Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Fuel-Optimal Continuous-Thrust Orbital
Rendezvous under Collision Avoidance Constraint
Richard Epenoy
[email protected]
Centre National d’Etudes Spatiales
18 avenue Edouard Belin 31401 Toulouse Cedex 9, France
Workshop on Advances in Space Rendezvous Guidance
October 30-31, 2013, LAAS-CNRS, Toulouse
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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 related work
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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 related work
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Dynamical equations
Optimal control formulation
Tschauner-Hempel equations in Hill’s frame
Keplerian motions - Small intersatellite distance
(a, e, v ): semimajor 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
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Dynamical equations
Optimal control formulation
State-constrained minimum-fuel rendezvous (1/2)
The problem to solve

Find u = argmin J(uu ) = −m(vf )



u



s.t.




u (v )



ẋx (v ) = A(v )xx (v ) + B(v )


m(v )


 ṁ(v ) = −c(v )kuu (v )k
(P)


kuu (v )k ≤ 1 v ∈ [v0 , vf ]






g (v , x (v )) ≤ 0 v ∈ [v0 , vf ]






x (v0 ) = x 0
h (xx (vf )) = 0




m(v0 ) = m0
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Dynamical equations
Optimal control formulation
State-constrained minimum-fuel rendezvous (2/2)
Collision avoidance: a second-order inequality state 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 indirect shooting methods
1
The control is bang-off-bang =⇒ the shooting function is not
C 1 and its Jacobian is singular on a large domain
2
State constraint =⇒ necessity to a priori define the number
and nature of the arcs constituting the optimal trajectory
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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 related work
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (1/4)
A logarithmic barrier approach [Bertrand and Epenoy 2002]†

Rv
Find u δ = argmin Jδ (uu ) = J(uu ) − δ v0f F (v , u (v ))dv



u



s.t.




u (v )



ẋx (v ) = A(v )xx (v ) + B(v )


m(v )
(P)δ > 0
ṁ(v ) = −c(v )kuu (v )k





g (v , x (v )) ≤ 0 v ∈ [v0 , vf ]






x (v0 ) = x 0
h (xx (vf )) = 0




m(v0 ) = m0
F (v , u (v )) = c(v ) (log (kuu (v )k) + log (1 − kuu (v )k))
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (1/4)
A logarithmic barrier approach [Bertrand and Epenoy 2002]†

Rv
Find u δ = argmin Jδ (uu ) = J(uu ) − δ v0f F (v , u (v ))dv



u



s.t.




u (v )



ẋx (v ) = A(v )xx (v ) + B(v )


m(v )
(P)δ > 0
ṁ(v ) = −c(v )kuu (v )k





g (v , x (v )) ≤ 0 v ∈ [v0 , vf ]






x (v0 ) = x 0
h (xx (vf )) = 0




m(v0 ) = m0
Method: Solve (P)δi , (i = 0, . . . , n) with δ0 > δ1 > · · · > δn > 0
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (1/4)
A logarithmic barrier approach [Bertrand and Epenoy 2002]†

Rv
Find u δ = argmin Jδ (uu ) = J(uu ) − δ v0f F (v , u (v ))dv



u



s.t.




u (v )



ẋx (v ) = A(v )xx (v ) + B(v )


m(v )
(P)δ > 0
ṁ(v ) = −c(v )kuu (v )k





g (v , x (v )) ≤ 0 v ∈ [v0 , vf ]






x (v0 ) = x 0
h (xx (vf )) = 0




m(v0 ) = m0
† R. Bertrand and R. Epenoy: “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
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (2/4)
An illustrative example

R2
R2
Find u δ = argmin Jδ (u) = 0 |u(t)|dt − δ 0 F (t, u(t))dt



u



s.t.




ẋ(t) = −x(t) + u(t)




|u(t)| ≤ 1 t ∈ [0, 2]




1

 x(0) = 0
x(2) =
2
F (t, u(t)) = log (|u(t)|) + log (1 − |u(t)|)
Shooting function
Sδ : z = λδ,z (0) 7−→ Sδ (z) = xδ,z (2) −
Richard Epenoy
1
2
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (3/4)
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Problem regularization (4/4)
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Indirect solution of state-constrained problems (1/2)
Existing penalty methods
Poor numerical efficiency in general
L. S. Lasdon, A. D. Waren and R. K. Rice: “An Interior Penalty Method for Inequality
Constrained Optimal Control Problems,” IEEE Transactions on Automatic Control, Vol.
AC-12, No. 4, pp. 388-395, 1967.
K. L. Teo and L. S. Jennings: “Nonlinear Optimal Control Problems with Continuous
State Inequality Constraints,” Journal of Optimization Theory and Applications, Vol.
63, No. 1, pp. 1-22, 1989.
A. Xing: “The Non-parameter Penalty Function Method in Constrained Optimal Control
Problems,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 4, No. 2,
pp. 165-174, 1991.
E. Polak, T. H. Yang and D. Q. Mayne: “A Method of Centers Based on Barrier
Functions for Solving Optimal Control Problems with Continuous State and Control
Constraints,” SIAM Journal on Control and Optimization, Vol. 31, No. 1, pp. 159179, 1993.
M. C. Bartholomew-Biggs: “A Penalty Method for Point and Path State Constraints
in Trajectory Optimization,” Optimal Control Applications and Methods, Vol. 16, No.
1-4, pp. 291-297, 1995.
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Indirect solution of state-constrained problems (2/2)
Feasible directions methods
R. Pytlak and R. B. Vinter: “A Feasible Directions Algorithm for Optimal Control
Problems with State and Control Constraints: Convergence Analysis,” SIAM Journal
on Control and Optimization, Vol. 36, No. 6, pp. 1999-2019, 1998.
R. Pytlak and R. B. Vinter: “A Feasible Directions Algorithm for Optimal Control
Problems with State and Control Constraints: Implementation,” Journal of Optimization Theory and Applications, Vol. 101, No. 3, pp. 623-649, 1999.
Use of saturation functions
Autonomous constraint g(x(t)) ≤ 0, necessity to use normal form coordinates
K. Graichen and N. Petit: “Incorporating a Class of Constraints Into the Dynamics of
Optimal Control Problems,”Optimal Control Applications and Methods, Vol. 30, No.
6, pp. 537-561, 2009.
Homotopy approach
Regularity assumptions plus necessity to manage different formulations of the PMP
A. Hermant: “Homotopy Algorithm for Optimal Control Problems with a Second-order
State Constraint,” Applied Mathematics and Optimization, Vol. 61, No. 1, pp. 85-127,
2010.
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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,
Rv

2
Find u 2σ,α, = argmin Jσ,α,
(uu ) = Jδn (uu ) + v0f ψσ,α, (g (v , x (v )))dv


u




s.t.




u (v )

 ẋx (v ) = A(v )xx (v ) + B(v )
m(v )


ṁ(v ) = −c(v )kuu (v )k







x (v0 ) = x 0
h (xx (vf )) = 0



m(v0 ) = m0
z
z 1 +
−→ Max 0,
1 +
ψσ,α, (z) = σlog 1 + exp
σ→0
σ
α − z
α − z
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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,
Rv

2
Find u 2σ,α, = argmin Jσ,α,
(uu ) = Jδn (uu ) + v0f ψσ,α, (g (v , x (v )))dv


u




s.t.




u (v )

 ẋx (v ) = A(v )xx (v ) + B(v )
m(v )


ṁ(v ) = −c(v )kuu (v )k







x (v0 ) = x 0
h (xx (vf )) = 0



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, pp. 1-29, 2009.
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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 → x 2σk ,αk ,k , mσ
, u 2σk ,αk ,k
k ,αk ,k
IF (αk ≤ αlim ) THEN
end = true
2
, u 2σk ,αk ,k
(xx δn , mδn , u δn ) ← x 2σk ,αk ,k , mσ
k ,αk ,k
ELSE
αk+1 = θαk n
o
1
σk+1 = Min σk , αqk+1
n R
o
αq2
IF Min k , vvf Max(0, g (v , x 2σk ,αk ,k (v )))dv > k THEN
0
σk
αqk2
k+1 = τ
σk
ELSE
k+1 = k
ENDIF
k =k +1
ENDIF
END WHILE
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Smoothing the bang-off-bang control
A new approach to deal with the state constraint
Convergence theorem [Epenoy 2011 in JGCD]
Assumption: An index k0 ≥ 0 exists such that, for k ≥ k0 ,
g v , x 2σk ,αk ,k (v ) ≤ 0 for v ∈ [v0 , vf ]. Then, the following results hold:
1
For all k ≥ k0 , k = k0
The sequence Jσ2k ,αk ,k u 2σk ,αk ,k k≥k is strictly decreasing and
0
converges toward Jδn (uu δn ) as k → ∞
u 2σk ,αk ,k k≥k admits a subsequence u 2σk ,αk ,k
such that
j
j
j
0
j
3
u 2σk ,αk ,k
→ u δn according to the weak-* topology on
j
j
j
j
L∞ [v0 , vf ] , R3 as j → ∞
4
→ x δn uniformly on [v0 , vf ] as j → ∞
x 2σk ,αk ,k
j
j
j
j
5
→ mδn uniformly on [v0 , vf ] as j → ∞
mσ2 k ,αk ,k
2
j
j
j
j
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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 related work
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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 )
(tf )

 
 dt

dt
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

A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Norm of the normalized thrust vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Normalized thrust vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Intersatellite distance
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
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
δn = 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
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Fuel consumption vs. iteration index k
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Minimum distance vs. iteration index k
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Norm of the normalized thrust vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Normalized thrust vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Intersatellite distance
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Relative position vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 1 - Relative velocity vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Norm of the normalized thrust vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Normalized thrust vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Intersatellite distance
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Relative position vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Statement of the test case
Unconstrained rendezvous
Rendezvous under collision avoidance constraint
Case 2 - Relative velocity vector
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Conclusion and related work
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 related work
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Conclusion
The smoothed exact penalty approach
Applied for the first time to an optimal control problem
Efficient to deal with the collision avoidance constraint
More theoretically based than other penalization techniques
Can be used to solve a large class of problems
Reference
R. Epenoy: “Fuel Optimization for Continuous-Thrust Orbital Rendezvous
with Collision Avoidance Constraint,” Journal of Guidance, Control and
Dynamics, Vol. 34, No. 2, pp. 493-503, 2011.
Application to other state-constrained problems
Minimum-fuel rendezvous in perturbed environment
Reentry trajectories under thermal flux constraint
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Related work
Feedback control under collision avoidance constraint
G. Lantoine and R. Epenoy: “Quadratically Constrained Linear Quadratic
Regulator Approach for Finite-Thrust Orbital Rendezvous,” Journal of
Guidance, Control and Dynamics, Vol. 35, No. 6, pp. 1787-1797, 2012.
Quadratically-constrained discrete-time LQR
I The mass of the chaser is assumed to be constant
I The cost function is quadratic
I Thrust arcs are approximated by impulsive velocity increments
I Use of Yamanaka-Ankersen transition matrix
I Quadratic control and state inequality constraints
State space partition in subregions associated with sets of active
constraints
High-order Taylor series expansions of the HJB equation on each
subregion
Offline explicit computation of the feedback control law
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous
Problem statement
Solving the path-constrained rendezvous problem
Numerical results - A rendezvous in Highly Elliptical Orbit
Conclusion and related work
Thank you for your attention
Richard Epenoy
A New Penalty Method for Path-Constrained Rendezvous