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CORRELATION OF MEASUREMENTS IN THE PROCESS OF SATELLITE CATALOG MAINTENANCE

Z. Khutorovsky

"Vympel" International Corporation, Moscow, Russia Summary The problem of satellite catalog maintenance in Russian information system is discussed at scientific conferences and workshops since 1992. The issues of general structure, characteristics and design of the algorithms of this system are discussed for different classes of satellites. The present paper also considers this problem and continues the discussion of methodical and algorithmic issues. Here we treat the key algorithm of catalog maintenance process - the procedure for allocation (correlation) of acquired measurements to satellite orbits. The presented detailed investigation of the problem has three levels - general theoretical, "system" and "realistic system" levels. Consideration covers all the classes of observable Earth satellites and all types of available observations. Introduction This paper treats the task of satellite catalog maintenance in the context of the problem of establishing correlation between the observations and the satellites. In fact, this problem covers only a part of the task of processing sensors' data with the goal of primary determination and tracking satellite orbits. However, correlation of observations with the objects, following the orbits of Earth satellites is the central and the most sophisticated part of this task. The principles of solving the general task of catalog maintenance for Russian information system, including the task of measurements' correlation are described in detail in the works 1-4. In particular, it is shown that the decomposition of the general task into separate processes is incorrect. The processes of primary orbit determination and tracking and their certain sub-processes, for example correlation of measurements with tracked satellites and orbit updating using the measurements are closely interconnected. However, separate consideration of the correlation problem is still useful from methodical point of view. The rationale is as follows. This approach simplifies the general task providing the possibility of more profound investigation of the problem of catalog maintenance. More clear understanding of the essence can be attained, leaving the insignificant details. Thus, for example we can clarify the causes of the difference between the correlation procedures used for primary orbit determination and the tracking, define the basic parameters, affecting the result and the techniques for accounting of the features of given information system. We acquire the capability of detailed analysis of the performance of certain system under different conditions, in particular, we can answer the question of the principal possibility of proper allocation of measurements to the satellites, which ultimately determines the tracking capabilities.

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We introduce three levels in our consideration of the correlation problem. First we solve the task in rather general formalization. We come to a theoretical algorithm, which does not take into account the features of certain information system. Even this stage of the investigation makes clear the difference between the processes of primary orbit determination and tracking, that determines different structure of decision-making procedures. The second stage, accounting of the specific features of the considered task, leads to the real algorithm. The "real" algorithm, however, turns to be unrealistic for available computers. Thus the third stage discusses the techniques for simplification of the procedure with nearly no losses. These techniques determine the feasible algorithm. At the second and the third stages we investigate separately primary orbit determination and tracking as well as the transition process. Theoretical algorithm In general, the algorithm, solving the problem of measurements-to-satellites correlation can be described as follows. Assume we have a set of measurements (observations) X = (xl, x2, ..., xn), acquired by all the sensors within the interval of their operation. Allocation (correlation) of these measurements to the satellites provides minimum to the functional

),,()...,,,(Ø)(

111121 ∑

=

Ψ=χ

χχk

l)k( aXa,aa (1)

where χ - certain allocation of measurements to the satellites;

)...,,,(21 lmllll xxxX = - the set of measurements, correlated with the l-th satellite according to

allocation χ; k(χ) - the number of satellites in allocation χ; m1 + m2 + … + mk = n - total number of measurements;

),( ll1 aXØ - the functional, defining the measure of closeness between the selected set of measurements Xl, and parameters al of the l-th satellite.

),( ll1 aXØ is a quadratic maximum likelihood functional having the form )),(())((),( -1

lllllllll1 aFXMaFXaXØ −′−= (2) constructed on the residuals between the measured Xl and calculated Fl (al) (on the basis of vector al) parameters of the l-th satellite.

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Ml - square matrix with dimensions 6ml x 6ml, which elements characterize integrated errors of observations and the propagation of the element sets to the times )...,,,

21 lmlll ttt of the measurements ;...,,,

21 lmlll xxx -1 and ' - denote inverse and transposed matrices; The functional (1) is minimized in all its arguments. To avoid "multiplication" of orbits, we exclude from the process the allocations X, for which the measurements of certain objects "inscribe" into the orbits of other ones. Doing this, we assume that at least one allocation exists, which does produce multiplication of orbits. In other words, we consider that the observations available for any object principally allow determination of its orbit with accuracy, making impossible correlation with "alien" measurements. This is so called "informativity" condition. In case the informativity condition is not satisfied, the proper (providing small errors) correlation of measurements to satellites looks impossible (on the basis of track data only). Thus the algorithm for measurements-satellites correlation means the search over the set of measurements. Now we shall assume that for certain 1-th satellite there is an orbit, determined using the measurements, "attached" already to this satellite, and the measurements from other satellites can not "inscribe" into this orbit. This is the situation we have in the tracking process. We will show that the decision-making algorithm can be significantly simplified in this case. For the functional ),,( lll axXØ of the form (2), constructed on the measurements Xl, already correlated with 1-th satellite, and a measurement x we have the following approximate relationshipF1:

min ψ l (X l ,x,a l ) _min ψ l (X l ,a l ) + ql (z) a l a l

(3)

where )a(fxz ll ˆ−= - residual between the new measurement x and its estimation using the parameters a of the orbit of 1-th satellite, acquired using previous measurements, which have the form

a l = arg min ψ l (Xl ,a l );(a l )

(4)

F1 If there is no time correlation in observation and prediction errors and the functional relationships )ˆ( ll af are linear, the approximate equality (3) becomes precise.

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fl(al) - is the functional relationship between parameters of the measurement x and orbital parameters al of the 1-th satellite; ql(z) - quadratic form:

,)ˆ()ˆ()(

1

zaafR

aafz

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ ′

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+′=l

lll

l

lll zq K (5)

where K and Rl, - covariation matrices of the errors x and la ;

l

llaaf

∂∂ )ˆ( - matrix of partial derivatives of parameters of the measurement with respect to orbital

parameters, calculated for the point la .

The matrix K +

∂fl (a l )∂a l

⎛⎝⎜

⎞⎠⎟

R l∂fl (a l )∂a l

⎛⎝⎜

⎞⎠⎟

′is the covariation matrix of the vector z, thus the

quadratic form (5) is the normalized distance between the observed and calculated (on the basis of previous measurements of the 1-th satellite) parameters of the measurement x. As follows from relationship (3) and the requirement that only the "own" measurements (acquired for 1-th object) can inscribe into the orbit of 1-th satellite, the values of ql(z) for the own measurements are significantly smaller than those for "alien" measurements (acquired for other objects). Thus the algorithm making decision on the correlation of measurement x with 1-th satellite means the calculation of quadratic form (5) and further comparing it with certain threshold. So, if the satellite has enough accurate orbit, the correlation of measurements with this object is essentially simplified. In case such orbit is not in place, we have the situation of primary orbit determination (orbit detection). In this situation, the search also can be simplified. Determination of the orbit that will be capable of correct selection of its measurements does not ultimately require that all the measurements must be used. Sometimes enough accurate orbits can be constructed on the basis of certain part (group) of measurements. The groups of measurements having this property are called complete. We are interested in complete groups of measurements with size ml1 « ml. If such groups exist, in the process of the search for measurements-to satellites arrangement the "depth" of the search for 1-th satellite does not exceed ml1. Real algorithm: tracking The correlation procedure, developed in the previous section is applicable to many tasks, since it is obtained under rather general assumptions regarding characteristics of the measurements and the objects moving in space. That's why we can call this procedure "theoretical algorithm". In this section, we try to incorporate in the theoretical procedure the specific features of the considered task, in particular, the characteristics of real measurements. Here we consider the

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situation of tracking, when the satellites already have the orbits providing efficient correlation of future measurements. Theoretical algorithm assumes that the errors of the measurements have Gaussian distribution with zero mean and known covariation matrixes. The actual situation is different. In reality, a small part of radar observations have abnormally great errors in certain parameters (range, azimuth, elevation angle, their rates) that do not correspond to the set r.m.s. values. Usually the distribution of abnormal errors is not known. We know only some parameters of this distribution, in particular the maximal values of abnormal errorsF2. In this situation, the direct application of theoretical algorithm will lead to unjustified amount of missed correlations (errors of the first type). We shall require that the error of the first type must be less than certain acceptable levelF3 for all possible realizations of abnormal errors. Then using the minimax approach developed in statistical decision theory, we obtain the following modification of the theoretical algorithm for the model of real errors1. Let N = {xn} and A = {xa} - be the sets of possible non-abnormal and abnormal components, and xn and xa - elements of these sets - certain combinations of non-abnormal and abnormal components of the measurement vector x. Let L = {l} - be the set of satellites for which the residuals zi with the parameters xi of the measurement x are acceptable, i.e.

,6...,,2,1,,22 =++≤ imaxxkz ixxi liiδσσ (6)

where

ixσ and lix

σ - r.m.s. deviations of non-abnormal errors of the observed and calculated

(using parameters of l-th satellite) components of the measurement x; maxxi ,δ - maximal values of possible abnormal components of the errors in the measurement x;

k - the constant, determined by acceptable probability of miss. Then the measurement x is considered uncorrelated if

NnxLl

ncq lnnl∈∈

>− *),,()(minmin αxx (7)

where lnx - is the evaluation of the vector of non-abnormal components xn on the basis of parameters of l-th satellite;

),( nkc α - the threshold depending on the set error of miss α and the dimension kn of the vector of non-abnormal components, providing the first minimum to (7).

F2 The paper1 describes the model of radar measurements that is not valid for only 0.1% of the real measurements. F3 In this case - less than 0.001.

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Otherwise the measurement x is correlated to the satellite l*, where

NnxLllnnlql

∈∈−= ),(minminarg* xx (8)

This algorithm can be used for correlation of radar measurements. Optical measurements usually do not have abnormal errors. Thus the real algorithm for correlation of optical observations nearly coincides with the theoretical one. For each mark )r,...2,1(),( =iii δα of the observation and the l-th satellite, the quadratic form )(i

lq is calculated using the formula (5). Among the cataloged satellites for which )(i

lq is less than the threshold, determined by the set error of the first type, the correlated one has the minimal value of )(i

lq . Real algorithm: primary orbit determination Now let us consider the case when the informativity condition is satisfied for the satellite, but there is no orbit providing correct correlation of the measurements. This is the situation of detection (primary orbit determination) that arises when we have long gaps in data acquisition or new objects arrived in space. Theoretical algorithm in this case assumes that the exhaustive search over the set of uncorrelated measurements Mume must be performed to find the best measurements-to-satellites arrangement. However, the existence of complete groups (with small sizes) of measurements allows reducing this search significantly. The features of real measurements provide the possibility to determine the minimal size of complete group. Let us consider the radar measurements. The characteristic feature of these measurements is unequal accuracy of the parameters of the measurement. The most frequent is the situation when only three parameters in the measurement have enough accuracy for selection of "alien" measurements. For the measurements of the radars with continuous radiation, these are "radar" coordinates: the range D, azimuth ε and elevation angle γ, for pulse radars - azimuth ε, elevation angle γ and radial velocityD . Hence one measurement cannot constitute a complete groupF4. Two observations separated for a revolution or more, usually allows determining three lacking parameters with accuracy, sufficient for good enough selection of other measurements. However, we cannot always be sure that two measurements belong to one and the same satellite with probability close to unity. Such a situation is typical after multi-element launches and break-ups. Two observations with significant errors in velocity components produced by

F4 If one measurement could constitute a complete group, no separate problem of making measurement-satellite correlation for the detection case would arise, since it would be reduced to the problem considered in the previous section.

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different fragments can very well inscribe into one orbit. Thus two measurements frequently does not constitute a complete group. Three observations for three different revolutions have much more chances to be a complete group, since linking of three observations belonging to different revolutions into one orbit in the case when they are produced by different satellites is much more difficult a task than for two observations. Thus, for example, it is unlikely that three measurements, among which the two boundary (regarding the timing) ones belong to one satellite and the remaining one - to the other, will be joined in one orbit. This is improbable due to the fact that when we calculate the position of the satellite for the moment of the "middle" measurement using the orbit determined on the basis of the "boundary" measurements, the resulting position determination error is of the order of position determination error for the time of boundary measurement (for the observations with close latitude arguments). Certainly we cannot state for sure that three measurements cannot be united in one orbit. The modeling revealed that in the case when the boundary measurements belong to different fragments of a break-up they are often united into one orbit and occasionally a measurement from some third fragment can be inscribed into it. However, such a situation is realistic only for the initial stage of the break-up, when shares of the observed fragments have not yet separated from one another and all of them are orbiting within one tube. Later the probability of such an event becomes small. Apart from that, when the time intervals between the measurements of the triplet are great, the number of revolutions between them cannot be accurately determined, resulting in the impossibility of creating enough accurate orbit. However, as shown in1, this is a rather rare event. Thus, the minimal possible size of complete group for radar measurements is three. Therefore, the algorithm for making measurement-satellite arrangements for primary orbit determination looks for three measurements in different revolutions, for which the inscribing orbit exists. When we say that the measurement and the orbit "inscribes" into each other we mean that the residuals between them, which in fact represent evaluations of measurements' errors, do not contradict with the model of these errors1. If such a triplet is discovered and the orbit inscribing the measurements is determined, the algorithm for correlating other uncorrelated measurements with this satellite is the same as the procedure, described in the previous section. Let us consider the case of optical measurements now. These measurements have the following characteristic feature. One measurement of optical-electronic sensors is normally sufficient for determination of four orbital parameters (out of six) accurately enough. These are right ascension, declination and their rates. Two observations of one satellite, acquired by two different stations (or by one station at different nights) produce an orbit, capable of correct selection of further measurements. On the other hand, two such measurements, produced by different satellites usually cannot be inscribed into one orbit.

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Thus, in distinction from radar measurements, even two optical measurements, acquired by different sensors (or by one at different nights) constitute a complete group. Therefore, the algorithm for orbit detection on the basis of optical measurements works as follows. Exhaustive search (over all possible pairs) is fulfilled to find the pair of measurements inscribing into the orbit. Then, by means of the procedure used in the tracking process, we correlate other uncorrelated measurements with this inscribing orbit. Disturbed informativity condition Disturbances of informativity condition (of local and short time character) sometimes occur in the course of tracking certain satellites and correlating measurements with them. Close approaches of satellites, gaps in data acquisition, arrival of the measurements, produced by not tracked satellites can be responsible for these disturbances. In these cases, the procedures described in previous sections must interact with one another. This section describes this interaction. Disturbance of informativity condition in the course of tracking can result in correlating "alien" measurements to the tracked satellite. In the process of orbit updating on the basis of correlated measurements, "alien" measurements are selected1. If a satellite has not many "alien" observations (not more than 10-30%), selection is often successful. Efficient selection in this case does not occur immediately but with certain delay, after other ("own") measurements arrive. Each separated "alien" measurement can be forwarded to re-correlation with all cataloged satellites, excluding those from which it is already separated (now or before). Previously described procedure can be used for correlation in this case. Such measurement sometimesF5 performs rather sophisticated journey before it becomes correlated to certain satellite or remains uncorrelated. In the cases of long gaps in data acquisition or significant share of alien measurements, the reliable selection of "own" measurements becomes impossible. Thus the break of tracking occurs and the satellite must be removed from this process. Removal of satellite from the tracking process in fact means that correlation of measurements with this satellite is prohibited and all the measurements previously correlated with this object are forwarded to correlation with other satellites. Certain part of these measurements can be further correlated with other satellites, the other part becomes uncorrelated and enters the primary orbit determination process. Further correlation of measurements to the satellite, which suffered break of tracking is possible only via primary orbit determination (detection) algorithm. This may happen as follows. Measurements from the lost satellite are incorporated into a certain primary determined orbit that is identified with this satellite. Then the orbit of the lost satellite is updated, the object is included into the tracking process and acquisition of observations begins.

F5 At the initial stage of detecting the fragments of multi-element launch or a break-up and also in case of a conglomeration of small-sized objects.

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Feasible algorithm: tracking Radar measurements Real algorithm, described previously takes into account the features of the considered task. However, direct calculations require significant computation effort. To make a decision on the correlation of one measurement, performed in time tme, the orbits of all the tracked satellites must be propagated to this time tme and 21 matricesF6 with dimensions from 3 x 3 up to 6 x 6 must be inverted. Only the propagation of N satellites for M measurements per day requires sMN τ⋅⋅ , where τ - the time needed for one propagation(s). Typical values of N, M and τ yield τ⋅⋅MN about several days5 and this is not acceptable. The times required for decision-making (implementing procedures (7) and (8)) are comparable with this value. Thus implementation of Real algorithm requires its simplification. Let us consider possible technique for reduction of time, consumed by propagation. For each measurement x, we perform rough selection of satellites that could not produce this measurement for sure. This selection is performed using simple criteria not involving precise prediction. At the first step, the measurement and the satellite are compared using the gates in inclination i and longitude of ascending node Ω. The measurement cannot belong to the satellite if

,2mod)(or Ω>−Ω−Ω−Ω>− cttcii obmeobobmeiobme π (9) where parameters memei Ω, of the six-dimensional measurement x are calculated using known kinematic relationships, and all the parameters of the satellite (indexed "ob") are taken from the catalog. If the values of i and Ω make us think that the measurement can be produced by the satellite, the second selection step is performed. The second step involves comparing (using the gates) the most accurate parameters of the measurement (azimuth and range) with the rough prediction of the orbit. Here for propagation we use polynomial extrapolation of cataloged orbit (with epoch tob) for time interval obme tt −=Δ . Coefficients of the polynomials λλλλλ ′′′′′′′′′′ ,,,,0 , ωω ′′′′′′′′′Ω′′Ω′′′′′′′ ,,,,,,,,,,, 00 eeeeLLLL for propagating elements1 ωλ and,,, eL Ω , and some functions of orbital elements

)cos,sin,cos,sin,cos,sin,1*( 2 ωωωω ==Ω=ΩΩ=Ω==−= cscsiciisiee are not calculated, but are retrieved from the catalog along with the elsets. These parameters are updated in the catalog together with the elsets. The predicted elsets are transformed into rectangular coordinate frame. The transformation uses known formulas of Keplerian theory with modifications reducing CPU time. The scheme of the procedure (prediction and transformation) is presented in the Appendix. F6 Corresponding to the number of possible combinations of non-abnormal components in the observation vector

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If the second step of selection reveals that the measurement can belong to the satellite, we use the decision function described in the real algorithm. Doing this we perform precise propagation of the cataloged orbit for the time tme. Under the conditions when the required probability of miss is less than the frequency of rough measurements by two orders of magnitude, the described scheme is not fast enough. The following technique can be used to enhance its computational performance. The measurements are processed in three passes. For the first pass, we use narrow gates Ωcci , , of the order of

21 ÷ , not accepting rough measurements. Thus we have correlated only enough accurate not abnormal measurements, which normally constitute the majority. The uncorrelated measurements (the minor part of the set of the measurements) are forwarded to the second pass, where they are selected with respect to Ω,i using the gates Ωcci , which are by the order of magnitude wider and thus accept the major share of rough and abnormal measurements. The measurements not correlated at the second step participate in the third pass, where no selection in Ω,i takes place. The scheme described above does not produce losses in the quality of the decisions, i.e., the probabilities of missing correlation and false correlation do not increase. Obviously implementation of this scheme will significantly accomplish computer code. However, the time consumed by propagation is reduced in the average by 1-2 orders of magnitude, and this gain justifies the complication. Now we will consider possible technique of simplifying the decision function with just insignificant losses in the quality of the decisions. Essential simplification of the decision function can be attained when we do not take into account correlations between the components of the vectors ,lnnn xxz −= that are used for making decisions on the correlation. In this case, the calculation and threshold checks for all the quadratic forms in (7), (8) can be replaced (with insignificant losses in the quality of decisions) by much more simple calculations and threshold checks for 6 components of zn. However, the components of vector zn, are significantly correlated. The root cause is that all the components of zn, corresponding to the most accurate components of xn, (there are not less than three of them), are under the major influence of the only one component of prediction error (meaning prediction of the cataloged orbit to the time of the measurement tme) - the along the track error, which usually significantly exceeds the errors of these components. These correlations must be taken into account indispensably, otherwise the probability of false correlation will increase. The account of the correlations can be done as follows. In the space of decision-making parameters )ˆ(1 lafxz −= we perform a non-linear transformation zz ~⇒ of stochastic vector z, that makes the components of the transformed vector z~ only slightly correlated. This transformation is the removal of temporal error δτ in the parameters of the satellite using the most informative component of the measurement (normally, the range).

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The transformation is performed as follows. Parameters of the satellite, aspiring to correlate with x, are propagated to the point, where the calculated and the observed values of the most informative componentF7 of the measurement x coincideF8. Denote: τδ ˆ - the difference between the time tpr, for which the propagation is fulfilled and the time tme of

the measurement x (evaluation of the temporal error in satellite orbit for the time of the measurement); )ˆ,ˆ( τδll af - components of the measurement for prediction time τδ ˆ+met , calculated using satellite's parameters. Then the vector z~ will have the form:

)~...,,~,~()ˆ),ˆ,ˆ((~62 zzzlll =−= τδτδafxz (10)

Vector z~ is six-dimensional since one of its components, used for removal of temporal error is zero and hence should not be accounted of. After that the decision function (7),(8) is significantly simplified. Decision on the correlation of the measurement x to the l-th satellite is made when at least one combination of the inequalities

)~(z~ 2222lii xxi k σσ +≤ , (11)

corresponding to possible composition xn, of not-abnormal components, is satisfied and the inequalities

22max,

~~lii xxii kx σσδ ++≤z (12)

are satisfied for other abnormal components of xa. If the measurement correlates with several satellites, we select the one corresponding to the minimal value of quadratic form

),~/(~~ 226

1

2lii xx

iil zq σσ +=∑

=

(13)

calculated using all the six components of the vector z~ . F7 i.e., the component, providing the highest accuracy of estimating parameterδτ . F8 If there are more than one such point within a revolution, we select the one, providing minimum of the absolute value of the residual for the other informative component (normally, azimuth, when the most informative is the range).

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In (11), (12), and (13) 2~lix

σ - is the r.m.s. value of the error of )ˆ,ˆ( τδll af - estimation of the i-th

component of the measurement on the basis of the parameters of l-th satellite. The advantage of this procedure is the small amount of the used "tuning" constants. For measurements' errors, we must set only the maximum values of the errors of abnormal components )6...2,1(max, =ixiδ . R.m.s. deviations

ixσ of the normal components of the errors are

the parameters that can be evaluated in the course of trackingl. Regarding the errors of orbit determination and prediction (for each satellite), we use only one parameter - the cross-track error acrσ , defined as ),,( eimax σσσ Ω , where ei σσσ ,, Ω - r.m.s. errors of determination of inclination, longitude of ascending node and eccentricity of the orbit. The along-the-track error crσ , which is involved (along with cross-error crσ ) in the calculation of

)ˆ,ˆ( τδll af is not a "tuning" constant. It is defined as )/(min 6 1 ixi xiσ= and does not depend on the

object. Parameter k determines the error of the first type (miss). In (11) and (12) for all decision-making parameters except τδ ˆ k = 3. For parameter τδ ˆ in (11) k varies within the limits 3 - 100 depending on the evaluation of the atmospheric parametersl, and the right side of (12) is equal to 50s. The described procedure is simple and physically clear. It is adequate to the model of real errors and hence provides that the error of the first type will not exceed 0.001. In the average, computer with computation rate about 1 mln operations per second can perform the correlation of one measurement with 10000 of cataloged satellites within ≈0.ls. Optical measurements Correlation procedure for this case is almost the same as for radar measurements. The difference is connected with the features of angular measurements: the model of the errors does not include abnormal components and both measured angles have equal accuracy. Active satellites The algorithms, described in the previous sections deal with passive satellites, i.e., those following the orbits, determined by natural forces only. However, there is a class of satellites which perform maneuvers or corrections in the course of their orbital motion. The share of these objects is not large - of the order of units percent of the total amount of tracked satellites. In some orbital regions, in particular in CEO, their share is by an order of magnitude greater. Correlation of measurements with active satellites requires special measures.

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If the characteristics of the operations performed in orbit (for example, the time of firing the engine and the components of velocity impulse) are known, no specific features in correlating the measurements arise. In this case, we will only have to account of this additional force in the prediction of the motion that is fulfilled for calculating the value )ˆ,ˆ( τδll af . However, this information usually is not available at the time of making decisions on the correlation of measurements. Thus we have to undertake other actions. Non-GEO region Here the decision-making procedure can take into account the fact that almost all the orbital operations result in the changes of orbital plane (increments of i and Ω) not exceeding 1-2 degrees. This situation allows to correlate rather accurate measurements for which the parameters i and Ω can be calculated with the errors, essentially smaller than these values. The major share of the real measurements of the objects after intensive maneuvers, that result in ceasing correlation of the "own" measurements according to the described "passive" scheme (the feasible algorithm described above), satisfy this condition. Let us consider this case separately. We will briefly describe decision-making algorithm. If a measurement is not correlated to any satellite according to "passive" scheme, it is preliminary linked to the closest active satellite A (on the basis of preliminary checks of the first pass). The decision on the final correlation of such a measurement is made in the course of updating the orbit A. We will consider two possible variants to understand the process. 1. Satellite A have not performed active operations, capable to disturb measurements' correlation according to "passive" scheme. In this case, the preliminary linked observation cannot inscribe into updated orbit, since a lot of measurements have been accumulated in the course of passive orbitingF9. In some time, the satellite following passive orbit will acquire its "own" measurement. When the orbit will be updated using the "own" measurement1, the considered measurement will be selected as an "alien" one according to the "passive" scheme described above. 2. The satellite A have performed intensive maneuver, after which the correlation using the "passive" scheme does not take place. In this case, similar to the previous one, the updating algorithm cannot inscribe the measurements after the maneuver into updated orbit. No decisions are made regarding these observations and they are accumulated.

F9 We make this assumption since otherwise no efficient correlation is possible.

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When the amount of the measurements incoming in succession and not inscribing into updated orbit (accounting of the time interval they cover) exceeds certain thresholdF10, the hypothesis of the maneuver is tested. In the process of these checks (we will not deal with the details here)F11 when the fact of maneuver is determined, the related parameters are evaluated and all the previously accumulated preliminary linked observations are finally correlated. When the maneuver result in significant change of orbital plane, the linking of the measurements does not take place. The measurements go straight to the orbital detection process where they are selected and the new (post-maneuver) orbit is determined. Then the new orbit is identified with the maneuvering satellite and the measurements are finally correlated. GEO region The account of the orbital corrections individually for each satellite is the characteristic feature of the algorithm. Almost all the active satellites perform orbital corrections in order to maintain the longitude of satellite projection (to the Earth surface) within certain limits, different for different objects. Part of the satellites aspires to maintain the direction from the satellite to certain point on the Earth surface (within some time intervals). When we are making decision on preliminary linking of measurement x performed at tme for each cataloged satellite, we use two closest orbits from its short (operative) archive of orbits, which timing tor1 and tor2 satisfy the condition: tor1 < tme < tor2 . The measurement is used to calculate the elements ime , Ωme and satellite projection λme, and on the basis of both archive orbits the parameters iob1, iob2, λob1, λob2 without prediction and parameters ,~,~,~,~

1121 obobobob ii ΩΩ propagated for time tme are calculated. The decision function has the form:

iameobmeob

iameobmeob

obamodmeobmodmeob

amodmeobmodmeob

ciiiiminiciiiimini

icmin

cmin

<−−=Δ<−−=Δ

<Ω−ΩΩ−Ω=ΔΩ

<−−=Δ

Ω

)~,~(~),(

/)~,~(

),(

21

21

2221

2221

ππ

λππ λλλλλ

(14)

where the first condition for i is checked only for the satellites that maintain the inclination close to certain value (normally near zero) and the second condition - for active objects not maintaining the inclination (for these satellites the evolution of the inclination is similar to the passive ones);

F10 This threshold is determined by the set values of the frequency of false determination of maneuvers and the time efficiency in detecting true maneuvers. F11 The algorithm is close to the procedure used in primary orbit determination, which is described in the next section.

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aiaa ccc Ω,,λ - threshold values; eecc aa ,4* += λλ - orbital eccentricity, ac λ* varies from 0.005 to 0.025 depending on the

inclination; iac = 0.005; acΩ varies from 0.005 to 0.035 depending on the inclination.

If several satellites aspire to be correlated with measurement x and the "distance" from one of them is significantly smaller than for the competing ones, the measurement is affiliated to the closest satellite. When these distances are comparable, no dominating hypothesis is in place and hence, we cannot make correlation decision providing acceptable error levels. In the GEO ring, such "stumping" situations occur much more frequently than for the other orbital regions. Thus, track measurements are not always sufficient for making correlation decisions. More than that, in case a satellite has performed intensive maneuver within the gap between available observations, the decision made by the algorithm described above also can be erroneous. Making decisions in disputable situations, and also for the cases of long gaps in data acquisition, may involve non-track data that can be acquired by optical-electronic sensors. Data processing practice demonstrated that these data can be efficient for correlating measurements of the active satellites. This problem is however beyond the scope of the present work, treating only the issues of processing track data, and requires special consideration. Feasible algorithm: primary orbit determination Radar measurements The realistic algorithm for measurement-satellite correlation in absence of accurate enough orbital data takes into account the features of considered task. However, direct calculations require excessive computation effort. The set of measurements simultaneously present in the uncorrelated measurements file Mume for ten days accumulation interval comprises thousands of them5. The search for triplets belonging to one satellite is not an easy task for this set, since the number of possible combinations of three measurements, including the new, not yet processed observation xF12 is of the order of millions. Reduction of the computation effort under the conditions when the abnormal and rough measurements constitute a small part of Mume can be achieved by implementation of techniques used in the tracking process, when sufficiently accurate a'priori orbit is available. At the first step, we exclude from Mume the measurements surely not belonging to the orbit that produced observation x. To fulfil the operation, we use parameters i, Ω, Τ which can be calculated from the measurement with rather small errors and (with account of these errors) have simple evolution.

F12 When the program operates in real-time scale by the time of processing measurement x for the previous observations from Mume, the task is already completed.

16

The remaining observations satisfy the conditions: Tmememeime cTTcttcii <−<Ω−−Ω−Ω<− Ωπ2mod

)( (15) where parameters without index refer to any measurement from Mume.

Ti ccc ,, Ω - selecting gates, which values are the functions of the involved sensors, RCSs (incorporated in the measurements), temporal distance between the measurements. Parameters of these functions can be chosen experimentally with provision for the probability of excluding the measurement on the same satellite that produced x: this probability must be less than the probability of abnormal and rough measurements in Mume. For Russian system, the minimal values are: Ti ccc ,, Ω for tme = t of the order of 0.01 day, maximum of ci for |tme - t| < 30 is by the order of magnitude higher, maximum values of Tcc ,Ω for tme ≠ t have the same level as the parameters Ω and Τ. At the second step, we select (out of the set Mume which we call the group of measurement x) the triplets of observations undoubtedly not belonging to one satellite. For this operation, we use the temporal parameter tu calculated from the measurement - the time, when the satellite passes (within the revolution of the measurement) the point with given value of latitude argument u. We exclude the triplets of observations x1, x2, x3, performed in times tl < t2 < t3, which do not belong to different revolutions, and also those for different revolutions, for which ,ˆ )2()2(

uuu ctt >− for any Ν∈ΜN and maxud kk ≤ (16) where )3()2()1( ,, uuu ttt - calculated for each of the three observations values of tu;

)3(2

)1(1

)2(ˆ uuu ttt αα += - estimation of )2(ut on the basis of )1(

ut and )3(ut , where NQNP /,/ 21 == αα ,

( ){ }uuu TttintP /)2()3( −= , Q = N-P, ( ) NttT uuu /)1()3( −= ; int {A} - entier part of A (rounded off); MN - the set of possible number of revolutions N between the boundary (regarding the timing) observations, i.e., MN = { }udkiiN ±±=+ ...,1,0;~ ;

( ){ }TttintN 3~/~

1−= ; T~ - evaluation of the period, calculated using the most accurate measurement of the triplet;

( ){ }TTNintkud~/~~δ= - indicator of uncertainty in the number of the revolutions between the

boundary measurements of the triplet; Tδ - evaluation of the absolute value of the maximum error of the estimation T ;

cu and kmax - parameters of the procedure.

17

Implementation of the algorithm for triplets' selection fixes the right observation x1 coinciding with the new uncorrelated observation x, and the search is performed by varying the following three parameters: the middle measurement x2, the left measurement x1 and the number of revolutions N between the boundary observations. In this case, we can arrange the search as follows. The left measurements are retrieved in the order of increasing timing starting from the most remote. For the fixed left measurement, the middle ones are selected providing that all three measurements are from different revolutions. Finally, we perform the search over possible values of N (for chosen measurements). Parameters of the algorithm cu and kmax are chosen experimentally accounting of required frequency of false decisions (that must not exceed the share of abnormal and rough measurements in Mume) and acceptable computation time, required for making final correlation decisions at the further, more elaborate stages. These requirements are contradictory and can hardly be satisfied simultaneously. Parameters cu and kmax are the functions of the sensors, RCSs, the number of measurements in the group, the number of new measurements in this group and the density of the measurements of the group in the spatiotemporal area { }tTiD ,,,Ω= . For routine operations, when the dominating share of the observed objects arrived due to recent events in space are already detected, with account of the features of Russian system the gate cu varies from 10s to 60s and kmax varies from 3 to 40. When the density of uncorrelated measurements in the region { }tTiD ,,,Ω= increases, the parameters cu and kmax should be reduced. At the third step, the selected triplets along with obtained evaluations of the number of revolutions between boundary measurementsF13 are checked for belonging to one satellite by construction of inscribing orbit. This is the most consuming part of the algorithm. We consider that the measurements x1, x2, x3 with the number of revolutions between x1 and x3 equal to N, cannot belong to one orbit, if

a )(min argˆwhere,)ˆ( aaa ΦΦ cp => Φ , (17)

where Φ (a) - functional of the least squares method, constructed on the measurements x1, x2, x3

1; a - estimation of the vector a of orbital parameters on the basis of measurements x1, x2, x3; Φp (a) - the sum of the squares of normalized residuals in Φ (a) for the most accurate components of the measurements x1, x2, x3 ; cΦ parameter of the algorithm which value depends on the temporal interval between the measurements of the triplet, perigee altitude and orbital eccentricity, maximum density of the measurements of the group in the spatiotemporal domain { }tTiD ,,,Ω= . For the conditions of Russian system, the values of cΦ are within 10 - 3000.

F13 Several acceptable evaluations of N can be calculated for one triplet.

18

The technique for determination of the minimum of Φ (a) is described in1. At the fourth step, the previous correlation decisions are updated. The procedure "bails out" (from the set Mume) the measurements inscribing into already determined orbit a . For this purpose, the described above tracking algorithm is used. Then on the basis of all the measurements (the three initial ones and the others correlated later), we perform calculations of updated orbit using the algorithm employed in satellite tracking1. After that, the final decision to correlate the measurements to one object is made. It is convenient to combine this decision with the decision on the final detection of the primary orbit and forwarding it to the preliminary tracking process1. This process includes monitoring of residuals Δ between the observations and determined orbit (for the most accurate coordinates), the number of measurements inscribedF14 into the orbit and the number of revolutions, for which the inscribed (regarding the residuals) measurements are in place. The parameters of the decision function (the thresholds cΔ differ for different parameters of the measurement, the threshold cme for the number of inscribed observations and the threshold crev for the number of revolutions with inscribed measurements) are chosen experimentally with provision that the frequency of "confirmations" of the orbitF15 cannot be less than the set level (usually about 0.8-0.9). Parameters cΔ, cme, crev are the functions of the sensor, the time interval of on-site tracking (related to the measurement), RCS, orbital parameters hp, e and ΔΤ of the satellite, the density of the measurements of the group in the spatiotemporal domain { }tTiD ,,,Ω= . Optical measurements The situation of optical measurements significantly differs from the radar measurements first of all by the fact, that the errors of real optical measurements usually do not include abnormal components and are well described by Gaussian distribution. Thus we can use the theoretical algorithm as a real one. Reduction of the computation time (needed for the search over the pairs) can be achieved, similar to the case of radar measurements, introducing preliminary selection with respect to i and Ω, using the rule (15). The gates ci and cΩ depend on many parameters, including the type of the orbit, the number of registrations in the observation and the time interval they cover. In this connection, it will be useful to use in their determination the correlation matrix of the errors of the initial orbital parameters calculated on the basis of measurement3. This matrix according to the least squares procedure is calculated simultaneously with the estimations of parameters. Regarding the conditions of Russian system, for GEO satellites ci normally does not exceed 0.01, and cΩ - 0.01/i, for non-GEO satellites ci and cΩ can be several times greater.

F14 i.e., the absolute value of the residuals Δ for all the most accurate coordinates are less than the respective thresholds cΔ. F15 By confirmation of the orbit, we mean either its further update by new observations or its identification with previously-tracked satellite.

19

The final decision for the selected pair is made according to the criteria of the theoretical algorithm. The measurements x1, x2 belong to one satellite if:

aaΦaΦ

)(min argˆwhere,)ˆ( cp =< a (18)

where Φ (a) - the least squares functional, constructed on the measurements x1, x2; a - evaluation of the vector a of orbital parameters on the basis of measurements x1, x2 ; cp parameter of the algorithm, which value depends on the time interval between the measurements, perigee altitude and orbital eccentricity. In practice, cp varies from 10 to 100. Correlation of the other measurements with the pair follows the same procedure as for the tracking process. Conclusion 1. The answer to the question, "what are the conditions providing the capability of tracking satellites on the basis of coordinate data ?" is given in rather general form. This condition coincides with the condition providing efficient allocation of measurements with respect to the satellites. The requirement is that the principal possibility must exist for determination of the orbit of any observable object (on the basis of available measurements) with accuracy providing that the errors of orbit determination and prediction for the times of the measurement will be smaller compared to the distances between different observed objects. 2. The condition providing the possibility for satellite tracking is determined in rather general form. The sense of this condition is that in the certain space domain, all the observable objects must have such orbits that the observations from other satellites cannot inscribe into them. 3. Rather general form of the theoretical algorithms for allocation of measurements to satellites are developed for the cases of tracking and detection. These algorithms do not depend on the features of information system. The essential fact is that the "true" allocation of measurements must exist and dominate over the competing ones. 4. The examples of account of the specific features of the observations conducted in Russia are presented and the "real" algorithms are developed. 5. The techniques for reduction of computation effort required by the "real" algorithms are considered. These techniques do not produce losses in the efficiency of the decisions. The developed algorithms can be implemented in real-time scale using the computers with computation rate of the order of a million standard operations per second. 6. The used approach is rather universal and can be applied to solving the problem of measurements' allocation for catalog maintenance in any information system.

20

References 1Z.N. Khutorovsky, V.F. Boikov, L.N. Pylaev, Low-Perigee Satellite Catalogue Maintenance, Near-Earth Astronomy (Space Debris), Russian Academy of Science, Institute of Astronomy, Moscow, 1998, pp. 34-101 (in Russian). 2Z.N. Khutorovsky, V.F. Boikov, L.N. Pylaev, Catalog Maintenance of Low-Earth-Orbit Satellites: Principles of the Algorithm, Journal of Guidance, Control, and Dynamics, Vol. 22, No. 6, 1999, pp. 745-758. 3Z.N. Khutorovsky, Monitoring of GEO Satellites in Russian Space Surveillance Center, Proceedings of the Third US/Russian Space Surveillance Workshop, US Naval Observatory, Washington DC, October 20-23, 1998, pp. 281-309. 4Khutorovsky Z.N. Satellite Catalog Maintenance, Space Studies, Vol. 31, Issue 4, 1993, pp. 101-114 (translated from Russian). 5G.Batyr, S.Veniaminov, V.Dicky, V.Yurasov, A.Menshicov, Z.Khutorovsky, The Current State of Russian Space Surveillance System and its Capability in Surveying Space Debris, Proceedings of the First Conference on Space Debris, Darmstadt, Germany, 5-7 April 1993, pp 43-47.

21

Appendix Calculation of Propagated Parameters in Rectangular Coordinate Frame

Input data:

.,,,,,*,,,,,,,,,,,,,,,,,,,,

0

00ωωωω

λλλλλcscscisieeeee

LLLLLΩΩΔ′′′′′′′′′

Ω′′Ω′′′′′′′′′′′′′′′′′′′′′

Output data: .,,,,,,,, zyxzyxzyx

Algorithm The calculations are performed following the formulas:

λλλλλλλ ~)))(((~0 ′′′′Δ+′′′Δ+′′Δ+′Δ+= is transformed to the interval (0,2π)

000

000

0

00

0

sin~sincos~coscossin~coscos~sinsin

)~sin~~cos

~1()~cos

~~sin~(

~sin)(~~cos)(

~))6/2/((~cos))6/2/((~sin)())(())6/2/((~cos))6/2/((~sin

)())((~

ssEssE

hkhkseeheek

scsccscs

eeeescsccscs

LLLLL

λλλλ

λλλλωδωδ

ωδωωδωωδωωωωδωωδωωδωωωωωδωδ

δδδδδδ

δ

+=+=

++−=+=+=

⋅−⋅+⋅−=⋅+⋅−⋅+=

′′Δ+′Δ=′′′Δ+′′Δ+′Δ=Ω⋅Ω−Ω⋅Ω+Ω⋅Ω−Ω=ΩΩ⋅Ω+Ω⋅Ω−Ω⋅Ω+Ω=Ω

Ω′′Δ+Ω′Δ=Ω′′′Δ+′′Δ+′Δ+=

cycle in i from 0 until m, for which 0000001.0<mEδ

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

−=−=

−−=−=

−=−=

−−=−=

+

+

+

iiiii

iiiii

iii

iii

iii

iiii

iii

iii

EEEEEEEEEE

EEEEEE

EssBAsE

EhEkBEhEkA

δδδδ

δδδδδδ

δδ

sinsincoscoscossincoscossinsin)24/5.0(1cos

)6/1(sin

/)(sin~

cos~

1cos~

sin~

1

1

22

21

333

2

)(~~cos)(

~~sin~~sin)(

~~cos~

)(~~cos)(~~sin~~sin)(~~cos~)~

~(

~)~

~(

~)~~~(sin~)

~~~(cos~*)~~/()~/(1

*)~1/(~*/)(**~/~cos~cossin~sin~~

zCzyCyxCxsiyzciyxyciyxxsiyzciyxyciyxx

xCkDyyChDxkBhEAyhBkEAx

eLDBACeABeeeeeLA

EEEEBBAA mmmm

µµµ

µδµ

−=−=−==Ω+Ω=Ω−Ω==Ω+Ω=Ω−Ω=

+=+−=−−=+−=

==+=⋅−======