Planning a Trip to the Moon? And Bac k?

Planning a Trip to the Moon? . . .. . . And Back?

John T. Betts

Boeing is a trademark of Boeing Management CompanyCopyright c© 2006 Boeing all rights reserved

Planning a Trip to the Moon?...And Back? 10/06

Typical Two Burn Orbit TransferEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Park Orbit: 150 nm circular, 28.5 deg inclinationMission Orbit: Geosynchronous Equatorial

Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06

Orbit Mechanics 101Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

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}∆v1

}∆vm

}∆v2

tI tm tF

Park Orbit

Mission Orbit

• Plane change most efficient at high altitude

• Three Burn Transfer Does This ∗

Eliminate Middle Burn and Use Moon For High Altitude ∆v

∗John T. Betts, “Optimal Three-Burn Orbit Transfer,” AIAA Journal, June, 1977


Lunar Swingby to Geosynchronous OrbitEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

‖∆V1‖ ‖∆V2‖ Total (fps)Hohmann 8056.67 5851.44 13908.12Swingby 10201.39 3518.72 13720.11


Lunar Swingby to Polar 24-hr OrbitEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology



Another Swingby to Polar 24-hr OrbitEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology



Swingby to Molniya Orbit (i = 116.6)Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology



Equations of MotionEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

ECI spacecraft state (r,v) and lunar state (rm,vm)

r = v

v =−µe

r3 r+gm

rm = vm

vm =−µ◦r3

mrm

where lunar gravitational perturbations on S/C are

gm =−µm

[1d3d+

1r3

mrm

]with d = r− rm


Analytic Two-Body PropagationEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Propagate from state (r◦,v◦) through angle ∆E to state (r,v).

σ◦ =1√µ

rT◦v◦

1a

=2‖r◦‖

−[

vT◦v◦µ

]

C = a(1− cos∆E) S =√

asin∆E

F = 1− C‖r◦‖

G =1√µ

(‖r◦‖S +σ◦C)

ρ = 1− ‖r◦‖a

r = ‖r◦‖+ρC +σ◦S

Ft =−√µ

r‖r◦‖S Gt = 1−C

rr = Fr◦+Gv◦ v = Ftr◦+Gtv◦

∆t =

√a3

µ

[∆E +

σ◦Ca√

a−ρ S√

a

]

State propagation is explicit, i.e.r = hr(r◦,v◦,∆E)v = hv(r◦,v◦,∆E)

Time change ∆t is implicit, from Kepler’s Equation.∆t = ht(r◦,v◦, t◦,∆E)


Equations of Motion–DAE FormulationEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Treat ∆E(t) as an algebraic (control) variable

Dynamics defined by the differential-algebraic (DAE) system

r = v

v =−µe

r3 r+gm

0 = ∆t−ht(r◦,v◦, t◦,∆E)

where lunar gravitational perturbations on S/C are

gm =−µm

[1d3d+

1r3

mrm

]with d = r−hr(r◦,v◦,∆E)


How To Solve A Hard ProblemEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Break A Hard Problem Into A Sequence of Easy Subproblems

• Newton’s MethodSolve a nonlinear constraint by solving a sequence of linear approximations;

• Nonlinear ProgrammingSolve a nonlinear optimization problem by solving a sequence of:◦ quadratic programming subproblems (an SQP Method) or◦ unconstrained subproblems (a Barrier Method)

• Optimal ControlSolve a sequence of NLP subproblems.

• Optimal Lunar SwingbySolve a sequence of ”Easier” subproblems.


Four Step Solution TechniqueEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Step 1: Three Impulse, Conic SolutionSolve small NLP with analytic propagation ignoring lunar gravity;

Step 2: Three-Body Approximation to Conic SolutionSolve “inverse problem” to fit three-body dynamics to conic solution;

Step 3: Optimal Three-Body Solution with Fixed Swingby TimeUse solution from step 2 to initialize optimal solution.

Step 4: Optimal Three-Body SolutionCompute solution with free swingby time, using step 3 as an initial guess.


Step 1: Three Impulse, Conic SolutionEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Optimization Variables (24)

(ro,vo,∆v1,∆Eo) : State at Park Orbit Departure

(ri,vi,∆v2,∆Ei) : State at Mission Orbit Arrival

(∆vL,∆EL) : Velocity Increment and Transfer Angle at Lunar Intercept

Park Orbit Conditionsrp = ro Position Continuity

vp = vo−∆v1 Impulsive Velocity Change

φp (rp,vp) = 0 Park Orbit Constraints

Mission Orbit Conditionsrm = ri Position Continuity

vm = vi +∆v2 Impulsive Velocity Change

φm (rm,vm) = 0 Mission Orbit Constraints


Step 1: Three Impulse, Conic SolutionEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Lunar Conditions

hr(ro,vo,∆Eo) = hr(ri,vi,∆Ei) Outbound/Inbound Position

hr(ro,vo,∆Eo) = hr(rL◦,vL◦,∆EL) Outbound/Lunar Position

hv(ri,vi,∆Ei) = hv(ro,vo,∆Eo)+hv(rL◦,vL◦,∆EL)+∆vL Velocity Change

Objective Minimize F = ‖∆v1‖+‖∆v2‖+‖∆vL‖

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$%

&&&&'

(ro,vo,∆v1)

(rL◦,vL◦)

(ri,vi,∆v2)

∆vL

∆Eo

∆Ei

∆EL

Park Orbit Mission Orbit


Step 2: Three-Body ApproximationEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Conic solution solvesr =−µe

r3 r

rm =−µ◦r3

mrm

notr =−µe

r3 r+gm

rm =−µ◦r3

mrm

“Fit” Three-Body Trajectory to Conic i.e. minimize

F = ∑k‖rk− rk‖2

subject to

r =−µe

r3 r+gm

rm =−µ◦r3

mrm

rLmin ≤ ‖r− rm‖

where rk is S/C position at k points on conic trajectory


Step 3: Fixed Swingby TimeEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Two Phases with

Three-Body Dynamics (ODE or DAE)

r =−µe

r3 r+gm

rm =−µ◦r3

mrm

.

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t = tL

&&&&'

(ro,vo,∆v1, tI)

(rL◦,vL◦)

(ri,vi,∆v2, tF)Park Orbit Mission Orbit



Phase 1: Outbound TransferAt (free) tI Satisfy Park Orbit Conditions

rp = ro Position Continuity

vp = vo−∆v1 Impulsive Velocity Change

φp (rp,vp) = 0 Park Orbit Constraints

At (fixed) tL Satisfy Lunar Conditions

‖r− rm‖ ≥ rLmin Closest Approach

(v−vm)T(r− rm) = 0 Lunar Flight Path Angle



Phase 2: Inbound TransferAt (fixed) tL

(r,v,rm,vm)(−) = (r,v,rm,vm)(+) State Continuity(rm,vm) = (rm,vm) Lunar State

At (free) tF Satisfy Mission Orbit Conditions

rm = ri Position Continuity

vm = vi +∆v2 Impulsive Velocity Change

tmax ≥ tF− tI Mission Duration

φm (rm,vm) = 0 Mission Orbit Constraints

Objective Minimize F = ‖∆v1‖+‖∆v2‖


Step 4: Optimal Three-Body SolutionEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Formulation as in Step 3except:

(a) Free Lunar Swingby Time tL(b) Lunar State Specified at (free) tI

rm = hr(r◦,v◦,∆E◦)vm = hv(r◦,v◦,∆E◦)


The Swingby SubproblemsEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

How To Efficiently Solve The Subproblems?

• Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 3 and 4

• Parameter Estimation (Inverse Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 2

• Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 1


Good Software/Algorithms to Solve NLPEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Find Variables xT = (x1, . . . ,xn)to minimize the Objective

F(x)

subject to Constraints

cL ≤ c(x)≤ cU .


Problems We Want to SolveEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Find Control Functions u(t) and/or parameters p to minimize

J =Z tF

tIw [y(t),u(t),p, t]dt

or

J = ∑k‖y(tk)− y(tk)‖2

subject to constraints over the domain tI ≤ t ≤ tF

y = f[y(t),u(t),p, t]0≤ g[y(t),u(t),p, t]

and boundary conditions


So What’s the Rub?Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

The NLP Works with a Finite Set of Variables x and Functions F(x), c(x)

...

But Optimal Control/Estimation is an Infinite Dimensional Problem;i.e. the functions u(t) and y(t)

How do we formulate the problem?


Shooting MethodsEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

“Eliminate” Infinite Dimensional Problem by solving

y = f[y(t),u(t), t] and/ory = f[y(t),u(t), t]0 = g[y(t),u(t), t]

The NLP involves the Finite Set of Boundary Values

BVP is very nonlinear—∆v at boundary, lunar gravity in “middle”ODE or DAE can be very unstable

ODE error control at suboptimal points—inefficientPath inequalities cumbersome (impractical?)

Shooting for Control⇐⇒ GRG for NLP


Discretization MethodsEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Variables

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[y(t),u(t)] x = [y1,u1, . . . ,yM,uM]+ .

Constraints

y = f[y(t),u(t), t] yk+1 = yk +hk2

(fk + fk+1)


An Optimal Control AlgorithmEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Direct Transcription Transcribe the optimal control problem into a nonlinear pro-gramming (NLP) problem by discretization;

Sparse Nonlinear Program Solve the sparse (SQP or Barrier) NLP

Mesh Refinement Assess the accuracy of the approximation (i.e. the finite dimen-sional problem), and if necessary refine the discretization, and then repeat theoptimization steps.

SNLP: Sequential Nonlinear Programming


Barrier or SQP?Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Step 1–Small, Dense NLP Subproblems

Mission Equatorial Polar MolniyaSQP-Newton (10,4) (16,10) (135,44)SQP-BFGS (24,19) (36,31) (186,96)Barrier-Newton (24,22) (57,55) (70,68)†Barrier-BFGS (242,241)† (58,56) (286,284)

Key: (Gradient Eval., Hessian Eval.) † No Solution

Some Sweeping GeneralizationsSQP Most Efficient and Robust

Quasi-Newton Hessian Too Slow for Large/SparseBarrier Method Lacks Robustness, Speed


Sequential Nonlinear ProgrammingEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Is an SQP better than Barrier for Sequential Nonlinear Programming?

Coarse Grid

xc = [y1,u1, . . . ,ym,um]+

Fine Grid

x f = [y1,u1, . . . ,yM,uM]+

NLP problem size grows—typically M > m

Question: How do we efficiently solve a sequence of NLP’s?Answer: Use coarse grid information to “Hot Start” fine grid NLP


Estimating Variables for SNLPEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

SQP Algorithmhigh order interpolation of coarse grid solution

consistent with discretization formula(e.g. collocation polynomial)

very good guess

Interior Point Algorithmmust be feasible⇐⇒ barrier algorithm perturbs guessnot consistent with coarse grid discretization formulaBarrier Algorithm Cannot Exploit a Good Guess!


Barrier vs SQP?Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Step 3 Subproblem, Polar Mission

Step 2 Trajectory Defines:Smooth Three-Body Trajectory Guess

Propagate Trajectory Using Variable Step Integrator To Define Initial Grid

SQPk M n m NGC NHC NFE ε Time (sec)1 594 7136 7715 18 10 3794 1×10−4 30.12 881 10580 11446 4 1 454 4×10−7 7.83 1113 13364 14462 4 1 454 1×10−8 10.6

Total 26 12 4702 48.6

Barrier†k M n m NGC NHC NFE ε Time (sec)1 594 7720 7715 328 319 112475 1×10−5 749.32 881 12985 12980 57 49 17637 2×10−7 233.73 1113 14090 14085 6 2 881 1×10−8 18.5

Total 391 370 130993 1001.6† Different Local Solution than SQP


Is Mesh Refinement Needed?Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Step 2 Inverse Problem Solution For Polar Mission

Step 1 Conic Trajectory Defines:1800 Residuals — 600 Equal ∆E incrementsInitial Trajectory Guess — Omit Point at Moon

k M n m NGC NHC NFE ε Time (sec)1 599 7188 7775 12 2 144 6×10−1 3.52 606 7272 7866 21 17 525 9×10−5 1.13 606 7272 7866 4 2 724 1×10−6 7.64 742 8904 9634 4 1 448 1×10−8 5.6

Total 41 22 1841 27.7

k Refinement No M Grid Pts n NLP varsm NLP cons. NGC Grad Eval NHC Hess EvalNFE Func Eval ε Disc. Error Time CPU


Velocity DiscontinuityEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology


Mesh RefinementEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Mesh Refinement “Smooths” Out Velocity DiscontinuityInverse Problem Approach Keeps Process Stable


DAE or ODE Formulation?Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Steps 3 & 4, Optimal Solution for Molniya Mission

M n NGC NHC Time (sec)630 7568 26 7 32.02807 9692 7 2 12.30940 11288 4 1 8.661190 14288 4 1 11.221190 14291 10 4 45.49

ODE Formulation Total Time = 109.69 sec

M n NGC NHC Time (sec)1097 8782 30 14 45.271530 12246 4 1 7.982056 16454 4 1 11.822056 16456 7 2 31.02

DAE Formulation Total Time = 96.09 sec

No clearcut difference in speed or accuracy


Summary and ConclusionsEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology

Optimal Lunar SwingbySignificant Performance Benefits for

Earth Orbital Missions with Large Plane Change

Swingby TrajectoryVery nonlinear boundary value problem

SQP More Robust, EfficientBarrier algorithm cannot exploit “good guess”.

Mesh RefinementCritical for stable solution.

Overall Approach Applicable to Many n-body Problems


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Planning a Trip to the Moon? And Bac k?