A NOVEL INITIAL ORBIT DETERMINATION ALGORITHM FROM DOPPLER ANDANGULAR OBSERVATIONS

C. Yanez, F. Mercier, and J-C. Dolado

CNES, 18 av. Edouard Belin, 31401 Toulouse Cedex 9, France, Email : carlos.yanez@cnes.fr

ABSTRACT

A novel numerical method for solving the initial orbit de-termination (IOD) problem is developed in the frame ofspace debris surveillance systems. Differently from clas-sical IOD methods, where three sets of angles or two po-sition vectors are used, the method presented in this pa-per makes use of N Doppler measurements along withtheir associated pairs of angular measurements comingfrom an Earth-based radar station. Numerical results arepresented for objects in three different Low Earth Orbits(LEO) showing the accuracy and performances obtainedby this method.

Key words: Initial Orbit Determination; Doppler Radar;Space Debris; Cataloguing; LEO objects.

1. INTRODUCTION

Space Situational Awareness (SSA) refers to the abilityto view, understand and predict the physical location ofnatural and manmade objects in orbit around the Earth.While the ability to view is made possible thanks toground and space based sensors (e.g. Radars, Telescopesor Lasers), the ability to understand and predict the phys-ical location of objects needs the determination of theirorbital state vectors, which uniquely determine the tra-jectory of the object in space.

For the determination of the orbital state of an object,mainly two different situations have to be considered. Ei-ther the object is already catalogued (i.e. it is regularlyobserved by sensors and an a priori state vector is avail-able) or the object has been newly discovered by a so-called surveillance sensor. In the first case, the a prioriorbital state of the object can be refined thanks to the newgathered observations following a Least-Squares (LS) oran Extended Kalman Filter (EKF) filtering approach. Inthe second case, an a priori orbital state has to be com-puted from the gathered observations, in order to predictthe position of the newly detected object at short termand to improve the accuracy of the orbit with the acquisi-tion of new observations from surveillance and/or track-ing sensors.

Present work is focused on the second problem, and, inparticular, we develop a method to estimate an initial or-bit from surveillance radar measurements. A distinctivefeature of such surveillance radars is that they are, for themost part, Doppler radars, which means that they providethe angular measurements and the Doppler shift (i.e. theradial velocity) of the observed target at each observingtime. While an extensive literature exists on algorithms toestimate an initial orbit from three pairs of angular mea-surements ([1], [2], [3]) or from two pairs of angular mea-surement and ranges (i.e., the Lambert’s problem, [4],[5], [6]), few literature exist on methods to estimate aninitial orbit from measurements coming from a Dopplerradar.

On this paper we introduce the developed method, itspractical implementation and its performance by meansof several test scenarios with simulated data of LEO ob-jects.

2. DESCRIPTION OF DOPPLER IOD ALGO-RITHM

In order to process surveillance radar data, it is necessaryto be able to solve the initial orbit determination problembased on its measurements. Contrary to a tracking radarfor which an a priori orbit is available, surveillance radarcan produce measurements from non-catalogued objects.

It is assumed, then, that no information on the state-vector of the object is available. Radar data comprises Nconsecutive observations (gathered in an observation arcor observation pass), not necessary equally time spaced.Each observation is composed of measurements on theline of sight L, defined by a pair of angles (for exam-ple, the right ascension α and the declination δ), andthe Doppler shift d of the unknown object, where d isa distance that depends on the type of radar1 and the dotrepresents the time derivative. In the case of a mono-static radar, this distance refers directly to the object

1We consider the cases of monostatic and bistatic radars. The formerrefers to a radar in which the transmitter and receiver are collocated, andthe latter to a radar in which the transmitter and receiver are in differentlocations, maybe hundreds of km away.

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)

Object (O)

Transmitter (T)

𝐿

Receiver (R)

𝜌𝑇𝑂 𝜌𝑅𝑂

Figure 1: Scheme of the geometry for a bistatic sensor.

range, dmonostatic = ρ. This equation assumes im-plicitly that the propagation time in the two-way tripfrom ground-based radar to orbiting object is negligibleand, in consequence, the epoch of emission and recep-tion is the same (see Appendix to take into account thisterm). On the other hand, for bistatic radars, this dis-tance corresponds to the transmitter-object-receiver dis-tance, dbistatic = ρTO +ρRO (see Figure 1). In this case,we will speak of object range to the distance between thereceiver and the object, which can be expressed as fol-lows:

ρ =d2 − |TR|2

2(|TR| · L+ d)(1)

The estimation of the derived distance measurementsd(t) is the main concern of the algorithm, so that theDoppler initial orbit determination can be reduced to awell-known Lambert’s problem between the boundariesof the observation arc. This estimation is performed us-ing optimization techniques. We begin by integrating2

the Doppler measurements between the initial time of theobservation arc t0 and a time t ≤ tf , where tf is the finaltime of the observation arc:

d(t) = d0 +

∫ t

t0

d dt. (2)

This equation assumes that the radar is taken measure-ments of the object continuously within a visibility pass.For LEO objects, this visibility pass covers typically afew minutes span. The evolution in time of the distanced and, consequently, the evolution in time of the objectrange, is known up to a constant of integration d0, whichcorresponds to the distance at the initial time. Somehowwe are transforming the Doppler radar problem into a dis-tance radar problem where an unknown bias on the dis-tance measurement applies.

The procedure for the determination of this constant isbased on the conservation, to first order, of the total or-bital energy. Actually, the energy of the orbit that passesthrough any two observations can be determined as a

2Within this work, numerical integration of the Doppler measure-ments is done through the Simpson’s 3/8 rule, based upon a cubic in-terpolation. Equally spaced in time measurements are, in consequence,considered in the simulations.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

-45

-30

-15

0

1 750 2 000 2 250 2 500 2 750 3 000

Std

. dev

iati

on

(km

2/s

2)

Orb

ital

en

erg

y (k

m2/s

2)

Initial distance d0 (km)

Figure 2: Specific orbital energy of the orbit passingthrough diffferent pairs of measurmeents as a function ofthe integration constant (black lines, left scale). Standarddeviation of the distribution of energies (grey line, rightscale). More pairs than those plotted are used in the com-putation of σ.

function of d0 by solving the associated Lambert’s prob-lem. Let ρi = ρ(ti, d0)Li and ρj be the position vectorsof the selected pair of points (i, j) and ∆t = tj − ti > 0the corresponding flight time, we can then solve the Lam-bert’s problem to get the specific orbital energy of thetwo-body problem, that is to say, εij(d0) = −µ/2a. Anideal two-body system is conservative, that is to say, thetotal energy of the problem is conserved. Furthermore, ifwe consider ideal (error-free) measurements, the curvesεij(d0) for any combination of two observations intersectinto a single point (see Figure 2). Thus, this point definesthe initial range d0 and, by extension, a preliminary esti-mate of the orbit. This point satisfies the energy integralof Kepler’s motion and it is, in this way, the optimal solu-tion. In a real case, however, the object will be subject todissipative forces (for LEO objects, the atmospheric dragis the most important one), and the measurements willnot be error-free but the sensor noise and bias should beconsidered. In this situation, a unique intersection in thegraph ε(d0) is not possible but, on the contrary, a distribu-tion of energies εij can be computed for any value of d0.The value of d0 that leads to the minimum standard devi-ation, σmin, in the distribution of energies is the optimalsolution. This distribution of energies depends directlyon the way the pair of measurements (i,j) are selected.The selection cannot be done at random since observa-tions needs to be handled equally (i. e. being selectedthe same number of times) in order not to introduce anyartificial bias in the solution. In Section 3.1, different ap-proaches are analysed.

Thus, the standard deviation of the distribution of en-ergies is a univariate function that only depends on theintegration constant, σ(d0). The minimization is per-formed by means of the Brent’s method [8] that doesnot require the use of derivatives. The optimizationsearch interval varies depending on the type of radar.For a monostatic radar we will search within the inter-val [hr/sin(el), ρmax], where hr is an altitude belowwhich reentry is considered imminent (' 120 km), el is

Scenario LEO 1 LEO 2 LEO 3Semi-major axis [km] 7198.0 6778.0 7578.0Eccentricity [-] 0.0 0.0 0.0Inclination [deg] 98.71 60.00 60.00

Table 1: Keplerian orbital elements defining the three ref-erence orbits.

the elevation and ρmax the maximum range of radar ac-quisition. For a bistatic radar, this search interval stays[2hr/sin(γ), 2ρmax], where γ = atan(2hr/|TR|).

3. TEST SCENARIOS

Three reference orbits are selected as base scenarios fortesting the developed method. These orbits are circular ataltitudes of 400, 820 and 1200 km depending on the case.Additional numerical data defining the orbits consideredare provided in Table 1. These scenarios only take intoaccount LEO objects (altitude below 2000 km [7]), sincethe observation of objects orbiting on higher orbits is typ-ically carried out with optical sensors. In all cases, theobservations were made from a ground station located inFrance at the following geodetic coordinates: longitude= 7.0 deg, latitude = 44.0 deg and alt = 1200.0 m. Then,only monostatic radar results are presented in this paper.Nevertheless, same scenarios have been executed consid-ering a bistatic radar (with a distance transmitter-receiverof 400 km), showing similar results. These three basescenarios are enriched by a sensitivity study concerningthe following parameters:

• Measurement noise. In order to simulate realisticdata, measurements are corrupted with a Gaussiannoise characterized by its standard deviation σ. Weconsider angular noise σang , equal in both direc-tions (azimuth and elevation), going from 1 mdegto 1 deg, and Doppler noise, σdop in the interval 1cm/s to 10 m/s. Two main cases are usually reportedas low and high noise case (see Table 2).

• Observation timespan ∆T . The duration of the ob-servation interval has a direct impact on the accuracyof the orbit estimate since it is related to the observ-able portion of the orbit. We consider observationintervals up to 10 minutes for LEO 1 and LEO 3scenarios that roughly corresponds to the first visi-bility interval, and up to 5 minutes for the LEO 2scenario, which is the one at a lower altitude (' 400km).

• Measurement acquisition frequency ∆t. We con-sider time separation between observations that goesfrom 1 to 30 seconds. These measurements are takenall over the observation interval.

Noise case Angular noise Doppler noiseσang (mdeg) σdop (cm/s)

Low noise 10.0 10.0High noise 100.0 100.0

Table 2: Gaussian standard deviations of the measure-ments noise defining two noise cases.

Results presented hereafter are all average values ob-tained from 1000 simulation runs.

3.1. Influence of the selection of measurements pairs

The aforementioned Doppler IOD method depends on theway the pair of measurements are selected to compute thedistribution of energies. Two approaches are envisaged:

• Consecutive: We take one observation and the fol-lowing one in such way as to have pairs of indexcovering the whole observation interval. We con-sider high frequency radar observations so that it ispossible to compute the velocity with a finite dif-ferences approximation, vi = (ri+1 − ri)/∆t. Wecompute then the orbital energy as εij(d0) = v2/2−µ/r. We have made the assumption that close obser-vations are not correlated to each other. In this worksensor noise is simulated with a Gaussian compo-nent added to the geometric value and, for each ob-servation, the Gaussian noise is computed randomly,so independently of other observations.

• Half-arc separation: The idea behind this approachis to mitigate the effect of the measurement noise.Thus, we take measurements separated by a longerinterval equal to the half of the observation inter-val (considering that observations are equally spacedin time). All the observations are considered oncein the computation. Moreover, Izzo method [6] isused for solving the Lambert’s problem instead ofthe simple finite differences scheme.

We have assessed both approaches with observations ofthe sun-synchronous case (LEO 1) covering a 10 mntimespan. It is worth noting that for close observations,the consecutive approach fails in providing a satisfac-tory orbit estimation (notice the error in semi-major axisshown in Figure 3). This behaviour can be explainedby the influence of measurement noise on the Lambert’sproblem resolution when the observed portion of the or-bit covers a tiny part (0.1% of the orbit for ∆t = 6 s)and, in consequence, the relation ∆α, ∆δ >> σang be-tween two measurements no longer stays. Half-arc sep-aration approach presents, on the contrary, a more sta-ble trend with respect to the time between measurements.We are solving in this case a Lambert’s problem withbounds separated around 5 mn (all pairs are separated by

0.1

1

10

100

1 10

Ave

rage

sm

a e

rro

r (k

m)

Time between measurements (s)

Figure 3: Average semi-major axis error for LEO 1 casewith an observation timespan ∆T = 10mn. Low noisecase (circles) and high noise case (triangles) results areobtained either with half-arc separation (black) or con-secutive approach (grey).

0.001

0.01

0.1

1

10 100 1000

Ave

rage

tim

e p

er

run

(s)

Number of observations

Half-arc separation

Consecutive

Figure 4: Average time per run in the LEO 1 case withan observation timespan ∆t = 10mn.

the same timespan) which correspond to 5% of the or-bit. For the half-arc separation approach, we identify twomain elements having an impact on the orbit accuracy:

1. Measurement noise. We can see in Figure 3 thatblack lines are almost parallel. The difference be-tween both lies in having increased the measurementnoise in one order of magnitude in the three com-ponents observed (two angular values and Doppler).Accuracy in semi-major axis roughly decreases byone order of magnitude too.

2. Number of measurements. This is directly relatedto the quantity of information that is available aboutthe orbit. When the observations pass from beingspaced 1 s to 30 s in the same timespan, we areequivalently passing from 600 to only 20 observa-tions. The less observations are available, the worsethe distribution of energies is represented.

We have compared the computational time required byboth approaches on a multi-core 2.66 GHz machine with

0 50 100 150 200 250 300−0.4

0.00.40.8

0 50 100 150 200 250 300−0.4

0.0

0.4

0 50 100 150 200 250 300Time(s)

−60

0

60

Figure 5: Residuals on azimuth [deg] (top), elevation[deg] (center) and radial velocity [m/s] (bottom) of aLEO 1 object with measurements corresponding to thehigh noise case taken within a ∆T = 5mn.

0 50 100 150 200 250 300−0.4

0.0

0.4

0 50 100 150 200 250 300−0.4

0.0

0.4

0 50 100 150 200 250 300Time(s)

−4

0

4

Figure 6: Residuals of Figure 5 when the initial orbitis refined with a least-squares filter. Dashed line corre-sponds to the values 3σ of the sensor noise.

32 GB RAM. Results are plotted in Figure 4 show-ing a negligible difference between the approaches em-ployed. We can thus state that the simple finite differ-ences scheme does not speed up the computation sig-nificantly. Therefore, half-arc separation approach is se-lected as optimal in terms of accuracy and computationaltime. Results contained in the rest of this work refer in-variably to this approach.

3.2. Scenario LEO 1 : sun-synchronous orbit

This scenario analyses the accuracy that can be obtainedfor an object in a sun-synchronous orbit at 820 km of alti-tude. If we look closely to the residuals that are obtainedafter applying the Doppler IOD method (see Figure 5),we notice that the estimates of the angular positions areroughly contained within the order of magnitude of theangular noise. However, estimates on the radial veloc-ity present an important drift with respect to the observa-tions. This fact indicates that the semi-major axis and ec-centricity estimates are not as accurate as other angular-related elements as the inclination or the argument of lat-

1 2 5 1010−2

100

102

104sm

aer

ror

[km

]

1 2 5 10Observation timespan (mn)

10−6

10−4

10−2

100

ecc

erro

r[-

]

Figure 7: Average semi-major axis error [km] (top) andaverage eccentricity error (bottom) as a function of ob-servation timespan ∆T . Circles correspond to the highnoise case and diamonds to the low noise case. Solidblack and dashed grey lines are the result of the DopplerIOD method before and after applying the LS filter, re-spectively. ∆t = 6 s.

itude. It is then necessary to refine the initial orbit bymeans of a least-squares filter in order to obtain an op-timal orbit estimate. The effect of this LS step can beseen in Figure 6, where residuals in all three componentsof the observation stays within the band ±3σ. This opti-mal behaviour of the residuals in Figure 6 is attained byconsidering the same dynamical model in the simulatedobservation and in the LS filter. In a real case, the dy-namical model used in the LS filter is essential to achievecomparable accuracies. For an object in the LEO regime,at least a dynamical model including osculating J2 effectsis needed.

Sensitivity against the observation timespan can be seenin Figure 7. We highlight the constant trend observed inthe accuracy obtained, which can be approximated to apower law. It is worth noting the remarkable improve-ment in the accuracy for longer timespans. This is dueto a twofold reason. We are not only increasing the ob-served portion of orbit (with the consequently increase ofknowledge about the shape of the orbit, i. e. the eccen-tricity), but we are at the same time increasing the num-ber of observations, which helps in building up a morereliable distribution of energies, εij , for the optimizationproblem.

Figure 8 plots the average semi-major error as a functionof measurement noise. It is worth noting the preponder-ance of the angular noise over the Doppler one, shownup by the horizontal contour lines. For example, if theangular noise is above 30 mdeg, there is no differenceif Doppler measurements are remarkable accurate withσdop = 1 cm/s or, on the contrary, are obtained with aless performant radar of 10 m/s noise. This behaviourchanges after applying the LS filter and the contour plotexhibits, as expected, a diagonal increase of the error (to-wards the upper-right corner of the figure), that is to say,if the noise of any observational component increases,

100 101 102 103

Doppler noise (cm/s)100

101

102

103

Ang

ular

nois

e(m

deg)

0.51.0

2.0

5.010.020.0

50.0100.0200.0500.0

Figure 8: Average semi-major axis error [km] as a func-tion of measurement noise. ∆T = 5mn and ∆t = 6 s.

100 101 102 103

Doppler noise (cm/s)100

101

102

103

Ang

ular

nois

e(m

deg)

0.10.2

0.5

1.0

2.0

2.0

5.0

5.0

10.0

Figure 9: Average semi-major axis error [km] as a func-tion of measurement noise after having refined the initialorbit with a LS filter. ∆T = 5mn and ∆t = 6 s.

the accuracy obtained decreases. The orbital plane is, ingeneral, well defined. The difficulty to both, recover theobject a few orbital periods later and enable a LS filterto converge with a measurement arc containing severalpasses, comes essentially from the error in semi-majoraxis. It is worth thinking of in terms of orbital period. Inthis orbital regime, an error of 100 m in semi-major axiscorresponds to a difference of about a tenth of second ina revolution, and in one day (' 14 revolutions) less than2 s gap. Nevertheless, a 10 km error in semi-major axisis translated in a difference of± 13 s in a revolution, and,in consequence, one day later the object will pass throughthe predicted region of sky about 3 minutes before or af-ter the estimate, with a non-negligeable angular shift withrespect to the line-of-sight prediction.

3.3. Scenarios LEO 2 and LEO 3 : low and highLEO objects

These two scenarios analyze the performance of the IODRadar for objects at 400 and 1200 km of altitude, respec-tively. Figure 10 shows the sensitivity of the solution with

0.5 1.0 2.0 5.010−2

100

102

104sm

aer

ror

[km

]

0.5 1.0 2.0 5.0Observation timespan (mn)

10−6

10−4

10−2

100

ecc

erro

r[-

]

1 2 5 1010−2

100

102

104

sma

erro

r[k

m]

1 2 5 10Observation timespan (mn)

10−6

10−4

10−2

100

ecc

erro

r[-

]

Figure 10: Average semi-major axis error [km] (top) and average eccentricity error (bottom) as a function of observationtimespan ∆T , for LE0 2 (left) and LEO 3 (right) scenarios. See Figure 7 for more details on the legend.

100 101 102 103

Doppler noise (cm/s)100

101

102

103

Ang

ular

nois

e(m

deg)

0.2

0.51.0

2.0

5.010.020.0

50.0

100 101 102 103

Doppler noise (cm/s)100

101

102

103

Ang

ular

nois

e(m

deg)

0.010.02

0.05

0.1

0.2

0.5

1.0 2.05.0

100 101 102 103

Doppler noise (cm/s)100

101

102

103

Ang

ular

nois

e(m

deg)

5.010.020.050.0100.0200.0500.0

1000.02000.03000.0

4000.0

100 101 102 103

Doppler noise (cm/s)100

101

102

103

Ang

ular

nois

e(m

deg)

0.1

0.20.5 1.0

2.0

5.0

5.0

10.0

20.0

Figure 11: Average semi-major axis error [km] as a function of measurement noise. ∆T = 5mn and ∆t = 6 s. LEO 2results are plotted at the top and LEO 3 results at the bottom. Results are presented before (left) and after (right) havingrefined the initial orbit with a LS filter.

respect to the observation timespan. Note that axes do notshare the same scale, as visibility periods for the lower al-titude case are limited to approximately 5 mn. In line withthe results from previous section, we notice an outstand-ing improvement of the acccuracy obtained for longertimespans, in the form of a constant decreasing trend thatcan be fitted to a power law. It is important, then, to have

access to a prolonged observation timespan that coversalmost the entire visibility period. Comparing both sce-narios, we observe that higher accuracies are obtained inthe LEO 2 case, which is the one at a lower altitude. Thisfact is explained by two complementary reasons. First,the observable portion of the orbit decreases, for a fixedobservation timespan, with the altitude of the object. For

Object (O)

Ground station (G)

𝐺 𝑡𝑂− 𝑑1 𝑐 𝐺 𝑡𝑂+ 𝑑2 𝑐

O 𝑡𝑂

𝑑1 𝑑2

𝑢1 𝑢2

Figure 12: Scheme of time propagation effect on the ge-ometry of the problem for a monostatic Doppler radar.

example, the percentage of observable orbit in 5 mn getsreduced from 5.4 to 4.6% as the altitude increases from400 to 1200 km. The extension of the observable orbitarc limits the attainable accuracy. And second, the posi-tion error induced by the angular noise depends directlyon the distance receiver-object, following the expressionσpos = ρROσang .

We can see in Figure 11 the average semi-major error as afunction of the measurement noise. We recover a similarbehaviour than that of the previous section. The strengthof the angular noise impact in the results is emphasized inthe LEO 3 scenario (higher altitude), as expected. In thatscenario, level curves are essentially horizontal, notinga minor influence of the Doppler noise on the orbit ac-curacy for the range of noise values explored. Again, theuse of the LS filter is necessary to greatly reduce the erroron the semi-major axis. In the configuration with the lessperformant radar, we obtain accuracies on the semi-majoraxis of the order of 5 and 20 km for LEO 2 and LEO 3scenarios, respectively. If we repeat the exercise in termsof orbital periods, we have that, for the LEO 2 regime,an error of 5 km corresponds to a difference of ±6 s ina revolution, and in one day (' 15.5 revolutions) around1m30s gap. Furthermore, in the LEO 3 scenario, an errorof 20 km in the semi-major axis induces a difference of±26 s in the orbital period. This causes, in one day (±13revolutions), an uncertainty of more than 5mn30s on thepredicted moment for the radar to track the object again(considering an optimistic angular shift contained withinthe field of regard of the radar). Hence, the importanceof re-observe the object in the early revolutions after itsidentification in order to achieve a good estimate on thesemi-major axis, that permit us to consolidate the objectorbit in the catalogue.

4. CONCLUSIONS

On this paper we have introduced a novel method tocompute the initial orbit of an orbiting object in theLEO regime from Doppler radar measurements, includ-ing monostatic and bistatic radar types. It is composedof two steps. First step, an optimization problem is builtup taking into account energetic considerations in order

to reduce the problem to a two-body Lambert’s problem.Second step, the initial orbit obtained previously is fittedto observations in a LS filter with a more complex dynam-ical model. The performance of this approach has beenassessed by the definition and analysis of three syntheticcases including LEO object at different orbital regimes.Furthermore, the robustness of the algorithm to a varia-tion on the quality and quantity of information has alsobeen assessed, by a sensitivity analysis on the measure-ments noise, the observation timespan and the time be-tween measurements. In addition, the accuracy of theproposed algorithm has been demonstrated through thedifferent simulation scenarios, which highlight the effi-ciency of the algorithm to estimate an initial orbit fromN observations coming from a Doppler radar.

APPENDIX : CONSIDERING PROPAGATIONTIME

The propagation time of the two-way trip between aground station and an object is modelled by the followingequations:

d1 = |D(tD)G(tD − d1/c)|, (3)d2 = |D(tD)G(tD + d2/c)|, (4)

d = (d1 + d2)/2, (5)

d = Doppler measurement, (6)

where tD is the date of signal reflection by the object, cis the speed of light and d1 and d2 are described in Fig-ure 12. These equations shall be solved completely, withan iterative scheme, for the reception time at the begin-ning and end of the Doppler signal. A solver dedicatedto this problem should be included in the LS filter stepto take into account the effects of geometric propagationtime. Similar equations can be derived for bistatic radars.

REFERENCES

1. Vallado D. A. (2007). Fundamentals of astrody-namics and applications. Third Edition. Microcosm,Hawthorne, CA.

2. Escobal P.R. (1965). Methods of Orbit Determination.John Wiley and Sons, Inc., New York

3. Gooding R. H. (1997). A new procedure for the so-lution of the classical problem of minimal orbit deter-mination from three lines of sight, Celest. Mech. Dyn.Astr., 66(4), 387-423

4. Lancaster E. and Blanchard R. (1969). A unified formof Lambert’s theorem. National Aeronautics and SpaceAdministration. Technical Note D-5368.

5. Gooding R. H. (1990). A procedure for the solutionof lambert’s orbital boundary-value problem, Celest.Mech. Dyn. Astr., 48(2), 145-165

6. Izzo D. (2015). Revisiting Lamberts problem, Celest.Mech. Dyn. Astr., 121(1), 1-15

7. Inter-Agency Space Debris Coordination Committee(2007). IADC Space Debris Mitigation Guidelines,IADC-02-01

8. Brent R. P. (1973). Algorithms for minimization with-out derivatives. Prentice-Hall series in automatic com-putation.