Onboard Terminal Area Energy Management Path Planning using Flatness Approach

2nd International ARA Days “10 Years after ARD”, Octobe r 21-23 2008, Arcachon, France. 1/27

Vincent Morio *, Franck Cazaurang*, Ali Zolghadri* and P. Vernis ��

*Automatic Control GroupIMS lab/University of Bordeaux

Francewww.laps.u-bordeaux1.fr/aria

�Guidance and Control DepartmentAstrium Space Transportation

Francewww.astrium.eads.net


Outline

� Motivations

� Trajectory planning by flatness and collocation

� Optimal convexification

� Application to the TAEM phase

� Future works


Motivations

• SICVER Project: Innovative Strategies for Guidance and Control of Experimental Launch Vehicles

• Increasing spacecraft autonomy/decreasing the ground level intervention load.

• Onboard trajectory planning provides a greater flexibility:

� to account for off-nominal conditions,

� to recover the vehicle from faulty situations.


Trajectory planning by flatness and collocation

• Flatness concept can be used to map the system dynamics to a lower dimension space

Flatness does not preserve convexity of the initial OCP• Convexification of the OCP in the flat-output space

• Transformation of the convex OCP into a Nonlinear Programming Problem (NLP)

Optimization of deformable geometric shapes by a genetic algorithmThe optimal solution of the initial OCP may be located on a constraint boundary

Minimum number of optimization variables in the Optimal Control ProblemMinimum number of optimization variables in the Optimal Control Problem


Trajectory planning by flatness and collocation


( )( )

==

)(),()(

)(),()(

tutxhty

tutxftxɺ� Consider the following NL model:

where Φ, Ψx, Ψu are smooth functions, z(αααα)(t), z(ββββ)(t) represent respectively the α-th and β-th time derivatives of z(t).

Definition: The system is differentially flat iffthere exists a set of differentially independent variables, called flat outputs, such that:

( ) mz t ∈R( ),)(),...,(),(),()( )( tutututxtz αɺΦ=

( )( )

== −

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

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

)1(

tztztztu

tztztztx

u

xβ

β

ψψ

ɺ

ɺsuch that

Equivalence between NL system trajectories and those of the trivial system (chain of pure integrators)

Trajectory planning by flatness and collocationDifferential flatness


� Consider the nonlinear model( ) ( ) ( )( ),x t f x t u t=ɺ ( ) ( ),n mt x t t u t∋ ∈ ∋ ∈ℝ ֏ ℝ ℝ ֏ ℝwhere

Trajectory planning by flatness and collocationOptimal Control Problem (OCP)

Trajectory planning by flatness and collocationOptimal Control Problem (OCP)

� We seek a trajectory of the system which minimizes the cost functional:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )0

0 0 0, , , ,ft

t f f f

t

J x u C x t u t C x t u t dt C x t u t= + +∫Subject to a set of initial, trajectory, and final constraints such that:

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

0

0 0 0 0 0 0

0

0 0 0 0 0

0

, , ,

,

, ,

,

, ,

, , ,

,

f

t t t t f

f f f f f f

t t t f

f f f f f

x t f x t u t t t t

l A x t B u t u

l A x t B u t u t t t

l A x t B u t u

L c x t u t U

L c x t u t U t t t

L c x t u t U

= ∈

≤ + ≤

≤ + ≤ ∈

≤ + ≤

≤ ≤

≤ ≤ ∈

≤ ≤

ɺ


� Reduction of the number of variables required in the OCP: the optimization variables become the flat outputs of the NL system

� Integration-free optimization problem: system dynamics is intrinsically satisfied

Trajectory planning by flatness and collocationOCP in flat output space

� The equivalent OCP in the flat output space is given by:

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

0 0 0 0

0

0 0 0 0

0

,

, ,

,

,

, ,

t t t f

f f f f

t t t f

f f f f

l A z t u

l A z t u t t t

l A z t u

L c z t U

L c z t U t t t

L c z t U

≤ ≤

≤ ≤ ∈

≤ ≤

≤ ≤

≤ ≤ ∈

≤ ≤

ɶ

ɶ

ɶ

ɶ

ɶ

ɶ

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

0

0 0minft

t f fz t

t

J z G z t G z t dt G z t= + +∫subject to:


Optimal Convexification

� The convexification problem (nonlinear constrained optimization) is solved by using a genetic algorithm in order to get a global optimum for the superquadrics volume

� Development of a Matlab software library: OCEANS

Optimal Convexification by Evolutionary Algorithm aNd Superquadrics

� May be used either in a “flat” framework or for typical nonconvex optimization problems

� Objective: Inner approximation of nonlinear trajectory constraints and cost functional by smoothly deformable geometric shapes

Generalities

Results in a convex OCP in the flat output space


� Generalization in n dimensions of superellipsoids (Barr, 1981)

Necessity to introduce additional convexity-preserving geometric transformations in ndimensions

Drawbacks:• Limited number of attaignable shapes• Symetric shapes

Optimal ConvexificationSuperquadrics


Advantages:• Compactness of the representation• Explicit parameterization of the shape


� n-D trigonometric paramaterization

=

−=

=

=

−

−

=−

−

=

−

− ∏

∏

nia

nia

ia

x

ii

n

kkii

n

kki

i

i

ki

k

1

1

11

1

1

1

1

sin

1,...,2cossin

1cos

θ

θθ

θ

ε

εε

ε

[ ]1,...,2

2,

2

1,

−=

−∈

=−∈

nisi

isi

i

i

ππθ

ππθ

2, ≤∀ ii εConvex iff

� n-D angle-center parameterization

( )

( ) ( )( )

=

−=

=

=

−−

−

=−−

−

=

∏

∏

nir

nirr

ir

x

ii

n

kkkii

n

kkk

i

11

1

111

1

1

sin

1,...,2cossin

1cos

θθ

θθθθ

θθ

with

( )

( )

1

1 1

12 2 2

1 1

1 2

2 2 2

1 1

11

cos sin

12,..., 1

cos sin

i

i i

i

i i

i i

r si i

a a

r si i n

a a

ε

ε ε

ε

ε ε

θ

θ θ

θ

θ θ+ +

= =

+ = = −

+


Optimal ConvexificationSuperquadricsSuperquadrics



3-D trigonometric parameterization

Variation of the number of anomalies

3-D angle-center parameterization

Variation of the number of anomalies


Optimal ConvexificationSuperquadricsSuperquadrics


More efficient sampling of the superquadric surface


Optimal ConvexificationGeometric transformations� The geometric transformations must preserve the convexity of the superquadrics

� n-D rotation:

� n-D translation:

� n-D linear pinching:


SuperquadricsOptimal Convexification

Superquadrics� The set containing of the sizing parameters needed to obtain a positioned, oriented and bended superquadric shape is given by:

Ψ

( ){ }111211111 ,,,,,,,,,,,,,, −+− ΦΦ=Ψ nnnnnn vvddaa …………… εε

sma roundness rotation. translation pinching

� Inside-outside function defined recursively: ( ) ( ),, ,n n nF x xΨ = Λ Ψ

( )( )( )

, 1

, 1

, 1

n

n

n

F x

F x

F x

Ψ <

Ψ = Ψ >

0=pv in the pinching direction

: x lies inside the shape

( )

( ) ( )( )

1 1

1

2

1

2 2

1 2,2

1 21 2

2

, , 1

,

1 1

, ,

1

k

k

k

n

p pp p

kn k n k

kk p

p

x xx

v va x a x

a a

xx x

va x

a

ε ε

ε

εε

−

−

−−

Λ Ψ = + + +

Λ Ψ = Λ Ψ + +

with

Inside-outside function property:

: x lies on the surface: x lies outside the shape


� Volume of a superquadric shape with n-D transformations

� n-D radial Euclidean distance

Radial distance from a point to the surface of the superquadric shape

( )( )1

20 0. 1

n

nd x F xε −−

= − or also, ( )( )1

20. 1

n

s nd x F xε −

= −

SuperquadricsOptimal Convexification

Superquadrics

0=pv in the pinching direction

where , 1, , 1,1 Rs t s t s tO − −ϒ = ϒ ϒ with and revolving-door gray code ,0

ss Oϒ = 0, 1t

tϒ =

( ) ( ) ( )( )

/2 2 1 2 1

0, 2 sin cosx y x y

B x y dx y

πφ φ φ− − Γ Γ

= =Γ +∫ is the beta function

( ) 1

0e t xx t dt

∞ − −Γ = ∫ is the gamma function

( )( )1

1 1

, 11 11 1

12 , 1

2 2

nkCard Cn nn

n n i i p n m m i ik ji m

k iV a a v Bε ε ε

−− −

− −= == =

+ Ψ = ϒ + ∑ ∑∏ ∏


Optimal ConvexificationConvexification problem

Problem: Consider a superquadric shape S of order n, parameterized by the set of parameters .

The optimization problem then consists in finding the optimal parameterscorresponding to the largest superquadric shape Sopt contained inside a feasible domain (supposed to be nonconvex), such that:

( ), 1

, 1,...,nl ui i i

F x

x x x i n

Ψ ≤ ≤ ≤ =

Ψ*Ψ

( )max nVΨ

Ψɶ

( ) ( )1

nn nV VΨ = Ψɶwhere : normalized volume

subject to:

( ),nF xΨ : inside-outside function



� Multi-population extended GA adapted to the convexification process

Optimal ConvexificationOptimization by Genetic Algorithm (GA)

Generate new

population

start stop

evaluation of offsprings

initialization


yes

mutation

recombination

fitness assigmentselection

no

reinsertion

migration

competition

optim criteria OK? best individuals


� Initial population: • small convex shapes inside the nonconvex feasible

domain• Random drawing of superquadrics and geometric

transformations tuning parameters in order to cover the whole search space

• Near-uniform sampling of the superquadric surface• Evaluation of initial population individuals (inside-outside

function constraint violation + volume)

� Genetic operators:• Selection: modified tournament selection operator• Crossover: simulated binary crossover (SBX)• Mutation: parameter-based mutation• Migration: complete net topology





� Assuming that some superquadrics have been found to satisfy the convexification sub-problem, the new optimal control problem can be written as:

� A convex cost functional can be found in the same way.

Optimal ConvexificationConvex Optimal Control Problem

( )( )( )( )( )

( )( )( )( )

0 0 0 0

0

0 0 0 0

*0

,

, ,

,

,

0 , 1, ,

t t t f

f f f f

n f

f f f f

l A z t u

l A z t u t t t

l A z t u

L c z t U

F z t t t t

L c z t U

≤ ≤

≤ ≤ ∈

≤ ≤

≤ ≤

≤ Ψ ≤ ∈

≤ ≤

ɶ

ɶ

ɶ

ɶ

ɶ

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

0

0 0minft

t f fz t

t

J z G z t G z t dt G z t= + +∫subject to:

Inside-outside function


Runway

Yrunway

Xrunway

Orbiterground track

TEP

Earth Horizon

HAC radius

NEP

Hypersonic phase

Zrunway

TAEM phase

Autolandingphase

Injection point

Application to the TAEM phase


� Missions: insertion in low-Earth orbit of of payloads and crews� First flight: 04/12/1981� Number of flights till 01/01/2007: 119� Cost: from 300 to 400 millions $ (2006) � 3 operational vehicles until 2010 (Shuttle retirement)

Main features

Max. L/D (for M < 3) ≈4

Weight (beginning of TAEM) 90 T

Wingspan 23.8 m

Reference area 250 m2

Max. roll rate 5 deg/s

Max. pitch rate 2 deg/s

Application to the TAEM phaseShuttle Orbiter


Application to the TAEM phaseEquations of motion� Assumptions: symmetric flight (β=0), flat Earth coordinates, and free trajectory duration λ ( ) ( ) ( ). .

. 'd d

d dtλ

τ= =, 0 1

tτ τλ

= ≤ ≤ : normalized time

( )

( )

2

2

1,

21

,2

D

L

D SV C M

L SV C M

ρ α

ρ α

= =

where and 0ref

h

Heρ ρ − = : simple density model

Application to the TAEM phaseEquations of motion

� The 3dof point-mass equations can be rewritten with respect to normalized time such that:

co s cos

s in co s

s in

x V

y V

h V

λ χ γλ χ γλ γ

′ = ′ = ′ =

sin

coscos

sin

cos

DV g

m

L g

mV V

L

mV

λ γ

µγ λ γ

µχ λγ

′ = − −

′ = −

′ =

Position: Velocity:


� However, all the states and inputs of the 3dof model can be rewritten as functions of the flat outputs , their derivatives, and the variable λ:( )hyx ,,

Application to the TAEM phaseFlatness property of the guidance model� Since β=0, the 3dof model is under-actuated and so, the system is not flat

• Additional equation: ( )2 ,1sin 0

2DSV C MV

gm

ρ αγ

λ′+ + =

cosarctan

cosgV

χ γµ γγ λ

′

= ′ +

( )0

1 1

2 cos

sinCL

am

a f M SV a

χ γαρ λ µ′

= −• Inputs:

2 2 21 2 3z z z

Vλ

′ ′ ′+ +=

( ) ( )( )

2 23 1 2 3 1 1 2 2

2 2 2 2 21 2 3 1 2

z z z z z z z z

z z z z zγ

′′ ′ ′ ′ ′ ′′ ′ ′′+ − +′ =

′ ′ ′ ′ ′+ + +

2 1 2 1

2 21 2

z z z z

z zχ

′′ ′ ′ ′′−′ =′ ′+

3

2 21 2

arctanz

z zγ

′ = ′ ′+

2

1

arctanz

zχ

′=

′

1 1 2 2 3 3

2 2 21 2 3

z z z z z zV

z z zλ

′ ′′ ′ ′′ ′ ′′+ +′ =′ ′ ′+ +

• States:


� Example: dynamic pressure constraint along the TAEM trajectory, expressed with respect to flat outputs

3 2 2 21 2 3

0

1

2ref

z

H z z zQ e Sρ

λ

− ′ ′ ′+ +=

� Nonconvex constraint: exponentially contracting spherical shape

� Simple GA tuning parameters have yield good results

Application to the TAEM phaseConvexification of trajectory constraintsConvexification of trajectory constraints

Application to the TAEM phaseConvexification of trajectory constraintsConvexification of trajectory constraints

Application to the TAEM phaseConvexification of trajectory constraints

� Each TAEM constraint can be convexified by using the same process


Application to the TAEM phaseConvexification of trajectory constraints


� LPV guidance controller so as to ensure the same level of performances along each TAEM/A&L trajectory

� Robustness and performances of the onboard path planner must be assessed

� Design of a fault-tolerant onboard path planner to take into account potential single/multiple actuator faults occurring during the TAEM/A&L flight segments.

Future works


THANK YOU FOR YOUR ATTENTION !

Onboard Terminal Area Energy Management Path Planning using Flatness Approach

Technology

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