F&G Taylor Series Solutions to the Stark Problem

F and G Taylor series solutions to the Stark problem, including general Sundman

transformations

Etienne Pellegrini, Ryan P. Russell, and Vivek Vittaldev

AAS/AIAA Astrodynamics Specialists Conference

Hilton Head, SC, 8/12/13

Examples of Stark trajectories, propagated using F&G Stark series

• Introduction

• The Sundman transformation

• The Stark Problem

• Derivation of the F&G Stark series solutions

• Numerical results – Study of the Sundman transformation – Runtime comparisons

• Conclusions and future work

Summary

Etienne Pellegrini - AAS/AIAA Astrodynamics Specialists Conference - 8/12/13 - Hilton Head, SC

• F&G series: – Taylor series of the f and g Lagrange functions

– Accurate way of propagating the 2-body problem Adapt the method to the Stark problem: 2-body motion

+ inertially constant perturbation

Introduction


• Why the Stark problem? – Good approximation for low-thrust

trajectories and slowly varying perturbations – “Simple” model, close to the 2-body

equations of motion – No TS solutions have been developed in

literature

r

v

r0

v0

• Use of the Sundman transformation – Efficient discretization schemes – Uniformizes the local truncation error (fixed step integrator

possible)

Goal of this work Obtain a fast integration scheme for the propagation of low-

thrust trajectories and/or slowly varying perturbations

Main contributions • Develop recursion formulas for the F&G Stark series &

demonstrate their superior computational efficiency

• Study the effects of the Sundman transformation on Taylor series solutions ; those results are valid for both the Stark

and Kepler problems

Introduction


• Introduced in 1912 by Karl F. Sundman. Used in astrodynamics because it reduces instability, and helps removing collision singularities. [Velez 1974, Nacozy 1976].

• The transformation slows time down as the particle gets

close to a singularity

• Generalized Sundman transformation:

• Modifies the equations of motion:

The Sundman transformation


The Sundman transformation


Effect of the Sundman transformation exponent on the discretization of the orbit

Clustering at apoapse Independent of periapse or

apoapse

Clustering at periapse

• Discovered in 1914 by J. Stark. Has applications in nuclear physics, but also in astrodynamics.

• 2-body motion + inertially constant perturbation

• Equations of Motion:

• Can be used to describe low-thrust trajectories [Rufer 1976, Yam et al. 2010, Lantoine 2011]

The Stark Problem


• Based on classic F&G derivation [Bate, Mueller,White 1971]

• Extra basis vector for out-of-plane motion

• Extra series for keeping track of time

• Have to compute 𝐹 , 𝐺 , 𝐻 , 𝑇

F&G Stark series


• Recursion – Differentiate

– Identify with

– Requires to be able to repeatedly differentiate 𝐹 , 𝐺 , 𝐻 , 𝑇

F&G Stark series: recursion equations










• Hamiltonian – One of the integrals of motion Change in Hamiltonian

represent the local error

• Benchmarks – 8th order Runge-Kutta-Fehlberg (RKF8)

• Fixed-step numerical integrator

– Modern Taylor Series (MTS) • Similar to F&G Stark series (TS of position and velocity wrt 𝜏) • Coefficients are the same

• Computed using Leibniz’s rule and Automatic Differentiation packages

• Fixed step

Numerical Results


RKF8

10-6

10-10

10-14 10-18

Cha

nge

in H

amilt

onia

n

Change in Hamiltonian vs. position on the orbit

Results: Sundman transformation


Eccentric Anomaly (rad) 0 1 2 3 4 5 6 0 1 2 3 4 5 6

Orbit: ellipse with a = 1 and e = 0.7

RKF8

10-6

10-10

10-14 10-18

Cha

nge

in H

amilt

onia

n

Change in Hamiltonian vs. position on the orbit



Eccentric Anomaly (rad) 0 1 2 3 4 5 6 0 1 2 3 4 5 6

F&G Stark series

10-2

10-6 10-10 10-14

Cha

nge

in H

amilt

onia

n Eccentric Anomaly (rad)

0 1 2 3 4 5 6

Orbit: ellipse with a = 1 and e = 0.7



Maximum change in Hamiltonian (over one revolution) vs. Sundman exponent

𝛼

Max

imum

cha

nge

in H

amilt

onia

n

• Nominal trajectory for timings – 1 revolution, tolerance = 10-12 – a = 1, e = 0.8, p = 0.001*[1,1,1] – 𝑑𝑡 = 𝑐𝑟 𝑑𝜏, 𝛼 = 1 – “Truth” trajectory propagated using F&G Stark order 21, quad

precision, 100 Segments

Timing results: Comparing F&G Stark to MTS and RKF8


Nominal trajectory, top view

0.8

0.4 0 -0.4

-0.8

Y

-2 -1.5 -1 -0.5 0 0.5 X

Nominal trajectory, side view

-0.8 -0.4 0 0.4 0.8 Y

8

4 0

Z

x10-3

• Compute truth using F&G Stark series at high order and quad precision

Results: Comparison algorithm


Truth

Prescribed accuracy: 𝜖 = 0.001

• For each TS order, increase # of segments until accuracy is met



F&G, 5 segments

𝜖 = 10

Truth


• For each TS order, increase # of segments until accuracy is met



F&G, 5 segments

F&G, 10 segments

𝜖 = 10

𝜖 = 3

Truth


• When specified accuracy is met, time the propagation



F&G, 5 segments

F&G, 10 segments

F&G, 50 segments Truth

𝜖 = 10

𝜖 = 3

𝜖 = 0.001


• For the RK: divide in segments AND in steps per segments until accuracy is met, then time the propagation



RKF8 50 segments

4 steps per segments

Truth

𝜖 = 0.001

• Software specifications – Implemented in Fortran – Compiler: gfortran v4.7.0 – F&G Stark coefficient files obtained using Matlab 15.01

• Hardware specifications – Processor: quad-core Intel Xeon W3550 – 3.07GHz clock-speed – 6GB RAM

Results: Setup details


Results: Speedups vs. RKF8


Speedup of the TS methods compared to the RKF integration when varying eccentricity

8 10 12 14 16 18 20 22 24 Order of the Taylor series



Speedup of the TS methods compared to the RKF integration when varying the Sundman exponent



Speedup of the TS methods compared to the RKF integration when varying the tolerance



Speedup of the TS methods compared to the RKF integration when varying the perturbation magnitude

• Demonstrated unexpected behavior of the TS methods for Sundman exponent of 1 – Classic 3/2 result is confirmed for the RKF8 – Results are valid for both the Stark and Kepler problems

• The F&G Stark series method is up to 50 times faster than a conventional RKF8. – Most efficient for orders between 10 and 15

discretization of about 20 – 40 segments per revolution – For orders above 18, the MTS method is more efficient

• Efficiency is tied to the (perturbed) 2-body problem EOMs – not as good for more complicated dynamics

• Future work – Application to the 1st and 2nd order State Transition Matrices

Conclusions & Future work


Thank you for your attention! Any questions?


• C. E. Velez, “Notions of Analytic vs. Numerical Stability as applied to the Numerical Calculation of Orbits,” Celestial Mechanics, Vol. 10, No. 4, 1974, pp. 405–422.

• P. Nacozy, “A Discussion of Time Transformations and Local Truncation Errors,” Celestial Mechanics, vol. 13, No. 4, 1976, pp. 495–501.

• C. H. Yam, D. Izzo, and F. Biscani, “Towards a High Fidelity Direct Transcription Method for Optimisation of Low-Thrust Trajectories,” 4th International Conference on Astrodynamics Tools and Techniques, 2010, pp. 1–7.

• G. Lantoine and R. P. Russell, “Complete closed-form solutions of the Stark problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 109, Feb. 2011, pp. 333–366, 10.1007/s10569-010-9331-1.

• D. Rufer, “Trajectory Optimization by Making Use of the Closed Solution of Constant Thrust- Acceleration Motion,” Celestial Mechanics, Vol. 14, No. 1, 1976, pp. 91–103.

• R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics. New-York, NY. Dover Publications, 1971.

References


Extra slides


• 6 “revolutions” • a = 1, e = 0.8, p = [0.01,0.01,0.01] • 𝛼 = 1 • Propagated using F&G, order 21

Example trajectory


Example trajectory, top view Example trajectory, side view

0.8 0.4 0 -0.4 -0.8

-1.6 -1 -0.5 0

Y

X -0.8 -0.4 0 0.4 0.8

0.4

0

-0.4

Z

Y



Speedup of the TS methods compared to the RKF integration when varying eccentricity

50

40

30

20

10

0

Spe

edup

ove

r RK

F8




50

40

30

20

10

0

Spe

edup

ove

r RK

F8


Speedup of the TS methods compared to the RKF integration when varying the Sundman exponent



50

40

30

20

10

0

Spe

edup

ove

r RK

F8


Speedup of the TS methods compared to the RKF integration when varying the tolerance



50

40

30

20

10

0

Spe

edup

ove

r RK

F8


Speedup of the TS methods compared to the RKF integration when varying the perturbation magnitude



• Initial conditions:




• First iteration:




• First iteration:

• Second iteration:

Technology

F&G Taylor Series Solutions to the Stark Problem