Akinori Harada*, Yuto Miyamoto*, Navinda Kithmal Wickramasinghe*

* Department of Aeronautics and Astronautics,Graduate School of Engineering, Kyushu University

Flight Trajectory Optimization Tool withDynamic Programming Developed for

Future Air Transportation System

The third ENRI International Workshop on ATM/CNS (EIWAC2013)Feb. 19-21, 2013, MIRAIKAN Hall, Tokyo, Japan

1. Introduction

� Background and objective

2. Trajectory Optimization by Dynamic Programming

� Definition of optimal control problem� Advantages and disadvantages of DP

3. Moving search space method, MS-DP

� Computational time reduction technique to overcome “Curse of Dimensionality”.

4. Conclusions

Outline2

Introduction3

Background

Airspace based operation isconventionally used today in AirTraffic Control (ATC).

Disadvantages� Bound to perform on the predetermined

flight route and altitude� Congestion at terminal airspace with

inefficient air traffic flow management

Following two advantages are expected by the realization of TBO.

�Reduction of fuel consumption, which is beneficial foreconomy and ecology.

�Enhancing air traffic capacity at terminal airspace andground handling capacity.

Trajectory Based Operations (TBO)

Introduction4

Free Flight which maximizes each aircraft performance is ideal for a futuristic air transportation system.

Objectives

Free Flight concept was proposed to improve the efficient usageof airspace, providing users the capability of selecting theoptimal flight route and airspeed.

�Developing a trajectory optimization tool for a jet passengeraircraft to generate fuel efficient flight trajectories.

�Proposing an effective method which reduces thecomputational time for trajectory design.

• Trajectory optimization is a key technology to realize an idealair transportation system in the foreseeable future.

• An efficient tool which provides plausible solutions is necessaryfor the trajectory optimization.

An example of 4D optimal trajectory 5

4D Optimal trajectory considering the presence of wind

[EN-41] A Study on Benefits Gained by Flight Trajectory Optimization forModern Jet Passenger Aircraft(Conference Room 2, 16:30～)

Optimal control problem6

( ) { }xaaTAS WVHRdt

d ++

= ψγφ

θsincos

cos

1

0

{ }yaaTAS WVHRdt

d ++

= ψγφcoscos

1

0

aTASVdt

dH γsin=

( ) ( ) aaaES mgDT

dt

dVm γψψγγ sincoscos −−=−−

Equations of motion

Performance index

∫=ft

tFFdtJ

0

(Total fuel consumption)

( )111 ,, Hθφ

( )222 ,, Hθφ

Optimal control problem is defined as deriving the optimal trajectory whichminimizes the total fuel consumption considering operational limitations.

： True air speed： Earth speed： path angle： Azimuth angle： Zonal wind： Meridional wind

TASV

ESVγψ

xW

yW

Dynamics of the aircraft is expressed by the timederivative of longitude , latitude , altitude andearth speed .

θ φ H

ESV

Grid system7

η

x

ζ

y

zTerminal point

Initial point

AltitudeCross range

Down range

Initial point Terminal point

① Calculation grid is defined by using state variables , , and .② Flight trajectory is presented by series of transitions between grid

points.� Control variables , , are derived between two grid points by

solving equations of motion with “difference approximation”.

� Fuel consumption is also calculated from the transition betweentwo grid points.

H ηV ς

aγ aψ T

tTcJFuel T ∆∆∆ == ( : Coefficient of specific fuel consumption)Tc

( ) ( ) aaaES mgDt

VmT γψψγγ

∆∆

sincoscos ++−−=

Advantages of DP8

� Global optimum� No iterative calculations

-Computational time can be estimated in advance.� Easy to handle inequality constraints on state

variables and control variables.� Easy to adapt for changing design conditions,

models and parameters.� Simple programming

Advantage; Global optimality9

The optimal trajectory obtained is equivalent to that solved as acombinatorial optimization problem for all transitions between the gridpoints.

Initial point

Terminal point

H

ηV

Optimaltrajectory

State variables: , , Independent variable:H ηV ς( :Downrange, : Crossrange )η ς

Advantage; No iterative calculation10

( )

∂∂

=

∂∂

+=∂

∂− tu

x

JxHf

x

JtuxL

t

J opt

u

opt

u

opt ,,,min,,min

Hamilton-Jacobi-Bellman (HJB) equation

( ) ( )[ ]JVHJVHJ kjihoptkk

kjihopt kkkkkk∆+=

+→++++ηςης ,,,min,,,

11111

H

ηV k 1+k

The optimal transition between two neighboring sections of independentvariable is governed by the Hamilton-Jacobi-Bellman optimality condition.Therefore, the calculation can proceed one way from the initial or final point.

( )kjihopt kkkVHJ ης ,,,

( )1,,,111 ++++ kjihopt kkk

VHJ ης

Advantage; Inequality constraints11

( ) ( ) ( ) ( )( ) 0,,...,,, 321 ≤fn xtxtxtxtxF

�State variable inequality constraints( ) ( ) ( ) ( )( ) 0,,...,,, 321 ≤fn ututututuG

�Control variable inequality constraints

maxmin HHH ≤≤

maxmin ηηη ≤≤maxmin ςςς ≤≤maxmin VVV ≤≤

max,min, aaa γγγ ≤≤

max,min, aaa ψψψ ≤≤

maxmin TTT ≤≤

H

ηV

H

ηV

It’s easy to handle inequality constraints on state variable and control variable.

Advantage; Design conditions, models and parameters.12

�Aircraft performance model

�Meteorological data

Tc

VcckFF

f

ktTASffnom cr

+=

2

1

,2 1

−=

4

313min

f

ftf c

hckFF

],max[ minFFFFFF nom=

The performance of an aircraft is calculated by using BADA (Base ofAircraft Data) model which is developed and maintained byEUROCONTROL.

Global Spectral Model (GSM) for Japan region provided by JapanMeteorological Agency (JMA) is used.

Pressure altitude

(12 surfaces)

Longitude (121 points)

Latitude ( 151 points)

DP has many advantages, however, it has also a major drawback.

Major drawback of DP13

� Curse of Dimensionality－Computational time and required memory increase

against the number of grid points.

1x

2x

1x3x

2x

n

m m

n

l

2nm× 22 lnm ××Total amount of computation: Total amount of computation:

The total amount of computation increases explosively with the square ofstate variable grid points number.

An effective method is proposed to solve this problem.

�1 dimension �2 dimension

�3 dimension 222 olnm ×××

Moving Search Space Method (MS-DP)14

①Partial search space is set around the initial referencetrajectory and the optimal trajectory is obtained in the space.

Terminal point

Initial point

Calculation process

②If the obtained solution is same as the reference trajectory, itis considered as the optimum solution. If not, the calculationprocess① is implemented again until the obtained solution isconverged.

Referencetrajectory

ReferencetrajectoryOptimal trajectory

Calculation example (Full search)15

An analysis with no wind conditions is implemented as an example.At first, the optimal trajectory is derived with all grid points.

RangeNumber of

divisionGrid

resolution

Down range 0~900 [km] 30 30 [km]

Altitude 3000~13,000 [m] 100 100 [m]

Velocity (CAS) 100~160 [m/s] 60 1 [m/s]

Calculation example (MS-DP)16

Gridresolution

Search point around the reference

Altitude 100 [m] ±10

Velocity (CAS) 1 [m/s] ±5

MS-DP method is applied to the same problem in order to reducethe computational time.

Comparison of results17

Exactly the same result as the full search case was obtained by the movingsearch space method with 18 iterations.

Full Partial

Computational time 1,321 [sec] 36 [sec]

Total amount of computation ( ) 3161101 2 ×× ( ) 18311121 2 ××× ( )

( )[ ]%5.2100

3161101

183111212

2

=×××

×××

[ ]%7.21001321

36 =×

� Computational time

� Total amount of computation

Full search Reduced search space (MS-DP)

The global optimum has been obtained with MS-DP.

1. A flight trajectory optimization tool useful for future airtransportation system research was developed.

2. DP is an easily-handled optimization method due to thesimplicity of including inequality constraints and designconditions, and the ability to avoid iterative calculations.� It can be considered that DP is suitable for the trajectory optimization of

passenger aircraft.

3. DP’s major drawback ‘The Curse of Dimensionality’ could berelaxed to some extent by the proposed moving searchspace method.� It provides the global optimum in most cases by adjusting the amount

of limits of the search space.

4. A related study has revealed that the developed DPoptimization tool is promising for the design of fuel efficientflight trajectories.

18

Conclusions