Prediction of Top of Descent Location forIdle-thrust Descents

Laurel Stell

NASA Ames Research Center

Background

Vertical profile from cruise to meter fix

40-120 nmi

10,0

00–

30,0

00ft

idle thrust

.

Top of Descent

(TOD)

meterfix

Idle-thrust descent reduces fuel consumption and emissions.

2

Example Meter Fix Time Errors for Operational Data

Numberof

Flights

−10 −5 0 5 10 150

10

20

30

40

50

Error (nmi)

Numberof

Flights

Top of Descent Path Distance

−30 −20 −10 0 10 20 300

10

20

30

40

50

Error (sec)

Meter Fix Crossing Time

Airbus 320B757B737−300B737−800

More than 90% of the meter fix crossing time predictionshave absolute error less than 30 sec.

3

Example Prediction Errors for Operational Data

−10 −5 0 5 10 150

10

20

30

40

50

Error (nmi)

Numberof

Flights

Top of Descent Path Distance

−30 −20 −10 0 10 20 300

10

20

30

40

50

Error (sec)

Meter Fix Crossing Time

Airbus 320B757B737−300B737−800

Less than 50% of the TOD location predictions haveabsolute error less than 5 nmi.

4

Outline of Presentation

Mathematical analysis and approximation of TOD locationbased on NASA trajectory predictorComparison of results with FMS test bench data and withoperational dataPossible applications

5

Mathematical Analysis

6

Treatment of Wind

For this mathematical analysis, wind is zeroIncorporating wind into the results is explained in paperAnalysis of operational data included wind

7

Kinematic Equations

dsdt

= Vt

dhdt

= γaVt

t : times : the ground path distance relative to the meter fixh : the altitude

Vt : the true airspeedγa : is the flight path angle with respect to the air mass

8

Kinetic Equation

mdVt

dt= T − D −mgγa

T : thrustD : dragm : aircraft massg : gravitational acceleration

9

All Equations for Vertical Profile

1g

dVt

dt=

T − Dmg

− γa

dsdt

= Vt

dhdt

= γaVt

Specify 2 out of 3: flight path angle γa, airspeed Vt , thrust T

10

Descent Procedure

bc bcbc

bc bc

cruise

.

Top of Descent

(TOD)constant Mach

constant

calibrated

airspeed (CAS)

deceleration

meter

fix• Idle thrust throughout descent• Known horizontal trajectory

11

Constant Mach and CAS Segments

1g

dVt

dt=

T − Dmg

− γa

dsdt

= Vt

dhdt

= γaVt

Vt (h) specified:

dsdh

=1γa

=

(1 +

12g

dV 2t

dh

)mg

T − D

12

Top of Descent Location

STOD = −∫ hfix

hcrz

(1 +

12g

dV 2t

dh

)mg

T − Ddh

− 1g

∫ Vfix

Vc

Vtmg

T − DdVt

STOD : TOD location as path distance relative to meter fixhcrz : cruise altitudehfix : meter fix altitudeVc : descent CAS

Vfix : meter fix CAS

GoalPolynomial approximation in terms of hcrz,Vc ,hfix,Vfix,W = mg

13

Top of Descent Location

STOD = −∫ hfix

hcrz

(1 +

12g

dV 2t

dh

)mg

T − Ddh

− 1g

∫ Vfix

Vc

Vtmg

T − DdVt

GoalPolynomial approximation in terms of hcrz,Vc ,hfix,Vfix,W = mg

13

Approach

Analyzed trajectories from NASA predictor to develop thepolynomial approximations

Test matrix with 9000 entries varyingcruise altitude hcrzcruise Machdescent speed Vcmeter fix altitude hfixmeter fix speed Vfixaircraft weight W

B737-700 and B777-200

14

Approximation for Constant Mach and Constant CAS

∫ hfix

hcrz

(1 +

12g

dV 2t

dh

)W

T − Ddh

≈ P(Vc ,W )

∫ hfix

hcrz

(1 +

12g

dV 2t

dh

)dh

≈ −(∆h)V(Vc)P(Vc ,W )

≈ −(∆h)× [linear function of Vc ,W ]

≈ linear function of Vc ,W ,∆h

Each of the last three lines gives a polynomial approximation,with decreasing accuracy but increasing simplicity.

15

Accuracy of Approximations for TOD Location

B737-700

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1

residual (nmi)

cumulative distribution function

approx 1

approx 2

linear

16

Accuracy of Approximations for TOD Location

B777-200

−15 −10 −5 0 5 10 150

0.2

0.4

0.6

0.8

1

residual (nmi)

cumulative distribution function

approx 1

approx 2

linear

16

Validation Against Real-World Data

17

FMS Test Bench Experimental Setup

Commercial FMS connected to flight simulatorSame aircraft types as in mathematical analysisTest matrix is (3 values of descent CAS Vc) × (3 values ofweight W )All other inputs the same in all runsRecorded TOD locations computed by FMS

18

FMS Test Bench Data Results

Least squares fit with a model linear in Vc and W gaveabsolute errors less than 2 nmi for both aircraft typesMuch more accurate than expected from previous results

Approximations above partly explain thisTest matrix does not adequately exercise predictor

19

Operational Data Collection Procedures

Collected data for flights arriving at Denver InternationalAirportSeptember 8–23, 2009United Air Lines and Continental Airlines participatedHumans in controller facility wrote down:

Flights that participated and whether descent interruptedSpeed profiles

Pilots wrote down aircraft weightExtracted cruise altitude and horizontal trajectory fromradar data

20

Operational Data Results

Analyzed about 70 descents each for Airbus 319/320 andB757-200 from one airlineLeast squares fit with a model linear in ∆h, Vc , and Wgave absolute errors less than 4 nmi for all except:

Four (6%) of Airbus descentsSix (8%) of B757 descents

21

Applications

22

Possible Uses of These Approximations

Design of FMS test bench experimentsSimplified sensitivity analysisKinematic predictor of TOD location

Dynamic machine learning of coefficients to handle de-icingsettings, for exampleContinuously updated error modelsOnly 5-10 coefficients; NASA predictor has 2000 entries inits table for B777 thrust

23

More About Kinematic Predictor

For “what-if” tool with data linkMight only consider changing Vc and horizontal trajectoryOnly need to assume STOD is linear in VcOnly need to estimate one coefficient

To fill in predicted trajectory between STOD and meter fixKinematic approach: straight line might be better thancurrent predictions with 10 nmi error in STODKinetic approach: modify W/(T − D) so that integratorgives STOD close to desired value

Replace with constantMultiply by constantAdd constant. . .

24

Conclusions

Need to improve prediction of vertical profile to enablefuel-efficient descents in congested airspaceLargest source of error is difference in W/(T − D)between FMS and ground predictorTOD location can be approximated well by a simplefunction of hcrz,Vc ,hfix,Vfix,WThese approximations might guide research to improve thepredictor

25