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Ground-based Estimation of the Aircraft Mass, Adaptive vs. Least Squares Method
R. Alligier D. Gianazza N. Durand
ENAC/MAIAA - IRIT/APO USA/Europe Air Traffic Management R&D Seminar, 2013
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 1 / 33
Introduction
altitude
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R Alllg1er, D G1anazza, N Durand (ENAC) Est1mat1on of the Aircraft Mass ATM 2013 2133
Introduction
altitude
£ ) e+ ; l / .
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R Alllg1er, D G1anazza, N Durand (ENAC) Est1mat1on of the Aircraft Mass ATM 2013 2133
Introduction
altitude
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£ ) e+ ; l / .
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R Alllg1er, D G1anazza, N Durand (ENAC) Est1mat1on of the Aircraft Mass ATM 2013 2133
Introduction
altitude • +
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R Alllg1er, D G1anazza, N Durand (ENAC) Est1mat1on of the Aircraft Mass ATM 2013 2133
An energy-rate oriented approach
Newton’s laws 1 dv 2
2 dt dt + g =
dz power (mass)
ener
g y-rate _
mass
f (m a ss)
_
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:
observed energy-rate = f (mass)
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 3 / 33
Objective
Previous work [Schultz et al., 2012] An adaptive method
Synthesized data Without noise
[Alligier et al., 2012] A least square method Real data No result on the mass estimation accuracy
In this work
Synthesized data generated using BADA 3.9 Gaussian noise added to the observed variables BADA 3.9 model of forces is used to estimate the mass Comparison of the mass estimation accuracies
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 4 / 33
1 Computing the power provided by BADA mass
2 The adaptive method [Schultz et al., 2012]
3 The least square method [Alligier et al., 2012]
4 Results
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 5 / 33
1 Computing the power provided by BADA mass
2 The adaptive method [Schultz et al., 2012]
3 The least square method [Alligier et al., 2012]
4 Results
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 6 / 33
A Point Mass Model
\
Weight
d V s . m. = Thr- 0 - m.g.sm(!)
" ATM 2013 7 / 33 R Alii W->r [1 1 l i.J i l ::J22:3 r J Durand (Et'-IAI I Estim ation of the A i rcraf t Mass
A simplified model (longitudinal+vertical)
VTAS.
d VTAS
dt + g. =
dt dz (Thr − D).VTAS
ener g y-rate
_
m p ow er
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, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 8 / 33
The BADA contribution
VTAS.
d VTAS
dt + g. =
dt dz (Thr − D).VTAS
ener g y-rate
_
m p ow er
mass
_
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 9 / 33
The BADA contribution
VTAS.
d VTAS
dt + g. =
dt dz (Thr − D).VTAS
ener g y-rate
_
m p ow er
mass
_
BADA model Max climb thrust: Thr = f (T , VTAS, z) Drag: D = f (T , VTAS, z, m)
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 9 / 33
The BADA contribution
VTAS.
d VTAS
dt + g. =
dt m dz (Thr − D).VTAS
ener g y-rate
_ p ow er
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, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 9 / 33
Equation at a given point
VTAS.
d VTAS
dt + g. =
dt m dz (Thr − D).VTAS
ener g y-rate
_ p ow er
mass
_ = f (T , VTAS, z , m)
BADA
model
_
Using radar and weather data, we know : T , VTAS, z, dz , dt dt
dVTAS
We want to adjust the mass m
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 10 / 33
Equation at a given point
VTAS. d VTAS dt + g. dt =
m dz
ener
g y-rate
_
(Thr − D). VTAS
p ow er
mass
_ = f ( T , VTAS , z , m)
BADA
model
_
Using radar and weather data, we know :
T , VTAS , z , dz , dVTAS dt dt
We want to adjust the mass m
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 10 / 33
Equation at a given point
VTAS. d VTAS dt
+ g. dt = dz
ener g y-rate
_
(Thr − D). VTAS
m p ow er
_ mass
Using radar and weather data, we know :
= f ( T , VTAS , z , m )
BADA
model
_
T , VTAS , z , dz , dVTAS dt dt
We want to adjust the mass m
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 10 / 33
Equation at a given point
VTAS. d VTAS dt + g. dt = dz
ener
g y-rate
_
(Thr − D). VTAS
m p ow er
_ mass
Using radar and weather data, we know :
= f ( T , VTAS , z , m )
BADA
model
_
T , VTAS , z , dz , dVTAS dt dt
We want to adjust the mass m
E ene
r g y-_rate
= f ( T , VTAS , z , m )
BADA
model
_
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 10 / 33
Equation at a given point
VTAS. d VTAS dt + g. dt = dz
ener
g y-rate
_
(Thr − D). VTAS
m p ow er
_ mass
Using radar and weather data, we know :
= f ( T , VTAS , z , m )
BADA
model
_
T , VTAS , z , dz , dVTAS dt dt
We want to adjust the mass m
E ene
r g y-_rate
= f ( T , VTAS , z , m ) =
BADA
model
_ P ( m )
m _ P , a known function
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 10 / 33
1 Computing the power provided by BADA mass
2 The adaptive method [Schultz et al., 2012]
3 The least square method [Alligier et al., 2012]
4 Results
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 11 / 33
The adaptive method [Schultz et al., 2012]
Principle We assume an initial guess m0
At each point i, the mass mi is estimated using mi− 1
Ei = Pi (mi )
m i
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 12 / 33
The adaptive method [Schultz et al., 2012]
Principle We assume an initial guess m0
At each point i, the mass mi is estimated using mi− 1
Ei = Pi (mi )
m i ⇔ mi =
Pi (mi ) E i
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 12 / 33
The adaptive method [Schultz et al., 2012]
Principle We assume an initial guess m0
At each point i, the mass mi is estimated using mi− 1
Ei = Pi (mi )
m i ⇔ mi =
Pi (mi ) E i
Pi (mi − 1 ) E i
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 12 / 33
The adaptive method [Schultz et al., 2012]
Principle We assume an initial guess m0
At each point i, the mass mi is estimated using mi− 1
Ei = Pi (mi )
m i ⇔ mi =
Pi (mi ) E i
Pi (mi − 1 ) E i
At each new point i, we have:
mi = Pi (mi − 1 ) E i
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 12 / 33
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) mi− 1 Ei − Pi (mi − 1 )
error on the
energy rat
_e
when using mi− 1
− 1
Introducing a sensitivity parameter βi :
mi = mi− 1 1 + βi P (m mi − 1
(
i i− 1 ) Ei − Pi (mi − 1 )
l −
i− 1 m
1
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 13 / 33
Logic of the sensitivity parameter β [Schultz et al., 2012]
mi = mi− 1 1 + βi P (m
mi − 1
(
i i− 1 ) Ei − Pi (mi − 1 )
l −
i− 1 m
1
Let ∆ E i = 1 gVTAS
(Ei − Pi (mi− 1)
\, β is updated using this rule: mi− 1
if i > 0 and ∆ E i > 0.0001
and ∆ E − ∆ E i
avg < 3 ∆ Eavg
then βi = max (0.205, βi− 1 + 0.05)
else βi = 0.005
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 14 / 33
Logic of the sensitivity parameter β [Schultz et al., 2012]
mi = mi− 1 1 + βi P (m
mi − 1
(
i i− 1 ) Ei − Pi (mi − 1 )
l −
i− 1 m
1
This mechanism increases robustness If ∆ E i repeatedly high in the same order of magnitude, β will increase, strengthening adaptation Isolated low or high ∆ E i has a lower impact on adaptation
The variation is limited
The variation is limited to 2% of the reference mass The estimated mass is kept within 80% and 120% of the reference mass
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 15 / 33
1 Computing the power provided by BADA mass
2 The adaptive method [Schultz et al., 2012]
3 The least square method [Alligier et al., 2012]
4 Results
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 16 / 33
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
At each point i, we have:
Pi (mi ) = E mi
i
However, the different masses are not independant variables ⇒ These equations above cannot be satisfied altogether (in general) Then, we search (m1, . . . , mn) minimizing:
n E (m1, . . . , mn) =
' \ "
i= 1
( P (m ) i i mi
− Ei 2
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 17 / 33
Relationship between the mi
fuel consumption BADA model of the fuel consumption:
dm dt
= −f (T , VTAS, z)
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 18 / 33
Relationship between the mi
fuel consumption BADA model of the fuel consumption:
dm dt
= −f ( T , VTAS , z )
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 18 / 33
Relationship between the mi
fuel consumption BADA model of the fuel consumption:
dm dt
= −f ( T , VTAS , z )
tn mi = mn + f (T (t ), VTAS(t ), z(t ))dt
ti
⇒ mi mn + ' \ " n− 1 f (t
k = i
k + 1 ) + f (t ) k (tk + 1 − tk ) 2
⇒ mi = mn + δi
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 18 / 33
Minimizing this error
The error function can be rewritten:
n E (m1, . . . , mn) = E (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δi
− Ei 2
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 19 / 33
Minimizing this error
The error function can be rewritten:
n E (m1, . . . , mn) = E (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δi
− Ei 2
Minimizing this error can be done by solving:
E /(m) = 0
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 19 / 33
Minimizing this error
The error function can be rewritten:
n E (m1, . . . , mn) = E (mn) =
' \ "
i= 1
( P (m + δ ) i n i 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, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 19 / 33
Minimizing this error
The error function can be rewritten:
n E (m1, . . . , mn) = E (mn) =
' \ "
i= 1
( P (m + δ ) i n i 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, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 19 / 33
Minimizing this error
The original error function:
n E (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δi
− Ei 2
⇒ Numerical issues solving E /(m) = 0
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 20 / 33
Minimizing this error
The original error function:
n E (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δi
− Ei 2
⇒ Numerical issues solving E /(m) = 0 An approximated error function:
n Eapprox (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δavg
− Ei 2
with: δavg = 1 �n δi n i= 1
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 20 / 33
Minimizing this error
The original error function:
n E (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δi
− Ei 2
⇒ Numerical issues solving E /(m) = 0 An approximated error function:
n Eapprox (mn) =
' \ "
i= 1
( P (m + δ ) i n i mn + δavg
− Ei 2
with: δavg = 1 �n δi n i= 1
⇒ Solving E /approx (m) = 0 leads to find roots of a polynomial of degree 4
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 20 / 33
1 Computing the power provided by BADA mass
2 The adaptive method [Schultz et al., 2012]
3 The least square method [Alligier et al., 2012]
4 Results
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 21 / 33
Synthesized aircraft trajectories
Each trajectory BADA 3.9 Trajectories start at altitude 12,000 ft Each 12 seconds, we observe: T ,VTAS, z, dz , dt dt
dVTAS
Trajectories of 4 minutes long (ie. 21 points) Each set of trajectories
Contains 1,000 trajectories of a given aircraft type Distribution of the parameters used to generate trajectories
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 22 / 33
parameter distribution CAS CASref + uniform([−30; 30]) Mach Machref + uniform([−0.03; 0.03]) ∆T uniform([−20; 20])
mass massref × uniform([0.8; 1.2])
Adding a Gaussian noise
Principle Given one variable among T ,VTAS, z, dz , dt dt
dVTAS and a standard deviation σ:
1 We draw errors from the Gaussian distribution For each observation of the choosen variable, the error is added to the observed value
2
For instance, if we have chosen T and σT = 2K : 1 We draw 1, 000 × 21 values from N (0, 2)
These 21, 000 values are added to the 21, 000 observations of T 2
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 23 / 33
Results: Noise on z
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
σHp [ft]
RM
S 10
0 ×
m
estim
ated
− m
actu
al
m
actu
al
÷
[%]
● ●
●
●
●
●
Weight Adaptation A320 A333 B744
Least Square ● A320 ● A333 ● B744
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 24 / 33
Results: Noise on T
● ●
●
●
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
σT [K]
RM
S 10
0 ×
m
estim
ated
− m
actu
al
m
actu
al
÷
[%]
●
●
●
●
●
●
●
Weight Adaptation A320 A333 B744
Least Square ● A320 ● A333 ● B744
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 25 / 33
Results: Noise on VTAS
● ● ●
●
●
●
0 10 20 30 40
0 1
2 3
σVa [kts]
RM
S 10
0 ×
m
estim
ated
− m
actu
al
m
actu
al
÷
[%]
● ●
●
●
● ● ●
●
●
Weight Adaptation A320 A333 B744
Least Square ● A320 ● A333 ● B744
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 26 / 33
Results: Noise on dVTAS dt
● ●
0.00 0.05 0.10 0.15 0.20
0 1
2 3
4
σdVa [kts/s] dt
RM
S 10
0 ×
m
estim
ated
− m
actu
al
m
actu
al
÷
[%]
●
●
● ●
●
●
●
Weight Adaptation A320 A333 B744
Least Square ● A320 ● A333 ● B744
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 27 / 33
Results: Noise on dz dt
●
● ●
0 100 200 300 400 500 600 700
0 1
2 3
4 5
σdHp [ft/min] dt
RM
S 10
0 ×
m
estim
ated
− m
actu
al
m
actu
al
÷
[%]
●
● ●
●
●
●
●●
●
●
Weight Adaptation A320 A333 B744
Least Square ● A320 ● A333 ● B744
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 28 / 33
Conclusion
accuracy Both methods gives a good estimation of the mass Least square method performs slightly better
Beyond accuracy
Adaptive method is simpler to implement than the least square method Adaptive method can use a black box model of the power Adaptive method needs more points to give a good estimate of the mass Adaptive method demands to tune the β sensitivity parameter
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 29 / 33
Further work
Comparing these two methods on real data Use the estimated mass in machine learning techniques
R. Alligier, D. Gianazza, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 30 / 33
Thank you, any questions ?
I
R All1g1er D G1anazza. N Durand (ENAC) Est1mat1on of the Aircraft Mass ATM 2013 31 /33
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.
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, N. Durand (ENAC) Estimation of the Aircraft Mass ATM 2013 32 / 33