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Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 Tokyo D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA February, 21 2013 D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation Universit Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 Tokyo February, 21 2013 1 / 146
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Page 1: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Mathematical Models for Aircraft Trajectory Design : ASurvey

EIWAC 2013 Tokyo

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron

Applied Mathematics Laboratory (MAIAA)French Civil Aviation University

Toulouse, FranceSchool of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, USA

February, 21 2013

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 1 / 146

Page 2: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 2 / 146

Page 3: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 2 / 146

Page 4: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 2 / 146

Page 5: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 2 / 146

Page 6: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 2 / 146

Page 7: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 3 / 146

Page 8: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectory Models

Aircraft Trajectory Features

Dimension Reduction Approaches

Front Propagation Approaches

Optimal Control Approaches

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 4 / 146

Page 9: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Classical representation

t x y z

t=a

t=b

Trajectory data is expressed as an ordered list of plots (no aircraftdynamics in such representation)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 5 / 146

Page 10: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectories as functional data

Trajectories are infinite dimension mathematical objects

Trajectories as mappings

t0 t1

γ( )t

Intuitive approach : a trajectory maps a bounded time interval [t0, t1]to the state space (R3 or R6).

Smoothness assumptions are made for trajectories (C 2).

Trajectories as shapes

The paths flown by aircraft are considered as curves in R3.

Such time independant trajectories are called shapes.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 6 / 146

Page 11: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectories as functional data

Trajectories are infinite dimension mathematical objects

Trajectories as mappings

t0 t1

γ( )t

Intuitive approach : a trajectory maps a bounded time interval [t0, t1]to the state space (R3 or R6).

Smoothness assumptions are made for trajectories (C 2).

Trajectories as shapes

The paths flown by aircraft are considered as curves in R3.

Such time independant trajectories are called shapes.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 6 / 146

Page 12: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectories as functional data

Trajectories are infinite dimension mathematical objects

Trajectories as mappings

t0 t1

γ( )t

Intuitive approach : a trajectory maps a bounded time interval [t0, t1]to the state space (R3 or R6).

Smoothness assumptions are made for trajectories (C 2).

Trajectories as shapes

The paths flown by aircraft are considered as curves in R3.

Such time independant trajectories are called shapes.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 6 / 146

Page 13: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectories as functional data

Trajectories are infinite dimension mathematical objects

Trajectories as mappings

t0 t1

γ( )t

Intuitive approach : a trajectory maps a bounded time interval [t0, t1]to the state space (R3 or R6).

Smoothness assumptions are made for trajectories (C 2).

Trajectories as shapes

The paths flown by aircraft are considered as curves in R3.

Such time independant trajectories are called shapes.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 6 / 146

Page 14: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectories as functional data

Trajectories are infinite dimension mathematical objects

Trajectories as mappings

t0 t1

γ( )t

Intuitive approach : a trajectory maps a bounded time interval [t0, t1]to the state space (R3 or R6).

Smoothness assumptions are made for trajectories (C 2).

Trajectories as shapes

The paths flown by aircraft are considered as curves in R3.

Such time independant trajectories are called shapes.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 6 / 146

Page 15: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Features

Notations

t=a

t=b

Trajectory ~γ : ~γ[a, b]→ E ([a, b] time interval, E : R3 or R6)

Trajectory length l(~γ) =∫ ba ‖~γ

′(t)‖dt

Parametrization by arclength : s(a, b)→ (0, l(~γ))s(t) =

∫ ta ‖~γ

′(x)‖dx (s

′(t) = ‖~γ′(t)‖ > 0 ∀t ∈ (a, b))

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 7 / 146

Page 16: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Features

Notations

t=a

t=b

Trajectory ~γ : ~γ[a, b]→ E ([a, b] time interval, E : R3 or R6)

Trajectory length l(~γ) =∫ ba ‖~γ

′(t)‖dt

Parametrization by arclength : s(a, b)→ (0, l(~γ))s(t) =

∫ ta ‖~γ

′(x)‖dx (s

′(t) = ‖~γ′(t)‖ > 0 ∀t ∈ (a, b))

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 7 / 146

Page 17: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Features

Notations

t=a

t=b

Trajectory ~γ : ~γ[a, b]→ E ([a, b] time interval, E : R3 or R6)

Trajectory length l(~γ) =∫ ba ‖~γ

′(t)‖dt

Parametrization by arclength : s(a, b)→ (0, l(~γ))s(t) =

∫ ta ‖~γ

′(x)‖dx (s

′(t) = ‖~γ′(t)‖ > 0 ∀t ∈ (a, b))

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 7 / 146

Page 18: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Features

Notations

t=a

t=b

Trajectory ~γ : ~γ[a, b]→ E ([a, b] time interval, E : R3 or R6)

Trajectory length l(~γ) =∫ ba ‖~γ

′(t)‖dt

Parametrization by arclength : s(a, b)→ (0, l(~γ))s(t) =

∫ ta ‖~γ

′(x)‖dx (s

′(t) = ‖~γ′(t)‖ > 0 ∀t ∈ (a, b))

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 7 / 146

Page 19: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Unit tangent vector

t=a

t=bτ

~τ(s) = ~γ′(s)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 8 / 146

Page 20: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Unit tangent vector

t=a

t=bτ

~τ(s) = ~γ′(s)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 8 / 146

Page 21: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Curvature

K (s) = ‖~γ′′(s)‖ = ‖~γ′ (t)∧~γ′′ (t)‖‖~γ′ (t)‖3

Aircraft trajectories have piecewise constant curvature.

Unit normal vector

t=a

t=bτ

ν

~ν(s) = ~γ′′

(s)K(s)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 9 / 146

Page 22: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Curvature

K (s) = ‖~γ′′(s)‖ = ‖~γ′ (t)∧~γ′′ (t)‖‖~γ′ (t)‖3

Aircraft trajectories have piecewise constant curvature.

Unit normal vector

t=a

t=bτ

ν

~ν(s) = ~γ′′

(s)K(s)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 9 / 146

Page 23: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Curvature

K (s) = ‖~γ′′(s)‖ = ‖~γ′ (t)∧~γ′′ (t)‖‖~γ′ (t)‖3

Aircraft trajectories have piecewise constant curvature.

Unit normal vector

t=a

t=bτ

ν

~ν(s) = ~γ′′

(s)K(s)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 9 / 146

Page 24: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Curvature

K (s) = ‖~γ′′(s)‖ = ‖~γ′ (t)∧~γ′′ (t)‖‖~γ′ (t)‖3

Aircraft trajectories have piecewise constant curvature.

Unit normal vector

t=a

t=bτ

ν

~ν(s) = ~γ′′

(s)K(s)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 9 / 146

Page 25: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Torsion

.

t=a

t=bτ

ν

β

~β(s) = ~τ(s) ∧ ~ν(s) ~β′(s) = T (s).~ν(s)

The real number T (s) is called the torsion of the curve at s andrepresents an obstruction for the curve to be planar.

T (t) = −det(~γ′(t),~γ

′′(t),~γ

′′′(t))

‖~γ′ (t)∧~γ′′ (t)‖2

Aircraft have piecewise constant torsion mainly in terminal area.

All the previous derivations rely on the fact that the first threederivatives of the trajectory are available.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 10 / 146

Page 26: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Torsion

.

t=a

t=bτ

ν

β

~β(s) = ~τ(s) ∧ ~ν(s) ~β′(s) = T (s).~ν(s)

The real number T (s) is called the torsion of the curve at s andrepresents an obstruction for the curve to be planar.

T (t) = −det(~γ′(t),~γ

′′(t),~γ

′′′(t))

‖~γ′ (t)∧~γ′′ (t)‖2

Aircraft have piecewise constant torsion mainly in terminal area.

All the previous derivations rely on the fact that the first threederivatives of the trajectory are available.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 10 / 146

Page 27: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Torsion

.

t=a

t=bτ

ν

β

~β(s) = ~τ(s) ∧ ~ν(s) ~β′(s) = T (s).~ν(s)

The real number T (s) is called the torsion of the curve at s andrepresents an obstruction for the curve to be planar.

T (t) = −det(~γ′(t),~γ

′′(t),~γ

′′′(t))

‖~γ′ (t)∧~γ′′ (t)‖2

Aircraft have piecewise constant torsion mainly in terminal area.

All the previous derivations rely on the fact that the first threederivatives of the trajectory are available.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 10 / 146

Page 28: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Torsion

.

t=a

t=bτ

ν

β

~β(s) = ~τ(s) ∧ ~ν(s) ~β′(s) = T (s).~ν(s)

The real number T (s) is called the torsion of the curve at s andrepresents an obstruction for the curve to be planar.

T (t) = −det(~γ′(t),~γ

′′(t),~γ

′′′(t))

‖~γ′ (t)∧~γ′′ (t)‖2

Aircraft have piecewise constant torsion mainly in terminal area.

All the previous derivations rely on the fact that the first threederivatives of the trajectory are available.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 10 / 146

Page 29: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Aircraft Trajectories Feature

Torsion

.

t=a

t=bτ

ν

β

~β(s) = ~τ(s) ∧ ~ν(s) ~β′(s) = T (s).~ν(s)

The real number T (s) is called the torsion of the curve at s andrepresents an obstruction for the curve to be planar.

T (t) = −det(~γ′(t),~γ

′′(t),~γ

′′′(t))

‖~γ′ (t)∧~γ′′ (t)‖2

Aircraft have piecewise constant torsion mainly in terminal area.

All the previous derivations rely on the fact that the first threederivatives of the trajectory are available.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 10 / 146

Page 30: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Trajectory Models

Aircraft Trajectory Features

Dimension Reduction Approaches

Front Propagation Approaches

Optimal Control Approaches

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 11 / 146

Page 31: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Explicit vs Implicit

Explicit

y = f (x)

Example 2D line y = a.x + bA curve may not have an explicit representation

Implicit

f (x , y) = 0

Example 2D circle x2 + y 2 − r 2 = 0

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 12 / 146

Page 32: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Parametric Form

Expresses the value of each spatial variables for points in terms of anindependent parameter u.

~p(u) =

x(u)y(u)z(u)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 13 / 146

Page 33: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Parametric Polynomial Curve

Consider a curve

~p(u) =

x(u)y(u)z(u)

A polynomial parametric curve of degree n is of the form :

~p(u) =n∑

k=0

~ck .uk

where each ~ck has independent x , y , z components : ~ck = [ckx , cky , ckz ]T

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 14 / 146

Page 34: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Advantages of the Parametric Polynomial Curve

Just needs to save a few control points

Local control of shape

Smoothness and continuity

Ability to evaluate derivatives

Stability

Ease of rendering

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 15 / 146

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Lagrangian Interpolation

Given n + 1 real numbers yi ,0 ≤ i ≤ n, and n + 1 distinct real numbersx0 < x1 < ... < xn, Lagrange polynomial of degree n associated with xiand yi is a polynomial of degree n solving the interpolation problem :

pn(xi ) = yi , 0 ≤ i ≤ n

Solution :

Ln(x) =n∑

i=0

f (xi )li (x)

where

li (x) =∏j 6=i

(x − xj)

(xi − xj)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 16 / 146

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Hermite Interpolation

Hermite interpolation generalizes Lagrange interpolation by fitting apolynomial to a function f that not only interpolates f at each knot butalso interpolates a given number of consecutive derivatives of f at eachknot. [

∂jH(x)

∂x j

]x=xi

=

[∂j f (x)

∂x j

]x=xi

for all j = 0, 1, ...,m and i = 1, 2, ..., k

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 17 / 146

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Runge phenomenon

Interpolation with high degree polynomial is risky...

Solution : Piecewise interpolation

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 18 / 146

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Piecewise Linear Interpolation

The simplest one

x0 xi xi+1xi−1 xn

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 19 / 146

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Piecewise Linear Interpolation

Given n + 1 real numbers yi ,0 ≤ i ≤ n, and n + 1 distinct real numbersx0 < x1 < ... < xn, we consider the n linear curves li (x) = aix + bi on theintervals [xi , xi+1] for i = 0, ...n − 1.

each li (x) has to connect two points (xi , yi ),(xi+1, yi+1)

yi = aixi + bixi yi+1 = aixi+1 + bixi+1

The resulting curves is not derivative.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 20 / 146

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Piecewise Quadratic Interpolation

x0 xi xi+1xi−1 xn

Initial slope

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 21 / 146

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Piecewise Quadratic Interpolation

We consider the n quadratic curves qi (x) = aix2 + bix + ci on the

intervals [xi , xi+1] for i = 0, ...n − 1.

Each qi (x) has to connect two points ((xi , yi ),(xi+1, yi+1)

yi = aix2i + bixi + ci

yi+1 = aix2i+1 + bixi+1 + ci

On each point the derivative of the previous quadratic has to be equalto the derivative of the next one.

2ai + bi = 2ai−1 + bi−1

For the first segment the term 2ai−1 + bi−1 is arbitrarily chosen. (thisaffects the rest of the curve).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 22 / 146

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Piecewise Quadratic Interpolation

We consider the n quadratic curves qi (x) = aix2 + bix + ci on the

intervals [xi , xi+1] for i = 0, ...n − 1.

Each qi (x) has to connect two points ((xi , yi ),(xi+1, yi+1)

yi = aix2i + bixi + ci

yi+1 = aix2i+1 + bixi+1 + ci

On each point the derivative of the previous quadratic has to be equalto the derivative of the next one.

2ai + bi = 2ai−1 + bi−1

For the first segment the term 2ai−1 + bi−1 is arbitrarily chosen. (thisaffects the rest of the curve).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 22 / 146

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Piecewise Quadratic Interpolation

We consider the n quadratic curves qi (x) = aix2 + bix + ci on the

intervals [xi , xi+1] for i = 0, ...n − 1.

Each qi (x) has to connect two points ((xi , yi ),(xi+1, yi+1)

yi = aix2i + bixi + ci

yi+1 = aix2i+1 + bixi+1 + ci

On each point the derivative of the previous quadratic has to be equalto the derivative of the next one.

2ai + bi = 2ai−1 + bi−1

For the first segment the term 2ai−1 + bi−1 is arbitrarily chosen. (thisaffects the rest of the curve).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 22 / 146

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Piecewise Cubic Interpolation

Also called Hermite Cubic Interpolation

xi xi+1xi−1

iyyi−1

yi+1

yi+2

xi+2

h

slope in islope in i+1

Ci (x) = aix3 + bix

2 + cix + di

Ci (xi ) = yi Ci (xi+1) = yi+1

C ′i (xi ) = y ′i =yi+1−yi−1

xi+1−xi−1C ′i (xi+1) = y ′i+1 = yi+2−yi

xi+2−xi

Moving a point do not affect all the curve

The curve is C 1 but not C 2.D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 23 / 146

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Curvature radius

R =1 +

(df (x)dx

) 32

|(d2f (x)dx2

)|

In order to have a continuous curverture one must force curves to be C 2.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 24 / 146

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Cubic Spline Interpolation

Piecewise cubic interpolation

Developped by General Motor in the 1950s.

xi−1

yi−1 yi+2

xi+1

iy

xi

yi+1

S i (t)

xi+2

Si (xi ) = yi Si (xi+1) = yi+1

S′i (xi ) = S

′i−1(xi+1) S

′i (xi+1) = S

′i+1(xi+1)

S′′i (xi ) = S

′′i−1(xi+1) S

′′i (xi+1) = S

′′i+1(xi+1)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 25 / 146

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Cubic Spline Interpolation

Si (x) for x ∈ [xi , xi+1]

Si (x) = σi6 .

(xi+1−x)3

xi+1−xi + σi+1

6 . (x−xi )3

xi+1−xi+ yi .

xi+1−xxi+1−xi −

σi6 .(xi+1 − xi )(xi+1 − x)

+ yi+1.x−xi

xi+1−xi −σi+1

6 .(xi+1 − xi )(x − xi )

where

σi =d2Si (x)

dx2

Such spline is also called natural spline because it represents the curve ofa metal spline constrained to interpolate some given points.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 26 / 146

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Bezier Approximation Curve

Bezier curves were first developped by automobile designers todescribe the shape of exterior car panels in the 1960s and 70s.

Given points ~P0 and ~P1, a linear Bezier curve is simply a straight linebetween those two points. The curve is given by

B(t) = ~P0 + t(~P1 − ~P0) = (1− t)~P0 + t~P1 , t ∈ [0, 1]

Bezier Curve with 2 points

P0

P1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 27 / 146

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Bezier Approximation Curve

Bezier curves were first developped by automobile designers todescribe the shape of exterior car panels in the 1960s and 70s.

Given points ~P0 and ~P1, a linear Bezier curve is simply a straight linebetween those two points. The curve is given by

B(t) = ~P0 + t(~P1 − ~P0) = (1− t)~P0 + t~P1 , t ∈ [0, 1]

Bezier Curve with 2 points

P0

P1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 27 / 146

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Bezier Approximation Curve

Bezier curves were first developped by automobile designers todescribe the shape of exterior car panels in the 1960s and 70s.

Given points ~P0 and ~P1, a linear Bezier curve is simply a straight linebetween those two points. The curve is given by

B(t) = ~P0 + t(~P1 − ~P0) = (1− t)~P0 + t~P1 , t ∈ [0, 1]

Bezier Curve with 2 points

P0

P1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 27 / 146

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Cubic Bezier curves

BÉZIER CURVE

P3

P2

P1

P0

P0P1

P2

Four points ~P0, ~P1, ~P2 and ~P3 in the plane or in higher-dimensionalspace define a cubic Bezier curve.

The curve starts at ~P0 going towards ~P1 and arrives at ~P3 comingfrom the direction of ~P2. Usually, it will not pass through ~P1 or ~P2 ;these points are only there to provide directional information.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 28 / 146

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Cubic Bezier curves

BÉZIER CURVE

P3

P2

P1

P0

P0P1

P2

Four points ~P0, ~P1, ~P2 and ~P3 in the plane or in higher-dimensionalspace define a cubic Bezier curve.

The curve starts at ~P0 going towards ~P1 and arrives at ~P3 comingfrom the direction of ~P2. Usually, it will not pass through ~P1 or ~P2 ;these points are only there to provide directional information.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 28 / 146

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Cubic Bezier curves

The polygon formed by connecting the Bezier points with lines,starting with ~P0 and finishing with ~Pn, is called the Bezier polygon(or control polygon).

The convex hull of the Bezier polygon contains the Bezier curve.

The start (end) of the curve is tangent to the first (last) section ofthe Bezier polygon.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 29 / 146

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Cubic Bezier curves

The polygon formed by connecting the Bezier points with lines,starting with ~P0 and finishing with ~Pn, is called the Bezier polygon(or control polygon).

The convex hull of the Bezier polygon contains the Bezier curve.

The start (end) of the curve is tangent to the first (last) section ofthe Bezier polygon.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 29 / 146

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Cubic Bezier curves

The polygon formed by connecting the Bezier points with lines,starting with ~P0 and finishing with ~Pn, is called the Bezier polygon(or control polygon).

The convex hull of the Bezier polygon contains the Bezier curve.

The start (end) of the curve is tangent to the first (last) section ofthe Bezier polygon.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 29 / 146

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Cubic Bezier curves

The explicit form of the curve is :

B(t) = (1− t)3~P0 + 3(1− t)2t~P1 + 3(1− t)t2~P2 + t3~P3 , t ∈ [0, 1].

B(t) =n∑

i=0

bi ,n(t)~Pi , t ∈ [0, 1]

where the polynomials

bi ,n(t) =

(n

i

)t i (1− t)n−i , i = 0, . . . n

are known as Bernstein basis polynomials of degree n.

A Bezier curve defined with n + 1 control points is of degree n.

So if there are many points one has to manipulate polynoms with highdegree ⇒ Basis-Splines

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 30 / 146

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B-Splines

Powerful tool for generating curves with many control points, B stands forbasis.

A single B-spline can specify a long complicated curve

B-splines can be designed with sharp bends and even “corners”

B-Spline interpolation is preferred over polynomial interpolationbecause the interpolation error can be made small even when usinglow degree polynomials for the spline.

Spline interpolation avoids the problem of Runge’s phenomenonwhich occurs when interpolating between equidistant points with highdegree polynomials.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 31 / 146

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B-Splines

Powerful tool for generating curves with many control points, B stands forbasis.

A single B-spline can specify a long complicated curve

B-splines can be designed with sharp bends and even “corners”

B-Spline interpolation is preferred over polynomial interpolationbecause the interpolation error can be made small even when usinglow degree polynomials for the spline.

Spline interpolation avoids the problem of Runge’s phenomenonwhich occurs when interpolating between equidistant points with highdegree polynomials.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 31 / 146

Page 59: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

B-Splines

Powerful tool for generating curves with many control points, B stands forbasis.

A single B-spline can specify a long complicated curve

B-splines can be designed with sharp bends and even “corners”

B-Spline interpolation is preferred over polynomial interpolationbecause the interpolation error can be made small even when usinglow degree polynomials for the spline.

Spline interpolation avoids the problem of Runge’s phenomenonwhich occurs when interpolating between equidistant points with highdegree polynomials.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 31 / 146

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B-Splines

Powerful tool for generating curves with many control points, B stands forbasis.

A single B-spline can specify a long complicated curve

B-splines can be designed with sharp bends and even “corners”

B-Spline interpolation is preferred over polynomial interpolationbecause the interpolation error can be made small even when usinglow degree polynomials for the spline.

Spline interpolation avoids the problem of Runge’s phenomenonwhich occurs when interpolating between equidistant points with highdegree polynomials.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 31 / 146

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Uniform B-Splines of Degree Zero

We consider a node vector ~T = t0, t1, ..., tn with t0 ≤ t1 ≤, ...,≤ tn andn points ~Pi .One want to build a curve ~X0(t) such that

~X0(ti ) = ~Pi

⇒ ~X0(t) = ~Pi ∀t ∈ [ti , ti+1]

~X0(t) =∑i

Bi ,0(t).~Pi

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 32 / 146

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Uniform B-Splines of Degree Zero

it =4

t i+1t i

B (t)i,0

X (t)0

0 1 2 3 5 6 7 8

Pi

9

1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 33 / 146

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Uniform B-Splines of Degree One

We are searching for a piecewise linear approximation :

~X1(t) =

(1− t − ti

ti+1 − ti

)~Pi−1 +

(1− t − ti

ti+1 − ti

)~Pi ∀t ∈ [ti , ti+1]

~X1(t) =∑i

Bi ,1(t).~Pi

it =4

t i+1t i

0 1 2 3 5 6 7 8

Pi

9

X (t)1

t i−1

i−1,1B (t)1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 34 / 146

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Uniform B-Splines of Degree Three

Developped at Boeing in the 70s.

One of the simplest and most useful cases of B-splines

Degree 3 B-Spline with n + 1 control points :

~X3(t) =n∑

i=0

Bi ,3(t).~Pi 3 ≤ t ≤ n + 1

For degree 3,Bi ,3(t) = 0 if t ≤ ti or t ≥ ti+4 So

~X3(t) =

j∑i=j−3

Bi ,3(t).~Pi t ∈ [j , j + 1], 3 ≤ j ≤ n

When a single control point Pi is moved, only the portion of thecurve ~X3(t) with ti < t < ti+4 is changed ⇒ local control.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 35 / 146

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Uniform B-Splines of Degree Three

Developped at Boeing in the 70s.

One of the simplest and most useful cases of B-splines

Degree 3 B-Spline with n + 1 control points :

~X3(t) =n∑

i=0

Bi ,3(t).~Pi 3 ≤ t ≤ n + 1

For degree 3,Bi ,3(t) = 0 if t ≤ ti or t ≥ ti+4 So

~X3(t) =

j∑i=j−3

Bi ,3(t).~Pi t ∈ [j , j + 1], 3 ≤ j ≤ n

When a single control point Pi is moved, only the portion of thecurve ~X3(t) with ti < t < ti+4 is changed ⇒ local control.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 35 / 146

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Uniform B-Splines of Degree Three

Developped at Boeing in the 70s.

One of the simplest and most useful cases of B-splines

Degree 3 B-Spline with n + 1 control points :

~X3(t) =n∑

i=0

Bi ,3(t).~Pi 3 ≤ t ≤ n + 1

For degree 3,Bi ,3(t) = 0 if t ≤ ti or t ≥ ti+4 So

~X3(t) =

j∑i=j−3

Bi ,3(t).~Pi t ∈ [j , j + 1], 3 ≤ j ≤ n

When a single control point Pi is moved, only the portion of thecurve ~X3(t) with ti < t < ti+4 is changed ⇒ local control.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 35 / 146

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Uniform B-Splines of Degree Three

Developped at Boeing in the 70s.

One of the simplest and most useful cases of B-splines

Degree 3 B-Spline with n + 1 control points :

~X3(t) =n∑

i=0

Bi ,3(t).~Pi 3 ≤ t ≤ n + 1

For degree 3,Bi ,3(t) = 0 if t ≤ ti or t ≥ ti+4 So

~X3(t) =

j∑i=j−3

Bi ,3(t).~Pi t ∈ [j , j + 1], 3 ≤ j ≤ n

When a single control point Pi is moved, only the portion of thecurve ~X3(t) with ti < t < ti+4 is changed ⇒ local control.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 35 / 146

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Uniform B-Splines of Degree Three

The basis functions have the following properties :

They are translates of each other i.e Bi ,3(t) = B0,3(t − i)

They are piecewise degree three polynomial

Partition of unity∑

i Bi (t) = 1 for 3 ≤ t ≤ n + 1

The functions ~Xi (t) are of degree 3 for any set of control points

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 36 / 146

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Uniform B-Splines of Degree Three

The basis functions have the following properties :

They are translates of each other i.e Bi ,3(t) = B0,3(t − i)

They are piecewise degree three polynomial

Partition of unity∑

i Bi (t) = 1 for 3 ≤ t ≤ n + 1

The functions ~Xi (t) are of degree 3 for any set of control points

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 36 / 146

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Uniform B-Splines of Degree Three

The basis functions have the following properties :

They are translates of each other i.e Bi ,3(t) = B0,3(t − i)

They are piecewise degree three polynomial

Partition of unity∑

i Bi (t) = 1 for 3 ≤ t ≤ n + 1

The functions ~Xi (t) are of degree 3 for any set of control points

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 36 / 146

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Uniform B-Splines of Degree Three

The basis functions have the following properties :

They are translates of each other i.e Bi ,3(t) = B0,3(t − i)

They are piecewise degree three polynomial

Partition of unity∑

i Bi (t) = 1 for 3 ≤ t ≤ n + 1

The functions ~Xi (t) are of degree 3 for any set of control points

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 36 / 146

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Uniform B-Splines of Degree Three

Bi−2,3(t) =1

h

(t − ti−2)3 if t ∈ [ti−2, ti−1]h3 + 3h2(t − ti−1) + 3h(t − ti−1)2 − 3(t − ti−1)3

if t ∈ [ti−1, ti ]h3 + 3h2(ti+1 − t) + 3h(ti+1 − t)2 − 3(ti+1 − t)3

if t ∈ [ti , ti+1](ti+2 − t)3 if t ∈ [ti+1, ti+2]0 otherwise

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 37 / 146

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Uniform B-Splines of Degree Three

B (t)2,3 3,3

B (t)B (t)1,3

1 2 3 54 6 7 8

B (t)4,3

2/3

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 38 / 146

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Homotopy Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 39 / 146

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Homotopy Trajectory Design

If we consider two (or more) references trajectories (γ1(t), γ2(t))joining thesame origine destination pair (past flown trajectories may be considered),one can create a new trajectory γ(α, t) by using an homotopy :

γ(α, t) =

γ(0, t) = γ1(t)γ(1, t) = γ2(t)

γ(α, t) = (1− α)γ1(t) + αγ2(t)

γ2

γ1

γα

B

A

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 40 / 146

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Functionnal Principal Component Analysis

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 41 / 146

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Functionnal Principal Component Analysis

Used for Stochastic Signal Compression (movies, image, voice)

The goal of principal component analysis is to compute the mostmeaninfugful basis to re-express a noisy data set (maximizeSNR,minimize redundancy).

If speed is suitable one must work in Sobolev space

Extraction of the Probability Density Function of PCA coefficients inorder to be able to randomly generate “flyable trajectories”.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 42 / 146

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Functionnal Principal Component Analysis

Used for Stochastic Signal Compression (movies, image, voice)

The goal of principal component analysis is to compute the mostmeaninfugful basis to re-express a noisy data set (maximizeSNR,minimize redundancy).

If speed is suitable one must work in Sobolev space

Extraction of the Probability Density Function of PCA coefficients inorder to be able to randomly generate “flyable trajectories”.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 42 / 146

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Functionnal Principal Component Analysis

Used for Stochastic Signal Compression (movies, image, voice)

The goal of principal component analysis is to compute the mostmeaninfugful basis to re-express a noisy data set (maximizeSNR,minimize redundancy).

If speed is suitable one must work in Sobolev space

Extraction of the Probability Density Function of PCA coefficients inorder to be able to randomly generate “flyable trajectories”.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 42 / 146

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Functionnal Principal Component Analysis

Used for Stochastic Signal Compression (movies, image, voice)

The goal of principal component analysis is to compute the mostmeaninfugful basis to re-express a noisy data set (maximizeSNR,minimize redundancy).

If speed is suitable one must work in Sobolev space

Extraction of the Probability Density Function of PCA coefficients inorder to be able to randomly generate “flyable trajectories”.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 42 / 146

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Optimization Approach

All the previous representations may be used in the following process

Reconstruction

Trajectory

Trajectory

Evaluation

Optimization

γ

X (parameters)

y=f(X)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 43 / 146

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Trajectory Models

Aircraft Trajectory Features

Dimension Reduction Approaches

Front Propagation Approaches

Optimal Control Approaches

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 44 / 146

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Method

Propagating front methods : General principle

Methods introduced by J.A. Sethian.

Figure: Curve propagating with speedF in normal direction.

Goal :

Track the motion of a front as itevolves.

How ?

We caracterize the position of thefront by the computation of thearrival time u(x , y) at each point(x , y).

⇒ Map of isocost.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 45 / 146

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Method

Propagating front methods

Fast Marching :

→ Isotropic problemThe speed of propagation F is thesame in any directions, it only de-pends on the position.

Ordered Upwind :

→ Anisotropic problemThe speed of propagation dependson position and direction of thepropagation.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 46 / 146

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Method

Fast Marching Method

Statement of the problem in the case of optimal path planning :(J.A. Sethian, 1998)

Let u(x) be the time where the front crosses the point x .

Computation of u → Solving the Eikonal equation :|∇u(x)|F (x) = 1 in Ω, F (x) > 0

Γ(u) = x |u(x) = u0,

where x is the position and F is the propagation speed.

To plan the optimal path γ(t) (back traking) :

dγ(t)

dt= − ∇u

||∇u||

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 47 / 146

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Method

Fast Marching Method

Statement of the problem in the case of optimal path planning :(J.A. Sethian, 1998)

Let u(x) be the time where the front crosses the point x .

Computation of u → Solving the Eikonal equation :|∇u(x)|F (x) = 1 in Ω, F (x) > 0

Γ(u) = x |u(x) = u0,

where x is the position and F is the propagation speed.

To plan the optimal path γ(t) (back traking) :

dγ(t)

dt= − ∇u

||∇u||

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 47 / 146

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Method

Numerical solving : Godonov Scheme

The principal idea is to construct the solution using only upwind values. For this,we divide all the mesh points in three sets :

Accepted : Set of points where the solution is known ;

Considered : Set of points which are adjacent to at least one Acceptedpoint ;

Far : Set of points where we do not have yet any information about thesolution.

Figure: Construction of the algorithm

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 48 / 146

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Method

Fast Marching Algorithm

Figure: Step 1 : Initialization

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 2 : Transfering → Considered

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 3 : Looking for the smallest value u(xi )

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 4 : Transfering → Accepted

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 5 : Transfering → Considered

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 6 : Looking for the smallest value u(xi )

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 7 : Transfering → Considered

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 8 : Recomputing the value u(xi )

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Fast Marching Algorithm

Figure: Step 8 : Recomputing the value u(xi )

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 49 / 146

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Method

Trajectory Models

Aircraft Trajectory Features

Dimension Reduction Approaches

Front Propagation Approaches

Optimal Control Approaches

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 50 / 146

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Method

Optimal Control for Trajectory Generation

Mainly used for time-parameterized of shapes.

Generating time-parameterized paths necessitates the incorporation ofthe aircraft dynamics.

The objective of optimal control theory is to determine the controlinput(s) that will cause a process to satisfy the physical constraints,while, at the same time, minimize (or maximize) some performancecriterion.

Feasibility of the trajectories is automatically ensured using thisapproach.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 51 / 146

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Method

Optimal Control for Trajectory Generation

Mainly used for time-parameterized of shapes.

Generating time-parameterized paths necessitates the incorporation ofthe aircraft dynamics.

The objective of optimal control theory is to determine the controlinput(s) that will cause a process to satisfy the physical constraints,while, at the same time, minimize (or maximize) some performancecriterion.

Feasibility of the trajectories is automatically ensured using thisapproach.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 51 / 146

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Method

Optimal Control for Trajectory Generation

Mainly used for time-parameterized of shapes.

Generating time-parameterized paths necessitates the incorporation ofthe aircraft dynamics.

The objective of optimal control theory is to determine the controlinput(s) that will cause a process to satisfy the physical constraints,while, at the same time, minimize (or maximize) some performancecriterion.

Feasibility of the trajectories is automatically ensured using thisapproach.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 51 / 146

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Method

Optimal Control for Trajectory Generation

Mainly used for time-parameterized of shapes.

Generating time-parameterized paths necessitates the incorporation ofthe aircraft dynamics.

The objective of optimal control theory is to determine the controlinput(s) that will cause a process to satisfy the physical constraints,while, at the same time, minimize (or maximize) some performancecriterion.

Feasibility of the trajectories is automatically ensured using thisapproach.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 51 / 146

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Method

Optimal Control for Trajectory Generation

Given initial conditions x0, final conditions xf ∈ X , and an initial timet0 ≥ 0, determine the final time tf > t0, the control input u(t) ∈ U andthe corresponding state history x(t) for t ∈ [t0, tf ] which minimize thecost function

J(x , u) =∫ tft0

L(x(t), u(t))dt,

where x(t) and u(t) satisfy, for all t ∈ [t0, tf ] the differential andalgebraic constraints.

x(t)− f (x(t), u(t)) = 0,C (x(t), u(t)) ≤ 0.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 52 / 146

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Method

Optimal Control for Trajectory Generation

Optimal control has its roots in the theory of calculus of variations,which originated in the 17th century by Fermat, Newton, Liebniz,etc...

It was not until the middle of the 20th century when the Sovietmathematician Pontryagin developed a complete theory that couldhandle such problem.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 53 / 146

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Method

Optimal Control for Trajectory Generation

Optimal control has its roots in the theory of calculus of variations,which originated in the 17th century by Fermat, Newton, Liebniz,etc...

It was not until the middle of the 20th century when the Sovietmathematician Pontryagin developed a complete theory that couldhandle such problem.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 53 / 146

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Method

Optimal Control for Trajectory Generation

Pontryagin’s celebrated Maximum Principle states that the optimalcontrol for the solution of the problem is given as the pointwiseminimum of the so-called Hamiltonian function, that is :

uopt = argminu∈UH(t, x , λ, u)

where H(t, x , λ, u) = L(x , u) + λT f (x , u) is the Hamiltonian, and λare the co-states, computed from

λ(t) = −∂H

∂x(x(t), λ(t), u(t)). (1)

subject to certain boundary (transversality) conditions on λ(tf ).

Numerical solution

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 54 / 146

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Method

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 55 / 146

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Strategic Planning

Continental Strategic Planning

Before take-off

Trajectory design for large segment (full trajectory)

Action on time and space

Large scale (30000-50000 aircraft)

Continental or Oceanic

Macroscopic congestion criterium

One must take into account uncertainties

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 56 / 146

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Strategic Planning

Uncertainties

t t + 10’ t + 20’

Trajectory prediction limitation Factors

1 Wind (~V = ~T + ~W )

2 Temperature, pressure (engine trust, drag d = 12 .cx .ρ.S .v

2)

3 Weight

On-board trajectory prediction

FMS in open loop : +−15Nm after one hour flight.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 57 / 146

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Strategic Planning Continental Strategic Planing

How much can we reduce congestion in the French Airspace ?Optimization Approach

EUROCONTROL

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 58 / 146

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Strategic Planning Continental Strategic Planing

How much can we reduce congestion in the FrenchAirspace ?

Approach based on optimization

What are our state space variables ?

2D Route + departure times (' 7000 flights).

What is our objective ?

Airspace congestion minimization

What are the constraints ?

Extra distance ≤ 10%

Time shift have to be limited (+− 45 minutes)

The optimization process has to take into account flight connexions(hubs) and equity between airline.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 59 / 146

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Strategic Planning Continental Strategic Planing

How much can we reduce congestion in the FrenchAirspace ?

Approach based on optimization

What are our state space variables ?

2D Route + departure times (' 7000 flights).

What is our objective ?

Airspace congestion minimization

What are the constraints ?

Extra distance ≤ 10%

Time shift have to be limited (+− 45 minutes)

The optimization process has to take into account flight connexions(hubs) and equity between airline.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 59 / 146

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Strategic Planning Continental Strategic Planing

How much can we reduce congestion in the FrenchAirspace ?

Approach based on optimization

What are our state space variables ?

2D Route + departure times (' 7000 flights).

What is our objective ?

Airspace congestion minimization

What are the constraints ?

Extra distance ≤ 10%

Time shift have to be limited (+− 45 minutes)

The optimization process has to take into account flight connexions(hubs) and equity between airline.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 59 / 146

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Strategic Planning Continental Strategic Planing

Mathematical Modeling

A pair of decision variable (δi , ri ) is associated with each flight n.

δi ∈ ∆n ri ∈ Rn

∆n = −δm,−δm + 1, ....,−1, 0, 1, ..., δp − 1, δpRn = r0, r1, r2, ..., rmax

(0, r0) : airline choice.

State point :

X =

[δ1 δ2 ... δk ... δNr1 r2 ... rk ... rN

]

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 60 / 146

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Strategic Planning Continental Strategic Planing

Objective function

Congestion Minimization

min y(X ) = mink=P∑k=1

((∑t∈T

W tSk

)φ × (maxt∈T

W tSk

)

maxt∈T W tSk

: is the maximum reported congestion.∑t∈T W t

Sk: is the sector cumulated congestion.

P is the number of elementary sectors, φ and ϕ are weight factors

max y1(X ) =y(Xref )

y(X )

(y1 = 2 means that the congestion has been divided by 2)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 61 / 146

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Strategic Planning Continental Strategic Planing

Simulation process

Initial

Flight Plans Flight Plans

Alternative Airspace

Sectors

Traffic

Simulator

Sampled

Trajectories

Genetic Algorithm

Best

Planning

Proposed Planning

Computation

Congestion

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 62 / 146

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Strategic Planning Continental Strategic Planing

Genetic Algorithm

TournamentSelection

λ

µ

POP(k)

CrossoverPc

MutationPm1−(Pm+Pc)

POP(k+1)

Nothing

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 63 / 146

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Strategic Planning Continental Strategic Planing

A Posteriori information

WBrW

Ar

B

A

S1

S2

S3

S4

S5

in out outin

Time

Sector congestion

Trend

Advance Delay

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 64 / 146

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Strategic Planning Continental Strategic Planing

State space

Congestion

Reported

Stochastic

Trend

r13

rnk r

N6

1∆ ∆n

1δ2

δnj δN

3

1∆

RnR1 RN

∆N

r

1

r r

n N

TN

T n1T

W W W

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 65 / 146

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Strategic Planning Continental Strategic Planing

Test Features and Parameters

One day of traffic 6381 flights (june, 21 1996)

89 elementary sectors with dynamic capacity

Pop size : 50

Generation number : 300

φ = 0.9 and ϕ = 0.1

Max time shift : + or - 45 mn

Alternative route with 10% extradistance

6 computation hours on Pentium 1Ghz

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 66 / 146

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Strategic Planning Continental Strategic Planing

Evolution of best planning with generations

One day of traffic with ' 7000 flights optimized with GA

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 67 / 146

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Multi-objective extension

Delays and extra-distances minimization

Delay on the ground : δs(i) = |t(i)− t0(i)|Delay on board : δr (i) = 3 ∗ (Tr (i)− Tr0(i))

Total delay : δ(i) = δs(i) + δr (i)

min y2 =N∑i=1

δ(i)2

(the square insure equity)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 68 / 146

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Multi-objective extension

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 69 / 146

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Strategic Planning Strategic Conflict Free Planing

Strategic Conflict Free PlanningOptimization Approach

FP7 4D-CO project

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 70 / 146

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Strategic Planning Strategic Conflict Free Planing

Strategic Conflict Free Planning

Consider the traffic over Europe (' 36000 flights)

Picture of Europe Traffic for One Day

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 71 / 146

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Strategic Planning Strategic Conflict Free Planing

Strategic Conflict Free Planning

We propose to design a gate-to-gate conflict free planning by addingwaypoints and/or by shifting the time on departure.Departure and arrival segments are added to En-Route segments.Optimal altitude profiles have been used.Time shift : +- 30 minutes.Waypoint constraints : max 10% extra distance

L/3 2L/3 L

D O

a

y

x

-­‐a

0 wx1,min wx1,max wx2,min wx2,max

wy1,max wy2,max

wy1,min wy2,min

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 72 / 146

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Strategic Planning Strategic Conflict Free Planing

Strategic Conflict Free Planning

Direct route planning induces ' 400000 interactions between trajectoires.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 73 / 146

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Strategic Planning Strategic Conflict Free Planing

Strategic Conflict Free Planning

This problem is NP HardOne point of the state space requests 2GO memory space.

⇒ Simulated Annealing (20 minutes computing 2.4 Ghz intel CPU)

0 0.5 1 1.5 2 2.5

x 104

0

2

4

6

8

10

12

14

16

18x 104

number of transitions

Bes

t obj

ectiv

e fu

ncito

n va

lue

Evolution of best objective function value

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 74 / 146

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Strategic Planning Strategic Conflict Free Planing

Strategic Conflict Free Planning

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Strategic Planning Oceanic Strategic Planning

Oceanic Strategic PlanningOptimization Approach

ENAC

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 76 / 146

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Strategic Planning Oceanic Strategic Planning

Oceanic Strategic Planning

Continental Airspace ⇒ Radar

Oceanic Airspace ⇒ Procedures based on oceanic tracks network

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 77 / 146

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How It Works Today ?

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Strategic Planning Oceanic Strategic Planning

Oceanic Network Structure

Ny

Nx11

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 79 / 146

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Network Limitation

Congestion Area

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Time Constraint for Oceanic Traffic

10 minutes

15 minutes 15 minutes

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Automatic Dependent Surveillance-Broadcast

One measure every second

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Time Constraint with ADSB

3 minutes 3 minutes

2 minutes

This new system increases the number of valid track changes and themaximum number of aircraft on the same track (wind optimal).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 83 / 146

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The model

Data : For each flight f ∈ F we know

Track fin the entry track

Track fout the exit track

t fin time of entrance in the trackFLf

in the input flight levelFLf

out the output flight level

Variables

x fi =

1 if flight f changes track at waypoint i0 otherwise

δf : time shift at track entry : t fin + δf

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 84 / 146

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Altitude ProfilesU

SA

EU

RO

PE

Altitude profiles will be considered as constraints.

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The model

ConstraintsNX−1∑i=1

x fi = |Track f

out − Track fin|

z fi =

1 if flight f changes flight level at waypoint i0 otherwise

NX−1∑i=1

z fi = |FLf

out − FLfin|

Objective functionNumber of conflicts on nodes (Cfn) and links (Cfl).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 86 / 146

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Induced Combinatorics

For each flight f we have the following

1 about 6 possible slots per flight.

2 an average of 4 track changes which have to be spread among the 10waypoint positions (= 210 options per flight)

3 the total number of options is about 1260.

For 500 flights we have 1260500 options.

No separability ⇒ Heuristic approach (EA)

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Coding

=d i

d id 1 d N

01 0 1 10 100δ t Ci

Level of congestion encountered by flight i

N number of aircraft

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Slicing Crossover

CROSSOVER

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 89 / 146

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Strategic Planning Oceanic Strategic Planning

Slicing Crossover

PARENT 1

SLICING

CROSSOVER

CHILD 1 CHILD 2

PARENT 2

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 90 / 146

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Strategic Planning Oceanic Strategic Planning

Mutation

01 0 10 10δ t Ci0 1

00 0 10 11δ t Ci0 1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 91 / 146

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Strategic Planning Oceanic Strategic Planning

Fitness Computation

Each aircraft trajectory is computed on the track network based on ;

Altitude profile

Aircraft speed

Track changes decision variables

Time delay at network entry (Max +/- 6x5=30 minutes)

Based on such simulation, we compute the conflicts on nodes (Cfn) and onlinks (Cfl).

fitness =1

0.01 + Cfn+

1

0.01 + Cfl

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 92 / 146

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Strategic Planning Oceanic Strategic Planning

Test Framework

387 aircraft trajectories from August 4th 2006 (USA → Europetraffic)

Evolutionary Algorithm parametersPop size 500

Genration number 1000Selection (λ = 6,µ = 2)

Proba Cross 0.5Proba Mut 0.1

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 93 / 146

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Strategic Planning Oceanic Strategic Planning

Results for Standard System

0

10

20

30

40

50

60

70

80

90

100

110

0 100 200 300 400 500 600 700 800 900 1000

Bes

t and

mea

n fit

ness

Generation

Simulation without ADSB

BestMean

Remaining conflicts on nodes : 609 (initially 1515)D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 94 / 146

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Strategic Planning Oceanic Strategic Planning

Results with ADSB Equiped Aircraft

0

20

40

60

80

100

120

140

160

180

200

0 100 200 300 400 500 600 700 800 900 1000

Bes

t and

mea

n fit

ness

Generation

Simulation with ADSB

BestMean

All conflict have been removedD. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 95 / 146

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Strategic Planning Oceanic Strategic Planning

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 96 / 146

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Pre-Tactical Planning

Pre-Tactical Planning

After take-off (1, 2 hours planning)

Features

2D route design and speed control (state space)

Congestion or weather areas avoidance (objective)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 97 / 146

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Pre-Tactical Planning

Pre-Tactical Planning

After take-off (1, 2 hours planning)

Features

2D route design and speed control (state space)

Congestion or weather areas avoidance (objective)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 97 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Wind Optimal Trajectory DesignFront Propagation Approach

Cap Gemini

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 98 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

What are our objectives ?

Currently

Using predefined air routes.

⇒ Proposed approach : Wind optimal route design.

⇒ New problem :

Optimization of aircraft trajectories based on weather conditions (wind)which avoid congestion areas (or bad weather phenomena, etc ...)

The optimization is based on Travel Time and (or) Fuel Consumption.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 99 / 146

Page 154: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Pre-Tactical Planning Trajectory Design in a Wind Field

What are our objectives ?

Currently

Using predefined air routes.

⇒ Proposed approach : Wind optimal route design.

⇒ New problem :

Optimization of aircraft trajectories based on weather conditions (wind)which avoid congestion areas (or bad weather phenomena, etc ...)

The optimization is based on Travel Time and (or) Fuel Consumption.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 99 / 146

Page 155: Mathematical Models for Aircraft Trajectory Design : A ... · Agenda Some Trajectory Models Strategic Trajectory Design Pre-Tactical Trajectory Design Tactical Trajectory Design Emergency

Pre-Tactical Planning Trajectory Design in a Wind Field

What are our objectives ?

Currently

Using predefined air routes.

⇒ Proposed approach : Wind optimal route design.

⇒ New problem :

Optimization of aircraft trajectories based on weather conditions (wind)which avoid congestion areas (or bad weather phenomena, etc ...)

The optimization is based on Travel Time and (or) Fuel Consumption.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 99 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Statement of problem

Inputs

Start point A,End point B ;

Constant aircraft speed ;

Wind forecast ;

Areas to avoid.

⇒ Goal : Connect the point A to the point B in order to minimize thetravel time.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 100 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Statement of problem

Inputs

Start point A,End point B ;

Constant aircraft speed ;

Wind forecast ;

Areas to avoid.

⇒ Goal : Connect the point A to the point B in order to minimize thetravel time.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 100 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Adaptation of the Fast Marching Method

Figure: Speed

−−→VGS =

−−−→VTAS +

−−→VW

with :

VTAS (True Airspeed) : speed of theaircraft relative to the airmass inwhich it is flying ;

VW (Wind Speed) ;

VGS (Ground Speed).

⇒ The aircraft ground speed is function of the direction !⇒ Anisotropic problem.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 101 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Adaptation of the Fast Marching Method

Figure: Speed

−−→VGS =

−−−→VTAS +

−−→VW

with :

VTAS (True Airspeed) : speed of theaircraft relative to the airmass inwhich it is flying ;

VW (Wind Speed) ;

VGS (Ground Speed).

⇒ The aircraft ground speed is function of the direction !⇒ Anisotropic problem.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 101 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Calculation of the speed function : F = ||−→F ||

Calculation of the aircraftspeed in the normaldirection.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 102 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Calculation of the speed function : F = ||−→F ||

Calculation of the aircraftspeed in the normaldirection.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 102 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Calculation of the speed function : F = ||−→F ||

Calculation of the aircraftspeed in the normaldirection.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 102 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Calculation of the speed function : F = ||−→F ||

Calculation of the aircraftspeed in the normaldirection.

Calculation of the cost u :

‖∇u‖ =1

||−→F ||

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 102 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Calculation of the speed function : F = ||−→F ||

Calculation of the aircraftspeed in the normaldirection.

Calculation of the cost u :

‖∇u‖ =1

||−→F ||

To plan the optimal path :

dX

dt= −−−→VW − VTAS

∇u

||∇u||D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 102 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Taking into account obstacles and weather conditions

‖∇u(x)‖ =1

F (x)

⇒ Change of the propagation speed according to obstacles :

‖∇u(x)‖ =1

((1− α(x))F (x))

with α(x) ∈ [0;α0] and 0 6 α0 < 1.

Interpretation :

α(x) = α0 : forbidden areasα(x) = 0 : free areas0 ≤ α(x) ≤ α0 penalized areas

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 103 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Example with obstacles

Figure: Obstacles (Forbidden areas then coefficient decreasing to 0.)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 104 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Example with obstacles

Figure: Optimal trajectory (green) without wind

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 104 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Example with obstacles

Figure: Wind

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 104 / 146

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Pre-Tactical Planning Trajectory Design in a Wind Field

Example with obstacles

Figure: Optimal trajectories : with wind and without wind.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 104 / 146

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Pre-Tactical Planning Light Propagation Algorithm

Wave Propagation Algorithm for Trajectory Design

Aircraft Trajectory Design in a Wind Field

Light Propagation Algorithm AIRBUS FMS Division

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 105 / 146

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Pre-Tactical Planning Light Propagation Algorithm

The light propagation method

The light propagation analogy

Light follows Geodesic in time thereby avoiding areas of high index.

Light propagation is controlled by the Descarte law.

Trajectory planning can be achieved by computing wavefronts.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 106 / 146

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Pre-Tactical Planning Light Propagation Algorithm

The light propagation method

The light propagation analogy

Light follows Geodesic in time thereby avoiding areas of high index.

Light propagation is controlled by the Descarte law.

Trajectory planning can be achieved by computing wavefronts.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 106 / 146

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Pre-Tactical Planning Light Propagation Algorithm

The light propagation method

The light propagation analogy

Light follows Geodesic in time thereby avoiding areas of high index.

Light propagation is controlled by the Descarte law.

Trajectory planning can be achieved by computing wavefronts.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 106 / 146

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Pre-Tactical Planning Light Propagation Algorithm

Principles of the light propagation method

Destination

Curent node

Origine

Geodesic computation (A∗ like algorithm or Triangle mesh algorithm)

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 107 / 146

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Experimental results 2D

Experimental results

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 108 / 146

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Experimental results 2D

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 109 / 146

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Tactical Planning

Tactical Planning

After take-off (horizon : 20 minutes))

Features

2D Route design (state space)

Collision avoidance (objective)

One must bring a proof for such algorithms

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 110 / 146

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Tactical Planning

Tactical Planning

After take-off (horizon : 20 minutes))

Features

2D Route design (state space)

Collision avoidance (objective)

One must bring a proof for such algorithms

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 110 / 146

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Tactical Planning

Tactical Planning

After take-off (horizon : 20 minutes))

Features

2D Route design (state space)

Collision avoidance (objective)

One must bring a proof for such algorithms

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 110 / 146

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Tactical Planning

Tactical Trajectory Design

Time extension of light Propagation Algorithm

Approach based on B-Splines

Approach based biharmonic navigation functions

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 111 / 146

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Tactical Planning Light Propagation Algorithm

Approach Based on LPA

Time extension for dynamic obstacles

sts

d

td

ts’

td’

s’

d’

Obstacle

Time

Space X

Space Y

Light has to propagate one way in time dimension

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 112 / 146

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Tactical Planning 2D+Time

Experimental results

A 2D + time algorithm version

The algorithm sequentially control conflicting aircraft.

The aircraft are represented by high index discs of radius the standardseparation.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 113 / 146

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Tactical Planning 2D+Time

7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 114 / 146

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7 Conflicting Aircraft

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 115 / 146

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Tactical Planning Traffic day

Conflict Resolution for a traffic day

How does it work ?

We compute aircraft trajectories for a day of traffic over France.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 116 / 146

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Tactical Planning Traffic day

Conflict Resolution for a traffic day

How does it work ?

We extract trajectories segments between t et t + 21 min.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 116 / 146

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Tactical Planning Traffic day

Conflict Resolution for a traffic day

How does it work ?

We identify clusters of conflict.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 116 / 146

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Tactical Planning Traffic day

Conflict Resolution for a traffic day

How does it work ?

We solve conflicts within each cluster using the light propagationalgorithm.

Modified trajectory segments

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 116 / 146

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Tactical Planning Traffic day

Conflict Resolution for a traffic day

How does it work ?

We reintroduce the new segments in the database and we recompute theremaining parts of trajectories.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 116 / 146

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Tactical Planning Traffic day

Conflict Resolution for a traffic day

How does it work ?

The time window is slid by 7 min. t ← t + 7.

Fraction of time window already flown by aircraft

Segment extracted in the next time window

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 116 / 146

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Tactical Planning Traffic day

Conflict Resolution for a full day of traffic

Numerical Results

The 8/12/2008 traffic day was tested with 8212 aircraft.

3344 clusters.

99% of clusters were resolved (the last % is due to aircraft already inconflict when algorithm starts ; could be solve initial time shifting

Number of modified trajectories is 1501.

Average extension distance= -4.41 Nm.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 117 / 146

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Tactical Planning Traffic day

Stochastic Extension

Open loop FMS error has been used for our simulation (+-15 Nm after 1Hour)

This algorithm has been extended with such uncertainties and is ableto manage 98% of the conflicts.

The remaining 2% have been solve by RTA setting (closed FMSmode).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 118 / 146

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Tactical Planning Traffic day

Tactical Trajectory Design

Time extension of light Propagation Algorithm

Approach based on B-SplinesCap Gemini

Approach based on biharmonic navigation functions

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 119 / 146

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Tactical Planning B-Spline Approach

Problem presentation

Our methodology

A combination of an optimization method and a smooth trajectorymodel : B-splines.

B-splines are controlled by the optimization method via theircontrol points

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 120 / 146

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Tactical Planning B-Spline Approach

Genetic Algorithm

Structure

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 121 / 146

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Tactical Planning B-Spline Approach

Trajectory model

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 122 / 146

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Tactical Planning B-Spline Approach

Semi-infinite programming formulation

minx

f (x)

s.t. g(x ; t) > α ∀t ∈ [t1, t2] (2)

where t is continuous, it is the semi-infinite parameter.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 123 / 146

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Tactical Planning B-Spline Approach

Semi-infinite programming formulation

Our objective function : relative distance increase.

Insure standard separation between each pair of aircraft at all time

c ij(u; t) = ‖γβi (u)(s(t))− γβj (u)(s(t))‖2 > τ ∀t ∈ [0, t ijmax ]

SIP is a local optimization method

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 124 / 146

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Tactical Planning B-Spline Approach

Results and comparison

32 aircrafts situation

Genetic Algorithm Semi-infinite programming.

Next : use GA to initialize control points for SIP

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 125 / 146

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Tactical Planning Bi-harmanic Approach

Tactical Trajectory Design

Time extension of light Propagation Algorithm

Approach based on B-Splines

Approach based on biharmonic navigation functionsCap Gemini

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 126 / 146

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Tactical Planning Bi-harmanic Approach

Collision-free trajectory planning using biharmonicnavigation functions

Objective

Create trajectories guaranteeing obstacle avoidance and enforcingATM constraints for several aircraft.

Constraints

1 Speed has to stay in a given range

2 Trajectories have be smooth

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 127 / 146

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Tactical Planning Bi-harmanic Approach

Navigation Function

Potential Field Analogy in order to compute the navigation function φ.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 128 / 146

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Tactical Planning Bi-harmanic Approach

Navigation function and navigation field

The navigation field is given by : −∇φ

Figure: Example of navigation field

With these navigation fields, we can be sure that :

any trajectory stays in the free space

any trajectory reaching the minimum stays at this minimum

There is no guarantee on the speed and trajectories may not be smooth ⇒Bi-Harmonic Functions.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 129 / 146

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Tactical Planning Bi-harmanic Approach

Navigation function and navigation field

The navigation field is given by : −∇φ

Figure: Example of navigation field

With these navigation fields, we can be sure that :

any trajectory stays in the free space

any trajectory reaching the minimum stays at this minimum

There is no guarantee on the speed and trajectories may not be smooth ⇒Bi-Harmonic Functions.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 129 / 146

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Biharmonic functions Theory

Mechanical stress field

Figure: The mechanical stress field

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 130 / 146

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Biharmonic functions Theory

Mechanical stress field

Figure: The mechanical stress field

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 130 / 146

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Biharmonic functions Theory

Mechanical stress field

Figure: The mechanical stress field

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 130 / 146

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Biharmonic functions Theory

Mechanical stress field

Figure: The mechanical stress field

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 130 / 146

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Biharmonic functions Theory

Mechanical stress field

Figure: Stresses representation

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 130 / 146

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Biharmonic functions Theory

Biharmonic functions : guideline

Solve 42F = 0 + boundary conditions

Compute the stresses by :

σxx = ∂2yyF (x , y) σyy = ∂2

xxF (x , y) σxy = −∂2xyF (x , y)

⇒ Tensor field

Compute the principal stresses(= eigenvalues)[σxx σxyσxy σyy

]⇒[σmin 0

0 σmax

]Compute the eigenvectors corresponding to σmin

⇒ Navigation field

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 131 / 146

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Biharmonic functions Theory

Fields with obstacle

Figure: With one obstacleFigure: For a more complex geometry

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 132 / 146

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Biharmonic functions Theory

Conclusions

Biharmonic Navigation Functions

Ensure conflict free trajectory design

With mathematical proof

With speed range constraint

With curvature constraint

May be used in tactical phase

Have to be extended to the stochastic framework ⇒ StochasticBiharmonic Functions

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 133 / 146

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Biharmonic functions Theory

Agenda

Some Trajectory Models

Strategic Trajectory Design

Pre-Tactical Trajectory Design

Tactical Trajectory Design

Emergency Trajectory Design

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 134 / 146

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Biharmonic functions Theory

On-Board A/C Optimal Trajectory Generation

Over 70% of fatal aviation accidents are in take-off/landing phases.

Cockpit emergency handling from crew can result in completelydifferent outcomes : Swissair Flight 111, US Airways Flight 1549

Landing in mountainous terrain (e.g., LinZhi airport in China),avoiding inclement weather, or other aircraft in the area requiresreliable obstacle avoidance.

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 135 / 146

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Biharmonic functions Theory

Aircraft Emergency Landing

Time is the most critical factor

Swissair flight 111 : 14minUS Airways flight 1549 : 3min

Fuel may be a limiting factor too

Challenges

Real-Time requirement

Convergence guarantees

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 136 / 146

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Biharmonic functions Theory

An Alternative

Use a hierarchical approach

Geometric planner

- State constraints, obstacles- Path generator

Motion planner

- Time parameterization- Trajectory generator

Key Idea : First find flyable path to avoid obstacles ; then find afeasible trajectory to follow along this path.

Requires the solution of optimal time parameterization (orvelocity generation) problem.

The latter is a one-dimensional optimal control problem that can besolved very efficiently !

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 137 / 146

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Biharmonic functions Theory

On-Line Optimal Trajectory Generation Schematic

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 138 / 146

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Biharmonic functions Theory

Initial Path Guess

Use Dubins paths with continuous descent

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 139 / 146

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Biharmonic functions Theory

Application to Real Test Cases

Swissair 111

US Air 1549

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 140 / 146

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Biharmonic functions Theory

Test Case 1 : Swissair 111

Swissair 111 (McDonnell Douglas MD-11) from JFK (NY) to Geneva(Switzerland).

On Wednesday, 2 September 1998, the aircraft crashed into theAtlantic Ocean southwest of Halifax International Airport (due to fireon Board).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 141 / 146

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Biharmonic functions Theory

Test Case 1 : Swissair 111

Swissair 111 (McDonnell Douglas MD-11) from JFK (NY) to Geneva(Switzerland).

On Wednesday, 2 September 1998, the aircraft crashed into theAtlantic Ocean southwest of Halifax International Airport (due to fireon Board).

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 141 / 146

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Biharmonic functions Theory

Test Case 1 : Swissair 111

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 142 / 146

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Biharmonic functions Theory

Test Case 1 : Swissair 111

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 143 / 146

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Biharmonic functions Theory

Test Case 1 : Swissair 111

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 144 / 146

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Biharmonic functions Theory

Test Case 2 : US Air 1549

VIDEO !

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 145 / 146

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Biharmonic functions Theory

Test Case 2 : US Air 1549

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 146 / 146

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Biharmonic functions Theory

QUESTIONS ?

D. Delahaye and S.Puechmorel and P.Tsiotras and E.Feron ( Applied Mathematics Laboratory (MAIAA) French Civil Aviation University Toulouse, France School of Aerospace Engineering Georgia Institute of Technology Atlanta, USA )Mathematical Models for Aircraft Trajectory Design : A Survey EIWAC 2013 TokyoFebruary, 21 2013 147 / 146


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