Comparison of Two Ground-based Mass Estimation Methods on Real Data - - PowerPoint PPT Presentation

Comparison of Two Ground-based Mass Estimation Methods on Real Data

• R. Alligier
• D. Gianazza
• M. Ghasemi-Hamed
• N. Durand

ENAC/MAIAA - IRIT/APO

International Conference on Research in Air Transportation, 2014

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 1 / 31

Objective

Context

Two mass estimation methods: Adaptive method [Schultz et al., 2012] Least square method [Alligier et al., 2013] A comparison was done on synthetic data [Alligier et al., 2013]

In this work: comparison on real data

Mode-C radar data + weather data Actual mass not known Comparison of the trajectory prediction accuracies

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 2 / 31

Objective

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 3 / 31

Objective

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 3 / 31

Objective

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 3 / 31

Objective

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 3 / 31

An energy-rate oriented approach

Newton’s laws

1 2 dv2 dt + g dz dt

• energy-rate

= power(mass) mass

• f(mass)

f is given by a physical model of the forces

Using past positions given by radar

We compute the observed energy-rate from radar data We search a mass such that:

• bserved energy-rate = f
• mass
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 4 / 31

1

Computing the power

mass provided by BADA 2

The adaptive method [Schultz et al., 2012]

3

The least square method [Alligier et al., 2012]

4

Experimental setup

5

Results

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 5 / 31

1

Computing the power

mass provided by BADA 2

The adaptive method [Schultz et al., 2012]

3

The least square method [Alligier et al., 2012]

4

Experimental setup

5

Results

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 6 / 31

A Point Mass Model

Path angle Weight Lift Thrust Drag

m.dVTAS dt = Thr − D − m.g.sin(γ)

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 7 / 31

A simplified model (longitudinal+vertical)

VTAS.dVTAS dt + g.dz dt

• energy-rate

= (Thr − D).VTAS m

• power

mass

z: altitude Thr (Thrust): thrust of the engines D (Drag): drag of the aircraft m: mass VTAS (True Air Speed): velocity in the air

dVTAS dt

: longitudinal acceleration

dz dt = VTAS.sin(γ): rate of climb

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 8 / 31

A physical model

VTAS.dVTAS dt + g.dz dt

• energy-rate

= (Thr − D).VTAS m

• power

mass

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 9 / 31

A physical model

VTAS.dVTAS dt + g.dz dt

• energy-rate

= (Thr − D).VTAS m

• power

mass

BADA model

Max climb thrust: Thr = f(T, VTAS, z) Drag: D = f(T, VTAS, z, m)

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 9 / 31

A physical model

VTAS.dVTAS dt + g.dz dt

• energy-rate

= (Thr − D).VTAS m

• power

mass

= f(T, VTAS, z, m)

• BADA model

BADA model

Max climb thrust: Thr = f(T, VTAS, z) Drag: D = f(T, VTAS, z, m)

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 9 / 31

Equation at a given point

• VTAS. dVTAS

dt

+ g. dz

dt

• energy-rate

= (Thr − D).VTAS m

• power

mass

= f ( T , VTAS , z , m )

• BADA model

Using radar and weather data, we know: T , z ,

dz dt , VTAS , dVTAS dt

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 10 / 31

Equation at a given point

• VTAS. dVTAS

dt

+ g. dz

dt

• energy-rate

= (Thr − D).VTAS m

• power

mass

= f ( T , VTAS , z , m )

• BADA model

Using radar and weather data, we know: T , z ,

dz dt , VTAS , dVTAS dt

The resulting equation: E

• energy-rate

= f ( T , VTAS , z , m )

• BADA model

= P ( m ) m

• P , a known function
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 10 / 31

1

Computing the power

mass provided by BADA 2

The adaptive method [Schultz et al., 2012]

3

The least square method [Alligier et al., 2012]

4

Experimental setup

5

Results

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 11 / 31

The adaptive method [Schultz et al., 2012]

Principle

Assuming an initial guess m0 At each point i, the mass mi is estimated using mi−1

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 12 / 31

The adaptive method [Schultz et al., 2012]

Principle

Assuming an initial guess m0 At each point i, the mass mi is estimated using mi−1 Ei = Pi

• mi
• mi
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 12 / 31

The adaptive method [Schultz et al., 2012]

Principle

Assuming an initial guess m0 At each point i, the mass mi is estimated using mi−1 Ei = Pi

• mi
• mi

⇔ mi = Pi

• mi
• Ei
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 12 / 31

The adaptive method [Schultz et al., 2012]

Principle

Assuming an initial guess m0 At each point i, the mass mi is estimated using mi−1 Ei = Pi

• mi
• mi

⇔ mi = Pi

• mi
• Ei

≃ Pi

• mi−1
• Ei
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 12 / 31

The adaptive method [Schultz et al., 2012]

Principle

Assuming an initial guess m0 At each point i, the mass mi is estimated using mi−1 Ei = Pi

• mi
• mi

⇔ mi = Pi

• mi
• Ei

≃ Pi

• mi−1
• Ei

At each new point i: mi = Pi(mi−1) Ei

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 12 / 31

Introduction of the sensitivity parameter β [Schultz et al., 2012]

The previous update formula can be rewritten: mi = mi−1         1 + mi−1 Pi(mi−1)

• Ei − Pi(mi−1)

mi−1

• error on the energy rate

when using mi−1

       

−1

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 13 / 31

Introduction of the sensitivity parameter β [Schultz et al., 2012]

The previous update formula can be rewritten: mi = mi−1         1 + mi−1 Pi(mi−1)

• Ei − Pi(mi−1)

mi−1

• error on the energy rate

when using mi−1

       

−1

Introducing a sensitivity parameter βi : mi = mi−1

• 1 + βi

mi−1 Pi(mi−1)

• Ei − Pi(mi−1)

mi−1 −1

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 13 / 31

Logic of the sensitivity parameter β [Schultz et al., 2012]

β low ⇒ point i has nearly no impact, mi ≃ mi−1 β is dynamically adjusted according ∆ ˙ Ei, . . . , ∆ ˙ Ei−p

Update rule [Schultz et al., 2012]

if i > 0 and ∆ ˙ Ei > 0.0001 and

• ∆ ˙

Ei − ∆ ˙ Eavg ∆ ˙ Eavg

• < 3

then βi = max(0.205, βi−1 + 0.05) else βi = 0.005

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 14 / 31

Logic of the sensitivity parameter β [Schultz et al., 2012]

mi = mi−1

• 1 + βi

mi−1 Pi(mi−1)

• Ei − Pi(mi−1)

mi−1 −1

This mechanism increases robustness

If ∆ ˙ Ei repeatedly high in the same order of magnitude, β will increase, strengthening adaptation Isolated low or high ∆ ˙ Ei has a lower impact on adaptation

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 15 / 31

Logic of the sensitivity parameter β [Schultz et al., 2012]

mi = mi−1

• 1 + βi

mi−1 Pi(mi−1)

• Ei − Pi(mi−1)

mi−1 −1

This mechanism increases robustness

If ∆ ˙ Ei repeatedly high in the same order of magnitude, β will increase, strengthening adaptation Isolated low or high ∆ ˙ Ei has a lower impact on adaptation

The variation is limited

The estimated mass is kept within 80% and 120% of the reference mass The variation is limited to 2% of the reference mass

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 15 / 31

1

Computing the power

mass provided by BADA 2

The adaptive method [Schultz et al., 2012]

3

The least square method [Alligier et al., 2012]

4

Experimental setup

5

Results

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 16 / 31

The least square method [Alligier et al., 2012]

Principle

All the points are considered at once Minimizes the sum of square error on the energy rate

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 17 / 31

The least square method [Alligier et al., 2012]

Principle

All the points are considered at once Minimizes the sum of square error on the energy rate For all point i, i ∈ 1; n, we have: Pi

• mi
• mi

= Ei

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 17 / 31

The least square method [Alligier et al., 2012]

Principle

All the points are considered at once Minimizes the sum of square error on the energy rate For all point i, i ∈ 1; n, we have: Pi

• mi
• mi

= Ei Masses m1, . . . , mn are not independent variables

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 17 / 31

The least square method [Alligier et al., 2012]

Principle

All the points are considered at once Minimizes the sum of square error on the energy rate For all point i, i ∈ 1; n, we have: Pi

• mi
• mi

= Ei Masses m1, . . . , mn are not independent variables ⇒ Equations cannot be satisfied altogether (in general)

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 17 / 31

The least square method [Alligier et al., 2012]

Principle

All the points are considered at once Minimizes the sum of square error on the energy rate For all point i, i ∈ 1; n, we have: Pi

• mi
• mi

= Ei Masses m1, . . . , mn are not independent variables ⇒ Equations cannot be satisfied altogether (in general) Then, we search (m1, . . . , mn) minimizing: E(m1, . . . , mn) =

n

• i=1

Pi(mi) mi − Ei 2

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 17 / 31

Relationship between the mi

fuel consumption

BADA model of the fuel consumption: dm dt = −f(T, VTAS, z)

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 18 / 31

Relationship between the mi

fuel consumption

BADA model of the fuel consumption: dm dt = −f( T , VTAS , z )

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 18 / 31

Relationship between the mi

fuel consumption

BADA model of the fuel consumption: dm dt = −f( T , VTAS , z ) mi = mn +

tn

• ti

f(T(t), VTAS(t), z(t))dt ⇒ mi ≃ mn +

n−1

• k=i

f(tk+1) + f(tk) 2 (tk+1 − tk) ⇒ mi = mn + δi

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 18 / 31

Minimizing this error

The error function can be rewritten: E(m1, . . . , mn) = E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 19 / 31

Minimizing this error

The error function can be rewritten: E(m1, . . . , mn) = E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2 Minimizing this error can be done by solving: E′(m) = 0

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 19 / 31

Minimizing this error

The error function can be rewritten: E(m1, . . . , mn) = E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2 Minimizing this error can be done by solving: E′(m) = 0 With the BADA model, Pi polynomial of the second degree

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 19 / 31

Minimizing this error

The error function can be rewritten: E(m1, . . . , mn) = E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2 Minimizing this error can be done by solving: E′(m) = 0 With the BADA model, Pi polynomial of the second degree ⇒ Solving E′(m) = 0 leads to find roots of a polynomial of degree at most 3(n − 1) + 4

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 19 / 31

Minimizing this error

The original error function: E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2 ⇒ Numerical issues solving E′(m) = 0

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 20 / 31

Minimizing this error

The original error function: E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2 ⇒ Numerical issues solving E′(m) = 0 An approximated error function: Eapprox(mn) =

n

• i=1

Pi(mn + δi) mn + δavg − Ei 2 with: δavg = 1

n

n

i=1 δi

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 20 / 31

Minimizing this error

The original error function: E(mn) =

n

• i=1

Pi(mn + δi) mn + δi − Ei 2 ⇒ Numerical issues solving E′(m) = 0 An approximated error function: Eapprox(mn) =

n

• i=1

Pi(mn + δi) mn + δavg − Ei 2 with: δavg = 1

n

n

i=1 δi

⇒ Solving E′approx(m) = 0 leads to find roots of a polynomial of degree 4

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 20 / 31

1

Computing the power

mass provided by BADA 2

The adaptive method [Schultz et al., 2012]

3

The least square method [Alligier et al., 2012]

4

Experimental setup

5

Results

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 21 / 31

Real data

Radar and weather data

Radar Mode-C from Paris Air Traffic Control Center Weather data from Météo France 4939 trajectories of A320 Trajectories of 12.5 minutes long (ie. 51 points) Each 15 seconds, we observe: T,VTAS, z, dz

dt , dVTAS dt

Estimate a mass with observed variables Problem: The actual mass is not known in our data

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 22 / 31

How to compare the accuracies ?

• t0 = 0

Hp0 = 18000 ft

15000 18000 20000 25000 30000 200 400 600

t [s] Hp [ft]

points

• future

past

Solution

Estimate the mass on past points and predict the future trajectory

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 23 / 31

1

Computing the power

mass provided by BADA 2

The adaptive method [Schultz et al., 2012]

3

The least square method [Alligier et al., 2012]

4

Experimental setup

5

Results

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 24 / 31

Trajectory prediction accuracy

Computing trajectory prediction

BADA 3.9 requires: initial mass: m = ˆ m(t = 0) speed profile: VTAS(t) = VTAS

(obs)(t)

method mean stdev mean abs rmse max abs mref

• 83

1479 1168 1482 5495 Adaptive

• 303

685 582 749 5535 Least Squares

• 532

653 631 843 6033

Table: Statistics on

• H(pred)

p

( ˆ mpast) − H(obs)

p

• at t = 600 s, in feet.
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 25 / 31

An example: trajectory with the largest error

Trajectory with the largest prediction error:

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 26 / 31

Relation between past error and future error

Quantile regression to estimate the 95 % quantile

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 27 / 31

Relation between past error and future error

Quantile regression to estimate the 95 % quantile

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 27 / 31

Relation between past error and future error

Quantile regression to estimate the 95 % quantile

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 27 / 31

Conclusion

Accuracy improvements

When compared to mref, the RMSE on the altitude is reduced by:

45 % for the Least Square method 50 % for the adaptive method

Adaptive method performs better Error on past points gives hints on the prediction error

Beyond accuracy

Adaptive method:

Simpler to implement Can be used with a black box model of the power

Least Square:

Do not have a β sensitivity parameter to be tuned

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 28 / 31

Future work and preliminary results

Machine learning techniques

Learn the mass ˆ mLS

future

Gradient Boosting Machine (GBM) with 10-folds cross-validation With all the available variables: arrival airport, aircraft operator, distance to go, etc. method mean stdev mean abs rmse max abs mref

• 83

1479 1168 1482 5495 Adaptive

• 303

685 582 749 5535 Least Squares

• 532

653 631 843 6033 GBM

• 68.8

452 337 457 5363

Table: Statistics on

• H(pred)

p

( ˆ mpast) − H(obs)

p

• at t = 600 s, in feet.
• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 29 / 31

Thank you, any questions ?

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 30 / 31

Alligier, R., Gianazza, D., and Durand, N. (2012). Energy Rate Prediction Using an Equivalent Thrust Setting Profile (regular paper). In International Conference on Research in Air Transportation (ICRAT), Berkeley, California, 22/05/12-25/05/12, page (on line), http://www.icrat.org. ICRAT. Alligier, R., Gianazza, D., and Durand, N. (2013). Ground-based estimation of aircraft mass, adaptive vs. least squares method. In 10th USA/Europe Air Traffic Management Research and Developpment Seminar. Schultz, C., Thipphavong, D., and Erzberger, H. (2012). Adaptive trajectory prediction algorithm for climbing flights. In AIAA Guidance, Navigation, and Control (GNC) Conference.

• R. Alligier, D. Gianazza, M. Ghasemi-Hamed, N. Durand (ENAC)

Estimation of the Aircraft Mass ICRAT 2014 31 / 31