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Discrete mechanics, optimal control and
formation flying spacecraft
Oliver Junge
Center for MathematicsMunich University of Technology
joint work with Jerrold E. Marsden and Sina Ober-Blobaum
partially supported by the CRC 376
Oliver Junge Discrete mechanics, optimal control and formation flying spacecraft p.1
Outline
I mechanical optimal control problem
I direct discretization of the variational principle(“DMOC”)
I applications: low-thrust orbital transfer, hovercraft,spacecraft formation flying
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Introduction
Optimal control problem
mechanical system, configuration space Q, to be moved from(q0, q0) to (q1, q1), force f , s.t.
J(q, f ) =
∫ 1
0
C (q(t), q(t), f (t)) dt → min
Dynamics: Lagrange-d’Alembert principle
δ
∫ 1
0
L(q(t), q(t)) dt +
∫ 1
0
f (t) · δq(t) dt = 0
for all δq with δq(0) = δq(1) = 0, Lagrangian L : TQ → R.
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Introduction
Equality constrained optimization problem
Minimize(q, f ) 7→ J(q, f )
subject toL(q, f ) = 0.
Standard approach:
I derive differential equations,
I discretize (multiple shooting, collocation),
I solve the resulting (nonlinear) optimization problem.
Here: discretize the Lagrange-d’Alembert principle directly.
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Discretization techniques: comparison
cost function + Ld’Ap
variation
cost function +EL
discretization
discrete cost function +discretized ode
Oliver Junge Discrete mechanics, optimal control and formation flying spacecraft p.5
Discretization techniques: comparison
cost function + Ld’Ap
variation
cost function +EL
discretization
discrete cost function +discretized ode
discretization
discrete cost function +discrete Ld’Ap
discrete cost function +DEL
variation
Oliver Junge Discrete mechanics, optimal control and formation flying spacecraft p.6
Discretization techniques: comparison
cost function + Ld’Ap
variation
cost function +EL
discretization
discrete cost function +discretized ode
discretization
discrete cost function +discrete Ld’Ap
discrete cost function +DEL
variation
finite differences,multiple shooting,collocation, ...
Oliver Junge Discrete mechanics, optimal control and formation flying spacecraft p.7
Discretization techniques: comparison
cost function + Ld’Ap
variation
cost function +EL
discretization
discrete cost function +discretized ode
discretization
discrete cost function +discrete Ld’Ap
discrete cost function +DEL
variation
finite differences,multiple shooting,collocation, ...
DMOC
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Discrete paths
Figure: J.E. Marsden, Lectures on Mechanics
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Discrete paths
Figure: J.E. Marsden, Lectures on Mechanics
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Discretization of the variational principleReplace
I the state space TQ by Q × Q,I a path q : [0, 1]→ Q by a discrete path
qd : 0, h, . . . ,Nh = 1 → Q,I the force f : [0, 1]→ T ∗Q by a discrete force
fd : 0, h, 2h, . . . ,Nh = 1 → T ∗Q.
Discrete Lagrangian Ld : Q × Q → R,
Ld(qk , qk+1) ≈∫ (k+1)h
kh
L(q(t), q(t)) dt,
virtual work
f −k · δqk + f +k · δqk+1 ≈
∫ (k+1)h
kh
f (t) · δq(t) dt,
f −k , f+k ∈ T ∗Q: left and right discrete forces.
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Discretizing the variational principleDiscrete Lagrange-d’Alembert principle
Find discrete path q0, q1, . . . , qN s.t. for all variationsδq0, . . . , δqN with δq0 = δqN = 0
δN−1∑k=0
Ld(qk , qk+1) +N−1∑k=0
f −k · δqk + f +k · δqk+1 = 0.
Forced discrete Euler-Lagrange equations
Discrete principle quivalent to
D2Ld(qk−1, qk) + D1Ld(qk , qk+1) + f +k−1 + f −k = 0.
k = 1, . . . ,N − 1.
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Boundary ConditionsI We need to incorporate the boundary conditions
q(0) = q0, q(0) = q0
q(1) = q1, q(1) = q1
into the discrete description.
Legendre transform FL : TQ → T ∗Q
FL : (q, q) 7→ (q, p) = (q,D2L(q, q)),
Discrete Legendre transform for forced systems
Ff +Ld : (qk−1, qk) 7→ (qk , pk),
pk = D2Ld(qk−1, qk) + f +k−1.
Discrete boundary conditions
D2L(q0, q0) + D1Ld(q0, q1) + f −0 = 0,
−D2L(qN , qN) + D2Ld(qN−1, qN) + f +N−1 = 0.
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The Discrete Constrained Optimization Problem
Minimize
Jd(qd , fd) =N−1∑k=0
Cd(qk , qk+1, fk , fk+1),
subject to q0 = q0, qN = q1 and
D2L(q0, q0) + D1Ld(q0, q1) + f −0 = 0,
D2Ld(qk−1, qk) + D1Ld(qk , qk+1) + f +k−1 + f −k = 0,
−D2L(qN , qN) + D2Ld(qN−1, qN) + f +N−1 = 0,
k = 1, . . . ,N − 1.Solution by, e.g., sequential quadratic programming.
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Implementation: quadrature
qk
qk+1
q(t)
q(t) q( tk+tk+1
2) = qk+qk+1
2
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Implementation: quadrature
Discrete Lagrangian
∫ tk+1
tk
L(q, q) dt ≈∫ tk+1
tk
L(q(t), ˙q(t)) dt
≈∫ tk+1
tk
L
(q
(tk + tk+1
2
), ˙q
(tk + tk+1
2
))dt
= hL
(qk + qk+1
2,qk+1 − qk
2
)=: Ld(qk , qk+1)
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Implementation: quadrature
Discrete forces
∫ (k+1)h
kh
f · δq dt ≈ hfk+1 + fk
2· δqk+1 + δqk
2
=h
4(fk+1 + fk)︸ ︷︷ ︸
=:f −k
·δqk +h
4(fk+1 + fk)︸ ︷︷ ︸
=:f +k
·δqk+1
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Example: Low thrust orbital transferSatellite with mass m, to be transferred from one circular orbitto one in the same plane with a larger radius. Number ofrevolutions around the Earth is fixed. In 2d-polar coordinatesq = (r , ϕ)
L(q, q) =1
2m(r 2 + r 2ϕ2) + γ
Mm
r,
M : mass of the earth. Force u in the direction of motion ofthe satellite.
Goal
minimize the control effort
J(q, u) =
∫ T
0
u(t)2 dt.
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Comparison to traditional scheme
10 15 20 25 30 35 40 45 500.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
number of nodes
obje
ctiv
e fu
nctio
n va
lue
EulerVariational
1 rotation
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Comparison to traditional scheme
5 10 15 20 25 30 35 40 45 500.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
number of nodes
obje
ctiv
e fu
nctio
n va
lue
1 rotationMidpointruleVariational
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Comparison to the “true” solution
5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
7
8
number of nodes
devi
atio
n of
fina
l sta
te
EulerVariational
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Comparison to the “true” solution
5 10 15 20 25 30 35 40 45 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
number of nodes
devi
atio
n of
fina
l sta
te
MidpointruleVariational
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Example: Reconfiguration of a group of hovercraftHovercraft
r
x
y
θ
f1
f2
Configuration manifold: Q = R2 × S1
Underactuated system, but configuration controllable.
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Reconfiguration of a group of Hovercraft
Lagrangian
L(q, q) =1
2(mx2 + my 2 + J θ2),
q = (x , y , θ), m the mass of the hovercraft, J moment ofinertia.
Forced discrete Euler-Lagrange equations
1hM (−qk−1 + 2qk − qk+1) + h
2
(fk−1+fk
2+ fk+fk+1
2
)= 0, (1)
k = 1, . . . ,N .
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Goal
Minimize the control effort while attaining a desired finalformation:
(a) a fixed final orientation ϕi of each hovercraft,
(b) equal distances r between the final positions,
(c) the center M = (Mx ,My ) of the formation is prescribed,
(d) fixed final velocities.
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Results
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Application: Spacecraft Formation Flying
I ESA: Darwin
I NASA: Terrestrial Planet Finder
I SFB 376 ”‘Massive Parallelism”’, Project C10: EfficientControl of Formation Flying Spacecraft
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The Model
I A group of n identical spacecraft,
I single spacecraft: rigid body with six degrees of freedom(position and orientation),
I control via force-torque pair (F , τ) acting on its center ofmass.
I dynamics (as required for Darwin/TPF): circularrestricted three body problem
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The Lagrangian
I Potential energy:
V (x) = − 1− µ|x − (1− µ, 0, 0)|
− µ
|x − (−µ, 0, 0)|,
µ = m1/(m1 + m2) normalized mass.
I kinetic energy:
Ktrans(x , x) =1
2((x1 − ωx2)2 + (x2 + ωx1)2 + x2
3 )
+
Krot(Ω) =1
2ΩTJΩ,
Ω: angular velocity, J : inertia tensor.
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The Control ProblemGoal: compute control laws (F (i)(t), τ (i)(t)), i = 1, . . . , n,such that
I within a prescribed time interval, the group moves from agiven initial state into a prescribed target manifold,
I minimizing a given cost functional (related to the fuelconsumption).
Target manifold:
1. all spacecraft are located in a plane with prescribednormal,
2. the spacecraft are located at the vertices of a regularpolygon with a prescribed center on a Halo orbit,
3. each spacecraft is rotated according to a prescribedrotation,
4. all spacecraft have the same prescribed linear velocity andzero angular velocity.
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Collision avoidanceArtificial potential
0 1 2 3 4 51
1.5
2
2.5
3
3.5
4
L = Ktrans + Krot − V −n∑
i,j=1i 6=j
Va(‖q(i) − q(j)‖).
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Halo orbits
0.999 1.0016 1.0042 1.0068 1.0094 1.012
−0.01
0
0.01
−12
−10
−8
−6
−4
−2
0
2
4
x 10−3
L2
x
E
y
z
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Result
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Hierarchical decomposition
same model for every vehicle → n identical subsystems,coupling through constraint on final state
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Hierarchical optimal control problem
minϕ1,...,ϕn
n∑i=1
Ji(ϕi) s.t. g(ϕ) = 0,
(g describes final state) with
J(ϕi) = optimal solution of 2-point bvp with ϕi
parametrizing the final state
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ParallelizationI solve inner problems in parallelI synchronization (communication) for iteration step in
solving the outer problemI implementation: PUB (“Paderborn University
BSP-Library”) – library to support development of parallelalgorithms based on the”Bulk-Synchronous-Parallel-Model”.
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Result
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Conclusion
I new approach to the discretization of mechanical optimalcontrol problems
I direct discretization of the underlying variational principle
I faithful energy behaviour by construction
Outlook
I convergence
I backward error analysis
I hierarchical decomposition in time: discontinuous(“weak”) solutions, Pontryagin-d’Alembert principle
I generalization to spatially distributed systems (PDEs)
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