Post on 13-Oct-2020
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
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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
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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
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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
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