An optimal control approach for minimum-fuel deployment of multiple - - PowerPoint PPT Presentation

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

An optimal control approach for minimum-fuel deployment of multiple spacecraft formation flying

Richard Epenoy

Richard.Epenoy@cnes.fr Centre National d’Etudes Spatiales 18, avenue Edouard Belin 31401 Toulouse Cedex 9 - France

CEA-EDF-INRIA School: Optimal control May 30 - June 1st 2007, INRIA Rocquencourt, France

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

Outline

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Problem statement

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Problem statement

Space dynamics equations

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Dynamics equations in cartesian coordinates

Two-body problem with perturbations and engine thrust ¨ − → r = −µ − → r − → r 3 + − → γp1 + − → γp2 µ: the Earth’s gravitational constant − → γp1: natural perturbative acceleration (geopotential disturbances, lunar and solar third body gravities, atmospheric drag, solar radiation pressure,...) − → γp2 =

− →

F m: perturbative acceleration caused by the thrust − → F : thrust vector of the engine m: mass of the satellite

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Orbital parameters (1/2)

Keplerian osculating elements a: semi-major axis e: eccentricity i: inclination ω: argument of perigee Ω: longitude of the ascending node v: true anomaly

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Orbital parameters (2/2) - Perturbative acceleration

Eccentric anomaly - Mean anomaly cos(E) = cos(v) + e 1 + e cos(v) sin(E) = √ 1 − e2sin(v) 1 + e cos(v) M = E − e sin(E) Perturbative acceleration in the (−

→ T ,− → N ,− → W ) local orbital frame

     − → γp = T− → T + N− → N + W − → W − → T = ˙ − → r ˙ − → r , − → W = − → r ∧ ˙ − → r − → r ∧ ˙ − → r , − → N = − → W ∧ − → T

S − → T − → N − → W

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Space dynamics equations in orbital parameters

Gauss equations                                             

˙ a(t) = 2V n2a T ˙ e(t) = 1 V

• 2 (e + cos(v))T − sin(v)

r a N

• ˙

i(t) = r cos(ω+v) n a2 √ 1−e2 W ˙ Ω(t) = r sin(ω+v) n a2 sin(i) √ 1 − e2 W ˙ ω(t) = 1 V e

• 2 sin(v)T +

2 e + (1 + e2)cos(v) 1 + e cos(v) N

• −

r cos(i) sin(ω+v) n a2 sin(i) √ 1−e2 W ˙ M(t) = n − √ 1 − e2 V e

• 2 sin(v)
• 1 +

e2 1 + e cos(v)

• T + cos(v)

1 − e2 1 + e cos(v) N

• with r =

a (1 − e2) 1 + e cos(v) , V =

• µ

2 r − 1 a

• and n =

µ a3

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Natural perturbations taken into account

Earth’s oblateness only - Atmospheric drag neglected The non-sphericity of the Earth yields gravitational perturbations:

1 The Earth’s gravity field representation in cartesian

coordinates is based on a spherical harmonic expansion

2 The Earth’s oblateness term J2 is the most important one

after the central term

3 The J2 term corresponds to a certain expression of the

perturbative accelerations T, N, and W

4 First effect of J2: short-period and long-period oscillations

with zero mean on the orbital parameters

5 Second effect: a secular effect, i.e. a linear drift, on Ω

(rotation of the orbital plane), ω and M

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Gauss equations with equinoctial elements and J2 (1/2)

State and control variables Let us consider a formation of n satellites xj: equinoctial orbital parameters for satellite Sj (j = 1, . . . , n) xj =         aj ex,j = ej cos (ωj + Ωj) ey,j = ej sin (ωj + Ωj) hx,j = tan (ij/2) cos (Ωj) hy,j = tan (ij/2) sin (Ωj) Lj = ωj + Ωj + vj         (aj, ej, ij, ωj, Ωj, vj): Keplerian osculating elements for Sj mj: mass of satellite Sj uj: normalized thrust vector for Sj in the (− → T , − → N , − → W ) frame

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Gauss equations with equinoctial elements and J2 (2/2)

State equations in compact form            ˙ xj(t) = f (xj(t)) + Fmax g (xj(t)) uj(t) mj(t) ˙ mj(t) = −Fmax uj(t) g0 Isp t ∈ [t0, tf ] (j = 1, . . . , n) Fmax: maximum thrust modulus of the n engines Isp: specific impulse of the n engines g0: acceleration due to gravity at sea level t0 and tf : fixed initial and final dates

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Problem statement

Optimal control formulation

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

The problem to solve

Minimum-fuel deployment                                          Min J (u1, . . . , un) = −

n

• j=1

mj(tf ) ˙ xj(t) = f (xj(t)) + Fmax g (xj(t)) uj(t) mj(t) ˙ mj(t) = −Fmax uj(t) g0 Isp uj(t) ≤ 1 t ∈ [t0, tf ] xj(t0) = xj,0 mj(t0) = m0 ψj (xj(tf )) = 0 (j = 1, . . . , n) φ (x1(tf ), . . . , xn(tf )) = 0

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Terminal conditions (1/2)

Non coupling conditions in our application (n = 4) ψj (xj(tf )) =       aj(tf ) − aj,f ex,j(tf ) − ex,j,f ey,j(tf ) − ey,j,f hx,j(tf )2 + hy,j(tf )2 − tan2 ij,f 2

• 

     (j = 1, . . . , n) where aj,f , ex,j,f , ey,j,f , and ij,f (j = 1, . . . , n) are given Remark ex,j,f = ey,j,f = 0 and ij,f = if (j = 1, . . . , n) in our application

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Terminal conditions (2/2)

Coupling conditions in our application

φ(x1(tf ),...,xn(tf )) =      φ1(x1(tf ),x2(tf ))

. . .

φn−1(xn−1(tf ),xn(tf ))      with, for j=1,...,n−1, φj(xj(tf ),xj+1(tf )) =       hx,j(tf ) hx,j+1(tf ) + hy,j(tf ) hy,j+1(tf ) − tan2 if

2

• cos(δΩj,f )

hx,j(tf ) hy,j+1(tf ) − hy,j(tf ) hx,j+1(tf ) − tan2 if

2

• sin(δΩj,f )

tan

• Lj+1(tf ) − Lj (tf )

2

• − tan

δLj,f

2

• 

    

Remark

φj(xj(tf ),xj+1(tf )) = 0 = ⇒   Ωj+1(tf ) − Ωj(tf ) = δΩj,f Lj+1(tf ) − Lj(tf ) = δLj,f modulo 2π  

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Space dynamics equations Optimal control formulation

Additional constraint - Balance of the fuel consumption

A nonlinear terminal inequality constraint θ (m1(tf ), . . . , mn(tf )) = −

n

• j=1

ζj log ζj log(n) ≥ θ where θ ∈ [0, 1] and ζj = mj(tf )

n

• k=1

mk(tf ) (j = 1, . . . , n) θ = 0: no constraint θ = 1: perfect balance of the consumption

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

The solution approach

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

The solution approach

Pontryagin’s Maximum Principle

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

Bang-off-bang controls

Optimal controls

u∗

j (t) = argmin

w≤1

• −

pmj (t) g0Isp w + pxj (t)T g(xj(t)) w mj(t)

• ,

t ∈ [t0,tf ] Let ρj(t) = Fmax   g(xj(t))T pxj (t) mj(t) + pmj (t) g0Isp  . If g(xj(t))T pxj (t) = 03,1, u∗

j (t) = −βj(t)

g(xj(t))T pxj (t) g(xj(t))T pxj (t) , with βj(t) =        if ρj(t) < 0 1 if ρj(t) > 0 w ∈ [0,1] if ρj(t) = 0 If g(xj(t))T pxj (t) = 03,1, u∗

j (t) is such that

u∗

j (t) = βj(t) Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

Numerical difficulties

Problem 1: limitation of the deployment duration in LEO Short-period oscillations due to J2 on the osculating parameters = ⇒ small integration stepsize and time consumption Problem 2: difficulties for the adaptive integration scheme Ordinary Differential Equations with discontinuous right-hand sides = ⇒ the integration scheme hardly reaches the desired accuracy Problem 3: difficulties for Newton’s method Nonsmooth shooting function with singular Jacobian matrix = ⇒ very small convergence radius for Newton’s method A solution for the last two problems Smoothing the control law

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

The solution approach

The continuation-smoothing method

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

Principles of the method (1/3)

Penalty - barrier approach                                          Min Jǫ (u1, . . . , un) = −

n

• j=1
• mj(tf ) + ǫ

tf

t0

F (uj(t)) dt

• ˙

xj(t) = f (xj(t)) + Fmax g (xj(t)) uj(t) mj(t) ˙ mj(t) = −Fmax uj(t) g0 Isp uj(t) ≤ 1 t ∈ [t0, tf ] xj(t0) = xj,0 mj(t0) = m0 ψj (xj(tf )) = 0 (j = 1, . . . , n) φ (x1(tf ), . . . , xn(tf )) = 0

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

Principles of the method (2/3)

The penalty approach

F 1(w) = w(1 − w) ≥ 0, ∀ w ∈ [0,1] If g(xj(t))T pxj (t) = 03,1, u∗

ǫ,j(t) = −βǫ,j(t)

g(xj(t))T pxj (t) g(xj(t))T pxj (t) , with βǫ,j(t) =            if ρj(t) ≤ −ǫ 1 if ρj(t) ≥ ǫ 1 2 + ρj(t) 2ǫ if ρj(t) ∈ [−ǫ, ǫ] If g(xj(t))T pxj (t) = 03,1, u∗

ǫ,j(t) is such that

u∗

ǫ,j(t)) = βǫ,j(t) Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

Principles of the method (3/3)

The logarithmic barrier approach

F 2(w) = log(w) + log(1−w) ∀ w ∈ ]0,1[ If g(xj(t))T pxj (t) = 03,1, u∗

ǫ,j(t) = −βǫ,j(t)

g(xj(t))T pxj (t) g(xj(t))T pxj (t) , with βǫ,j(t) = 2ǫ 2ǫ − ρj(t) +√ ρj(t)2 + 4ǫ2 If g(xj(t))T pxj (t) = 03,1, u∗

ǫ,j(t) is such that

u∗

ǫ,j(t)) = βǫ,j(t) Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Pontryagin’s Maximum Principle The continuation-smoothing method

Continuation procedure - Convergence results

Continuation procedure ǫ1 ≥ ǫ2 ≥ · · · ≥ ǫN until

• Jǫk+1(u∗

ǫk+1) − Jǫk

• u∗

ǫk

• ≤ (ǫk − ǫk+1) η

Proposition 1 Jǫ1

• u∗

ǫ1

• ≥ Jǫ2
• u∗

ǫ2

• ≥ · · · ≥ JǫN
• u∗

ǫN

• ≥ J (u∗) ≥ J
• u∗

ǫk

• k = 1...N

Proposition 2 lim

ǫ→0 Jǫ (u∗ ǫ ) = J (u∗) and lim ǫ→0 J (u∗ ǫ ) = J (u∗)

Reference

• R. Epenoy, R. Bertrand: ”New smoothing techniques for solving bang-bang optimal

control problems - Numerical results and statistical interpretation” Optimal Control Applications and Methods, Vol. 23, No. 4, pp. 171-197, 2002

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Numerical results - A deployment in Low Earth Orbit

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Numerical results - A deployment in Low Earth Orbit

Statement of the test case

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Initial and final conditions (1/2)

Initial configuration a0 = 7029.48 km e0 = 1.27 × 10−3 i0 = 98.08 deg ω0 = 214.51 deg Ω0 = 209.80 deg mj(t0) = 120 kg (j = 1, . . . , 4) v2(t0) − v1(t0) = 0.0163 deg v1(t0) − v3(t0) = 0.0163 deg v3(t0) − v4(t0) = 0.0163 deg Engines characteristics Fmax = 4 N and Isp = 210 s

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Initial and final conditions (2/2)

Final configuration aj,f = 7031 km (j = 1, . . . , 4) ej,f = 0 (j = 1, . . . , 4) ij,f = 98.08 deg (j = 1, . . . , 4) δΩ1,f = 0.0 deg δΩ2,f = +0.47 deg δΩ3,f = 0.0 deg δL1,f = +0.41 deg δL2,f = −0.14 deg δL3,f = −0.41 deg d = 50 km d′ ≈ 100 km

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Numerical results - A deployment in Low Earth Orbit

Deployment over seven days - Seven local solutions found

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

The seven solutions

Consumption and distribution features sol c (kg) (1 − θ0) ∆mmax (kg) 1 10.62 2.0 × 10−6 5.83 × 10−1 2 8.84 1.7 × 10−5 1.97 3 8.54 7.5 × 10−5 4.06 4 7.34 1.4 × 10−11 1.61 × 10−3 5 7.12 1.5 × 10−7 1.60 × 10−1 6 7.05 1.2 × 10−7 1.38 × 10−1 7 6.58 4.7 × 10−8 8.61 × 10−2 c =

n

• j=1

(m0 − mj(tf )), ∆mmax = max

1≤k<j≤n |mj(tf ) − mk(tf )|

and θ0 = θ (m1(tf ), . . . , mn(tf ))

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

The best solution obtained - Solution no 7 - 27 maneuvers

Differential parameters between S2 and S3 vs time (argument of true latitude: α = ω+v)

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Influence of J2

Secular effect of J2 in near-circular orbits (e ≃ 0)                                          ˙ a(t) = 0 ˙ e(t) = 0 ˙ i(t) = 0 ˙ ω(t) ≃ 3 4 R2 a7/2 √µJ2(4 − 5sin(i)2) ˙ Ω(t) ≃ −3 2 R2 a7/2 √µJ2cos(i) ˙ M(t) ≃ n + 3 4 R2 a7/2 √µJ2(2 − 3sin(i)2) R : the Earth’s equatorial radius J2 : the second zonal term of the Earth potential

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

The best solution obtained - Space dynamics interpretation

The strategy Two combined effects for obtaining the targeted difference in Ω at t = tf (δΩ2,f = 0.47 deg), between S2 and S3:

1 The differential drift in Ω, due to J2, that relates to an

inclination difference between the orbits ∆i ≈ +0.08 deg, caused by out-of-plane and opposed thrusts on both satellites

2 The instantaneous effect on Ω due to the same thrusts,

located at both beginning and end of the deployment In addition, a non significant semi-major axis difference ∆a ≈ +90 m, resulting from very small tangential maneuvers creates the difference in true longitude δL2,f = −0.14 deg at t = tf

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

An example of alternative solution - Solution no 5

Differential parameters between S2 and S3 versus time

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

The alternative solution - Space dynamics interpretation

The strategy Two combined effects for obtaining the targeted difference in Ω at t = tf (δΩ2,f = 0.47 deg), between S2 and S3:

1 The differential drift in Ω, due to J2, that relates to

A semi-major axis difference between the orbits ∆a ≈ −45 km, caused by tangential and opposed thrusts on both satellites An inclination difference ∆i ≈ +0.05 deg, inferior to the value

• btained for the best solution, due to out-of-plane thrusts

2 The instantaneous effect on Ω due to the thrusts

In addition, the semi-major axis difference enables to create the difference in true longitude δL2,f = −0.14 deg at t = tf

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Evolution of the best solution versus tf (1/2)

Evolution of the ∆i between S2 and S3 versus tf

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Evolution of the best solution versus tf (2/2)

Space dynamics interpretation

1 The maximum value of ∆i is obtained for tf = 16.5 days 2 The differential drift in Ω due to J2 and caused by ∆i

continuously grows with tf

3 The fuel consumption decreases as tf grows, which shows that

the strategy benefits more and more from the J2 effect

4 For tf ≤ 16.5 days, the direct effet of the thrusts on Ω is

predominant over the differential drift due to J2 for obtaining δΩ2,f = 0.47 deg at t = tf

5 For tf = 16.5 days, both effects are equal 6 For tf ≥ 16.5 days, the differential drift effect is predominant 7

lim

tf →∞ ∆i = 0

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Numerical results - A deployment in Low Earth Orbit

Balancing the fuel consumption

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Taking into account the inequality constraint

A continuation method θ (m1(tf ), . . . , mn(tf )) ≥ θ is replaced by θ (m1(tf ), . . . , mn(tf )) = η with η ∈ [θ0, θ] θ0: value of θ at a local minimum of the unconstrained problem Optimality concerns - Nonconvex problem If θ0 is sufficiently closed to θ, we hope that the continuation process does not converge to a solution of the initial problem with constraint θ (m1(tf ), . . . , mn(tf )) = θ that would not be optimal for our inequality constrained problem

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

Numerical results - An example

sol c (kg) (1 − θ0) ∆mmax (kg) 1 10.62 2.0 × 10−6 5.83 × 10−1 2 8.84 1.7 × 10−5 1.97 3 8.54 7.5 × 10−5 4.06 4 7.34 1.4 × 10−11 1.61 × 10−3 5 7.12 1.5 × 10−7 1.60 × 10−1 6 7.05 1.2 × 10−7 1.38 × 10−1 7 6.58 4.7 × 10−8 8.61 × 10−2 32 8.72 1.7 × 10−5 1.88 Solution sol3 is brought by continuation to a solution denoted sol32 with a similar fuel balance than sol2. The resulting solution is based on the same ”strategy” than sol3.

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

Conclusion and future prospects

1

Problem statement Space dynamics equations Optimal control formulation

2

The solution approach Pontryagin’s Maximum Principle The continuation-smoothing method

3

Numerical results - A deployment in Low Earth Orbit Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption

4

Conclusion and future prospects

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

Conclusion

Concerning the method Efficient extension of the continuation-smoothing technique to the multi-satellite framework Local minima A globalization technique allows to find out the problem’s local minima Fuel balance The method enables to accurately balance the fuel consumption among the satellites

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

Future prospects

Additional constraint - Collision avoidance The collision risk between the satellites exists = ⇒ necessity to take into account a collision avoidance constraint = ⇒ nonconvex state constraint difficult to handle through PMP Other type of applications Deployment in a High Elliptical Orbit for example = ⇒ necessity to take into account additional perturbative forces: gravitational influence of the Moon and the Sun, solar radiation pressure, ... = ⇒ necessity to regenerate the costate equations with ADIFOR∗

∗C. Bischof, A. Carle, G. Corliss, A. Griewank, P. Hovland: ”ADIFOR - Generating

Derivative Codes from Fortran Programs” Scientific Progr., no. 1, pp. 1-29, 1992

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

Recent and forthcoming publications

Note technique CNES

J.-B. Thevenet, R. Epenoy: ”Trajectoires en consommation minimale pour le d´ eploiement d’une formation de satellites - Techniques de commande optimale” Note Technique du CNES No 151, Mars 2007 (in french)

EUCASS 2007

J.-B. Thevenet, R. Epenoy: ”An optimal control approach for minimum-fuel deployment of multiple spacecraft formation flying” To appear in the proceedings of the 2nd European Conference for Aerospace Sciences (EUCASS), July 1-6 2007, Brussels, Belgium

Journal of Guidance, Control and Dynamics

J.-B. Thevenet, R. Epenoy: ”Minimium-fuel deployment for formation flying of satellites - An optimal control approach” Submitted to Journal of Guidance, Control and Dynamics

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment

Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects

Thank you for your attention

Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment