), and the EuropeanAviation Safety Agency (EASA) to certify UASsfor use in civilian airspace. When UASs were onlyoperated by the military, unguided parachutes andcrash landings were how forced landing situationswere dealt with, as they don’t operate over heavilybuilt up areas. These are unacceptable in civilianairspace, due to the danger posed to the public andtheir property. To mitigate these issues, systemsneed to be created that make the UASs muchsafer than they have been.

Operating UAS in civilian airspace raises theconcern in the event of an emergency that re-quires the UAS to perform a forced landing, whatcontingency measures will be in place to mitigatepublic safety and aircraft survivability concerns.As UASs tend to primarily be single engined fixedwing aircraft, they are as vulnerable to forcedlandings due to engine failure as any single en-gine General Aviation (GA) aircraft. Emergency

636 J Intell Robot Syst (2014) 73:635–653

landing after an engine failure was the secondlargest killer of pilots in 2001 in the US. Out of298 fatal accidents involving GA aircraft, 46 werecaused by engine failure, second only to loss ofaircraft control [8], which shows how serious anissue engine failure is.

A system needs to be developed to enable anUAS to safely deal with an engine failure andperform a forced landing. The system will needto be able to identify landing sites and decidewhich it can land in, then guide itself to that safelanding site.

Work has been conducted on site identificationusing a number of image processing tools, testedwith a downwards facing camera on a Cessna 172over flying a number of potential landing sites.To identify a suitable field, the algorithm looksfor open spaces which are clear of obstacles, andidentifies if it has an appropriate surface type andslope. Each of these requirements are analysedseparately and fused together to give a final mapof candidate landing sites [5]. As a follow on from[5, 6] lays out a framework for using the im-age processing techniques to determine the finalsuitability of a site.

Optimal flight paths for forced landing aircrafthave been an area of previous study. Eng [4] uses3D Dubins curves to plan the most efficient tra-jectory for an aircraft attempting a forced landing.Adler et al. [1] approaches the planning of anoptimum flight trajectory from an optimal controlpoint of view. By using glide equations and mo-tion primitives, they are able to compute an opti-mal path from a six-dimensional optimal controlproblem.

In some cases the aircraft may not be that closeto a landing site, and may have to glide some dis-tance to reach one. It may not be directly obviouswhich sites are in range. As well as glide perfor-mance, there is also wind, and a landing approachto consider. Reachability analysis of landing siteshas been neglected in previous works, and so willbe addressed in this paper. It will be achieved byusing a list of known landing sites, and the forcedlanding system will calculate which of those sites iswithin glide range of the aircraft. With knowledgeof the current state of the aircraft, wind profile andby using gliding and aircraft equations of motion,a maximum glide range can be obtained. This

combined with a glide approach circuit will givea full list of reachable sites.

Engine failure after take off and the aircraft’sglide performance has been studied. This was inorder to determine if the aircraft should landahead or make the turn to land on the runway[11]. These basic set of equations can be used, andextended to get a full set of equations for landingsite reachability in a forced landing scenario.

The emphasis of this paper will be on UAS,but as equations are generic to fixed wing aircraft,they can be used as an aid for civilian GA pilots aswell. This is why the examples used in this paperuse a Cessna 182. To perform the calculationson another aircraft is as simple as changing theparameters.

In Section 2 the equations of a gliding air-craft are derived. Section 3 presents how theglide range equations are formulated. In Section 5winds effect on glide range with a known windprofile is shown. In Section 6 a definition of a land-ing descent circuit, and how the reachability of afield can be determined. Section 7 shows a forcedlanding situation, and performs the reachabilityanalysis on a set of landing sites, and the resultsare presented.

2 Gliding Equations

A single engined aircraft with a failed engine hasno thrust to propel itself, and thus is effectivelya glider. As the aircraft is now forced to land,it is very important that its glide range is knownand maximised to present more landing locationoptions. This section will detail the equations forgliding performance and how the range of theaircraft is calculated in level and turning flight,also showing how to obtain best glide ratio. Manyof these equations can be found in text books suchas [7].

2.1 Straight Line Performance

Only Equivalent Airspeed (EAS) will be consid-ered and other airspeeds will first be convertedto EAS as the performance of a gliding aircraft isaffected by its altitude but EAS is altitude inde-pendent. EAS is the airspeed at sea level in the

J Intell Robot Syst (2014) 73:635–653 637

International Standard Atmosphere (ISA) atwhich the dynamic pressure is equal to the dy-namic pressure at the True Airspeed (TAS). Thismeans the best speed for glide is at a fixed EASat any altitude. It is also useful as the reading In-dicated Airspeed (IAS) from the airspeed sensorwill be approximately its EAS.

An equation is needed that enables rate ofsink of the aircraft (Vs) to be calculated for anyairspeed. This can be done using the known rela-tionship between rate of sink, drag, airspeed, andweight in Eq. 1, and by assuming that the dragpolar of the aircraft is parabolic:

Vs = DVW

(1)

where W is the aircraft’s weight, D is drag, andV is the airspeed.

An aircraft’s drag can be calculated using aparabolic polar in Eq. 2 with the dimensionlesscoefficients of lift and drag:

CD = CDo + kC2L

πAr(2)

where CD is the coefficient of drag, CDo isthe aircraft’s profile drag or drag at zero lift,and k is the induced drag factor determined bythe aircraft wing dimensions, configuration, theReynolds Number and Mach Number. Ar is theaircraft’s wing aspect ratio.

By combining Eqs. 1 and 2, the equation forrate of sink based on airspeed for a given aircraftis:

Vs = AV3 + BV

(3)

where A, and B are co-efficients based on theaircraft’s parameters

A = 0.5ρ0SCDo

W(4)

B = kπAr

2Wρ0S

(5)

where ρ0 is the density of air at sea level, and S isthe wing area.

2.2 Speed to Fly

It is important for an aircraft to fly at a speed thatwill maximise the range it can travel. These arebased on an aircraft flying through still air:

γ = Vs

V= AV2 + B

V2(6)

where γ is the glide ratio of the aircraft.By differentiating Eq. 6 in terms of V, and

finding the minimum value of γ by equating it tozero gives:

dγdV

= 2AV − 2BV3

= 0 (7)

The speed to fly for maximum glide ratio (Vio)and the corresponding maximum glide ratio areshown in Eqs. 8 and 9 respectively:

Vio =(

BA

) 14

(8)

γmax = 2A(

BA

) 12

(9)

2.3 Speed to Fly in Wind

The previous section describes the optimum speedto fly in still air, but depending on meteorologicalconditions, this will rarely be the case. It is im-portant to factor wind into the best glide speed,as an incorrect best glide speed may significantlyreduce the glide distance. It can be shown, bygraphical analysis using a glider’s hodograph, thatoptimum glide velocity increases in a head windand decreases in a tail wind. By making somesimple assumptions, a simpler set of equations canbe formulated which is shown in [2].

Assuming that the glide angle (θg) is small, us-ing the small angle approximation the glide anglecan be approximated:

θg = Vs

V + vw(10)

where vw is the wind speed relative to the aircraft’sdirection of travel, and a tail wind is negative.


By substituting Vs = V(CL/CD) and assuminga parabolic polar, glide angle can be calculated by:

θg = CDo + kC2L

CL(1 + vw

V

) (11)

Using a technique similar to the one shown inthe previous section, by taking the derivative ofEq. 11 with respect to V and setting this derivativeto zero gives a fifth-order polynomial equation:

V̂5 + 3

2v̂V̂4 − V̂ − v̂

2= 0 (12)

where the nondimensional velocities V̂ and v̂ aredefined as:

V̂ = VVio

, v̂ = vw

Vio. (13)

Although there is no analytical way to solvethis equation for V̂, it is shown in [2] that it canbe represented as an infinite series. Taking theterms up to the forth order from the infinite series,means an accurate closed solution can be formed:

V̂ = 1− v̂

4

(1− 7

8v̂+ 5

8v̂2 − 167

512v̂3 + 1

16v̂4

). (14)

This closed solution gives the airspeed to flyneeding only the optimum speed to fly in zerowind conditions and the head or tail wind speed.

2.4 Turning Performance

In straight and level flight, the weight componentsare along the aircraft’s longitudinal and verticalaxis and do not experience any normal loadinggreater than unity. In turning flight these aremarkedly different as the aircraft will be in abanked attitude, where now it’s weight will havea component along the lateral axis as well. Therewill also be centripetal acceleration due to a turnrate about the global vertical axis.

The full equations of motion for turning flightin the aircraft’s longitudinal, lateral, and verticalaxis respectively are as follows:

mdVdt

= −mgsin(θ)− D + Tcos(θ) (15)

mgcos(θ)sin(φ) = mVψ̇cos(θ)cos(φ) (16)

mgcos(θ)cos(φ) − L − Tcos(θ)

= −mVψ̇cos(θ)sin(φ) (17)

where θ , φ, ψ are the aircraft’s roll, pitch and,yaw angle respectively. ψ̇ is the turn rate of theaircraft, shown in [7].

For the engine out case, T = 0. At a constantspeed (best glide) for any reasonable aircraft, theglide angle will be small. A small angle approxi-mation can be used, and the above equations canbe simplified as:

0 = −mgsin(θ)− D (18)

tan(φ) = Vψ̇g

(19)

mgcos(φ)− L = −mVψ̇sin(φ) (20)

Rearranging Eq. 20 in the form of LW , which is

equivalent to the Normal Load Factor (n) on theaircraft, gives:

n = Lmg

= cos(φ)+ Vψ̇g

sin(φ). (21)

Using Eqs. 19 and 21 can be reduced to Eq. 22then simplified as:

n = cos(φ)+ tan(φ)sin(φ) (22)

n = sec(φ). (23)

Figure 1 shows the sink rate of a gliding Cessna182 across a range of airspeeds and bank angles.Each curve is referred to as a glide polar curve,or a hodograph. Each point along the curve cor-responds to a particular L/D, and co-efficientof lift.

To achieve the best glide range the aircraft isflown at a speed which corresponds to a particularCL. In a turn the forces change, so to achievebest glide range, the speed will have to increaseto maintain the same CL. As L/D is constant at aparticular CL the drag will increase as well, whichhas the effect of increasing the aircraft’s rate ofsink. In straight and level flight L = nW but nis unity. In a coordinated turn n will be greater,so greater lift will be required to maintain a co-ordinated turn. The speed of the aircraft will haveto be greater to achieve this same coefficient oflift. The aircraft’s speed will have to be increasedby a factor of n

12 as lift is proportional to V2 from


Fig. 1 Hodograph of aCessna 182 for variousangles of bank

Eq. 24. This needs to be applied to the speed to fly,if the optimum speed is to be maintained, whichmeans Vθ is the new optimum speed to fly in theturn. This has the effect of shifting the glide polarto a higher airspeed.

CL = L0.5ρV2S

(24)

Vφ = Vsec12 (φ) (25)

The sink rate of the aircraft will also increasein a turn. Shown by a shift of the glide polar toa higher rate of sink. As lift has increased by nto maintain L/D drag will increase by n as. Bysubstituting W in Eq. 1 for L/n:

Vsφ = DVnL

(26)

As DVL = Vs and using Eq. 25 to convert V to

Vφ the new sink in a turn can be determined:

Vsφ = Vssec32 (φ) (27)

where Vsφ is the sink in a turn.

3 Glide Range

With the basic equations of motion of the aircraftdetermined, these can be used to construct a set ofequations to determine the maximum glide range

of an aircraft in any direction. The problem needsto be broken down into two parts, the initial turnto get the aircraft onto the correct course to thelanding site, and the level gliding descent. Theheight loss in the level gliding descent can becalculated from glide ratio shown in Section 2.1,but the height loss over the turning phase is notknown, and will be defined here.

3.1 Turning Phase

In a turn there are two things a gliding aircraft canchange: its speed and its roll angle. Both affect theaerodynamic efficiency of the aircraft, so increas-ing or decreasing its sink, and affecting the turnradius thus affecting the extra distance traveledin the turn. The height loss in a turn is derivedhere, a similar derivation is performed in [11]. Theequations in that paper are in co-efficient form,whereas here they are in the more useful velocityform.

By assuming that an aircraft obtains its desiredroll angle instantaneously, the path that the air-craft takes during the turning phase can be sim-plified to an arc along a circle.

The arc of length Larc, is the path the aircraftwill take during a turn, as shown in Fig. 2. The arclength is defined as:

Larc = Rdψ (28)


Fig. 2 Path of aircraft inthe turning phase

where dψ is the total change in heading of theaircraft in the turning phase, which is in radiansbound between ±π .

The radius R is calculated based on the bankangle and forward velocity of the aircraft from[11]:

R = V2φ

gtan(φ).(29)

The height loss within a turn is needed (�hturn).By multiplying the aircraft’s glide ratio by thedistance the aircraft has flown the loss of heightcan be calculated.

�hturn = LarcVsφio

Vφio(30)

where Vφio is the maximum glide ratio speed ina turn, having adjusted Vio for the extra normalloading using Eq. 25, and Vsφio is the correspond-ing rate of sink at that speed.

By using Eq. 3 to obtain the aircraft’s sink inlevel flight, multiplying it by Eq. 27 to get thesink in a turn, and dividing it by the aircraft veloc-ity, the glide ratio can be calculated. Using theseequations the final value of change in height ina turn based on velocity, turn angle, and aircraftparameters can be calculated as

�hturn = dψVφio

gtan(φ)

(AV3

io + BVio

)sec

32 (φ) (31)

3.2 Total Glide Range

Equations from both the turning, and the straightand level phases of flight can now be brought to-gether to construct a set of equations to calculatethe glide distance of the aircraft given an initialheight and heading that the aircraft needs to turnon to. To determine the range, the coordinatesof the end points on the turning circle and thestraight gliding descent need to be calculated.

If these calculations are performed between ±πthen a maximum glide range area is formed andanything within the area can be reached. Thisformulation is clearly shown in Fig. 3, with theaircraft at its origin.

For these set of equations the height lost inthe turn must not exceed the initial height of theaircraft.

The chord length (C) of the turn circle shownin Fig. 2 is defined in Eq. 32. This is obtained byusing the turn radius and total turn angle with thesine rule:

C = V2φ

gtan(φ)sin(|dψ |)

sin(π−|dψ|

2

) (32)

The location of the coordinates at the end ofthe turn are shown in Eq. 33.

x0 = Csin(π − dψ

2

)

y0 = Ccos(π − dψ

2

) (33)

where x0, y0 are the coordinates at the end of theturning phase, relative to the initial position andheading of the aircraft.

The next phase is straight and level flight. Tomaximise range, the aircraft will fly at its maxi-mum glide angle speed Vio. The sink and glideratio are known at this speed from Section 2.1.As the initial altitude of the aircraft is known and

Fig. 3 Maximum glide range formulation representation,including the turning phase and the straight glide phase


the height it will lose in the turn calculated fromEq. 31, the remaining height can be used with theglide ratio to calculate the remaining range

Dglide = (hinit −�hturn)Vio

Vsio(34)

where Dglide is the total ground distance traveledin the level phase of flight, and hinit is the initialhight of the aircraft.

The final coordinates can be found using knownangle relationships shown in Fig. 3 and by addingthe coordinates at the end of the turn

x = Csin(π − dψ

2

)+ Dglidesin(dψ)

y = Ccos(π − dψ

2

)+ Dglidecos(dψ). (35)

where x, y are the coordinates at the end of theglide, relative to the initial position and headingof the aircraft.

The total straight line distance the aircraft canglide at a particular turn angle is Rglide

Rglide =√

x2 + y2. (36)

4 Turn Including Roll Rate

In Section 3 it was assumed that the aircraft canroll to a desired roll angle φ instantaneously.Equations are presented here that take accountfor non-instantaneous roll angle attainment. Air-craft dynamics in a turn are very complex, andmodeling it fully for aileron, rudder, elevator de-fection, and angular acceleration would introducemore error in modeling than would be presentfrom a more simplistic model. The extension pro-posed will include an instantaneous constant rollrate (φ̇) until the desired roll angle is reached.

If the heading (ψ), and the airspeed of theaircraft are known over the turning phase of flightthe location of the aircraft can be calculated. Theturning phase will be broken down into threephases. The first phase is the initial roll to thedesired roll angle, this will be at a constant (φ̇),the yaw rate (ψ̇) will not be constant. Once theaircraft has reached the desired roll angle the nextphase of constant φ, and ψ̇ begins. In the final

phase aircraft rolls back to level at the same φ̇which is simply the reverse of the first phase.

While the airspeed of the aircraft in the turn isknown, ψ at a given time, and both are needed forposition. Using Eq. 19 and substituting φ for φ̇τ ,an equation for ψ̇ in relation to time (τ ) can bederived

ψ̇ = g tan(φ̇t + φint)

Vφ

(37)

where φint is the initial roll angle of the aircraft,which is only relevant when the aircraft is rollingback to level flight. To find (ψ) at time τ Eq. 37must be integrated with respect to time shown inEq. 38, its indefinite solution is shown in Eq. 39.This equation will be referred to as f (τ ).

ψ = gVφ

∫tan(φ̇τ + φint)dt (38)

f (τ ) = ψ = g

Vφ̇ln | sec(φ̇τ + φint)| + C (39)

Each phase of flight will take a certain amountof time to complete, these times are required inorder to determine the three definite integralsneeded to calculate ψ across all three phases.Each phase will use a slightly different equationfor ψ . In phase one φ will saturate after time τ1,this is calculated using the desired steady stateroll angle (φss), and the constant rate of roll theaircraft uses to achieve this is shown

τ1 = φss

φ̇(40)

The equation for ψ in phase one is Eq. 39 but asthe aircraft has not began to turn so φint = 0.

The time for the second phase is from τ1 toτ2 + τ1. As this phase of the flight is at a constantknown ψ̇ the equation forψ is much simpler, ψ̇ssτ .The time τ2 is calculated by finding the headingchange of the aircraft during the second phase anddividing by the steady state yaw rate ψ̇ss. ψ̇ss iscalculated

ψ̇ss = g tan(φss)

Vφ

(41)

The final equation for τ2 is shown

τ2 = δψ − 2 f (τ1)

ψ̇ss(42)


Fig. 4 Heading in a turnwith an initial roll rate of15 ◦/s to a maximum of50◦ roll angle

δψ is the total heading change of the wholeturn.

The time for the third phase is the same asthe first from τ2 + τ1 to τ2 + 2τ1 the aircraft willroll back to level at the same roll rate as beforejust in the opposite direction. The equation for ψis Eq. 39 as the aircraft is at φss, φint = φss. Thefull set of equations for aircraft heading over theentire turn is shown in Eq. 43.

ψ=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

g

φ̇Vln | sec(φ̇τ )| for τ < τ1

ψ̇τ for τ1 ≤ τ ≤ τ1 +τ2g

−φ̇Vln | sec(−φ̇τ + φss)| for τ1 +τ2 ≤ τ

≤ 2τ1 +τ2

(43)

As an example, the heading of the aircraft overa turn is shown in Fig. 4. This is for an aircraftflying at 45 m/s, with an initial roll rate of 15 ◦/s, toa steady state of 50◦ bank angle for a total headingchange of 120◦.

During turns with a smaller heading change, orlower roll rates the aircraft may not have enoughtime to achieve φss before it has to roll back tolevel flight by the time it has reached its newheading shown in Fig. 5. In this case a new set ofequations are needed to calculate ψ over a turn.The condition in which these new set of equationsare needed if δψ

2 < f (τ1) is satisfied.In this case there are only two phases of the

turn, the roll in and the roll out, as there now is nosteady state. A new τ1 is needed, which is half thetime of the whole turn corresponds to a heading

Fig. 5 Heading in a turnwith an initial roll rate of5 ◦/s turn to a newheading of 20◦


change of δψ

2 . By rearranging Eq. 39 in terms ofτ , and replacing ψ with half the total requiredheading change δψ

2 , τ1 can be calculated, this isshown

τ1 =asec

(±e

�ψVφ φ̇2g

)

φ̇. (44)

In Eq. 44 the modulus of sec is taken, whicheffectively loses the sign information, meaning| sec(φ̇τ )| has two solutions between 0 and π . Bypassing both answers back through Eq. 39 thisallows the correct solution to be identified. Iff (τ1) = �ψ

2 then that is the correct value of τ . Thefull equation for the whole turn is shown in Eq. 45.

ψ =

⎧⎪⎪⎨⎪⎪⎩

g

φ̇Vln | sec(φ̇τ )| for τ < τ1

g

−φ̇Vln | sec(−φ̇τ + φint)| for τ1 ≤ τ ≤ 2τ1

(45)

where φint is the achieved φ at τ1, which is φ̇τ1.Using either method heading is now known

at any given time throughout the turn, using Vφ

the change in x, y position can be calculated byintegrating Eq. 46 in respect to time.

y = Vφ

∫cos(ψ)dτ

x = Vφ

∫sin(ψ)dτ (46)

Unfortunately in the roll in and out phase∫cos(ln | sec(φ̇τ |) has no analytical solution, so a

numerical integration method is required shown.

yτt = yτt−1 + Vφ

τt − (τt−1)cos(ψ)

xτt = xτt−1 + Vφ

τt − (τt−1)sin(ψ) (47)

This turn model is clearly more complex thanthe previous one, and thus much more computa-tionally expensive. Its use needs to be justified bycomparing it with the simpler model presented inSection 2.

Using the same example as Fig. 4, the simplerconstant ψ̇ model was run in the same initial stateand compared to the model presented in this sec-tion. The heading over time for both these modelsare shown in Fig. 6, the paths are shown in Fig. 7.

It can be seen that the aircraft which attains itbank angle instantly is able to achieve the turn tois desired heading much faster, and travels muchless distance. This will mean that by using thesimpler model, it will over estimate the overallglide range of the aircraft. The difference betweenthese models become smaller the faster its rollrate is.

As in Section 3.2 the loss of height in the turnis needed. The equations in that section assumethe aircrafts path is that of an arc, and that φ isconstant leading to a constant Vs in a turn. This isno longer the case. It would be possible to replaceφ with φ̇τ in Eq. 31 and integrate in terms of τfor the threes phases of the turn to calculate Vs.

Fig. 6 Heading in a turnwith an initial roll rate of15 ◦/s to a maximum of50◦ roll angle comparingtwo models for aircraftin a turn


Fig. 7 Path of aircraft turning though 120◦ with initial rollrate of 15 ◦/s to a maximum of 50◦ roll angle

Due to complex transitional aerodynamic effectsduring roll in and roll out the height loss of anreal aircraft would be different to that calculated.The drag would be higher due to deflections fromall three sets of control surfaces. To calculate theheight loss, the distance in the turn is multipliedby the glide ratio of the aircraft in this turn at φss

like in Eq. 30. The distance in the turn is simplythe airspeed of the aircraft multiplied by the totaltime in the turn, and the glide ratio is calculated inthe same way as in Eq. 31.

�hturn = (2τ1 + τ2)

(AV3

io + BVio

)sec

32 (φss) (48)

This accounts for the longer distance traveledby the aircraft, and while it does not account forthe different Vs across the whole turn, it is areasonable approximation.

5 Wind Effect’s on Glide Range

As wind has a large influence on how far theaircraft can glide, this needs to be factored in tothe glide range equations.

For a steady wind velocity it is simply an ad-dition to the glide range equations. By assum-ing the transformation between the ground andwind frame for velocity is Vg = V + vw, then Dg =Vt + vwt where Dg is the ground distance covered.By calculating the time the aircraft takes in the

turn and in the level gilding descent by multiplyingthese times by the wind speed the aircraft’s extradisplacement relative to the ground is known.This can simply be added to the max glide rangecoordinates equation Eq. 35.

Wind is not constant across altitude, so thismodel is not sufficiently accurate. In the north-ern hemisphere the wind will turn anticlockwise(known as backing) as it approaches the ground,and weaken. According to [10], from 2,000 ft tothe ground the wind weakens by 1

3 , and backsby 30◦, during the day over land. Given a knownwind profile with height (h) where the wind inthe north and east directions are functions ofheight (u = f (h), v = f (h)) winds effects can becalculated.

To extend this to factor in a wind profile withheight, the profile needs to be integrated withrespect to time. The equation for wind speedwith height needs to be converted to wind speedwith time. The sink and the time taken by anaircraft in the turning phase and the gilding phasetime can be related to height by h = −Vst + hinit

where hinit is the initial height of the aircraft at thestart of the turning phase

u = f (−Vst + hinit) = f (t,Vs)

v = g(−Vst + hinit) = g(t,Vs) (49)

where u, v are wind speeds in the east and northdirection.

By integrating the newly calculated windprofile in relation to time, the change of aircraftposition in x (�x) and y (�y) due to wind can becalculated. As there are two phases of flight withdifferent Vs, the definite integral needs to be split.In the turning phase of flight Vs is Vsφ , and thelimits of the integral are between the time takenfor the turn (tturn) and zero. In the non turninggliding phase of flight Vs is Vsio, and the limitsof the integral are between the time taken for thewhole glide (ttot) and tturn. Shown in Eq. 50.

�x =∫ ttot

tturn

f (t,Vsio)dt +∫ tturn

0f (t,Vsφ)dt

�y =∫ ttot

tturn

g(t,Vsio)dt +∫ tturn

0g(t,Vsφ)dt (50)

where ttot = tturn + tglide.


By adding Eqs. 50–35 the maximum glide rangefor a known wind profile over the altitude of thedescent can be calculated.

6 Reachable Landing Site

Thus far only maximum glide range of the aircrafthas been calculated, which does not account for anapproach and safe landing. A landing descent pathneeds to be defined and factored into the maxi-mum glide range, to determine the true reachabil-ity of a landing site.

In a forced landing, pilots fly particular descentpaths to maximise the chance of making a landingsite safely. A particular descent path will be cho-sen in this section, which defines the point whichthe aircraft has to join the descent circuit, and theheight it must join it at.

The high key low key technique has been cho-sen [9]. This is a forced landing circuit techniquewhich requires an aircraft to be at particular loca-tions at certain heights, thus the key points in thecircuit can be mathematically defined based on itsglide performance. This circuit is conservative asit is designed for the use by human pilots, but isa good starting point for being able to factor in aforced landing circuit into the reachability analy-sis. Any descent path can be defined, and usingthis technique means it is easily interchangeable

for any path that provides a joining location andheight.

The high key low key technique, and all its keyheights are shown in Fig. 8. The circuit can be inthe clockwise or anti clockwise direction.

There are five points on the descent path,Downwind (DW), End Base 1 (EB1), EB2, De-cision Height 1 (DH1), and DH2. The prefix a, crefer to either an anti-clockwise, or clockwise cir-cuit direction. The aircraft’s azimuth to the centerline of the runway (AzCl) governs which point theaircraft joins, which will be explained later in thissection. The coordinates of each point along thepath need to be calculated, all points are relativeto the Initial Aiming Point (IAP) which is halfway down the landing site, which is aligned withnorth. These unrotated relative coordinates willbe rotated to the orientation of the runway (�),and relative to the aircraft’s location later. Thex, y coordinates of these key locations in bothdirection relative to the IAP are as follows where152.4 m is 500 ft:

DH2 =[

0152.4γ

](51)

aDH1 =[

R152.4γ + R

](52)

aEB2 =[

R + γ (152.4 − 2�h)152.4γ − R

](53)

Fig. 8 Clockwise andanti-clockwise circuitdefinition


Table 1 Given aircraft azimuth to the runway, the stage ofthe descent circuit the aircraft will join

Relative bearing Part of circuit to join

330◦ to 30◦ Final30◦ to 120◦ Anti-clockwise base120◦ to 180◦ Anti-clockwise downwind180◦ to 240◦ Clockwise base240◦ to 330◦ Clockwise downwind

aEB1 =[

2R + γ (152.4 − 2�h)152.4γ − 2R

](54)

aDW =[

2R + γ (152.4 − 2�h)−152.4γ − 2R

](55)

cDH1 =[ −R

152.4γ + R

](56)

cEB2 =[−R − γ (152.4 − 2�h)

152.4γ − R

](57)

cEB1 =[γ (152.4 − 2�h)

152.4γ − 2R

](58)

cDW =[γ (152.4 − 2�h)

152.4γ − 2R

](59)

All these points are determined from the dis-tance the aircraft will travel to lose height to landat the IAP. Working backwards from the Initalaiming point at DH2 the distance in y is the dis-tance the aircraft travels to lose 500 ft or 152.4 m.all other points are determined in the same wayby working from the previous point, and havingknowledge of the glide ratio, the radius and lossof height in a turn for a given angle of bank.

The coordinates of each descent path pointneed to be rotated by runway orientation (�) andmake relative to the aircraft. This is performedusing a rotation matrix Eq. 60.[

x′y′

]=

[cos� − sin�sin� cos�

][xr

yr

](60)

Fig. 9 Five differentdescent path join pointsbased on the aircraft’sazimuth to the center lineof the runway


Table 2 Parameters forthe Cessna 182 skylane

Parameter Value

Cdo (clean) 0.0245S 16.16 m2

Ar 7.48k 1.3408W 13793 NA 1.758×10−5B 5381.2

where xr, and yr are coordinates relative to thecenter of the runway in meters, and x′ and y′ arethe rotated coordinates of a point relative to theaircraft.

Which part of the descent circuit the aircraftjoins depends on the aircraft azimuth to the centerline of the runway. The range of angles whichcorrespond to a given join point is shown inTable 1. A visual representation of this is shownin Fig. 9. The join point will be referred to asx′

join, y′join.

With a known runway location, descent path,and aircraft performance, the possibility of land-ing at a given runway can now be calculated.The runway is reachable if the glide range of theaircraft exceeds the range to the join point.

A search needs to be performed on the set ofknown runways to find the reachable ones. L is aset of reachable landing sites, and r is landing sitebeing evaluated for reachability. If the distance to

the join point calculated by√

x′2join + y′2

join is lessthan the glide distance to that point then thatlanding site is not reachable, as shown⎧⎨⎩

r ∈ L for√

x′2join + y′2

join < Rglide

r �∈ L for√

x′2join + y′2

join > Rglide

(61)

7 Simulation

To demonstrate and validate the algorithms pre-sented in this paper, the theory is compared tomaneuvers and forced landing scenarios in X-Plane flight simulator.

X-Plane has a plethora of different aircraft thatcan be flown. It works off an aircraft dynamicsmodel which is based on blade element theory,which means the aircraft will have representativedynamics through the whole flight envelope. Thisprogram is used throughout industry for simu-lation as it’s output and inputs can be manip-ulated through plugins, which means it can in-terface with other programs and devices. Manyaccurate aircraft models can be downloaded, alsothey can be modified, or whole new planes de-signed from within the program. This makes theprogram very flexible. “XPlane has received FAAcertification as a training simulator when usedwith certain hardware configurations because ofits high fidelity simulation of flight model andvisualization” [3] which is why it is able to be usedto test and validate algorithms. X-Plane is unableto model a wind profile as discussed in Section 5,so only a static wind model is used.

A plugin has been developed where all aircraftdata like attitude position etc. are sent over anetwork port the software can receive controlinputs like throttle, elevator etc. over the samenetwork port. This has enabled the aircraft tobe controlled externally by separate computer. InSimulink a controller is setup to command thespecific maneuvers required, and is also used tolog data.

The simulation shown is preformed on a Cessna182. The Cessna has been modified with a variable

Fig. 10 Comparisonbetween theoretical andX-Plane heading turnmodel


Fig. 11 Comparisonbetween theoretical andX-Plane roll angle turnmodel

Fig. 12 Comparisonbetween theoretical andX-Plane roll rates

pitch prop, so it can be feathered to stop the propwind milling during a forced landing scenario.Wind milling props courses an extra level of com-plexity, that has not been modeled. The Cessna182 parameters are shown in Table 2.

The validity of the extended turn model is firstinvestigated. A turn through a heading of 130◦at a 10 ◦/s rate of roll is performed in X-Plane,and compared to the theoretical values. Shownin Fig. 10 are the headings of the aircraft withtime. Shown in Fig. 11 are the roll angles for bothmodels. Shown in Fig. 12 are the roll rates for bothmodels. Shown in Fig. 13 is the most importantcomparison of the actual flight paths of theorycompared to theory.

The curves for heading over time are verysimilar, as is the roll angle. It can be seen thatthe roll rate does not match as well, as angular

Fig. 13 Comparison between theoretical and X-Planepaths


Fig. 14 Glide distance ofa Cessna 182 from 800 mheight, wind at 270◦ 10knots. Both theoretical,and X-Plane paths

acceleration was not modeled, which can be seenas X-Plane does not attain its demanded roll rateinstantaneously. This has not had much effect onthe heading or the actual roll angle. The path ofthe aircraft matched well, but the path length wasslightly shorter due to a tightening of the turn dueto a slightly higher roll rate on the roll out seenin Fig. 12. This caused the target heading to bereached sooner.

The maximum glide range will now be consid-ered. A Cessna 182 at 800 m height flying north,with wind from 270◦ blowing steady at 10 knots.The speed to fly in still air for maximum glide ratioVio is 46.1 m/s. The speed to fly in wind dependson the direction the aircraft is traveling relative tothe wind with the maximum speed being 51 m/sfor a full tail wind component, and the minimum43.2 m/s with full head wind. The angle of bankof the aircraft while in the turn phase is 10◦. Thisroll angle is lower than a pilot would use in anemergency but is used here to extended the turnso it is clearly demonstrated.

Using equations from Section 3.2 the maximumglide range to 500 ft height is plotted in Fig. 14. Tocompare the theoretical glide range with the ac-

tual glide range the aircraft in X-Plane was flownin multiple directions and the paths recorded.Four paths where flown, one with no turn and on aheading of 000◦ magnetic, two more with 90◦ turnsto 090◦ and 270◦ magnetic. Finally a 180◦ turn tothe right to 180◦ magnetic.

It can be seen in Fig. 14 the theoretical gliderange and the simulated one are quite similar, thetheoretical slightly over predicts the glide range.Much of the error is in the 180◦ turn, where thereis an over prediction of 600 m. The equationsassume two Vs speeds, one in the turn and onein the straight and level glide, but Vs of the ac-tual aircraft is not steady at these two phasesespecially in the transitional turning phase. Turnangle, rate, and airspeed all need to be controlled,and as they are coupled they are hard to controlaccurately while doing a perfectly co-ordinatedturn. It is these transitional effects and imperfectcontrol which mean that Vs is unsteady. In anactual forced landing scenario the turns would beat a higher roll angle, and so would constitutea smaller proportion of the fight so any errorscaused during this phase will become much lessrelevant. A more robust controller is needed


Fig. 15 Three knownlanding sites, with sixdescent circuit join points

so not to introduce unwanted and unmodeleddynamics.

A forced landing scenario will now be con-sidered. A Cessna 182 flying out of East Mid-lands Airport (EMA) in Leicestershire UK,

travels north 6km to an altitude of 1,088 m, justwhen the aircraft is about to leave EMA’s controlzone it suffers from complete and sudden enginefailure. The wind is from 270◦, at 10 knots. Thesystem has been programmed with three landing

Fig. 16 Reachability ofthree landing sites duringa forced landing situation


Fig. 17 Path of theCessna 182 during aforced landing on runway27 at EMA

sites is shown in Fig. 15. From map data the heightabove mean sea level (amsl) for each landing siteshown next to the field. Using equations fromSection 6 the landing descent circuits are plot-ted for both landing directions at each site, withthe heights amsl of the six possible join pointsshown.

To determine which sites are reachable, andwhich requires least glide distance, the glide dis-tance to each of the six join points are calculated.Shown in Fig. 16 is this scenario. Four concentriccurves are shown in Fig. 16 representing maximumglide range to a height of 0 ft (outer curve), 500 ft,1,000 ft and 2,000 ft (inner). Down the left handside is the reachability of each descent circuit,white indicating it is reachable, and black unreach-

able. The distance displayed for each circuit isthe excess reachability, which is the extra glidedistance available once the join point has beenreached by the aircraft. Circuit 1, and 4’s joinheight is 1,000 ft and are both within the 1,000 ftglide range curve, meaning they are reachable.While circuit 3 is not reachable as it is not withinthe inner curve which is glide distance to 2,000 ft.In this case the four reachable circuits are 4, 2, 6, 1two of which are circuits for the runway at EMA.In this case the emergency landing site is circuit4, which is Runway 27 at EMA. The aircraft willhave 2.9 km extra glide distance for an large safetymargin, it is a nice safe tarmaced landing surfacewith emergency crews at the ready, and is in thedirection of the wind.

Fig. 18 Altitude profileof the Cessna 182 duringa forced landing onrunway 27 at EMA


Having used the system to determine where tomake a forced landing in to, the aircraft turns tothe correct heading to join on clockwise base forRunway 27, which can be seen in Fig. 17. As it has2.9 km of excess glide range the final leg of the cir-cuit is extended, and full drag flaps are extendedclose to the runway. The altitude profile is shownin Fig. 18. The turn can be seen at the start of Fig.18 as the gradient is steeper which counts for thehigher Vs in a turn. This gradient steepens closeto the end due to the flaps being fully extended.How a landing at a given site is performed is up toa path planner, which mentioned before there area few available.

8 Conclusion

A method for calculating the reachability of land-ing sites in the presence of a known wind profileusing gliding equations for an aircraft with aknown drag polar has been presented. A max-imum glide range can be defined by separatingthe glide into a turning phase and the straightand level glide descent. If the wind profile vari-ation with height is known mathematically, theglide range can be adjusted to account for thesignificant effect of wind. A descent circuit isdefined to guide the aircraft on a glide path for asafe landing on a field, based on the high key lowkey technique used by human pilots. This descentcircuit is used to give a height, and a location theaircraft must be able to glide to in order for thatsite to be reachable.

An extension to the turn model is presented,which accounts for roll rate. While more com-putationally expensive it provides a much moreaccurate turn model especially at low rates of roll.

The equations presented can be used on anyconventional fixed wing aircraft, meaning thatthey are not confined to exclusive use by UAS.This system could be extended to assist pilots inforced landing situations, as the high work loadmay impede their ability to identify all reachablelanding sites.

This descent circuit is conservative, and it wasdesigned for human pilots to make sure they donot lose sight of the field, enabling them to visu-

ally survey the landing site, and keep maximumlanding options available. A UAS may performmaneuvers much more precisely, and will not losesight of the field. There is a need to developa more representative descent path formula tofollow, which will not require such a wide andwasteful circuit.

One assumption in this study is that the windprofile is known, and also the local effects ofthe wind has not been taken into account. Onemethod to investigate the influence of local effectsin different scenarios and landscapes on the maxi-mum glide range, so an appropriate safety margincan be taken into account when identifying reach-able landing sites.

To validate some of the equations presented, asimulation environment was set up using the X-Plane flight simulator. The extended turn modelmatched closely to the simulation. Total gliderange while matched well for glides with a lowinitial heading change, glides with a higher head-ing change where not as accurate. Real aircraftdynamics are very complex and many assump-tions have been made to simplify the equations.The generic nature, and minimal data needed touse these glide calculations mean that the smalldrop off in performance is more than accept-able for keeping these equations so flexable anduseful.

To test the algorithms for winds effect on glidewith a known wind profile, a X-Plane plugin needsto be developed to change the wind direction andstrength as an aircraft descends.

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Documents

Reachability Analysis of Landing Sites for Forced Landing of a UAS