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
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France2/24
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
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France3/24
TAEM Re-entry Mission (1/2)
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
Orbiter
ground track
TEP
NEP
Hypersonic phase
Zrunway
TAEM phase
Autolanding
phase
Injection point
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France4/24
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)
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
HAC3
HAC1
TEP
HAC4
Preliminary
dissipation S-turn
Xrunway
Zrunway
Yrunway
HAC acquisition
S-turn
HAC2
NEP
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France5/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France6/24
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 = 1.5 (interpol + full airbrake deflexion)
M = 2 (ref)
M = 2 (interpol)
M = 2 (interpol + full airbrake deflexion)
M = 3 (ref)
M = 3 (interpol)
M = 3 (interpol + full airbrake deflexion)
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 = 0.5 (interpol + full airbrake deflexion)
M = 1.5 (ref)
M = 1.5 (interpol)
M = 1.5 (interpol + full airbrake deflexion)
M = 2 (ref)
M = 2 (interpol)
M = 2 (interpol + full airbrake deflexion)
M = 3 (ref)
M = 3 (interpol)
M = 3 (interpol + full airbrake deflexion)
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France7/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France8/24
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…
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• 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
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France9/24
Brief Review of Flatness Theory (2/3)
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).
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France10/24
Brief Review of Flatness Theory (3/3)
Path tracking
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• 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
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France11/24
Guidance Scheme Design (1/7)
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• 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
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France12/24
Guidance Scheme Design (2/7)
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• 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
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France13/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France14/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France15/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France16/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France17/24
Guidance Scheme Design (7/7)Tracking design
Flat outputs derivatives along the vector field
The corrected control inputs can be computed as:
with
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France18/24
Guidance Performances Assessment (1/5)Animation using Matlab Virtual Reality Toolbox (nominal case)
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France19/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France20/24
Guidance Performances Assessment (3/5)Monte Carlo simulation results (100 cases)
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France21/24
Guidance Performances Assessment (4/5)Monte Carlo simulation results (100 cases)
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France22/24
Guidance Performances Assessment (5/5)Monte Carlo simulation results (100 cases)
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France23/24
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
• TAEM Re-entry Mission
• ARES-H Vehicle
• TAEM Guidance Problem
• Brief Review of Flatness
• Guidance Scheme Design
• Guidance Performances
• Perspectives
17th IFAC Symposium on Automatic Control in Aerospace, June 25-29, 2007, Toulouse, France24/24
Thank you for your attention