TRAJECTORY COMPUTATION INFRASTRUCTURE BASED ON BADA AIRCRAFT PERFORMANCE MODEL

Eduardo Gallo, Javier López-Leonés, Miguel A. Vilaplana, Francisco A. Navarro, Boeing Research & Technology Europe, Madrid (Spain)

Angela Nuic, EUROCONTROL Experimental Centre, Brétigny sur Orge (France)

Abstract A more predictable Air Traffic Management

(ATM) system based on automation requires a precise Trajectory Computation Infrastructure (TCI). The accuracy of the computed trajectories does not only depend on the TCI trajectory integration capabilities, but also on the input received from the associated Aircraft Performance Model (APM), the operational instructions defining each trajectory, and the atmospheric data.

EUROCONTROL Experimental Centre (EEC) conducts a number of activities in the domain of APMs, which are performed within the scope of Base of Aircraft Data (BADA). EEC has beesupported by Boeing Research & Technology Europe (BR&TE) in the definition, development, implementation, and evaluation of an advanced APM suitable for the stringent requirements of future trajectory prediction tools. This new APM will become version 4.0 of BADA, released and maintained by EEC. BR&TE has also supported EEC in the design and development of an advanced bidimensional (vertical plane) TCI intended to exploit all the BADA 4.0 APM capabilities.

n

This TCI accepts either a BADA 4.0 or a BADA 3.x APM, together with all kinds of winds, as well as temperature and pressure deviations over the International Standard Atmosphere (ISA).

The TCI requires five simultaneous operational instructions to compute any trajectory segment. These instructions model the commands issued by the flight deck, and comprise the “aircraft intent” (AI). Three of them set the position of the landing gear, high lift devices, and speed brakes, while the other two determine the aircraft motion in the vertical plane.

Accepted instructions include different speed, energy, altitude, vertical speed, path angle, and throttle laws, and even optimum speed laws

intended to maximize certain flight characteristics, such as range or cost.

This paper presents the TCI capabilities and interfaces. It describes the different elements involved in the trajectory computation process, placing special emphasis in the instructions comprising the AI and how they can be combined, as well as the equations describing the aircraft motion and their integration. It also includes an example of the trajectory computation process.

Introduction The rapidly growing levels of air traffic are

putting the current Air Traffic Management (ATM) system under increasing pressure. It is expected that by the year 2020 the ATM system should be capable of managing at least twice as much traffic as it does today to accommodate increasing demand, with a higher level of safety, and diminished environmental impact [1].

These requirements cannot be achieved without new operational concepts that result in significant system efficiency improvements. The performance of the ATM system is closely linked to its predictability, as increasing levels of uncertainty prevent the evolution towards a more automated and strategic system. As a result, safety is ensured in the current highly unpredictable environment at the expense of sacrificing efficiency.

A more efficient ATM system needs to be more predictable, allowing higher automation. As shown in Figure 1, such a system capable of adjusting to the users’ preferences shall rely on model-based applications, and in particular on a trajectory predictor that permits precise trajectory computation. The trajectory prediction capability will thus play a key role in any future ATM system [2, 3, 4]. Such a trajectory-based ATM system can only be enabled by accurate aircraft performance knowledge, as the flow down of

1.C.4-1

requirements (blue arrows of Figure 1) must mathe flow up of limitations

tch (red arrows).

Figure 1. Trajectory Based ATM System

Base of Aircraft Data (BADA) is an Aircraft Performance Model (APM) developed and maintained by EUROCONTROL Experimental Centre (EEC). It is intended for trajectory computations related to ATM research, as well as for ground non safety critical operations planning.

The current 3.x versions of BADA [5, 6] provide acceptable accuracy for an elevated number of aircraft types1 when the aircraft speed schedule is restricted to a narrow interval within the operational envelope and the atmospheric conditions do not deviate substantially from the standard ones.

Boeing Research & Technology Europe (BR&TE) has supported EEC in the definition and development of BADA 4.0 by means of the project AMEBA2, with the objective of obtaining a more realistic APM with higher accuracy over the complete operational envelope and in any atmospheric conditions. A detailed analysis of the model and its results can be found in [7].

This paper is a continuation of [6] and [7] and focuses on the experimental bidimensional (vertical plane) Trajectory Computation Infrastructure (TCI) developed to demonstrate and exploit the capabilities of the new BADA 4.0 APM.

By accepting a wide array of possibilities in the definition of the flight command instructions and atmospheric conditions, this TCI places stringent requirements on the underlying APM; any

1 Includes turbojet, turboprop, and piston powered aircraft. 2 Advanced Model Engineering for BADA.

APM deficiency, either in its capabilities, applicability, or accuracy [8], would undermine the validity of the computed trajectories. In fact, and with the exception of tridimensional (lateral motion) capabilities, this TCI offers at least as much flexibility as the trajectory computation capabilities on board any current commercial airliner.

This paper is structured in four sections. The first one introduces the different TCI components, including short descriptions of the aircraft and earth models. Specific sections are allocated for detailed descriptions of the accepted instructions that describe the aircraft intent, as well as for the equations of motion and their integration. A generic example covering some of the introduced concepts is then presented.

Trajectory Computation Infrastructure

The top half of Figure 2 represents the process that leads to an actual aircraft trajectory. It starts with the pilot (either human, computer or the collaboration of both) controlling the aircraft via certain flight commands, which are the input received by the different aircraft control systems. The flight commands are entered with the objective of obtaining a certain aircraft motion, represented in the figure by the planned trajectory (thin solid line connecting way points).

The aircraft response to the flight commands is in last instance implemented through aerodynamic configuration changes combined with propulsive (throttle lever) and aerodynamic (elevator) control actions3, resulting in an actual trajectory (thick solid line), which also depends on the given atmospheric conditions (wind, pressure, and temperature).

The bottom half of Figure 2 represents an abstraction of the above described process by means of a trajectory predictor, employed to obtain a predicted trajectory that should resemble the actual trajectory flown by the aircraft. A detailed description of the different elements involved in the trajectory prediction domain is given in [9]. The

3 As this TCI is restricted to the vertical plane, the aileron and rudder controls are not taken into account.

1.C.4-2

process starts with a mathematical representation of the flight commands, called aircraft intent (AI) [10, 11], which specifies how the aircraft should be operated during the time interval for which the trajectory is being computed.

The AI instructions, together with the initial conditions4, are the input to the TCI, which is divided into three different modules. Its key component is the Trajectory Engine (TE), responsible for executing the trajectory computation process by means of the integration of the equations of motion in combination with the AI instructions. The trajectory computation process requires an APM to provide models for the different aircraft performance aspects that appear in the equations. The computation infrastructure also requires an earth model (EM) to provide the values of the wind, pressure and temperature present in both the equations of motion and the APM models.

Figure 2. Context of Trajectory Prediction

The TE relies on a collection of resolution strategies and numerical recipes to integrate the equations describing the aircraft motion into the predicted trajectory (thick dashed line), which anticipates the actual trajectory of the real flight. The deviations between the actual and the predicted

] d AI

4 The initial conditions are the mathematical representation of the aircraft motion state at the beginning of the trajectory.

trajectory result from the differences between the real elements involved in the trajectory generation process and the models representing them.

The TCI described in this paper contains all these modules. They are described below, with special emphasis on the AI and the TE. As this TCI is an experimental tool not intended for operational use, MatLab® has been used for its implementation, hence facilitating the development and comprehension at the expense of reduced computation speed.

Aircraft Performance Model The APM provides the means to predict the

aircraft response to given command and control inputs based on its position, airspeed, and the atmospheric pressure and temperature. A three degrees of freedom kinetic APM provides the following:

• Actions. Computes the forces acting on the aircraft (lift L, drag D, thrust T, and weight W), together with the fuel consumption F that affects the aircraft mass and thus its weight.

• Limitations. The hypotheses, equations, and results involved in the actions models are only valid within certain limits or borders outside of which they no longer reflect the real aircraft behavior. The limitations model ensures that the results are realistic and represent a valid approximation to the aircraft performances.

• Operations. Provides the velocity at which the high lift devices and the landing gear are deployed or retracted.

Although both BADA 3.x and 4.0 APMs [5, 6, 7] are accepted by the TCI, the improved capability, applicability, and accuracy [8of the later result in a wider array of accepteinstructions (see next section), increased protection against operational envelope violations, and higher accuracy in the resulting trajectories.

1.C.4-3

Earth Model The EM provides the atmospheric properties

(pressure, temperature, and wind) at each point of the aircraft flight, accounting for the changing characteristics of the atmosphere. The atmospheric properties may vary with the aircraft position, and also with time as the meteorological and seasonal conditions change. These variations, as well as their influence over the aircraft performances and motion, may be significant.

The wind vector w is usually found in the meteorological data expressed in LLS5 as function of the position and time.

LLS LLS1 3w ,w f (r,h, t= )

nts is

(1)

where r is the horizontal distance, h is the geometric altitude, t is the time, and the sub indexes 1 and 3 represent the horizontal (forward) and vertical (down) directions6, respectively.

The temperature and pressure deviations at mean sea level ΔT and Δp define the specific atmosphere that occurs at a given horizontal position and time.

T, p f (r, t)Δ Δ = (2)

Once an atmosphere is defined, the model predicts the temperature T and pressure p values at different altitudes:

T,p f ( T, p,h)= Δ Δ (3)

Aircraft Intent The AI is the main input to the TCI

(see Figure 2) as it models the way in which the aircraft is to be operated during flight. It represean abstraction of the way in which the aircraft instructed to behave by the flight deck. In the same way that the aircraft exhibits a unique trajectory as a result of the flight deck input, the aircraft intent shall be formulated in such a way that the ensuing aircraft motion is unambiguously determined, resulting in a unique computed trajectory.

5 Two different reference systems are employed in this paper: Local Level System (LLS) and Wind Fixed System (WFS). 6 Index 2 is generally reserved for the lateral direction, which is not employed in this bidimensional (vertical plane) TCI.

BR&TE has developed and defined a formal language to express and share the aircraft intent, known as Aircraft Intent Description Language (AIDL) [10, 11]. The AI accepted by this TCI represents an AIDL dialect (or subset), as it makes use of many but not all of its features.

The AI is based on two concepts, which are introduced here and described in detail in the following sections:

• Instructions are the minimal indivisible pieces of information that capture basic commands, guidance modes, and control inputs at the disposal of the flight deck to direct the aircraft behavior. Each instruction closes a degree of freedom in the equations of motion.

• Operations are groups of instructions that univocally determine the aircraft motion for the period of time for which they are active. Each valid operation is composed of five individual instructions, which together close all five degrees of freedom of the equations of motion in the vertical plane, converting them into a well posed mathematical problem with a unique solution. Each operation also contains one or more termination conditions, which determine the instant at which the operation stops being active.

The TCI accepts an AI made up by a series of operations, one executed immediately after the other. When one operation finalizes, the TE stops its integration, and starts that of the new operation formed by different instructions.

Instructions Table 1 shows the possible instructions

grouped in two different profiles. Each instruction is composed by its name or label (Table 1), a parameter, and a target, as described below.

1.C.4-4

Table 1. Aircraft Intent Instructions

Profile Label Meaning Longitudinal Motion

SL Speed Law OPTSL Optimum Speed Law EL Energy Law AL Altitude Law VSL Vertical Speed Law PAL Path Angle Law TIL Throttle Input Law

Aerodynamic Configuration

LGV Landing Gear Value HLV High Lift Value SBL Speed Brakes Law

Instruction Parameters

The instruction label taken from Table 1 is complemented with a parameter that identifies the specific motion or configuration aspect targeted by the instruction7 or the specific purpose of the instruction89.

Table 2 shows some of the possible parameters accepted by the different instructions labels10. Some labels accept a single possible parameter, while others contain several valid possibilities.

The table also includes the units in which the parameter value provided by the instruction target (as explained below) is expressed.

Table 2. Instruction Parameters

Label Meaning Parameter Units SL Mach number M dimless

calibrated airspeed

CAS kt

true airspeed TAS kt OPTSL max.

endurance cruise

MEC N/A

max. range MRC N/A

7 As an example, an SL (speed law) instruction may refer to the Mach number, in which case the parameter is M, or to the true airspeed, in which case the parameter is TAS. 8 As an example, a TIL (throttle input law) instruction may refer to the MCMB (maximum climb) or LIDL (low idle) power plant regimes. 9 As an example, the OPTSL (optimum speed law) instruction combined with the parameter MRC (maximum range cruise) means that the aircraft shall employ a Mach law that maximizes the flight range. 10 Table 2 is intended to show the most important parameters, but it is not a comprehensive list of all allowed possibilities.

Label Meaning Parameter Units cruise economy cruise

ECON N/A

max. rate of climb

MROC N/A

min. rate of descent

MROD N/A

economy climb

ECONCMB N/A

economy descent

ECONDES N/A

EL energy share factor

ESF dimless

acceleration factor

AF dimless

AL pressure altitude

HP ft

geometric altitude

h ft

VSL rate of climb HP fpm vertical speed h fpm

PAL aerodynamic path angle

AER deg

geometric path angle

GEO deg

TIL max. climb regime

MCMB N/A

low idle regime

LIDL N/A

high idle regime

HIDL N/A

throttle parameter11

δT dimless

thrust coefficient12

CT dimless

thrust T N LGV landing gear

parameter13 δLG dimless

HLV high lift parameter14

δHL dimless

SBL speed brake parameter15

δSB dimless

11 The throttle parameter δT is an abstraction of the throttle lever position. 12 CT = T / (WMTOW δ), where δ is the atmospheric pressure ratio and WMTOW is the aircraft maximum take off weight. 13 δLG is an abstraction of the landing gear position. 14 δHL is an abstraction of the high lift devices position.

1.C.4-5

Instruction Targets The combination of the instruction label and its

parameter (Table 1 and Table 2) identifies the instruction meaning, but needs to be complemented with a target to determine the specific effect of the instruction16. Depending on the instruction label and parameter, there are three possible target types:

• Extrinsic Laws. They consist of uni or multi dimensional mathematical functions that relate the value of the desired instruction parameter with other variables. Laws act as black boxes, in which one or several variables (generally part of the state vector r17) act as inputs18, and the value of the parameter as output. They are applicable to SL (speed law), EL (energy law), AL (altitude law), VSL (vertical speed law), PAL (path angle law), and SBL (speed brakes law) instructions19, together with those TIL (throttle input law) instructions in which the parameter is not a power plant regime.

• Values. The LGV (landing gear value) and HLV (high lift) instructions require a scalar to indicate the position towards where their respective configuration parameters shall move if they are not already there20. Some power plant regimes, such as MCMB (maximum climb), also require a scalar to specify the assumed temperature deviation.

• Intrinsic Laws. They are analogous to the extrinsic laws described above, with the only difference that they are provided by

15 δSB is an abstraction of the speed brakes position. 16 As an example, an SL (speed law) instruction with parameter M (Mach number) requires the target law 0.7 (constant) to complement it, indicating that the aircraft shall fly at a constant Mach number of 0.7. 17 The state vector is defined in the Trajectory Engine section. 18 The user can select among several predefined laws: constant, linear, step, gaussian, and interpolating tables, among others. 19 As an example, the AL (altitude law) instruction with parameter HP (pressure altitude) complemented with the 30000 + 5 t linear law indicates that the aircraft shall climb from FL300 at the rate of 5 ft/s. 20 As an example, the LGV (landing gear value) instruction with target 1 means that the aircraft shall extend the landing gear (if it is retracted) or leave it as it is (if it is extended), while 0 means it shall retract it.

the APM instead of by the user. In these cases, the combination of instruction label and parameter is sufficient to completely identify its purpose. All OPTSL (optimum speed law) instructions and those TIL (throttle input law) containing power plant regimes result in the APM providing the instruction target law.

Operations Each operation is composed of five individual

instructions. There shall always be three instructions belonging to the aerodynamic configuration profile (one each for the landing gear, high lift devices, and speed breaks), and two instructions belonging to the longitudinal motion profile (refer to Table 1).

Instruction Compatibility Not all combinations between longitudinal

motion profile instructions result in a well posed mathematical problem with a unique solution. As a result, only certain combinations are valid. The compatibility between pairs of longitudinal motion instructions is shown in Table 3, where “ ” represents compatibility and “ ” incompatibility. The compatibility matrix is intended to avoid the formulation of ill-posed mathematical problems that could give rise to no solution or multiple ones.

Table 3. Instructions Compatibility Matrix

SL EL AL VSL PAL TIL SL EL AL VSL PAL TIL

The OPTSL (optimum speed law) instructions have their own logic of incompatibilities, shown in Table 4. These instructions result from the offlinresolution of an optimization problem, which requires them to be coupled with very specific instructions21.

e

21 As an example, an OPTSL (optimum speed law) instruction with MEC (maximum endurance cruise) parameter can only be combined with an AL (altitude law) or VSL (vertical speed

1.C.4-6

Table 4. Optimal Speed Law Compatibility Matrix

OPTSL parameter

Compatible instructions

MEC AL or VSL MRC AL or VSL ECON AL or VSL MROC TIL MCMB MROD TIL LIDL or TIL HIDL ECONCMB TIL MCMB ECONDES TIL LIDL or TIL HIDL

Operation Termination Conditions

In addition to five instructions, each operation also contains a termination condition that triggers when the instructions defining an operation stop being applicable and the ones belonging to the next operation replace them. The termination conditions are algebraic equations involving different variables22, and can be combined with logical OR operators to obtain more complex conditions23. At least one termination condition, or end trigger, shall be defined for each operation.

Table 5. End Trigger Parameters

Code Meaning Units t time s m mass kg r horizontal distance nm HP pressure altitude ft h geometric altitude ft TAS true airspeed kt CAS calibrated airspeed kt M Mach number - ROC rate of climb fpm TA transition altitude24 N/A

law) instruction, whereas ECONCMB (economy climb) can only be coupled with a TIL (throttle input law) MCMB (maximum climb regime) instruction. 22 Typical examples are to conclude the operation at a given time, altitude, airspeed, or horizontal distance. 23 A possible termination condition for a climb operation is that combining a given altitude with a given rate of climb. The TE finishes the operation either if the aircraft reaches a certain altitude or if its rate of climb falls below a given threshold. 24 Only applicable to ascent operations at constant CAS followed by constant Mach (both in MCMB regime), or descent operations at constant Mach followed by constant CAS (both in LIDL or HIDL regimes). It triggers when the speed at

End triggers are composed by a parameter, which identifies the variable that is inspected by the TE at each integration step, and a law, which is a black box function like those described for instruction targets. Table 5 lists some of the possible end trigger parameters accepted by the TE.

Instruction Definition This section provides a detailed definition of

each of the possible instructions, with specific emphasis in the OPTSL (optimum speed laws).

Longitudinal Motion Profile The longitudinal motion profile encompasses

those instructions that govern the aircraft motion in the vertical plane. In a real flight, this is controlled from the flight deck with the elevator and the engine throttle. The instructions are the following:

The speed law (SL) instructions may refer to M, CAS, or TAS. The TCI accepts any possible target law of either of these parameters with the state vector r variables plus time. It is compatible with AL, VSL, PAL, and TIL instructions.

The maximum endurance cruise (OPTSL MEC) instruction sets the Mach number that provides maximum endurance as function of the weight coefficient CW

25. It is mostly employed during holding maneuvers to maximize the time that an aircraft can remain aloft with a given fuel load. It is compatible with AL and VSL instructions, although the most common case is when the aircraft is assigned to a flight level. Its mathematical form is26:

FT MEC * W

dC (M,C ) 0 M M (C )dM

= → =

(4)

The maximum range cruise (OPTSL MRC) instruction sets the Mach number that provides maximum range as function of the weight coefficient CW and the wind Mach number MW

27. It is used to maximize the distance that an aircraft can

both segments coincides. TA is the only end trigger condition parameter that does not require an associated law. 25 CW = W / (WMTOW δ). 26 CF = (F LHV) / (δ θ0.5 WMTOW a0), where CF is the fuel coefficient, θ is the atmospheric temperature ratio, LHV is the fuel lower heating value, and a0 is the standard speed of sound at mean sea level. 27 MW = w1

LLS / a, where a is the speed of sound.

1.C.4-7

cover with a given fuel load, and is compatible with AL and VSL instructions. Its mathematical form is:

FF T W T

dCC (M,C ) (M M ) (M,C ) 0dM

− + =

(6)

The economy cruis SL ECON) instruction sets the Mach minimizes the t t

e with

(5)

) MRC * W WM M (C ,M→ =

e (OPT number that

otal flight costs as function of the weighcoefficient CW, the wind Mach number MW, and thcost index coefficient CCI

28, and is compatible AL and VSL instructions. The flight costs are evaluated as the sum of the time and fuel costs. Its mathematical form is:

FF T W

dCC (M,C ) (M M− + T CI) (M,C ) C 0dM

+ = (7)

)

The maximum rate of clinstruction sets the Mach number that the a

,

MB.

t

ECON * W W CIM M (C ,M ,C→ = (8)

imb (MROC) maximizes

ircraft surplus power29 when climbing asfunction of the maneuver weight coefficient CnW

30

the atmospheric pressure ratio δ, and the temperature deviation ΔT. It shall only be combinedwith the maximum climb regime TIL MCIndependently of what the name implies, and based on its definition, the MROC speed law does notcontinuously maximize the aircraft rate of climb nor minimize the time to climb between two differenaltitudes. Its mathematical form is:

[ ] MROC *d M(T D) 0 M M (− = → = nWC , , T)

dMδ Δ (9)

The minimum rate of descent (MROD) instruction sets the Mach number that minimizes the aircraft deficit power hen descending as func ,

plies,

31 wtion of the maneuver weight coefficient CnW

the atmospheric pressure ratio δ, and the temperature deviation ΔT. It shall only be combined with the low or high idle regimes (TIL LIDL or HIDL). Independently of what the name im

of

28 CCI = (CI LHV) / (δ θ0.5 WMTOW a0), where CI is the cost index. 29 The climbing surplus power is defined as the product of the airspeed by the difference between the thrust and drag forces. 30 CnW = n CW, where n is the load factor. 31 The descending deficit power is defined as the product of the airspeed by the difference between the drag and thrust forces.

and based on its definition, the MROD speed law does not continuously minimize the aircraft rate descent nor maximize the time to descend betweentwo different altitudes. Its mathematical form is:

[ ] MROD * nWd M(D T) 0 M M (C , , T)

dM− = → = δ Δ (10)

The economy climb (ECONCMB) instruction sets a specific CAS/Mach speed profile (containin two SL instructions wi eters CAS and M) that minimizes the total costs of the flight climb an cruis

osts of the

fligh ly be

o

patible with TIL instructions.

is

t is com

S ith

epted. It is co

instr

ed th param

e phases. It shall only be combined with the maximum climb regime TIL MCMB.

The economy descent (ECONDES) instruction sets a specific Mach/CAS speed profile (contained in two SL instructions with parametersM and CAS) that minimizes the total c

t descent and cruise phases. It shall oncombined with the low or high idle regimes (TIL LIDL or HIDL).

The energy law (EL) instructions may refer tthe ESF or AF. The TCI accepts target laws of either parameter with the state vector r variables plus time. It is com

The altitude law (AL) instructions may refer to HP or h. The TCI accepts any possible target lawof either of these parameters with time or mass. Itcompatible with SL and TIL instructions.

The vertical speed law (VSL) instructions may refer to the rate of climb or the vertical speed32. Target laws of these parameters with the state vector r variables plus time are accepted. I

patible with SL and TIL instructions.

The path angle law (PAL) instructions may refer to the geometric γ or the aerodynamic γTApath angles. Target laws of these parameters wthe state vector r variables plus time are acc

mpatible with SL and TIL instructions.

The TIL instructions associated with engine regimes set one of the following power plant control laws: MCMB, LIDL, or HIDL. They are compatible with all other longitudinal motion

uctions except those noted in Table 4.

32 The rate of climb is the pressure altitude HP differential with time, while the vertical speed is that of geometric altitude h.

1.C.4-8

The TIL instructions with direct throttle inpumay refer to the thrust force T, thrust coefficienor throttle parameter δT. They can be combined with

t t CT,

all o

ways be the position

speed

sets

ion. If the initia

the

e operation. If the initia

eration. The TCI the

CI and responsible for the integration of the system of

ribing the aircraft motion

in

ent

vector at that point turns into

m

ircraft forces, and wind d vertical plane.

me is other

ion (14), so it can be ebraic

ther longitudinal motion instructions except those noted in Table 4, and accept target laws based on the state vector r variables plus time.

Aerodynamic Configuration Profile All three instructions present in the

aerodynamic configuration profile shall alpresent in each operation. They define of the landing gear, high lift devices, andbrakes, influencing the aerodynamic response of theaircraft, and hence the resulting motion.

The landing gear value (LGV) instruction the desired value of the landing gear configuration parameter δLG for the length of the operat

l value differs from the desired one, the aircraft will automatically extend or retract the landing gear to the desired position. The APM operations model determines the speed at which the landing gear varies its position.

The high lift value (HLV) instruction sets desired value of the high lift devices configuration parameter δHL for the length of th

l position differs from the desired one, the aircraft will automatically extend or retract the high lift devices to the desired position. The APM operations model determines the speed at which the high lift devices vary their position.

The speed brakes law (SBL) instruction defines the value of the speed brakes control parameter δSB for the length of the op

accepts any possible target law of δSB withstate vector r variables plus time.

Trajectory Engine The TE is the nucleus of the T

differential equations descin the vertical plane together with the algebraic equations defined by the aircraft intent, resultingthe computed trajectory.

Each operation is integrated as an independentity. When the termination condition of a given operation is met, the state

the next operation initial conditions, and the integration resumes with the new instructions.

Equations of Motion The equations of motion employed result fro

the following hypotheses and simplifications:

• Aircraft motion, acontained in a fixe

• Zero airspeed bank angle (μTAS=0). • Constant airspeed yaw angle

(dχTAS/dt=0). • Thrust force parallel to airspeed. • The path angle γ variation with tiTAS

small when compared with theterms in equatremoved, converting it into an algexpression.

The resulting kinematic, motion, and mass variation equations are the following:

LLSTAS TAS 1

dr v cos wdt

= γ + (11)

LLSdh v sin w= γ − TAS TAS 3dt (12)

WFS1dv dw (13) TAS

TAST D Wsin mdt dt

⎛ ⎞− − γ = +⎜ ⎟

⎝ ⎠WFS3

TASdwL W cos m

dt− γ = − (14)

dm Fdt

= (15)

where vTAS is the true airspe anaerodynamic path angle.

Wh n an SLTE replaces equation (13) with the energy

ed d γTAS is the

e or EL instruction is present, the

formulation to facilitate its resolution: WFSdwdh ⎛ ⎞ LLSTAS1

3T D m w ESFdt dt W

= − − −⎜ ⎟⎝ ⎠

(16v ESF )

TAS TASv dv1ESF ; AF= = 1 AF g dh+

(17)

Resolution The TE solves the system of dif

algebraic equations (DAE) composed bequations (obtaining the aircraft performances from

ferential y the above

1.C.4-9

the APM, and the atmospheric properties and wind from

time of any other aspect of the

The five AI equations are necessary to close the e

k

the a

ly

relat

s in

n.

the EM) and the five AI instructions.

When combined with the initial conditions, its solution provides the variation of the state vector r variables with time for the length of the operation:

[ ](t)r TASr(t), h(t), v (t), m(t)= (18)

The same APM, EM, and AI equations can then be employed to determine the variation with

aircraft trajectory.

xisting five degrees of freedom, which correspond to the five ways in which the flight deccan affect the vertical plane motion: by modifying

erodynamics (landing gear, high lift devices, and speed brakes), by controlling the power plantswith the throttle lever, or by moving the elevator. The last two can be controlled directly or through a control law to achieve a certain objective, represented by the different AI instructions.

The philosophy within the TE is the reductionof the DAE problem into different sets of ordinary differential equations (ODEs) by analyticaloperating with and merging all the algebraic

ionships into the differential equations.

Depending on the combination of longitudinal instructions (see Table 6), this reduction resultsix different ODE integrators. They all use theMatLab® ode113 solver to obtain the solutio

Table 6. Different ODE integrators

# Longitudinal Motion Instructions 1 SL or OPTSL with TIL 2 SL or OPTSL with AL or VSL 3 SL with PAL 4 EL with TIL 5 AL or VSL with TIL 6 PAL with TIL

Once have been integrated, the TE ide atic, and dynamic asp of mpled at equispaced values of altitude, horizontal distance, or time33.

all operations prov s all geometric, kinemects the trajectory sa

33 Prior to the integration process, the user must select the

The resulting data can then be employed to creatcustom plots or reports.

The TCI has been developed under an object oriented approach in whi

variable (altitude, horizontal distance, or time) with which each operation shall be sampled, and at which interval. Typical

e

ch each class has clearly diffe

initia

nd

stem requires the ate vector variables plus

time

rentiated responsibilities. Objects containing instances of the EM, APM, AI, and resulting trajectory can be stored as XML files for later use.

The TCI has an associated Graphical User Interface (GUI) that enables the user to set the

l conditions and to load, save, create, and modify EM, APM, and AI objects to obtain the required TCI inputs, to launch the integration, ato analyze34 and save the resulting trajectories.

Initial Conditions The resolution of the ODE sy

initial values of the st35:

[ ]0(t ) r(t ), h(t ), v (t ), m(t )=r 0 0 TAS 0 0 (19)

The TE internally checks the vainitial conditions with respect to the AI instructions. Not

traje

This section contains an example of the trajectory computation process using the TCI covering the different modules described above.

The selected APM is a BADA 4.0 model corresponding to a commercial airliner37, while the

lidity of these

all initial values of the state vector componentsmay be required, as in some cases (depending on the AI) they are not independent from each other36.

The TE employs the initial conditions provided by the user to integrate the first operation in a

ctory. It then takes the final values of the state vector variables in one operation as the initial conditions for the integration of the next operation.

Example

devices, landing gear, and

peed law) instructions overwrite the

fined.

values are to sample at every 1000 ft for ascent or descent operations, and at every 10 nm for cruise operations. 34 By means of plots or data reports. 35 The initial positions of the high liftspeed brakes are optional. All of them are set to their retracted position if not specified. 36 As an example, all SL (sinitial value of the true airspeed. This may result in discontinuities if the aircraft intent is not carefully de37 This paper does not identify the specific aircraft.

1.C.4-10

EM is characknot

terized by a constant tailwind of 20

38

instr er

s combined with a constant temperature deviation of +10 deg K. The initial mass and altitude are 65000 kg and 10000 ft respectively .

The AI contains six operations intended toshow some of the different possibilities accepted. Figure 3 shows the main characteristics ofeach operation. The first two rows show the label and parameter of the two longitudinal motion profile instructions39, while the third shows the applicable aerodynamic configuration profile

uctions40. The fourth row identifies the triggparameter.

SL CAS

SL M

AL HP

TIL MCMB

PAL AER

HLV

LONG PROF 1

AL HP

TILID

L L

PAL AER

TIL LIDL

VSL HP

SL M

SBL

OP 1 OP 2 OP 3 OP 4 OP 5 OP 6

LONG PROF 2

CONFIG

TRIGGER M HP M r CAS t

EM ΔT= 10 K w1LLS = 20 kt

TIL MCMB

SL CAS

SL M

AL HP

TIL MCMB

PAL AER

HLV

LONG PROF 1

AL HP

TILID

L L

PAL AER

TIL LIDL

VSL HP

SL M

SBL

OP 1 OP 2 OP 3 OP 4 OP 5 OP 6

LONG PROF 2

CONFIG

TRIGGER M HP M r CAS t

EM ΔT= 10 K w1LLS = 20 kt

TIL MCMB

Figure 3. Example Instructions

A brief description of each operation follows:

1. Constant CAS of 280 kt and maximum climb regime until reaching Mach number 0.65. High lift devices at five degrees.

2. Increase the Mach number at a pace of 0.01 for every 1000 ft maintaining a ROC of 200 fpm until reaching FL350. Retract the high lift devices to obtain clean configuration.

38 The initial value of all other state vector variables is zero. 39 A different cocontai ed in Table 1 to facilitate its ide

lor has been assigned to each instruction ntification.

l er 0.79. Clean

configuration.

4. Descend by 120 fpm maintaining the 0.79 Mach number for 500 seconds in clean configuration.

5. Descend at a constant aerodynamic path angle of – 3 deg with the power plants at the low idle regime for 50 nm in clean configuration.

6. Continue descending at – 3 deg and low idle regime but fully deploy the speed brakes to quickly decelerate until 180 kt CAS.

Figure 4 shows the resulting trajectory rate of climb ROC versus pressure altitude HP. It can be observed that the ROC drops steadily with altitude during the first operation, remains constant at 200 fpm for the second, and is zero (horizontal flight) during the third. It is then a constant – 120 fpm during the fourth and higher (although decreasing as the aircraft decelerates) during the constant γTAS fifth and sixth operations.

n40 Only those configuration instructions that imply non clean aerodynamics are displayed. Operations 2 through 5 contain clean aerodynamics, but the high lift devices are deployed in operation 1 and the speed brakes are used in operation 6.

3. Accelerate at 35000 ft of altitude by using the maximum climb regime untireaching Mach numb

Figure 4. Rate of Climb versus Altitude

1.C.4-11

Figure 5 shows the flight geometry in a pressure altitude versus horizontal distance plot. Altitude increases rapidly although at diminisrate for the first operation41 and then at a constant

hing

short

.

pace for the second. It is then constant for the third operation and decreases in the fourth, to diminish much faster in the last two operations42

Figure 6. Mach Number versus Time

Conclusions This paper describes the bidimensional

(vertical plane) Trajectory Computation Infrastructure (TCI) jointly developed by Boeing Research & Technology Europe (BR&TE) and EUROCONTROL Experimental Centre (EEC).

traje

By accepting all kinds of weather conditions and employing the aircraft intent (AI) abstraction

the .

s,

ercial aircraft.

Figure 5. Altitude versus Horizontal Distance

The trajectory Mach number variation with time is shown in Figure 6. It increases with altitude during the first constant CAS operation and continues growing at a constant pace with altitude during the second. It then quickly accelerates during the third to reach 0.79, which is maintained during the fourth. The Mach then diminishes during the constant γTAS descent of the fifth and sixth operations. The speed brakes deployment at the last operation augments the loss of speed.

The TCI is intended to provide a means for ctory computation using the BADA Aircraft

Performance Model (APM). Although it accepts the current 3.x version of BADA, its main purpose is to demonstrate the capabilities of the new advanced BADA 4.0 APM [7].

to charaprovides xibility intypes of trajectories it is capable of computingWith the exception of tridimensional capabilitiethis flexibility is at least as high as that on boardany current comm

I m ior in the veans of the combination of two

instructions capable of representing speed, evertical speed, path angle, and throttle

laws. Specific optimum speed laws intended to

cterize the flight deck commands, this TCI the user with significant fle

aIn ddition to the aerodynamic configuration, the A ertical plane by m

nergy, altitude,

odels the aircraft behav

maximize or minimize certain flight characteristics (such as range or total costs) are also accepted.

41 The diminishing slope of Figure 5 for the first operation matches with the decreasing ROC of Figure 4. 42 The constant slope of Figure 5 for the fifth and sixth operations is caused by the constant path angle instruction.

1.C.4-12

1.C.4-13

References [1] EUROCONTROL Long Term Forecast of Flights (2004-2025), www.eurocontrol.int/%20statfor/gallery/content/public/forecasts

[2] JPDO Concept of Operations for the Next Generation Air Transportation System, version 2.0, June 2007.

[3] SESAR D2 Feedback Report, March 2007.

[4] Swierstra, Sipke, S. Green, 2003, “Common Trajectory Prediction Capability for Decision Support Tools”, 5th USA/EUROCONTROL ATM R&D Seminar, Budapest, Hungary.

[5] EUROCONTROL, 2004, User Manual for the Base of Aircraft Data (BADA) Revision 3.6, EEC Note No. 10/04.

[6] Nuic, Angela, C. Poinsot, M. Iagaru, E. Gallo, F. Navarro, C. Querejeta, 2005, “Advanced Aircraft Performance Modeling for ATM: Enhancements to the BADA Model”, 24th Digital Avionics Systems Conference, Washington D.C., USA.

[7] Gallo, Eduardo, F. Navarro, A. Nuic, M. Iagaru, “Advanced Aircraft Performance Modeling for

Results”, 25th Digital Avionics ce, Portland OR, USA.

CARE-

E,

Intent Description Language as Key Enabler for d Operations”, 26th Digital Avionics

l A., E. Gallo, F. Navarro, S.

escription of Aircraft Intent”, 24 gton

ATM: BADA 4.0Systems Conferen

[8] Key Performance Indicators for APMs. COURAGE Project, CARE-TP (Collaborative ATM Research in EUROCONTROL – TrajectoryPrediction). BR&TE, DFS, and Avtech. 2005.

[9] Domain Analysis. COURAGE Project, TP (Collaborative ATM Research in EUROCONTROL Trajectory Prediction). BR&TDFS, and Avtech. 2005.

[10] López-Leonés, Javier, M. Vilaplana, E. Gallo, F. Navarro, C. Querejeta. “A Standard Aircraft

Trajectory BaseSystems Conference, Dallas TX, USA.

[11] Vilaplana, MigueSwierstra, 2005, “Towards a Formal Language for the Common D th

Digital Avionics Systems Conference, WashinD.C., USA.

Email Addresses Eduardo Gallo: eduardo.gallo@boeing.com

Javier López-Leonés: javier.lopezleones@boeing.com

Miguel A. Vilaplana: miguel.vilaplana@boeing.com

Francisco A. Navarro: francisco.a.navarro2@boeing.com

Angela Nuic: angela.nuic@eurocontrol.int

26th Digital Avionics Systems Conference October 21, 2007