AE554 Applied Orbital Mechanics

Orbit Propagation

Egemen İmre

229/12/2007

Numerical Integration

� Solution of ODEs

� Many problems can be solved

� Generic methods exist

� Methods can be tailored for accuracy requirements (more later)

� Orbit Propagation: Integration of equations of motion

� Find position and velocity in time

“When your only tool is a hammer, every problem looks like a nail.”

Anonymous

329/12/2007

Propagation: Analytical vs. Numerical� Analytical

� Insight into the underlying dynamics

� Very fast

� Simplifications obligatory

� Complicated application of perturbing forces

� Very difficult (if not practically impossible) to include certain perturbations (3rd body etc)

� Numerical

� Fairly straightforward application of higher order geopotentials and other perturbativeforces

� Less insight into the dynamics

� Speed depends heavily on prediction timeframe and accuracy

� Long term stability depends primarily on integration technique

429/12/2007

Accelerations

Position and Velocity

integration

Anatomy of a Propagator

529/12/2007

Numerical Integration Basics

� The function f(t) is known (discrete values or continuous)but function Y may be too complicated to evaluate.

� Shown is first order approximation, all the second order and higher terms are neglected.

)(

)(

)(

2

0

1

0

tObtaY

tfY

tfy

k

i

ii

t

t

∆++∆≈

=

=

∑

∫

=

y

timet0

t1

∆t

629/12/2007

Numerical Integration: Concepts and Issues� In theory, as timestep goes to zero, integration error goes to zero as

well.

� In reality, there is a limit where further reduction in stepsize has the opposite effect!

� Error sources:

� Round-off error

� Truncation errors

� All are artefacts of “finite precision”

� Float vs. double vs. long double

� Function of stepsize (function evaluations)!

� Error

� Order in Taylor polynomial and appriximation comparable

729/12/2007

Numerical Integration: Concepts and Issues� Computational load

� Desktop computers and CPU farms

� Onboard applications?!

� 8bit &16bit vs 32bit & 64bit

� Force/function evaluations are expensive!

� Avoid them like plague!

� Principle of GIGO: Garbage in-garbage out

� Easy to set up a bad integration scheme

� Loss of precision (big number+small number)

� Verification?

� No truth model!

829/12/2007

Simple Numerical Integration Schemes (Explicit)

� Midpoint/rectangle rule

� Single force evaluation

� Zeroth order approx

� Assume function is constant within the step

� Trapezoid rule

� Double force evaluations

� First order approx

� Assume function changes linearly within the step

929/12/2007

Simple Numerical Integration Schemes (Explicit)

� Simpson’s rule

� Three force evaluations

� Second order approx

� Approximate the function with quadratic polynomials

� Composite rule

� n+1 force evaluations

Though we assumed equally spaced “substeps”, this is not necessarily the case!

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Adaptive Schemes

� Previous schemes work well if the function is “well-behaved” and “smooth”.

� If this is not the case (such as a singularity or a “spike”) can be totally ignored by the numerical scheme.

� Solution: Adaptive scheme that “watches its step”.

� Estimate the rate of change of f(x)

� Check future values of f(x)

� Lower order scheme and compare

� Set a smaller timestep to capture the fast-changing behaviour

� Extra function evaluations and added complexity

� Existing schemes can be modified to work “adaptively”

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The Story So Far...Applications to Orbit Propagation� Any of these methods can be used to integrate the

equations of motion

� Integrate acceleration/force >> velocity update

� Integrate velocity >> position update

� For higher the accuracy for a given duration:

� Smaller timesteps (more total force evaluations)

� Higher order integration (more total force evaluations)

� Find an optimal scheme

� Highly elliptic orbits: fast and slow dynamics at perigee and apogee

� Adaptive schemes ideal

1229/12/2007

Runge-Kutta Methods

� Runge-Kutta (RK) and adaptive RK schemes immensely popular!

� Single step method (no previous steps required)

� Ubiquitous 4th order RK scheme (RK4)

� Weighted average of four sample points per timestep

� 4th order scheme

� Error per step is O(h5)

� Total accumulated error is O(h4)

� Can be generalised to nth order

� Euler and midpoint are special cases of RK methods

1329/12/2007

Runge-Kutta Methods

� RK methods are versatile and easy to use, hence the popularity, particularly of higher order versions

� Not quite cheap!

� 4th order: 4 function evaluations per step

� 8th order: 13 function evaluations per step

� 10th order: 17 function evaluations per step

� RK8(7)-13 mnemonic (also a popular scheme)

� 8th order integration

� 7th order “embedded” scheme for “adaptivity”

� 13 force evaluations per step

1429/12/2007

Runge-Kutta-Nyström Method

� Rather than integrating acceleration twice to find the position update, integrate directly in a single step

� Solution of second order differential equations

� Particularly useful if the acceleration does not depend on the velocity (geopotential but NOT drag!)

� Save one function evaluation per timestep on RK6(4)!

� “Embedded” adaptive schemes available

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Multistep Methods

� Adams-Bashforth and Adams-Bashforth-Moulton

1. Prediction: Estimate the solution at ti+1

2. Evaluation: Evaluate the fi+1 at ti+1

3. Correction: Apply correction to the solution at ti+1

4. Evaluation: Evaluate the fi+1 at ti+1 using the improved solution

� Attempt to minimise the number of function evaluations

� Iterative and (sort of) implicit method

� High-order explicit multistep methods (AB) more prone to “instability” than implicit methods (ABM)

� Computational errors dominate the result

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More Multistep Methods

� Störmer-Cowell Methods

� Multistep methods for direct integration of second order differential equations (like RKN)

� Predictor-corrector scheme

� Gauss-Jackson or Second Sum Methods

� Modified Störmer-Cowell method (similar to ABM)

� Probably the best fixed-stepsize multistep method

� Very good stability properties in comparison to AB methods

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Extrapolation Methods (Bulirsch-Stoer)� Single stepsize method

� Richardson extrapolation: Extrapolate to zero stepsize!

� Wacky idea – but it works!

� Use (a slightly modified) midpoint rule (with stepsize h)

� Evaluate it again with a different stepsize (h’)

� The two results using the two stepsizes can be extrapolated to another result that happens to have a stepsize of zero!

� Zero stepsize = zero error! (in theory, that is!)

� Use a polynomial extrapolation scheme

1829/12/2007

Extrapolation Methods (Bulirsch-Stoer)� Can be extended to any order via linear combinations of

results from different stepsizes

� When integrating over a stepsize of H, try stepsizes hi

)4for2(48... 32, 24, 16, 12, 8, 6, 4, 2, 2 ≥=== − innnn

Hh ii

i

i

� Similar to RK methods, a stepsize control can be devised by comparing the neighbouring estimates.

� When integrating over a stepsize of H, try stepsizes hi

1929/12/2007

Evaluation of the Methods

� Speed vs accuracy

� Slowed down by number of function evaluations

� If function evaluations are not that expensive, then arithmetic operaitons inside the scheme becomes important too

� Even with the adaptive methods, eccentric orbits are more expensive to calculate for a given accuracy

� Output points hamper the performance

� Use existing results as much as possible

� Montenbruck tests!

2029/12/2007

Conservation Principles

� The previous methods are good as general-purpose tools to solve ODEs

� But they have an utter disregard to the nature of the problem!

� Conservation laws regarding the problem – not quite conserved!

� No conservation, so what?!

� Results diverge from the truth!

� Similar to a dissipative system where no dissipation should occur!

� Semimajor axis decay and eccentricity change.

2129/12/2007

Conservation Principles

� Many problems in astrodynamics has conservation properties

� N-body problem

� Simulating the Solar System or a galaxy (stability?)

� Satellite orbit in a non-dissipative environment

� Non-LEO orbits

� For a satellite:

� Hamiltonian system (and Hamiltonian = Energy)

� Conservation of energy (as long as non-dissipative)

� Conservation of angular momentum (spherically symmetric force field i.e., no tesseral harmonics)

2229/12/2007

Hamiltonian: the concept

Hamiltonian = Energy (more or less!) – defines the motion

r = position, v = velocity (strictly speaking, momenta)

)(2

1)(),(),(

2rrrvrv UvRKH +=+=

� K = Keplerian (simple)

� R = higher order geopotentials (complicated!)

2329/12/2007

� Equations of Motion (Hamilton’s Equations)

� This definition conserves the energy. (symplecticness!)

� Separable to Keplerian and higher order geopotentials

Hamiltonian Dynamics

vr

r

rrv

v&& −=

∂

∂=

∂

∂==

∂

∂ )(UHH

2429/12/2007

Hamiltonian Dynamics

� Hamilton’s equations demonstrate a way to calculate the accelerations

� Conservation of energy

rr

vv

&&&

∂

∂+

∂

∂=

HHH = 0

vr

r

rrv

v&& −=

∂

∂=

∂

∂==

∂

∂ )(UHH

2529/12/2007

Hamiltonian Dynamics: Keplerian Case� Derive the equations for the Keplerian case (K is the

Keplerian Hamiltonian):

rvv

rrr

&&& ==∂

∂−==

∂

∂

−=

K

r

K

rvK

3

2

2

1

µ

µ

� Conservation of angular momentum:

03

=×+

−×=×+×=

×=

rrrrvrvrh

vrh

&&&&&

r

µ

2629/12/2007

Conserved Quantities: Reality

� For a more complicated potential (or, equivalently, force model) than Keplerian, conservation principles differ slightly.

� It can be shown that energy is constant for an axisymmetricgeopotential (i.e., zonal harmonics only). z component of the angular momentum is also constant.

� Potential is not time varying (latitudinal bands)

� For tesseral harmonics, energy variation is periodic

� Rotating longitudinal bands

� Third body effects induce periodic variations in force (or potential)

� Drag is a dissipative force and no conservation is possible

� But quite small!!

2729/12/2007

Symplecticness

� Symplecticness - Areas in phase space are conserved i.e., the conservation of the constants of motion (in practice!).

� Left: Forward Euler (Should be rotation only but distortion creeps in!)

� Right: Implicit Midpoint (Symplectic – Phase space area conserved!)

(Iserles and Zanna 1996)

p p

q q

2829/12/2007

Symplectic Integration

� Just another integration scheme – but with a twist!

� Unlike classical integrators, conserved quantites are conserved

� What we actually want is time-reversibility!

� Motion between (r,v)t0 and (r,v)t1 is a canonical transformation.

� Symplectic integrators better than classical integrators:

� An order of magnitude faster

� More reliable in long term

� No secular energy drift – can have fairly large local errors but better long term characteristics

2929/12/2007

Symplectic Integration

� Hamiltonian can usually be split into two parts (H=H0+H1)

� Kinetic + potential energy

� Kepler + J2

� Each part corresponds to a force (i.e., motion) via Hamilton’s equations

� We consider them in isolation

� The approximated motion corresponds to a Hamiltonian which is slightly different than the real Hamiltonian

� The error is a “surrogate Hamiltonian”

� The error is a “conserved quantity”!

� No “secular” errors – for a quasi-periodic system

3029/12/2007

Symplectic Integration: Leapfrog

� 2nd Order symplectic integration: Leapfrog

� For this example, applied to a Keplerian case

� Generalised leapfrog has an error of O(h3) in one step and O(h2) when repeated over a duration.

H1

12/1

2/1

1

2/1

2

1

)(

2

1

2

1

++

++

+

+=

∂

∂−=

+=

iii

iii

iii

h

Kh

h

vrr

r

rvv

vrr

Motion due to H0 (PE)

Motion due to H1 (KE)

3129/12/2007

Symplectic Integration: Higher Order Schemes� If H1 is smaller than H0, i.e. H1=O(ε) then the global error

becomes O(εh2)!

� Possible to derive higher order schemes

� Possible to derive adaptive schemes

� Stepsize must be kept constant!

� Time becomes another element of the position vector –with corresponding momentum term!

H1

H0

3229/12/2007

Conservation of Energy and Propagated Orbit

3329/12/2007

Energy: Hamiltonian

Accelerations: Hamilton’s Equations

Position and Velocity

Symplectic integration

Anatomy of a Symplectic Propagator

3429/12/2007

Symplectic Propagator: SPSAT

� SPSAT: Symplectic Orbit Propagator

� Integration scheme!

� Force model:

� Exact Keplerian dynamics

� “36x36” geopotential model

� Harris-Priester drag model (diurnal variations and stuff)

� Luni-solar attraction

� A bit dated WGS84 and very dated GEM10B geopotential models

� Simple conversion between rotating frame and inertial frame (fortesseral harmonics)

� Validation and verification is nigh on impossible!

� Set progressively smaller stepsizes

� Conservation of energy – fragile!

3529/12/2007

SPSAT Results – Orbit Stability

� Absolute position accuracy with respect to integration stepsize – near-circular case (36x36)

� Truth: 2000steps/orbit

3629/12/2007

SPSAT Results – Eccentricity

� Absolute position accuracy (200 steps/orbit) – near-circular and elliptic (e=0.35?) cases (36x36)

3729/12/2007

SPSAT Results – Absolute Energy

� Normalised absolute energy (200 steps/orbit) – near-circular and elliptic cases (36x36)

� Tesserals cause oscillation

3829/12/2007

SPSAT Results – Geopotentials

� Absolute position accuracy (100 steps/orbit) – near-circular case

� Truth: 1000 steps/orbit

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4029/12/2007

� http://www.physics.drexel.edu/courses/CompPhys/Integrators/leapfrog/

� http://lec.ugr.es/~julyan/papers/rkpaper/node9.html