Download - 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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
Lunar Swingby to Polar 24-hr OrbitEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
‖∆V1‖ ‖∆V2‖ Total (fps)Hohmann 8113.34 8610.82 16724.17Swingby 10225.16 3440.44 13665.61
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Another Swingby to Polar 24-hr OrbitEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
‖∆V1‖ ‖∆V2‖ Total (fps)Hohmann 8113.33 8610.77 16724.10Swingby 10251.24 3459.66 13710.90
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
Swingby to Molniya Orbit (i = 116.6)Engineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
‖∆V1‖ ‖∆V2‖ Total (fps)Hohmann 546.99 39682.76 40229.76Swingby 10392.39 4826.32 15218.71
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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 =1õ
(‖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)
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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)
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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.
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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.
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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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$$
$%
&&&&'
(ro,vo,∆v1)
(rL◦,vL◦)
(ri,vi,∆v2)
∆vL
∆Eo
∆Ei
∆EL
Park Orbit Mission Orbit
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
Step 2: Three-Body ApproximationEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
Conic solution solvesr =−µe
r3 r
rm =−µ◦r3
mrm
notr =−µe
r3 r+gm
rm =−µ◦r3
mrm
“Fit” Three-Body Trajectory to Conic i.e. minimize
F = ∑k‖rk− rk‖2
subject to
r =−µe
r3 r+gm
rm =−µ◦r3
mrm
rLmin ≤ ‖r− rm‖
where rk is S/C position at k points on conic trajectory
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Step 3: Fixed Swingby TimeEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
Two Phases with
Three-Body Dynamics (ODE or DAE)
r =−µe
r3 r+gm
rm =−µ◦r3
mrm
.
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$%
t = tL
&&&&'
(ro,vo,∆v1, tI)
(rL◦,vL◦)
(ri,vi,∆v2, tF)Park Orbit Mission Orbit
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
Step 3: Fixed Swingby TimeEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
Step 3: Fixed Swingby TimeEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
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‖
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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◦)
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
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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 .
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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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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?
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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
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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)
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
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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
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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
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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!
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
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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
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Velocity DiscontinuityEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
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Mesh RefinementEngineering and Operations Technology | Phantom Works E&IT | Mathematics and Computing Technology
Mesh Refinement “Smooths” Out Velocity DiscontinuityInverse Problem Approach Keeps Process Stable
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06
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
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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
Copyright c© 2006 Boeing all rights reserved Planning a Trip to the Moon?...And Back? 10/06