Discrete mechanics, optimal control and formation ying spacecraftpatrick/BIRS/Junge.pdf ·...

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


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.


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.


Discretization techniques: comparison

cost function + Ld’Ap

variation

cost function +EL

discretization

discrete cost function +discretized ode




variation

cost function +EL

discretization


discretization

discrete cost function +discrete Ld’Ap

discrete cost function +DEL

variation




variation

cost function +EL

discretization


discretization



variation

finite differences,multiple shooting,collocation, ...




variation

cost function +EL

discretization


discretization



variation

finite differences,multiple shooting,collocation, ...

DMOC


Discrete paths

Figure: J.E. Marsden, Lectures on Mechanics


Discrete paths

Figure: J.E. Marsden, Lectures on Mechanics


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.


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.


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.


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.


Implementation: quadrature

qk

qk+1

q(t)

q(t) q( tk+tk+1

2) = qk+qk+1

2



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)



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


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.


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


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


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


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


Example: Reconfiguration of a group of hovercraftHovercraft

r

x

y

θ

f1

f2

Configuration manifold: Q = R2 × S1

Underactuated system, but configuration controllable.


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 .


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.


Results


Application: Spacecraft Formation Flying

I ESA: Darwin

I NASA: Terrestrial Planet Finder

I SFB 376 ”‘Massive Parallelism”’, Project C10: EfficientControl of Formation Flying Spacecraft


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


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.


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.


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)‖).


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


Result


Hierarchical decomposition

same model for every vehicle → n identical subsystems,coupling through constraint on final state


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


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”.


Result


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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