ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010).

[EN­103] Development of Trajectory Prediction Model+

Yutaka Fukuda*, Masayuki Shirakawa*, Atsushi Senoguchi**Air Traffic Management Department
Electronic Navigation Research Institute (ENRI)
Tokyo, Japan[fukuda | shirakawa | senoguchi]@enri.go.jp

Abstract: This paper describes a trajectory prediction model and prediction accuracy error analysis for the climb anddescent phases of flight. This model predicts trajectories based on aircraft performance, airline operation, the navigation
database, and weather forecasts. In order to attain the required accuracy for operational use, uncertainty factor analysis
based on actual flight operation environments is important. Predicted trajectories using the model are compared withrecorded flight data. Error factors from the aircraft speed model and error factors from weather forecasts were analyzed
for ground speed predictions. BADA (Base of Aircraft Data) from EUROCONTROL was used as the aircraft speed
model. The numerical forecast model of the Japan Meteorological Agency (JMA) is used for weather forecasts. As aresult, the speed difference between the aircraft speed model and actual operation were larger than the weather forecast
error. The operational speed is smaller than the aircraft speed model in most samples. It is estimated that more fuelsaving flights are selected in actual flight.
Keywords: Air Traffic Management, Trajectory Based Operation, Trajectory Prediction Model
The Electronic Navigation Research Institute (ENRI) is
developing a trajectory prediction model for TBO [5][9]. It generates precise four-dimensional trajectories(4DT) using aircraft performance data and weather
forecast data, etc. ENRI compares trajectories
generated from the prediction model with operationaldata, and carries out prediction error analysis.

1. INTRODUCTION
Trajectory Based Operation (TBO) is considered to bea key concept of Air Traffic Management (ATM) into
the future. Trajectory is a description of the movementof an aircraft, both in the air and on the ground,including position, time and, through calculation,
speed and acceleration [1]. A trajectory is generated
based on the expectations of the flight operator inconsideration of various elements, such as aircraft
performance and weather conditions. It is modified to
avoid hazards such as bad weather and conflict withother aircraft. All flight phases from departure toarrival can be uniformly managed by TBO, and in thisway, flight operation efficiency can be improved.

This paper describes the trajectory prediction modeland prediction error analysis on the climb and descentphases of flight. Firstly, the method of trajectory
prediction is shown. Secondly, the paper analyzes theerror factors that influence trajectory prediction
accuracy. Trajectories predicted by the trajectory
prediction model are compared with the trajectories ofoperational data. Error factors from weather forecasts
and the aircraft speed model are analyzed.

NextGen (The Next Generation Air Transportation
System) of the United States and SESAR (The Single
European Sky ATM Research Program) of Europe aim
to achieve TBO [2], [3]. CARATS (Collaborative
Actions for Renovation of Air Traffic Systems) is
being discussed in Japan [4]. TBO is one of the target
concepts.

2. METHOD OF TRAJECTORY PREDICTION
Figure 1 shows a block diagram of the trajectory
prediction model. It uses aircraft performance data, theairline operation model, a navigation database, andweather forecast data, etc. A Total Energy Model(TEM) is used as the aircraft model [10]. The TEM
equates the rate of work done by forces acting on theaircraft to the rate of increase in potential and kinetic
energy.

Aircraft have a Flight Management System (FMS),
which generates the optimum trajectory for fuelconsumption and flight time. Trajectory optimization islimited to individual aircraft. It does not work for all
aircraft. Aircraft equipped with the latest FMS havehighly accurate trajectory control. But old aircraft do
not have this function. Therefore, a ground-based
trajectory prediction and control system for all aircraftis required for TBO and overall aircraft optimization.

The aircraft performance data includes the flightenvelope of each aircraft model (maximum speed,
minimum speed, etc.), aerodynamics (wing area anddrag coefficient), engine thrust, fuel consumption, etc.
The airline operation model includes standard altitude,
speed, and weight during the climb, cruise, and95

Y. Fukuda, M. Shirakawa, A. Senoguchi

Navigation Database

Airline Operation Model

Aircraft Performance

AltitudeLat. Lon of Waypoints

Data

Speed

Route

Flight Envelope

WeightDept./Arv. Route

Aerodynamics

Energy Condition

TrajectorydhV(T − D ) VTAS = mg
+ mVTAS TAS
dt
dt

Predictor

Weather Forecast
Wind, Temp.

Surveillance Data

4D Trajectory
Actual Trajectory
Evaluation, Prediction Error Analysis
Figure 1.

Block Diagram of Trajectory Prediction Model

Distance S

size of the small segment is decided with an aircraft’s
wind-influenced changes in GS.

D

The aircraft usually flies at a constant Indicated Air
Speed (IAS) or Calibrated Air Speed (CAS) and Mach
number. The trajectory prediction model converts the
CAS or Mach number into True Air Speed (TAS). GS
is calculated, taking into account the influence of wind
aloft.

s
VGND(s)O

Figure 2. Flight Time Prediction

Eq. (2) shows the relationship between TAS VTAS anddescend phases. BADA (Base of Aircraft Data) by
EUROCONTROL is used for this data. The navigation
database provides positional data for routes andwaypoints. ARINC424 data is used as a navigation
database [11]. Surveillance data includes an aircraft’s
current position and speed. It is used to monitor andupdate trajectories.

CAS VCAS [10].
μ
⎡
1
⎫⎤
⎧⎛
⎡
⎤⎞
μ
⎢ 2 P ⎪⎜ (P0 )ISA ⎢⎛ μ (ρ 0 )ISA
⎟
⎪⎥
2⎞
⎥
⎟
⎜
−
VTAS = ⎢
1
V
1
1
+
−
⎬⎥
⎨⎜1 +
CAS
⎟
⎟
⎜ 2 (P )
⎢
⎥
μρ
P
⎟
0 ISA
⎠
⎪⎥
⎪⎜⎝
⎢
⎣⎝
⎦⎠
⎭⎦
⎩
⎣

stands for International Standard Atmosphere, ( P0 ) ISA
is the ISA pressure at sea level = 101325 Pa, and( ρ0 ) ISA is the ISA air density at sea level = 1.225 kg/m3.

Figure 2 is given by Eq. (1).

1

0

VGND ( s )Tp = ∫

ds(2)

where P is the pressure at altitude, ρ is the air
density at altitude, γ is the isentropic expansion
coefficient for air = 1.4, and μ = (γ − 1) / γ . ISA

In order to calculate the flight time to a waypoint fromthe present position, Ground Speed (GS) and distancealong the route are used. The total flight time iscalculated as the sum of small segment flight times.
Total flight time Tp from origin O to destination D inS

1/ 2

Air pressure P and density ρ are given as functionsof temperature.

(1)g

P = ( P0 ) ISA [T / T0 ] KT R−

where S is the distance from origin O to destination D,s is small segment distance, and VGND ( s ) is GS at thegρ = ρ 0 [T / T0 ]− K R −1
Tsmall segment. The distance can be calculated
accurately by using the RNAV (Area Navigation) route[12]. The accuracy of GS predictions is important. The

(3)
(4)

where T is the temperature at altitude (K), T0 is thetemperature at sea level, ρ 0 is the air density at sea

96

ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010).
Agency (JMA). They are defined at a threedimensional grid point (latitude, longitude, andaltitude) in the atmosphere. A numerical forecast
model of the Global Scale Model (GSM) or the Meso
Scale Model (MSM) are used. The grid point data isinterpolated into continuous four-dimensional data,using latitude, longitude, altitude and time.

True North

φW Wind
TAS
vector

Track

VTAS

Angle

φT

vectorW

DriftφDAngle3. COMPARISON OF PREDICTION AND
MEASUREMENT

VGND
GS vector
Figure 3.

3.1 Comparison of climb phase
Figure 4 through Figure 8 show an example of climb
trajectory comparison. The horizontal axis is the flighttime. The predicted speed of a medium type aircraft iscompared with its measured speed. For the calculationof prediction data, reduced climb thrust is used. This isfrom 85 % to 100 % of standard climb thrust for fuelsaving, depending on the aircraft’s weight. The CAS or
Mach number is obtained from the speed of the BADA
airline operation model. It is converted into TAS using
forecast temperature. GS is calculated by adding theforecast wind. The weather forecast data is interpolated
from the MSM delivered before the aircraft departure
time. The actual altitude and weight is used in
calculating the prediction. The measured speed isrecorded by the aircraft. Though the precision of themeasurements are not known, it is assumed to be true
because it is measured and calculated by the aircraft.

GS Calculationlevel, R is the real gas constant for air, g is gravity
acceleration, KT is the ISA temperature gradient withaltitude, − g K T R = 5.25583.
Eq. (5) shows the relationship between TAS VTAS andMach number M .

VTAS = M γ ⋅ R ⋅ T(5)The transition altitude is the boundary between CASbased operation and Mach-based operation. When thealtitude is above the transition altitude, the Mach
number is used to calculate TAS. On the other hand,when the altitude is below the transition altitude, CAS
is used.

Figure 4 shows the measured altitude profile of theaircraft. Figure 5 compares measured and predicted
CAS. Predicted CAS is based on the BADA airline
operationmodel.
Predicted
CASchangesinstantaneously at 4,000 ft, 5,000 ft, 6,000 ft, and10,000 ft (1 ft = 0.3048 m). Measured CAS changes
from 250 kt to 290 kt in 42 s transition time (1 kt =0.514 m/s). Acceleration rate is about 0.95 kt/s. CAS isrestricted to less than 250 kt below 10,000ft by Air
Traffic Control (ATC) procedures. There is a 10 kt
CAS difference between the predicted CAS of 300 kt
and the measured CAS of 290 kt above 10,000 ft.
Aircraft cross the CAS and Mach transition altitude at
11:48. The CAS decreases with constant Machoperation above transition altitude.

The GS is calculated by TAS and the wind vector(Figure 3). GS is calculated by Eq. (6) [7].

VGND = VTAS cos φD + W cos(φW − φT )(6)

where W is Wind Speed (WS), φW is Wind Direction(WD), φT is track angle, and φ D is drift angle.
is calculated by Eq. (7).

⎛ W⎞
sin(φW − φT ) ⎟⎟⎝ VTAS⎠

φD = sin −1 ⎜⎜

φD

(7)

TEM is used for movement calculation in the climb
and descend phases.
(T − D)VTAS = mgdVdh
+ mVTAS TAS
dt
dt

(8)

Figure 6 compares measured and predicted GS. TAS
and GS increase while the aircraft climbs with constantCAS. This is why air density decreases from low
altitude to high altitude. Predicted GS corresponds wellto measured GS.

where T is thrust, D is drag, m is aircraft mass, his altitude, and t is time. The rate of work done byforces (thrust and drag) acting on the aircraft equals therate of increase in potential and kinetic energy.
Potential energy corresponds to altitude, and kinetic
energy corresponds to TAS. For example, the descent
rate at constant CAS and idle thrust is calculated bysolving Eq. (2) and (8) with parameters for thrust, drag,and CAS.

Figure 7 compares measured and predicted WS along
an aircraft’s trajectory. Measured WS changes up anddown more than predicted WS. The predicted WS is in
the mid range of the measured WS. The difference
between measured WS and predicted WS is less than
10 kt.

WD, WS, and temperature are calculated using
weather forecasts from the Japan MeteorologicalFigure 8 shows the error factor of GS from WS, WD,
temperature, and CAS. This data is calculated from a

97

40

40,00035,00030,00025,00020,00015,00010,0005,000
0Error Factor (kt)

Altitude （ｆｔ）

Y. Fukuda, M. Shirakawa, A. Senoguchi

11:30

11:35

11:40 11:45
Time (hh:mm）

11:50

20
0
-20
-40

WS
Temp

-60
-80
11:30

11:55

350
300
250
200
150
100
50
0
11:30

Measured
Predicted
11:35

11:40 11:45
Time (hh:mm）

11:50

11:55

GS(kt)WS (kt)

11:55

25
20
15
10
5
0
-80 -60 -40 -20 0
20
GS Error (kt)

40

60

80

Figure 9 GS Error Distribution (CLB)
the measured GS is smaller than the predicted GS. It isestimated that more fuel saving flight is selected in
actual operation, because slower GS saves fuelconsumption.

Measured
Predicted
11:35

11:40
11:45
Time (hh:mm）

11:50

11:55

3.2 Comparison of descent phase
Figure 10 to Figure 14 show an example of descent
trajectory comparison for the same aircraft. The speed
calculation is the same as for the climb. The thrust isnominal descent thrust. It is almost the same as idle
thrust.

Figure 6 GS (CLB)
35
30
25
20
15
10
5
0
11:30

11:50

30

Figure 5 CAS (CLB)
600
500
400
300
200
100
0
11:30

11:40
11:45
Time (hh:mm)

Figure 8 Error Factor (CLB)

The number of ACCAS(kt)Figure 4 Altitude (CLB)

11:35

WD
CAS

Measured
Predicted
11:35

11:40
11:45
Time (hh:mm）

11:50

11:55

Figure 7 WS (CLB)
partial derivative of Eq. (6) [7], [8]. This represents thedegree of influence of each factor on GS error. Air
temperature and WD are small. WS is less than 10 kt.
CAS is about -15 kt after 11:38. This means that themeasured CAS is smaller than the predicted CAS. The
CAS error factor is larger than the wind forecast error
factor. The average GS error in the climb phase is 5.34 kt in this sample.

Figure 10 shows the measured altitude profile of theaircraft. Figure 11 compares measured and predicted
CAS. Predicted CAS changes instantaneously at
10,000 ft, 6,000 ft, 3,000 ft, 2,000 ft and 1,500 ft.
Measured CAS reduces gradually earlier than predicted
CAS mostly. Measured CAS reduces from 290 kt to
250 kt about 1 min. Deceleration rate is 0.64 kt/s. In
CAS transition period, CAS error is large. There are 10
kt CAS difference between predicted CAS of 290 kt
and measured CAS of 280 kt above 20,000 ft.
Figure 12 compares measured and predicted GS. TAS
and GS decrease while the aircraft descends withconstant CAS. Measured and predicted GS are almost
the same. CAS differences at CAS transition periods
affect GS differences. The predicted GS corresponds
well to the measured GS.
Figure 13 compares the measured and predicted WS
along the aircraft’s trajectory. Measured WS changes
up and down more than predicted WS. The predicted
WS is in the mid range of the measured WS. The

Figure 9 shows a histogram of average GS error. The
number of aircraft in the sample is 84. The histogramof GS error shifts to a negative value. This means that98

40,000
35,00030,00025,00020,00015,00010,0005,000
040Error Factor (kt)

Altitude （ｆｔ）

ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010).

12:20

12:25

12:30 12:35
Time (hh:mm）

12:40

20
0
-20
-40
-80
12:20

12:45

Measured
Predicted
12:25

12:30 12:35
Time (hh:mm）

12:40

12:45

12:40

12:45

25
20
15
10
5
0

600

GS (kt)

12:30
12:35
Time (hh:mm)

30

Figure 11 CAS (DES)
500-80 -60 -40 -20 0
20
GS Error (kt)

400

Figure 15 GS Error Distribution (DES)

300
200
100
0
12:20

Measured
Predicted
12:25

12:30 12:35
Time (hh:mm）

12:40

12:45

35
30
25
20
15
10
5
0
12:20

Measured
Predicted
12:25

12:30
12:35
Time (hh:mm）

40

60

80

of GS error shifts to a negative value. This means thatthe measured GS is smaller than the predicted GS. It isestimated that speed reduction is assigned by ATC for
sequencing and spacing to the destination airport andmore fuel saving flight is selected in actual operation.
The aircraft speed setting is decided in consideration ofvarious conditions, such as aircraft weight, fuel costs,
weather conditions, and delays. The speed is
influenced by the cost index (CI) of FMS [13]. In theanalysis, the measured GS was smaller than thepredicted GS in many samples.

Figure 12 GS (DES)

WS (kt)

12:25

WD
CAS

Figure 14 Error Factor (DES)The number of AC

CAS (kt)

Figure 10 Altitude (DES)
350
300
250
200
150
100
50
0
12:20

WS
Temp

-60

12:40

12:45

Figure 13 WS (DES)
difference between the measured and predicted WS isless than 10 kt.
Figure 14 shows the error factor for GS from WS, WD,
temperature, and CAS. Air temperature and WD are
small. WS is less than 10 kt. of the CAS factor is large
at CAS transition periods and at high altitude. The
CAS error factor is larger than the weather error factor.
The average GS error in the descent phase is +0.44 kt
in this sample.

ICAO recommend that speed changes in the cruise
phase of more than 5 % of TAS from their given flightplan shall be reported to an air traffic services unit [14].
In the future, it will be useful to acquire the intended
speed of all flight phases for trajectory prediction with
SWIM (System Wide Information Management). The
Mach number and CAS information can be acquired bydata communication with the DAPs (Downlink
Aircraft Parameters) function of SSR (Secondary
Surveillance Radar) mode S and ADS-B (Automatic
Dependent Surveillance-Broadcast). It is useful to take
into account such surveillance information for
trajectory prediction.
With regard to weather forecasts, monitoring current
wind and temperature is important. Currently, someaircraft have a wind information downlink function. If
forecast errors compared with the aircraft's monitored
weather data increase in some areas, trajectory
prediction accuracy decreases in the areas. In this case,measures are taken such as weather forecast update

Figure 15 shows a histogram of average GS error. The
number of aircraft in the sample is 85. The histogram99

Y. Fukuda, M. Shirakawa, A. SenoguchiInternational Workshop on ATM/CNS (EIWAC)
2009, March 2009.

delivery or error margin expansion for trajectory
prediction.

[7] Shirakawa M, Fukuda Y and Senoguchi A.

"Aircraft Trajectory Prediction Error Analysis".
IEICE Technical Report SANE 2009-167，Feb.
2010.

4. SUMMARY
This paper has provided a trajectory prediction modeland prediction accuracy error analysis on the climb anddescent phases of flight. Trajectories predicted by the
trajectory prediction model are compared with the
trajectories of operational data.

[8] Fukuda Y, Shirakawa M and Senoguchi A.

"Development and Evaluation of Trajectory
Prediction Model". 27th International Congress of
the Aeronautical Sciences, Sept. 2010.
[9] Shirakawa M, Fukuda Y and Senoguchi A. "A

Error factors due to the aircraft speed model and error
factors due to weather forecasts were analyzed for GS
prediction. As a result, the Mach number and CAS
difference between the airline operation model andmeasurement data were large in most cases where the
GS difference was large. Airline operation model error
was larger than weather forecast error. The measured
CAS is smaller than that of the airline operation modelin most cases. It is estimated that more fuel saving
flight is selected in actual operation.

Study on Aircraft Vertical Trajectories". 48th
Aircraft Symposium，Dec. 2010.
[10] Eurocontol Experimental Center. "User Manual

for the Base of Aircraft Data (BADA), revision3.7". EEC Technical/Scientific Report No.20080003, March 2009.
[11] "Navigation Systems Data Base", ARINC

Specification 424-18, 2005.
[12] Fukuda Y, Shirakawa M. "Analysis of RNAV

Detailed analysis is being considered for future study
when weather forecast error is large. Climb anddescent rate analysis is required to define the position
uncertainty of 4DT. The development of algorithms tomodify trajectories in order to resolve conflicts and toadjust speed for CTA (Controlled Time of Arrival) isfor future study.

Departures and Arrivals Using Track Data".
APISAT-2009, Nov. 2009[13] Rumler W, Gunther T, Fricke H, Weißhaar U,"Flight Profile Variations due to the Spreading
Practice of Cost Index Based Flight Planning",
ICRAT-2010, June 2010.
[14] "International Standards Annex 2, Rules of the Air

Tenth Edition", ICAO, July 2005.

ACKNOWLEDGMENTS

COPYRIGHT

The authors would like to express thanks to all parties
who cooperated in this research.

Copyright Statement
The authors confirm that they, and/or their company or
institution, hold the copyright of all original material
included in this paper. They also confirm they haveobtained permission, from the copyright holder of any
third party material included in their paper, to publish
it as part of their paper. The authors grant full
permission for the publication and distribution of theirpaper as part of the EIWAC 2010 proceedings or as
individual prints from the proceedings.

REFERENCES[1] ICAO, "Global Air Traffic Management

Operational Concept", ICAO Doc 9854AN/458,
2005[2] Joint Planning and Development Office. "Conceptof Operation for the Next Generation Air
Transportation System Ver.3.1", April 2010[3] SESAR Joint Undertaking, "European Air Traffic

Management Master Plan. Edition 1". March 2009[4] Working Group on Future Air Traffic System.

CARATS (Collaborative Actions for Research of
Air Traffic Systems). 2010[5] Shirakawa M, Fukuda Y and Senoguchi A.

"Trajectory predictions with the aircraft
performance models". IEICE Technical Report
SANE 2008-99，Jan. 2009.
[6] Fukuda Y, Shirakawa M and Senoguchi A. "Studyon Trajectory Prediction Models". ENRI

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