Optimal trajectory generation framework for in-flight

Optimal trajectory generation framework for in-flight

Optimal trajectory generation framework for inflight applications
Rafael Fernandes de Oliveira
TCC4 – Autonomous Systems,
Image & Signal Processing
Motivation
the work integrates into the CLEANSKY
european project, which targets significant
reductions in the aviation environmental
impact
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Motivation
Fuel burn at design range
Average fuel
fuelburn
burn for
for new
new jet
jet aircrafts,
aircraft, 1960-2008
Average
1960-2008
100
Annual Improvement
Period
Seat-km T on-km
1960s
2.3%
3.6%
1970s
0.6%
-0.1%
1980s
3.5%
2.5%
1990s
0.7%
0.9%
post-2000 0.0%
0.3%
1960s
1970s
seat-km
1980s
75
1990s
ton-km
post-2000
50
25
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Year
ICCT (2009). "Efficiency Trends for New Commercial Jet Aircraft, 1960 to 2008."
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08
Background
MISSION
PLANNING
planning & Scheduling
resource Allocation
sequencing
time scale ≈ 1 hr
PATH
PLANNING
navigation
weather avoidance
collision avoidance
time scale ≈ 1 min
TRAJECTORY
GENERATION
guidance, waypoint navigation
time scale ≈ 1s
TRAJECTORY
FOLLOWING
control & stabilization
time scale ≈ 0.1s
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Background
aircraft fuel efficient trajectories can be
defined in terms of
• optimal climb and descent profiles
• cruise altitude and speed
• modified for weather and
wind conditions
Fuel cons. vs speed [knots.kg/s]
finding optimal trajectories is a well-known problem
studied since the beginning of aviation
1200
1000
800
600
400
200
0
0.8
4
0.6
3
0.4
2
0.2
Mach
0
1
0
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Altitude [ft]
4
x 10
Trajectory planning using optimal control
Find the controls and system
dynamics that results in the
best possible trajectory …
𝒖 𝒕
⇓
𝒙 𝒕 = 𝒇 𝒙 𝒕 ,𝒖 𝒕 ,𝒕
… while respecting all
necessary constraints
𝑥𝑙𝑏 𝑡 < 𝑥 𝑡 < 𝑥𝑢𝑏 𝑡
𝑓 𝑥 𝑡 , 𝑢 𝑡 , 𝑡 ≈ 𝑚𝑜𝑑𝑒𝑙
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Flight Planning
flight plan is translated into a series of intermediate actions to
be fulfilled by the aircraft and a set of rules to be followed
ground track
flight profile
flight rules
take off from RWY 18
take off from RWY 18
keep speed below
250 KIAS until FL100
cross DF197
retract flaps to 1 at F
speed
keep speed below
Mach 0.82
fly to DF160
retract flaps to 0 at S
speed
fly to ROSIG
climb to cruise
altitude
keep Mach above
buffeting-onset
speed
fly to DF201
fly to DONAB
fly to SOBRA
keep bank angle
under 35°
(25°for TO/LD)
vertical acceleration
under 5ft/s²
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Flight Planning
the set of actions and rules is parsed into a multiple phase
optimal control problem, and solved using a pseudospectral
colocation method
14000
12000
Altitude [ft]
10000
8000
6000
4000
2000
0
0
5
10
15
Distance [NMI]
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20
25
Aircraft equations of motion
full flight
𝜑=
𝑉 ⋅ cos 𝛾 ⋅ cos 𝜓
𝑅0 + ℎ
𝑉 ⋅ cos 𝛾 ⋅ sin 𝜓 ⋅ sec 𝜑
𝜆=
𝑅0 + ℎ
vertical plane
𝑠 = 𝑉 ⋅ cos 𝛾
ℎ = 𝑪𝑹
𝑉=
ℎ = 𝑪𝑹
𝑉=
𝑇ℎ𝑟𝑢𝑠𝑡(Γ) − 𝐷𝑟𝑎𝑔 ℎ, 𝑉, 𝜙
− 𝑔 ⋅ sin(𝛾)
𝑚
𝑇ℎ𝑟𝑢𝑠𝑡(Γ) − 𝐷𝑟𝑎𝑔 ℎ, 𝑉, 𝜙
− 𝑔 ⋅ sin(𝛾)
𝑚
Γ = 𝜞𝑪
𝑚 = −𝐹𝑢𝑒𝑙𝐹𝑙𝑜𝑤 𝑇ℎ𝑟𝑢𝑠𝑡, 𝑉, ℎ
𝜓=
𝑔 sin(𝜙)
⋅
𝑉 cos(𝛾)
Controls:
Γ = 𝜞𝒄
𝜙 = 𝝓𝒄
𝑚 = −𝐹𝑢𝑒𝑙𝐹𝑙𝑜𝑤 𝑇ℎ𝑟𝑢𝑠𝑡, 𝑉, ℎ
𝑪𝑹 - vertical acceleration
𝜞𝒄 - throttle dot
𝝓𝒄 - roll acceleration
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Aircraft performance
config
phase
𝑪𝑫 𝟎
𝑪 𝑫𝟐
𝑪𝑳𝒎𝒂𝒙
clean
cruise
0.0267
0.0387
1.5998
1
initial
climb
0.0230
0.0440
2.2681
1+F
take off
0.0330
0.0410
2.5131
2
approach
0.0380
0.0419
2.8591
full
landing
0.0960
0.0371
3.0776
based on BADA 3.10
data for > 100 aircrafts
3
2.5
L
2
C
extrapolated from flight
data, errors are
minimized for
operational region
1.5
clean
1
1+F
2
full
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
C
D
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0.35
0.4
Aircraft Performance
4
Minimum
x 10
4
3.5
fuel flow is function of
speed and thrust
Altitude [ft]
thrust is function of
altitude
3
2.5
2
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
Mach
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0.7
0.8
Aircraft Performance v2
neural networks are used to map the
cloud of points into a continuously
differentiable function
𝑇ℎ𝑟𝑢𝑠𝑡𝑀𝐶𝑅 = 𝑓 𝑀𝑎𝑐ℎ, 𝐴𝑙𝑡
5
x 10
2.5
Net Thrust [N]
2
1.5
1
Cumulative distribution [%]
0.5
Net Thrust
1
0
0
0
5000
0.5
0.2
0.4
mean = 0.35% ± 0.64%
10000
0.6
15000
0.8
1
Altitude [m]
0
0
0.5
1
1.5
2
2.5
Relative Error [%]
3
3.5
4
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Mach
Aircraft Performance v2
4
Minimum
x 10
4
4
x 10
4
3.5
3
2
Minimum
3
1.5
2.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mach
Altitude [ft]
Altitude [ft]
3.5
2.5
2
1.5
1
better representation
of fuel consumption
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
Mach
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0.7
0.8
Optimization criteria
TIME
FUEL
NOX
CO
HC
𝐽𝑇𝑖𝑚𝑒 𝑡 = 𝑡𝑓
𝑡𝑓
𝐽𝐹𝑢𝑒𝑙 𝑡, 𝒙, 𝒖 =
𝐽𝑁𝑂𝑋 𝑡, 𝒙, 𝒖 =
𝐽𝐶𝑂 𝑡, 𝒙, 𝒖 =
𝐽𝐻𝐶 𝑡, 𝒙, 𝒖 =
𝑡0
𝑡𝑓
𝑡0
𝑡𝑓
𝑓𝑒𝑛𝑔𝑖𝑛𝑒 𝒙 𝑡 , 𝒖 𝑡
𝐸𝐼𝑁𝑂𝑋 𝒙 𝑡
𝐸𝐼𝐶𝑂 𝒙 𝑡
𝑡0
𝑡𝑓
𝑡0
𝐸𝐼𝐻𝐶 𝒙 𝑡
𝑑𝑡
⋅ 𝑓𝑒𝑛𝑔𝑖𝑛𝑒 𝒙 𝑡 , 𝒖 𝑡
𝑑𝑡
⋅ 𝑓𝑒𝑛𝑔𝑖𝑛𝑒 𝒙 𝑡 , 𝒖 𝑡
𝑑𝑡
⋅ 𝑓𝑒𝑛𝑔𝑖𝑛𝑒 𝒙 𝑡 , 𝒖 𝑡
𝑑𝑡
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Aircraft Emissions
0.15
NOx [kg/s]
3
2
1
0
150
0.1
0.05
0
150
400
100
50
Thrust [KN]
0 0
400
100
200
50
Thrust [KN]
FL
200
0 0
FL
-3
x 10
1
HC [kg/s]
0.01
CO [kg/s]
AEM3 estimates the
emission index, given in
grams of pollutant per
kilogram of burnt fuel (g/kg),
corrected for atmospheric
conditions and engine
setting
Fuel Flow [kg/s]
Advanced Emission Model 3
(AEM3) from EUROCONTROL
0.005
0
150
400
100
50
Thrust [KN]
200
0 0
FL
0.5
0
150
400
100
50
Thrust [KN]
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200
0 0
FL
Optimization criteria
NOISE
on station
𝑁
𝐽𝐿𝐴𝑚𝑎𝑥 𝑡, 𝒙 𝑡 , 𝒖 𝑡
= log10
𝐿𝐴𝑚𝑎𝑥𝑖
10 10
𝑖=1
𝑁
AWAKENINGS
𝐽𝑁𝑜𝑖𝑠𝑒 𝑡, 𝒙 𝑡 , 𝒖 𝑡
=
𝑃𝑜𝑝𝑖 ⋅ 0.0087 ⋅ 𝑆𝐸𝐿 𝑡, 𝒙 𝑡 , 𝒖 𝑡
− 50.5 dB
𝑖=1
from the 1997 study by the Federal Interagency
Committee on Aviation Noise (FICAN)
models the overall sleep disturbance related to
the Indoor Sound Exposure Level (SEL)
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1.79
Aircraft Noise
Doc.29 model used to
calculate SEL and LAmax
𝑙
ℎ
𝛽
𝑑
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Population Dataset
20k population
grid converted
into 500 clusters
Gallego F.J., 2010, A population density
grid of the European Union,
Population and Environment
Page 18
October 19th 2011
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Optimization criteria
regularization of the solution to avoid bang-bang control inputs
and numerical noise
𝐽 𝑡, 𝒙 𝑡 , 𝒖 𝑡
= … + 𝛼𝐶𝑅 ⋅
𝑡𝑓
𝑡0
2
𝐶𝑅 𝑑𝑡 + 𝛼Γ ⋅
𝑡𝑓
𝑡0
2
Γ𝑐 𝑑𝑡 + 𝛼𝜙 ⋅
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𝑡𝑓
𝑡0
2
𝜙𝑐 𝑑𝑡
Results
fixed range: 400 NMI
targets: emissions, fuel and time
4
Altitude [ft]
4
x 10
3
2
1
time
0
0
50
100
fuel
150
nox
200
co
250
hc
300
Distance [NMI]
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350
400
Distance [NMI]
TAS [knots]
Results
500
400
300
200
100
0
time
50
100
fuel
150
nox
200
co
250
300
Distance [NMI]
time [s] fuel [kg] NOX [kg]
time
3224
3324
130,00
fuel
3968
2716
95,90
NOX
4580
3042
84,90
CO
3698
2916
112,20
HC
3379
3384
126,90
CO [kg]
12,30
19,60
24,50
10,80
11,90
HC [kg]
2,62
4,04
5,05
3,72
2,57
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hc
350
400
Departure from RWY 18C at AMSTERDAM SCHIPHOL
53% reduction in awakenings and 15%
higher fuel consumption
Hoofddorp
Amsterdam
Bussum
Aalsmeer
Uithoorn
Nieuw-Vennep
Hilversum
fuel
awakenings
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EDDF RWY 18 departure
558 A320 departures
from RWY 18 over
one year period
real trajectories
recorded from ADS-B
tracking
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EDDF RWY 18 departure
Assumptions
• initial mass estimated from distance to final destination plus
extra fuel and 85% PAX occupation
• iterative inverse dynamics calculation to estimate necessary
thrust, yaw and bank angles
• aircraft configuration based on S and F speeds
• wind taken from NOTAM reports for time of departure
• same ground track for optimized trajectories, only optimize
vertical profile and speed schedule
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savings
EDDF RWY 18 departure
Altitude [ft]
15000
awakenings
NOX
fuel
29.4%
22.9%
11.3%
10000
5000
0
0
5
10
15
20
25
Distance [NMI]
400
TAS [knots]
350
300
original
time
fuel
NO
250
200
150
0
X
awakenings
5
10
15
20
Distance [NMI]
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25
TAS [knots]
EDDF RWY 18 departure - awakenings
sleep disturbance
(no. people)
400
original
1819
350
time
1423
300
fuel
1420
NOX
1286
awakenings
1284
original
time
fuel
NOX
250
200
awakenings
150
0
5
10
15
Distance [NMI]
20
25
and then cut thrust
while flying over
people
trajectory climbs
faster while far from
population clusters
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EDDF RWY 18 departure – mean potential for savings
awakenings
33.9%
NOX
fuel
25.1%
10.2%
from an average of 558 flights
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Conclusions
Optimization based approaches have the potential to reduce environmental
impact of flying from existing aircrafts
Translation of flight problem into an OCP, to be solved using existing optimization
tools and framework (GPOPS/SNOPT/WORHP): no need to build custom
solver, well-studied problem, continued improvement in solvers
Challenges:
how to best implement the interface between Pilot/ATM and the trajectory
optimization and to integrate a self-defined trajectory into the ATC scenario
how to communicate changes and receive updates (the more information we
have, the closer to the real optimal we can be)
how to certify
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Thanks for your attention!
questions?
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