hypersonic reentry & flatness theory - application to medium l/d entry vehicle

This document is the property of EADS SPACE Transportation and shall not be communicated to third parties and/or reproduced without prior written agreement.Its contents shall not be disclosed. © - EADS SPACE Transportation - 2006

1st International ARA DaysArcachon

3-5 July 2006

Hypersonic Reentry & Flatness TheoryApplication to medium L/D Entry Vehicle

Vincent MORIO – Franck CAZAURANGLAPS - University of Bordeaux 1

Philippe VERNISGuidance & Control SystemsEADS-ST France



3-5 July 2006

CONTENTS

� MEDIUM L/D VEHICLE AND MISSION REQUIREMENTS

� FLATNESS-BASED HYPERSONIC REENTRY GUIDANCE

� Differential Flatness

� Flat Modelling of Longitudinal Dynamics

� Trajectory Planning & Tracking

� Guidance Performances Assessment

� CONCLUSION AND PERSPECTIVES



3-5 July 2006

MEDIUM L/D VEHICLE & MISSION REQUIREMENTS

� Atmospheric re-entry: 3 flight phases

� from 120 km down to Mach 2 Hypersonic phase

� from Mach 2 down to Mach 0.5 TAEM phase

� below Mach 0.5 Auto-Landing phase

Hypersonic re-entry � 120 km high down to Mach 2 gate

TAEM phase (winged/lifting bodies)� from Mach 2 gate down to Mach 0.5 gate

Auto landing

� from Mach 0.5 gate down to touch-down



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Design constraints

g-load 2 g

heat flux (nose) 500 kW/m2

heat load N.A.

dynamic pressure 6 kPa

a.o.a profile 40 deg ± 3 deg

MEDIUM L/D VEHICLE & MISSION REQUIREMENTS

� Reentry vehicle

� Medium L/D hypersonic lifting-body demonstrator

Estimated main features

L/D ~ 0.7

mass 2000 kg

Sref 6 m2

max roll rate 15 deg/s

max pitch rate 2 deg/s

Pre-X AREV



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FLATNESS-BASED GUIDANCE SCHEME

� Introduced by M. Fliess, J. Levine, P. Martin and P. Rouchonin 1992 in a differential algebraic context.


( )( )

==

)(),()(

)(),()(

tutxhty

tutxftxɺ� Consider the following nonlinear system:

The system is said to be differentially flat if there exists a set

of variables z(t) ∈∈∈∈ ℝm which are differentially independent,

called flat outputs, of the form : ( ))(),...,(),(),()( )( tutututxtz αɺΦ=

such that( )( )

== −

)(),...,(),()(

)(),...,(),()()(

)1(

tztztztu

tztztztx

u

xβ

β

ψψ

ɺ

ɺ

where Φ, Ψx, Ψu are smooth functions, z(αααα)(t), z(ββββ)(t) are respectively

the α and β order time derivatives of z(t).

No integration process needed



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reference flat outputs

Trajectory planning (open-loop)

- Interpolation polynomials - Open-loop optimization

reference flat outputs



� Guidance scheme global design:

Trajectory tracking (closed-loop)

- Nonlinear controller- Linear controller

estimated flat outputs

corrective terms

Dynamic Inversion

- Flat modeling using formal calculus

controlinputs

RLV motion

- Nonlinear model



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� Trajectory Planning

� case of a back-up entry at Istres (but TAEM not considered)

� reference trajectory designed using the ARD guidance scheme



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( )( ) ( )

( )

( ) ( )( ) ( )

( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

+=

−+=

−−=

=

=

=

θψγµγ

αψ

γµαγ

γα

ψγφ

θψγθ

γ

tansincossincos

,

coscos,

)(

sin,

)(

coscos)(

cos

sincos)(

sin)(

R

V

mV

ML

V

g

R

V

mV

MLt

gm

MDtV

R

Vt

R

Vt

VtR

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

( ) ( )

( ) ( )

( )

+==

=

=

22

2

2

,2

1,

,2

1,

T

TT

D

L

RhRg

MSCVMD

MSCVML

µµ

αρα

αρα

with


� Flat modelling of PRE-X longitudinal dynamics

� We consider the well-known full dynamics of the vehicle(without Coriolis, Euler and J2 terms):



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� Using candidate flat outputs R, θ and Φ, the system is not flat : it has been proved using the ruled manifold criterion in thestudy of T. Neckel, C. Talbot & N. Petit.


( )( )

( ) ( )

( ) ( ) ( )

−+=

−−=

==

γµαγ

γαγγ

coscos,

)(

sin,

)(

cos)(

sin)(

V

g

R

V

mV

MLt

gm

MDtv

Vtx

Vth

ɺ

ɺ

ɺ

ɺ

� Nonlinear system with four states (R,x,V,γ) and two control inputs α and µ

� The longitudinal dynamics of the vehicle is given by thefollowing set of equations:

Uncoupled in-plane and out-of-plane dynamics



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cz

α

M = 1.5

M = 40

40° 50°30° 37° 43°

Mi



� In the case of this study, we consider the subsequent polynomials for lift and drag aerodynamic coefficients:

� ai and bi are nonlinear functions obtained by interpolatingaerodynamic coefficients for various Mach numbers Mi

within the range [1.5; 40]

( ) ( ) ( ) ( ) ( )( ) ( ) ( )αα

αααα.,

...,

10

33

2210

iiiX

iiiiiZ

MbMbMC

MaMaMaMaMC

+=+++=



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� Choosing candidate flat outputs as altitude and curvilinear abscissa, the longitudinal dynamics can be rewritten as follows :

22

21

2

1

21

sin;arctan

;

zzh

Vz

z

zxzh

ɺɺ

ɺ

ɺ

ɺ+==

=

==

γγ



22

21

221122

21

2121

21

..;

..

;

zz

zzzzV

zz

zzzz

zxzh

ɺɺ

ɺɺɺɺɺɺɺ

ɺɺ

ɺɺɺɺɺɺɺ

ɺɺɺɺ

+

+=

+−

=

==

γ

� The states derivatives can also be extracted from these expressions :



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� One may extract literal open-loop expressions of inputs µ and αw.r.t. flat outputs z1, z2 and their time derivatives

NOTE: atmospheric density is interpolated in CIRA88 atmospheric table toimprove flatness-based guidance scheme accuracy



( ) ( )

( )Mb

SV

gVmMb

1

20..

sin...2

ργ

α

++

−=

ɺ

( )

+−+=

∑=

VRz

Vg

MaSV

m

T

k

ik

γγαρ

µ cos

...

.2arccos

1

2

3

0

ɺ

( )222111 ,,,,, zzzzzzf ɺɺɺɺɺɺαα =

( )222111 ,,,,, zzzzzzf ɺɺɺɺɺɺµµ =



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� Nonlinear regulation scheme via typical PID feedback

and


� Trajectory tracking

� Now we can extract expressions of µc and αc w.r.t. referenceflat outputs and their time derivatives :

( )( ) ( )

+==

==

=

γγγ

γ

cos.sin.

sin.

11

1

1

ɺɺɺɺ

ɺɺ

VVvz

Vhz

hz

c

( )( ) ( )

−==

===

γγγ

γ

sin.cos.

cos.

22

2

2

ɺɺɺɺ

ɺɺ

VVvz

Vxz

xz

c

( )( )

−

+−

++=

d

d

Td

dddddd

dcXddc

ddcZddc V

g

Rh

VVg

m

MCVSv

MCVS

m γγγαρ

γαρµ 2

2

12cos.sin.sin.

.2

,...

cos.,...

.2arccos

( ) ( ) ( )( )

+++−= dcdc

ddd

dc gvv

VS

mMb

Mbγγ

ρα sincos.

..

.211220

1

( )cddcddc vzzvzzf 222111 ,,,,, ɺɺµµ =

( )cddcddc vzzvzzf 222111 ,,,,, ɺɺαα =



3-5 July 2006


� Trajectory tracking

� v1c(t) and v2c(t) are linear corrective terms such that :

∑=

−−

+

=

1

0)(

2)(

2

)(1

)(1

2

1

2

1

)()(

)()(

)(

)(

)(

)(

iiesti

id

iesti

id

i

i

d

d

c

c

tztz

tztz

q

p

tv

tv

tv

tv

where )()( 11 tztv dd ɺɺ= and )()( 22 tztv dd ɺɺ=

� Then the system is asymptotically stable if the roots of the previous polynomials are located in the left complex half-plane.

� Using the typical approximation ω0.tr ≈ 3 for a second order system, where ω0 is the bandwidth and tr is the rise time of thestep response, we obtain :

−−

+

−−

+

=

)()(

)()(.6

.6

)()(

)()(9

9

)(

)(

)(

)(

22

11

2

2

1

1

22

11

22

21

2

1

2

1

tztz

tztz

t

t

tztz

tztz

t

ttv

tv

tv

tv

estid

estid

r

r

estid

estid

r

r

d

d

c

c

ɺɺ

ɺɺ

ξ

ξ



3-5 July 2006

Monte-Carlo Analysis

� EADS-ST SITHAR Simulation Testbed (5 dof)

RLV motion

2nd ordermodel

ideal or constant offsets

Attitude Control

Guidance

Navigation

GNC manager

out-of-plane guidance

in-plane guidance

rotating planet

J2 terms

aerodynamicuncertainties

atmosphericuncertainties

vehicle design uncertainties

∆Ca∆Cn

∆m ∆αe.g

∆ρ∆T

wind

initial kinematicconditions

∆X ∆V


� Guidance performances assessment



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Guidance

� in-plane guidance activation dynamic pressure > 400 Pa� in-plane guidance deactivation Mach number < 2

� PID controllers parameters have been chosen to follow a fixed-shape linear profile defined w.r.t. range to go in order to improve accuracy when vehiclecontrollability increases

� Aerodynamic coef. polynomials cross the 40 deg a.o.a. breakpoint� lateral corridor ± 5 deg for all velocities� a.o.a range [37 deg; 43 deg] for all velocities� Guidance step 0.5 s

Navigation idealControl ideal

� GNC tuning



3-5 July 2006



� Cumulated causes Monte Carlo – 1000 runs

� Nominal trajectory � Missrange at Mach 2 gate : 0.75 km� All mechanical constraints OK

requirement mean value standard deviation

max value

max heat flux < 500 kW/m² 418.4 kW/m² 4.2 kW/m² 429 kW/m²

max load factor < 2 g 1.7 g 0.012 g 1.74 g

max dynamic pressure < 6 kPa 5.3 kPa 0.05 kPa 5.5 kPa

max heat load N.A. 394.5 MJ/m² 4.23 MJ/m² 404.9 MJ/m²

re-entry duration N.A. 1436.6 s 1.9 s 1441 s

black-out duration N.A. 1212.2 s 1.04 s 1214.5 s

skip deviation no skip 0 m 0 m 0 m

missrange at Mach 2 < 20 km 5.1 km 4.2 km 20.7 km



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Overshoots on heat flux are limited by bounding the angle-of-attack within the range [37 deg; 43 deg]




commanded angle-of-attack (deg)heat flux at stagnation point (kW/m²)



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The number of roll reversals depends on the lateral logic.It varies from 3 (few cases) to 5 (majority of cases)




commanded bank angle (deg) altitudes of roll reversals (km)



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� The flatness-based guidance scheme applied to a lifting-bodyseems able to cope with mission requirements and designconstraints, provided that both the angle-of-attack and thebank angle are controlled

� Monte Carlo simulations show that it meets the 20 km 3D-offsetrequirement at TAEM handover, but it is particularly sensitive tomodelling errors compared to the ARD-like guidance schemeextended to handle lifting/winged bodies

� On-going research work to apply a similar algorithm to the TAEM guidance of a winged-body (but with coupled longitudinal and lateral motions) discloses promising results

CONCLUSIONS & PERSPECTIVES

hypersonic reentry & flatness theory - application to medium l/d entry vehicle

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