On the design of a flatness-based algorithm for the terminal area energy management guidance of a...

17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France1/24

Vincent Morio – Alexandre Falcoz – Franck Cazaurang IMS lab. UMR 5218 CNRS / Université Bordeaux 1

Philippe VernisASTRIUM Space Transportation

On the Design of a Flatness-based

Guidance Algorithm for the

Terminal Area Energy Management

of a Winged-body Vehicle


Outline

TAEM Re-entry Mission

ARES-H Vehicle

TAEM Guidance Problem

Brief Review of Flatness Theory

Flatness-based Guidance Scheme

Guidance Performances & Robustness

Conclusion & Perspectives

• TAEM Re-entry Mission

• ARES-H Vehicle

• TAEM Guidance Problem

• Brief Review of Flatness

• Guidance Scheme Design

• Guidance Performances

• Perspectives


TAEM Re-entry Mission (1/2)


• ARES-H Vehicle





• Perspectives

Orbiter

ground track

TEP

NEP

Hypersonic phase

Zrunway

TAEM phase

Autolanding

phase

Injection point


TAEM Re-entry Mission (2/2) Scenario based on the Russian Buran strategy:

• Preliminary dissipation S-turn (if needed)

• HAC acquisition S-turn (constant bank angle)

• HAC homing phase (straight path, full lift-up bank angle)

• HAC tracking (time-varying bank angle)


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• Perspectives

HAC3

HAC1

TEP

HAC4

Preliminary

dissipation S-turn

Xrunway

Zrunway

Yrunway

HAC acquisition

S-turn

HAC2

NEP


ARES-H Vehicle (1/2) Demonstrator of a Hypersonic glider

Designated as HERCULES in the FLPP plan

Potential candidate to be the future European Next Generation

Launcher (NGL)

Assigned to the aeroshape validation including TPS

subassemblies and the GNC strategy from orbit until landing

Estimated main features

L/D (for M < 2) between 2 and 3

mass 2000 kg

length 5.9 m

Sref 8.97 m2

max roll rate 15 deg/s

max pitch rate 2 deg/s


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• Perspectives


ARES-H Vehicle (2/2)

0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

45

Cx =f(alpha,M,deltax)

M = 0.5 (ref)

M = 0.5 (interpol)

M = 0.5 (interpol + full airbrake deflexion)

M = 1.5 (ref)

M = 1.5 (interpol)


M = 2 (ref)

M = 2 (interpol)

M = 2 (interpol + full airbrake deflexion)

M = 3 (ref)

M = 3 (interpol)


0 10 20 30 40 50 60-5

0

5

10

15

20

Cz=f(alpha,M,deltax)

M = 0.5 (ref)

M = 0.5 (interpol)


M = 1.5 (ref)

M = 1.5 (interpol)


M = 2 (ref)

M = 2 (interpol)


M = 3 (ref)

M = 3 (interpol)


Explicit aerodynamic model needed for flatness approach

Simplified model obtained by least-squares polynomial

interpolations of aerodynamic coefficients and for

different Mach numbers

Retained model: 2rd and 3rd order for and respectively


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• Perspectives


TAEM Guidance Problem Dissipate the kinetic and potential energy of the vehicle from

TAEM Entry Point (TEP) down to Nominal Exit Point (NEP)

Align the vehicle with the extended runway centerline to

enable a safe auto-landing

2 kinds of requirements:

- Path constraints (load factor, dynamic pressure)

- Final constraints (kinematics at NEP)

Design constraints

Max load factor < 2 g

Max dynamic pres. < 10 kPa

NEP constraints

Exit point velocity 150 m/s

Exit point altitude 5 km

Distance to runway 10 km

Heading headwind landing


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• Perspectives


Brief Review of Flatness Theory (1/3)

Introduced by M. Fliess, J. Levine, P. Martin and P.

Rouchon in 1992 in a differential algebraic context.

Initially introduced for nonlinear systems, but extended

later to systems with delays, infinite dimensional systems,

linear systems, etc…


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• Perspectives

The main contribution of this theory is focused on the

determination of a set of particular outputs (so-called flat

outputs) such that the input-to-state linearization is ensured

without some unobservable nonlinear dynamics



Nonlinear Dynamic Inversion (NDI)

Consider the following nonlinear system:

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

variables which are differentially independent, called

flat outputs, of the form :

such that

where , x, u are smooth functions, z()(t), z()(t) are respectively

the and order time derivatives of z(t).


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• Perspectives



Path tracking


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• Perspectives

We use the linear trivial model resulting from the flatness

property of the vehicle (obtained by differentiation of the flat

outputs):

The corrected control inputs can be simply chosen as:

where is the characteristic number associated with a flat output

The functions are then forced to follow

second order reference models such as


Guidance Scheme Design (1/7)


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• Perspectives

Global guidance scheme using coupled longitudinal and

lateral dynamics

Inversion of the full dynamics of the vehicle in local flat

Earth coordinates

Linear tracking controller based on a trivial model


Guidance Scheme Design (2/7)


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• Perspectives

6 d.o.f implementation under Matlab/Simulink®

Attitude control 2nd order performance model

and ideal navigation considered

Sufficient to provide guidance performances

preliminary results


Guidance Scheme Design (3/7) Reference path computed off-line by an ASTRIUM path planner

• Constant-width HAC radius

• S-turn performed with

• Piecewise L/D profile wrt. flight segments

• Total length of the trajectory about 70 km with TEP

defined at location

from runway center


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• Perspectives


Guidance Scheme Design (4/7)Nonlinear RLV model

Nonlinear model with states and controls

Equations of motion in flat Earth coordinates (assumption

of a non-rotating Earth) linked to the runway center

Position: Velocity:

with and


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• Perspectives


Guidance Scheme Design (5/7)Full NDI of the RLV model

No integration

process needed

Using candidate flat outputs the RLV

model is flat

State variables and control inputs can be written w.r.t

these flat outputs and their time derivatives

Position Velocity


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• Perspectives


Guidance Scheme Design (6/7)Full NDI of the RLV model

Controls w.r.t flat outputs and their time derivatives

Bank angle

Speed-brake

Angle-of-attack

with


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Guidance Scheme Design (7/7)Tracking design

Flat outputs derivatives along the vector field

The corrected control inputs can be computed as:

with


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• Perspectives


Guidance Performances Assessment (1/5)Animation using Matlab Virtual Reality Toolbox (nominal case)


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• Perspectives


Guidance Performances Assessment (2/5)Monte Carlo simulation results (100 cases)

Initial dispersions at TEP (3 values)

downrange offset ±10 km

crossrange offset ±10 km

altitude offset ±3 km

velocity offset N.A.

flight path offset ±3 deg

heading offset ±3 deg

Aerodynamic model (3 values)

lift coefficient ±5 %

drag coefficient ±5 %

vehicle dispersions (3 values)

overall mass ±5 %

Atmospheric dispersions (3 values)

atm. density at sea level ±3 %

horizontal wind magnitude 10 m/s

horizontal wind direction ±60 deg


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• Perspectives



Rqmt Mean Min Max

Downrange offset N.A 20.3 m 51.7 m 0 m 232.9 m

Crossrange offset N.A -14.5 m 11.5 m 0 m 6.5 m

Altitude offset N.A 16.4 m 30.9 m 0 m 124.1 m

3D offset N.A 54.7 m 40.4 m 0 m 264.4 m

Max dynamic pres. < 10 kPa 6.27 kPa 0.33 kPa 5.8 kPa 7.45 kPa

Max load factor < 2 g 1.6 g 0.13 g 1.31 g 1.9 g

Duration N.A 254.4 s 7.8 s 232 s 272 s


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Full respect of mechanical constraints with margins left

The flatness-based TAEM guidance scheme seems able to

cope with all mechanical and path constraints

3D missrange to targeted point below 120 m in 95% of the

simulated cases

Max 3D missrange of 264.4 m mainly due to severe initial

dispersions


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Control inputs close to their

nominal profile for most of the

simulated cases

Limited saturations occur

when counteracting severe

kinematics dispersions at the

beginning of the phase


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• Perspectives


Conclusion & Perspectives

Next step (currently under investigations) is to use flatness

approach to generate fully constrained TAEM trajectories in real-

time

TAEM is a critical unpowered atmospheric re-entry phase

which has to be performed with limited control means while

managing mechanical and kinematics constraints

Monte Carlo analyses have shown that flatness can be a

promising tool to treat complex guidance problems

It has been proved that the RLV coupled nonlinear model is

flat. A flatness-based guidance scheme has been designed with

coupled horizontal and vertical motions


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• Perspectives


Thank you for your attention

On the design of a flatness-based algorithm for the terminal area energy management guidance of a...

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