Fuel-Optimal Trajectories for Continuous-Thrust Orbital
Rendezvous with Path Constraints
Richard Epenoy
To cite this version:
Richard Epenoy. Fuel-Optimal Trajectories for Continuous-Thrust Orbital Rendezvous with
Path Constraints. SADCO A2CO, Mar 2011, Paris, France.
HAL Id: inria-00577164
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Submitted on 16 Mar 2011
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Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Fuel-Optimal Trajectories for Continuous-Thrust
Orbital Rendezvous with Path Constraints
Richard Epenoy
[email protected]
Centre National d’Etudes Spatiales
18, avenue Edouard Belin 31401 Toulouse Cedex 9, France
Aerospace Applications of Control and Optimization,
ENSTA ParisTech, Paris, March 2, 2011
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Outline
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Dynamical equations
Optimal control formulation
Problem statement
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Dynamical equations
Optimal control formulation
Problem statement
Dynamical equations
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Dynamical equations
Optimal control formulation
Tschauner-Hempel equations in Hill’s frame
Keplerian motions - Small intersatellite distance
(a, e, v ): semi-major axis, eccentricity and true anomaly of the
target satellite
X (v ), Y (v ), Z (v ): relative coordinates of the chaser




x1 (v )
X (v )
 x2 (v )  = (1 + ecos(v ))  Y (v ) 
x3 (v )
Z (v )
x4 (v ), x5 (v ), x6 (v ): derivatives of xi (v ), (i = 1, ..., 3) w.r.t. v
m(v ): mass of the chaser at true anomaly v
u(v ): normalized thrust vector of the chaser at true anomaly v
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Dynamical equations
Optimal control formulation
Problem statement
Optimal control formulation
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Dynamical equations
Optimal control formulation
State-constrained minimum-fuel rendezvous (1/2)
The problem to solve

Find u = argmin J(u) = −m(vf )



u



s.t.




u(v )


ẋ(v ) = A(v )x(v ) + B(v )


m(v
)



ṁ(v ) = −c(v ) ku(v )k
(P)


ku(v )k ≤ 1 v ∈ [v0 , vf ]






g (v , x(v )) ≤ 0 v ∈ [v0 , vf ]





x(v0 ) = x0
h (x(vf )) = 0




m(v0 ) = m0
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Dynamical equations
Optimal control formulation
State-constrained minimum-fuel rendezvous (2/2)
Collision avoidance constraint
p
x1 (v )2 + x2 (v )2 + x3 (v )2
g (v , x(v )) = 1 −
≤ 0 v ∈ [v0 , vf ]
dmin (1 + ecos(v ))
Key parameter
dmin : minimum safety distance between the chaser and the target
Main issues for shooting methods
The control is bang-off-bang =⇒ numerical difficulties
State constraint =⇒ the number and location of constrained
arcs must be defined beforehand in order to build the MPBVP
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Solving the path-constrained rendezvous problem
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (1/2)
A logarithmic barrier approach
Rv

Find u δ = argmin Jδ (u) = J(u) − δ v0f F (v , u(v )) dv



u


 s.t.




u(v )


ẋ(v ) = A(v )x(v ) + B(v )


m(v )


ṁ(v
)
=
−c(v
)
ku(v
)k
(P)δ


ku(v )k ≤ 1 v ∈ [v0 , vf ]





g (v , x(v )) ≤ 0 v ∈ [v0 , vf ]






x(v0 ) = x0
h (x(vf )) = 0



m(v0 ) = m0
F (v , u(v )) = c(v ) (log (ku(v )k) + log (1 − ku(v )k))
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (2/2)
Convergence results - No state constraint
Jδ (u δ ) → J(u) as δ → 0
u δ → u for the weak-* topology on L∞ [v0 , vf ] , R3 as δ → 0
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (2/2)
Reference
R. Epenoy and 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, 2002, pp. 171-197.
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Solving the path-constrained rendezvous problem
A new approach to deal with the state constraint
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
The smoothed exact penalty method
The penalized problem (P)2σ,α,
R vf

2
2
Find
u
=
argmin
J
(u)
=
J
(u)
+

δ
σ,α,
σ,α,
v0 ψσ,α, (g (v , x(v ))) dv


u



s.t.




u(v )



 ẋ(v ) = A(v )x(v ) + B(v ) m(v )
ṁ(v ) = −c(v ) ku(v )k





ku(v )k ≤ 1 v ∈ [v0 , vf ]





h (x(vf )) = 0

 x(v0 ) = x0


m(v0 ) = m0
z z
ψσ,α, (z) = σlog 1 + exp
1 +
−→ Max 0,
1 +
σ→0
σ
α − z
α − z
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
The smoothed exact penalty method
The penalized problem (P)2σ,α,
R vf

2
2
Find
u
=
argmin
J
(u)
=
J
(u)
+

δ
σ,α,
σ,α,
v0 ψσ,α, (g (v , x(v ))) dv


u



s.t.




u(v )



 ẋ(v ) = A(v )x(v ) + B(v ) m(v )
ṁ(v ) = −c(v ) ku(v )k





ku(v )k ≤ 1 v ∈ [v0 , vf ]





h (x(vf )) = 0

 x(v0 ) = x0


m(v0 ) = m0
G. Liuzzi and S. Lucidi: A Derivative-Free Algorithm for Inequality Constrained
Nonlinear Programming via Smoothing of an l∞ Penalty Function, SIAM Journal on
Optimization, Vol. 20, No. 1, 2009, pp. 1-29.
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Algorithm
1
Let 0 < q1 < q2 < 1, 0 < αlim < α0 , 0 > 0, σ0 > αqlim
, 0 < θ < 1, 0 < τ < 1
Let k = 0, end = false
WHILE (end = false)
2
Solve problem (P)2σk ,αk ,k → xσ2 k ,αk ,k , mσ
, uσ2 k ,αk ,k
k ,αk ,k
IF (αk ≤ αlim ) THEN
end = true
2
, uσ2 k ,αk ,k
(x δ , mδ , u δ ) ← xσ2 k ,αk ,k , mσ
k ,αk ,k
ELSE
n R
o
αq 2
IF Min k , vvf Max 0, g (v , xσ2 k ,αk ,k (v )) dv > k THEN
0
σk
αqk2
k+1 = τ
σk
ELSE
k+1 = k
ENDIF
αk+1 = θαk n
o
σk+1 = Min
k =k +1
ENDIF
END WHILE
1
σk , αqk+1
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Convergence results
Lemma
Let α > 0, z < α and > 0 be given. Then, (σ → ψσ,α, (z))
is strictly increasing on ]0, +∞[
Let z < 0, > 0 and σ > 0 be given. Then, (α → ψσ,α, (z))
is strictly increasing on ]0, +∞[
Convergence theorem - See Epenoy (2011) in JGCD
Jσ2k ,αk ,k uσ2k ,αk ,k → Jδ (u δ ) as k → ∞
uσ2k ,αk ,k → u δ according to the weak-* topology on
L∞ [v0 , vf ] , R3 as k → ∞
xσ2k ,αk ,k → x δ uniformly on [v0 , vf ] as k → ∞
mσ2 k ,αk ,k → mδ uniformly on [v0 , vf ] as k → ∞
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Numerical results - A rendezvous in Highly Elliptical Orbit
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Numerical data - SIMBOL-X project
a = 106246.9753 km
Fmax = 0.1 N
e = 0.798788
Isp = 220 s
v0 = 3.317940017547 rad
t0 = 0.0 s
vf = 3.349161118514 rad
tf = 8000.0 s


 

X (tf )
X (t0 )
−100 m
 Y (tf )
 Y (t0 )  =  −100 m 
Z (tf )
Z (t0 )
−100 m



dX
dX
(t0 )
(t )


 dt

 dt f
0.0 m/s
 dY




 dY
(t0 )  =  0.0 m/s 

(t )

 dt

 dt f
0.0 m/s
 dZ

 dZ
(t0 )
(tf )
dt
dt
Richard Epenoy
m0 = 960.0 kg



500 m
 =  500 m 
500 m


 
0.0 m/s

 
 = 0.0 m/s 

0.0 m/s

Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Numerical results - A rendezvous in Highly Elliptical Orbit
Unconstrained rendezvous
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Norm of the normalized thrust vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Normalized thrust vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Intersatellite distance
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Numerical results - A rendezvous in Highly Elliptical Orbit
Rendezvous under collision avoidance constraint
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Parameters of the algorithm
Smoothing parameter: The same as for the unconstrained problem
δ = 9.879x10−9
Case 1: dmin = 50.0 m
q1 = 0.3
q2 = 0.6
αlim = 4.73x10−4
α0 = 0.98
0 = 0.0999
σ0 = 1.0
θ = 0.99
τ = 0.9
Case 2: dmin = 140.0 m
q1 = 0.3
q2 = 0.6
αlim = 1.37x10−4
α0 = 0.98
0 = 0.0157
σ0 = 1.0
θ = 0.99
τ = 0.9
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Fuel consumption vs. iteration index k
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Minimum distance vs. iteration index k
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Norm of the normalized thrust vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Intersatellite distance
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Normalized thrust vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Relative position vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Relative velocity vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Norm of the normalized thrust vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Intersatellite distance
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Normalized thrust vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Relative position vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Relative velocity vector
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Conclusion and future prospects
1
Problem statement
Dynamical equations
Optimal control formulation
2
Solving the path-constrained rendezvous problem
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
3
Numerical results - A rendezvous in Highly Elliptical Orbit
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
4
Conclusion and future prospects
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Conclusion
The smoothed exact penalty approach
Applied for the first time to an optimal control problem
Efficient to deal with the collision avoidance constraint
Implies solving a sequence of state constraint-free problems
More theoretically based than other penalization techniques
Can be used to solve a large class of problems
A just published paper
R. Epenoy: Fuel Optimization for Continuous-Thrust Orbital Rendezvous
with Collision Avoidance Constraint, Journal of Guidance, Control and
Dynamics, Vol. 34, No. 2, March-April 2011.
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Future prospects
Minimum-fuel rendezvous in perturbed environment
Necessity to modify the dynamical equations
Low Earth Orbit applications =⇒ J2 term of the Earth’s potential
Application to other state-constrained problems
Reentry trajectories under thermal flux constraint
Interplanetary trajectories with minimum flyby altitude constraint
Toward closing the loop
Model Predictive Control methodology
Neighboring extremal paths in the presence of state constraints
Hamilton-Jacobi-Bellman equation for state-constrained problems
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and future prospects
Thank you for your attention
Richard Epenoy
Fuel-Optimal Trajectories for Path-Constrained Rendezvous