PRECISE ORBIT PROPAGATION FOR SPACE DEBRIS OBJECTS USING THEHERMITE INTEGRATION SCHEME

J. Rodmann, D. Hampf, T. Hasenohr, L. Humbert, W. Riede, F. Sproll, and P. Wagner

German Aerospace Center (DLR), Institute of Technical Physics, 70569 Stuttgart, Germany, Email: jens.rodmann@dlr.de

ABSTRACT

We present a numerical integration technique based onthe Hermite scheme, a self-starting, implicite predictor-corrector method originally developed and widely usedfor gravitational N-body systems. The Hermite schemetakes the acceleration and its first time derivative to pre-dict future position and velocity vectors from previousvalues. The beauty of the method relies on the fact thatthe second and third derivatives can be explicitely calcu-lated from the acceleration and its first derivative alone.These are then used iteratively to correct the object’s statevector, yielding an integration method with fourth-orderglobal error. The method can be used with constant orvariable timesteps. For constant timesteps the Hermiteintegrator is time-symmetric, and shows no secular errorin the semi-major axis and eccentricity. The code can in-tegrate a large number of objects in parallel, with eithershared or individual timesteps. This can be applied forpredicting future states of a catalogue of space objects, orto propagate an object’s state uncertainty (covariance) ina realistic manner.

Key words: space debris; astrodynamics; numerical inte-gration method.

1. INTRODUCTION

DLR’s Institute of Technical Physics is actively develop-ing laser-based optical tracking methods to determine 3Dpositions of LEO space debris objects to within a fewmetres. For Space Situational Awareness applicationslike collision avoidance or re-entry analyses, any initialhigh-precision orbit needs to be propagated, taking intoaccount the various gravitational and non-gravitationalforces perturbing the object’s orbit. A key prerequisite forthat is an accurate, fast, and versatile integration methodallowing the prediction of trajectories from initial condi-tions.

The integrator can be envisaged as the ”beating heart of adynamical simulation” [7]. Several numerical integrationmethods for the computation of satellite orbits have beendeveloped, see [11] for an overview and a discussion on

the advantages and disadvantages of the various methods.A special class of integrators are time-symmetric. Theypreserve first integrals (e.g. energy, angular momentum)for a conservative system [7]. Therefore, they are veryuseful for long-term simulations of dynamical systems.For a Keplerian orbit these integrators show no secularerror in the semi-major axis and eccentricity when usingconstant timesteps.

Section 2 describes the Hermite integration scheme andthe timestep selection. In Section 3 we assess the accu-racy and performance of the Hermite scheme by compar-ing our results with the semi-analytical solution of theKepler problem for various eccentricities. In Section 4the applicablity of using the Hermite integrator for short-term propagation for ∼100 orbits is discussed. Finally,in Section 5 we show results from a long-term simula-tion covering 3 million orbits, corresponding to severalhundreds years for LEO orbits.

2. HERMITE INTEGRATION SCHEME

The Hermite scheme is a direct integration method withpredictor-corrector algorithm to solve the N -body prob-lem. The method has found widespread applications forthe numerical simulation of the dynamical evolution ofmany-body systems [8, 9, 5, 14, 2].

For an object the momentary acceleration a0 and the firstderivative with respect to time a0 are used to compute thefuture position r(t) and velocity v(t) at a desired time tfrom the corresponding instantaneous values r0 and v0 attime t0 < t. In the first step, new position and velocityvectors are predicted for the next time step t by expandingr0 and v0 into Taylor series. The positive time differencet− t0 is written as ∆t.

rp(t) = r0 + v0∆t+1

2a0∆t2 +

1

6a0∆t3 (1)

vp(t) = v0 + a0∆t+1

2a0∆t2 (2)

Higher-order derivatives of the acceleration have to beconsidered to improve the accuracy of the force polyno-

Proc. 7th European Conference on Space Debris, Darmstadt, Germany, 18–21 April 2017, published by the ESA Space Debris Office

Ed. T. Flohrer & F. Schmitz, (http://spacedebris2017.sdo.esoc.esa.int, June 2017)

mial. Making a Taylor series ansatz, one writes

a(t) = a0 + a0∆t+1

2a(2)0 ∆t2 +

1

6a(3)0 ∆t3 (3)

a(t) = a0 + a(2)0 ∆t+

1

2a(3)0 ∆t2 , (4)

where a(n)0 denotes the n-th derivative of a with respect

to time evaluated at t = t0. The quantities a0, a0 are al-ready known, while a(t) and a(t) are evaluated using thepredicted values for the particle’s position rp and velocityvp. We therefore designate the left-hand sides of Equa-tions 3 and 4 with ap and ap, respectively. Solving thelatter equation for a(2)

0 and substituting it in Equation 3,we obtain [9]:

a(3)0 = 12

a0 − ap

∆t3+ 6

a0 + ap

∆t2;

re-inserting this expression in Equation 3 yields

a(2)0 = −6

a0 − ap

∆t2− 2

2a0 + ap

∆t.

That means the higher-order derivatives of the accelera-tion can be explicitly calculated in terms of ap and ap.

The predicted position and velocity of particle i can nowbe refined by expanding Equations 1 and 2 up to fourthorder to corrected position and velocity:

rc(t) = rp(t) +1

24a(2)0 ∆t4

= rp(t)− 1

4(a0 − ap)∆t2 − 1

12(2a0 + ap)∆t3

vc(t) = vp(t) +1

6a(2)0 ∆t3 +

1

24a(3)0 ∆t4

= vp(t)− 1

2(a0 − ap)∆t− 1

12(5a0 + ap)∆t2,

where a(2)0 and a

(3)0 have been substituted with the com-

posite expressions derived previously. The remainder ofthe Taylor series is proportional to O(∆t5). Taking thetime derivative, we see that the error in the force calcula-tion is ofO(∆t4). After re-grouping, the corrector reads:

vc(t) = v0 +1

2(a0 + ap)∆t+

1

12(a0 − ap)∆t2

rc(t) = r0 +1

2(vc + v0)∆t+

1

12(a0 − ap)∆t2.

The sequence of, first, evaluation of the acceleration andits first derivative and, second, correction of the positionand velocity can be iterated to increase the accuracy of theorbit integration. We typically use three (3) iterations.

The code can integrate an ensemble of objects in paral-lel, with either shared or individual timesteps. We use

100

1000

10000

100000

5 6 7 8 9 10 11 12 13

Func

tion

calls

Accuracy (in digits)

D1 (e=0.1)D4 (e=0.7)

Figure 1. Performance diagram for the Hermite integra-tion scheme, showing two test cases from [4].

quantized so-called block timesteps, where the timesteplengths can only be negative powers of two ∆tk = 2−k

[10]. This ensures that objects with the same timestepshare the same simulation time. In this way, predic-tions and computations of forces and first derivativescan be done in parallel. Furthermore, this approach en-sures co-eval output times, which is useful for generatingephemerides for a large sample of objects, e.g. a cata-logue.

Hermite schemes with order sixth or eighth order havebeen developed for even higher precision [12].

3. TEST CASES

A set of test problems for (unperturbed) Kepler orbitshave been defined in [4], with eccentricities ranging from0.1 to 0.9 in steps of 0.2 (tests D1 to D5). All orbits are in-tegrated for 20/2π ≈ 3.18 revolutions and the final state(two-dimensional position and velocity vectors) are com-pared to the analytical result found by iteratively solvingKepler’s equations.

The numerical accuracy achieved at the end of an integra-tion is defined as

ε = − log

√√√√ 4∑k=1

(xk − xk)2,

where xk are the components of the state vector deter-mined by numerical integration, and xk the analytical ref-erence orbit.

In Figure 1 we plot the numerical accuracy achieved withthe fourth-order Hermite scheme for two test cases witheccentricities of e = 0.1 (D1) and e = 0.7 (D4). For thelow-eccentricity case (nearly-circular orbit), an accuracyof∼9 is obtained after about 2500 function calls, compa-rable to the RKF7 method of [3]. About 10 times morefunction calls are needed for the high-eccentricity orbit toreach the same accuracy.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 20 40 60 80 100

Posi

tion

erro

r at 1

000

km [m

]

Number of orbits

Figure 2. Errors in position due to the numerical methodfor an integration covering 100 orbits.

0

5x10-6

1x10-5

1.5x10-5

2x10-5

2.5x10-5

3x10-5

0 20 40 60 80 100

Velo

city

err

or a

t 100

0 km

[m/s

]

Number of orbits

Figure 3. Errors in velocity due to the numerical methodfor an integration covering 100 orbits.

4. SHORT-TERM INTEGRATIONS

Propagation times of a few days and up to a week arerelevant for SSA use cases like conjunction analysis, sen-sor tasking, etc. Here we demonstrate the applicability ofthe Hermite integration method for forecasting LEO or-bits for these timescales. We chose a stop time of 200π,i.e. 100 orbits. For a typical RSO in LEO with∼100 minorbital period the simulation covers ∼7 days.

The great majority of RSOs is on nearly circular or-bits, with 90% of the objects having eccentricities < 0.1.Therefore, we selected an orbit with an eccentricity ofe = 0.1, similar to test case D1 above. The evolutionof the errors of position and velocity vectors for an orbitwith semi-major axis a = RE + 1000 km are shown inFigures 2 and 3.

As is expected from the secular drift in the argument ofpericentre [6], the errors in positions and velocity show alinear trend. The error in position after 100 orbits are wellbelow one metre, proving that the Hermite integrator canbe used for precise orbit determination based on laser-range measurements with metric accuracy. We found thevelocity error to be∼0.03 mm/sec. Both position and ve-locity errors oscillate with one orbital period with grow-ing amplitude.

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1.5 -1 -0.5 0 0.5 1

y

x

e=0.1e=0.3e=0.5e=0.7e=0.9

Figure 4. Position vectors in the orbital plane for thecomplete 3.2 million revolutions. Test orbits with initialeccentricities of 0.1, 0.3, 0.5, 0.7, and 0.9 are shown.

5. LONG-TERM INTEGRATIONS

Long-term integrations are required for analysing the or-bital decay due to atmospheric drag, investigating reso-nant effects or chaotic motion, or studying space-debrispopulations. Here the superior properties of the Hermiteintegration scheme, that is conserving integral of motionsas well as semimajor axis and eccentricity, will becomeapparent. We integrated the Kepler orbits D1 to D5 for3.2 million revolutions, corresponding to ∼600 years fora LEO orbit.

Figure 4 displays the position vectors, illustrating that theoverall orbital geometry is not changing over more than3 million revolutions. Semimajor axes and eccentricitiesshow no secular drift, and can be considered constant forall practical purposes. This can also be seen in the plotof the relative energy error ∆E/E0, see Figure 5. Whilethe relative energy error undergoes periodic changes, nosecular drift is present.

As discussed above the argument of pericentre ω under-goes a slow linear drift, cf. Figure 6. This is in con-trast to high-order multistep integrators which generallyhave quadratic errors in the angle variables [6]. The errorgrows with eccentricity.

-7

-6.95

-6.9

-6.85

-6.8

-6.75

-6.7

-6.65

-6.6

-6.55

0 500000 1x106 1.5x106 2x106 2.5x106 3x106

log 1

0(∆

E/E 0

)

Number of orbits

Figure 5. Change of the relative energy error over theintegration time for a set of 5 objects on D1 to D5 orbits.

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

0 500000 1x106 1.5x106 2x106 2.5x106 3x106

log 1

0 (∆

Ay/

Ay,

0)

Number of orbits

e=0.1e=0.3e=0.9

Figure 6. Relative error of the y-component of theLaplace-Runge-Lenz vector.

6. SUMMARY

We have demonstrated that the Hermite integrationscheme is a very useful numerical method for orbit propa-gation. Especially due to its time-symmetry the integratorhas several desirable properties that make it highly usefulfor long-term simulation of orbital dynamics up to sev-eral millions of revolutions. For a conservative system,energy and angular momentum show no secular errors.As a consequence, both semimajor axis and eccentricityof an object are not drifting away from their initial values.

The Hermite scheme provides excellent accuracy onthe propagated state vectors for short-term integrations,which are important for several applications and use casesin the domain of Space Situational Awareness (uncer-tainty propagation, conjunction analysis, or re-entry pre-dictions). For example, a 7-day propagation of a typicalLEO orbit results in a numerical position error below 1metre.

The code can integrate an ensemble of objects in par-allel, with either shared or individual timesteps. It cantherefore be applied to propagate a catalogue of space ob-jects and/or to perform realistic uncertainty propagationvia Monte-Carlo or sigma-point methods.

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