MANEUVER DETECTION OF SPACE OBJECT FOR SPACE SURVEILLANCE
Jian Huang*, Weidong Hu, Lefeng Zhang
$756WDWH.H\/DE1DWLRQDO8QLYHUVLW\RI'HIHQVH7HFKQRORJ\&KDQJVKD+XQDQ3URYLQFH&KLQD (PDLOKXDQJMLDQ#QXGWHGXFQ ABSTRACT on the technique of a moving window curve fit. Holzinger & Scheeres presented an object correlation Maneuver detection is a very important task for and maneuver detection method using optimal control maintaining the catalog of the orbital objects and space performance metrics [5]. Kelecy & Jah focused on the situational awareness. This paper mainly focuses on the detection and reconstruction of single low thrust in- typical maneuver scenario where space objects only track maneuvers by using the orbit determination perform tangential orbital maneuvers during a relative strategies based on the batch least-squares and long gap. Particularly, only two classical and extended Kalman filter (EKF) [6]. However, these commonly applied orbital transition manners are methods are highly relevant to the presupposed considered, i.e. the twice tangential maneuvers at the maneuvering mode and a relative short observation gap. apogee and the perigee or one tangential maneuver at In this paper, to address the real observed data, we only an arbitrary time. Based on this, we preliminarily consider the common maneuvering modes and estimate the maneuvering mode and parameters by observation scenarios in practical. For an orbital analyzing the change of semi-major axis and maneuver, minimization of fuel consumption is eccentricity. Furthermore, if only one tangential essential because the weight of a payload that can be maneuver happened, we can formulate the estimation carried to the desired orbit depends on this problem of maneuvering parameters as a non-linear minimization. Therefore, choices in the modes of least squares problem. To obtain a more sensible and orbital maneuver are limited. The thrust imposed on accurate result, the prior knowledge is incorporated the tangential direction is an efficient maneuvering into the iterative solving process calculated by SOCP mode for minimizing fuel consumption, which is algorithm. Finally, the performance and efficiency of commonly applied in the process of various orbital our method are validated by the theoretical analysis maneuvers [7]. In particular, the maneuvering positions and some observations. are usually chosen at the apogee and perigee point [7]. The studies in this paper are based on all the above- 1 INTRUDUCTION mentioned hypothetical maneuvering modes. Nowadays, there are at least 20000 trackable objects in Furthermore, due to the limited coverage of the sensors, earth orbit and among them 1300 have the capability of the small and regular observation gap couldn’t be performing orbital maneuver [1]. When the orbital guaranteed, which may be from hours to many days. maneuver occurs unexpectedly during the gaps, how to Therefore, it is more meaningful and expected that the detect and reconstruct the abnormal event will directly performance of the proposed maneuver detection affect the capability of space situational awareness. method will be little affected by the observation gap, Particularly, the available observations collected by the especially the large gap. Certainly, we also hope to current space surveillance systems, such as the AFSSS reconstruct the maneuvering parameters precisely (US Air Force Space Surveillance System) [2], which when the maneuver event occurs during a small gap. has a long gap between the neighboring observations, This paper presents a novel and stepwise algorithm to are generally discrete in the spatial-temporal domain. handle these problems as follows. In section 2, the The problem of maneuver detection during observation maneuvering modes and maneuvering parameters are gaps brings much more difficult, in contrast with that estimated preliminarily according to the change of commonly encountered in real-time tracking semi-major axis and eccentricity. If there is only one applications. tangential maneuver during the gap from the result, we Regarding the problem of maneuver detection, the formulate the estimation problem of maneuvering corresponding methods are the varieties with respect to parameters as a non-linear least squares problem. Thus, different modes of orbital maneuvering and detection a more precise result can be obtained by solving the metrics. Storch estimated the maneuvering parameters constrained non-linear least squares iterative process of a collocated satellite in geosynchronous orbit by using SOCP (Second Order Cone Programming) using nonlinear least squares [3]. In [4], the energy per algorithm in Section3. Section 4 is the simulation unit mass was computed to detect a space event based results and performance analysis. Section 5 is the
______Proc. ‘6th European Conference on Space Debris’ Darmstadt, Germany, 22–25 April 2013 (ESA SP-723, August 2013) conclusion. 'DH 32 ' cos I HHDDH (2) 'DD 1 H2 2 H'H 2 ESTIMATION OF MANEUVERING MODE When we want to obtain the sensible values of the As we know, the orbits often have the unexpectable maneuvering true anomaly I and velocity 'Y , the changes due to undertaking the various tasks. The most solution of Eq. 2 must satisfy: frequent orbital maneuvers are implemented to keep 1cddosI 1 (3) the spacecraft at the prearranged orbit, especially the semi-major axis D and eccentricity H which determine However, Eq. 3 can’t be always satisfied, i.e. only one the orbital shape. Generally, these two orbital tangential maneuver can’t simultaneously adjust the parameters are controlled more rigorously than any semi-major axis and eccentricity arbitrarily. Therefore, other parameters, and they are usually not allowed to multiple maneuvers should be applied, one efficient fluctuate with respect to the time. The common and mode of which is that twice tangential maneuvers at efficient maneuvering mode is to exert an additional the apogee and perigee point which could minimize the tangential velocity at a proper point or the perigee and fuel consumption. apogee point [7]. However, one tangential maneuver can’t simultaneously adjust the semi-major axis and 2.2 Twice tangential maneuvers at the eccentricity arbitrarily, while two tangential maenuvers perigee and apogee at the apogee and perigee can do that. They are both the common applying maneuvering modes. Therefore, Most of the orbits of space objects have small the maneuvering mode and the corresponding eccentricities and can be approximated as circle orbits. maneuvering parameters can be estimated preliminarily Thus, the item H2 in Eq. 1 can be omitted. Eq. 1 can be according to the change of semi-major axis and transformed into: eccentricity. 2 12 °QD' 2 12HIcos H'Y ® (4) 2.1 One tangential maneuvering mode 2 12 ¯°QD'HY 2 12HIcos H cos IH ' If the observed semi-major axis and eccentricities changed obviously at different times, the maneuver Firstly, the 'Y is used to adjust the eccentricity H at detection should be taken into account. Based on the the perigee or the apogee, i.e. I 0D or I 180D . hypothesis of one tangential maneuvering mode, we This maneuvering velocity is denoted as . should reconstruct the maneuvering velocity as well as 'Y H judge that where the maneuver occurs and whether D When I 0 , we can get ''YQ DH 2 . When only one tangential maneuver can realize this change of H D these parameters. I 180 , ''YQ H DH 2 . According to the perturbation motional formula, a After the adjusting of eccentricity H , the semi-major tangential impulse velocity 'Y exerted at an arbitrary axis D should be modified a value 'D to adapt the point I during the orbit will simultaneously change final change. 'D includes two parts: the first is the value of the semi-major axis 'D and eccentricity original changed value 'D ; the second is the deviation 'H as Eq. 1: dD H caused by the adjusting of H . Thus, the current D 2 12 value is 'DD ' dD H . Therefore, when I 0 , 'DY 2 ' ° 12HIcos H D ° QH1 2 'D 'D H 1 DH' ; when I 180 , 'D 'D ® (1) 2 H 1 DH' . In addition, when the signs of 'D and ° 21 H 2 12 'HY 12HIcos H cos IH ' D ¯° QD 'H are the same, we choose I 0 ; while they are different, we choose I 180D . Accordingly, the value Where QD P 3 , P 3.98600436u1014PV 3/ 2 is the of 'D will be reduced and the fuel consumption is earth gravitational constant, I is the true anomaly. saved. Once the 'D and 'H are known, we can calculate the In order to adjust the 'D , we exert an equivalent maneuvering parameters I and 'Y . Dividing the each once at the apogee and second equation by the first equation at Eq. 1, we can tangential velocity 'Y D obtain that: perigee respectively. In this case, the eccentricity H will not be changed while adjusting D according to Eq.
4. We can obtain that 'Y D QD' 4 . Finally, twice tangential maneuvers at the perigee and apogee ) m 350 k can be utilized to achieve the arbitrary change of ( eccentricity H and the semi-major axis D . The de tu
ti 300 concrete formats are as follows: al l ta
(1) When the sign of 'D and 'H are the same, bi 250 2 7 12 17 'DD ' 1 H D'H, and the first tangential impulse Or Time(circle) ty
D ci 0.01 'Y1 is exerted at I 0 , the second tangential impulse i D ntr 'Y2 is exerted at I 180 , where 0.005 ecce l
ta
QD''H QDDD ' 1 H 'H bi 'YQ 0
° 1 Or ° 24 4 2 7 12 17 ® (5) Time(circle) QD'' D 1 H D'H °'YQ )LJXUH2EVHUYDWLRQVDWWKHILUVWWLPH ¯° 2 44 7DEOH$QDO\VLVUHVXOWVRIWKHILUVWREVHUYDWLRQV (2) When the sign of 'D and 'H are different, Circle 1 to Circle 2 Circle 2 to Circle 16 'DD ' 1 H D'H, and the first tangential impulse Maneuver Maneuver Maneuver Maneuver D velocity position velocity position 'Y1 is exerted at I 180 , the second tangential (m/s) I(°) (m/s) I(°) D Mode 1 X X X X impulse 'Y2 is exerted at I 0 , where -15.5 180 -17.4 0 Mode 2 QD''H QDDD ' 1 H 'H -15.5 0 3.9 180 °'YQ1 From Tab. 1, we can see that the change of observed ° 24 4 ® (6) orbits from circle 1 to circle 2 can be achieved by QD'' D 1 H D'H °'YQ exerting one -15.5m/s maneuvering velocity at the ¯° 2 44 apogee and perigee respectively, while only one tangential maneuver can never realize it. For the Therefore, the common maneuvering modes for second maneuver from circle 2 to circle 16, it can be adjusting the eccentricity H and the semi-major axis D realized by exerting a -17.4m/s impulse velocity at the have been introduced. When we get the observations at perigee and then a 3.9m/s impulse at the apogee. In different time, the maneuvering mode and parameters addition, the mode 1 still has no solution. Commonly, can be reconstructed according to the value of 'D and it is impossible that the velocity is reduced firstly, and 'H . Then, the sensibility and availability of the results then increased immediately. The second increase of can be further analyzed. velocity may be induced by the observation noise. Thus, the second maneuver can be equivalent to one 2.3 Analysis of observed data tangential maneuver with about -15m/s impulse To verify the effectiveness of our presented maneuver velocity. Therefore, the preliminary analysis result can detection method, some typical orbital observations of be derived: there may be three times continual Shenzhou Spaceship are selected for maneuver maneuvers at the perigee and apogee points. In detection. addition, the tangential impulse velocity exerted at each time is about -15m/s. The observations are displayed in the form of the The second observations are shown in Fig. 2. orbital period, i.e. the times of full circle where the m) k
space object turns around the observation station. The ( 350 first data contains a neighbouring observation, and only de tu the concerned altitude and eccentricity are plotted in ti 300 al Fig. 1. l ta
bi 250 0 5 10 15 20
We take the aforementioned two maneuvering modes Or Time(circle) to analyze the first observed data. The results are listed -3 y t
i x 10
in Tab. 1, where X denotes no solution; Mode 1 and c 2 Mode 2 represent the one tangential maneuvee and ri ent twice tangential maneuvers at the perigee and apogee, c 1 respectively. ec al t 0 0 5 10 15 20 Orbi Time(circle)
)LJXUH2EVHUYDWLRQVDWWKHVHFRQGWLPH The same maneuver detection method is implemented position vector and velocity vector of space object at to analyze the second observations. The results are the time W and W , respectively. I È J È I È J are shown in Tab. 2. 0 1 1 1 1 1 r v È È È the functions of 0 , 0 and WWP 0 . I2 J2 I2 7DEOH$QDO\VLVUHVXOWVRIWKHVHFRQGREVHUYDWLRQV r v J2 are the functions of P , Pc and WW1 P [8]. Circle 2 to Circle 17 Circle 17 to Circle 18 Assuming that the observation noise is zero mean Maneuver Maneuver Maneuver Maneuver n n Gaussian white noise, (0,40 ) and (0,41 ) denote velocity position velocity position (m/s) I(°) (m/s) I(°) the observation noises at the time W0 and W1 ,
Mode 1 -31.96 90.04 X X respectively, where 40 and 41 are the noise -15.99 0 3.30 180 Mode 2 covariance. -15.96 180 3.29 0 The two maneuvering modes are both available to ª 222222º Let 4401 GLDJ ¬V UUU,,,,,VVVVVYYY¼ in this conform with the observations from circle 2 to circle paper. Thus, the observed orbital elements in the 17. The maneuvering velocity in mode 1 is about the presence of noise is denoted by total of twice maneuvering velocities in mode 2. 77 7 ªºrvcc77ªºr 77v n ªºrvcc77 However, the thrusters should not always change the ¬¼00,, ¬¼ 00(0,4 0), ¬¼11, magnitude of impulse while concerning the stableness 7 ªrv77º n and difficulty of control and manufacture. So, the ¬ 11, ¼ (0,41 ) . maneuvering velocity in mode 2 more approximates the analysis results of the first observations and may be The orbital maneuvering process can be represented by sensible. In addition, the observations from circle 17 to the following equation: circle 18 are inclined to suggest that no maneuver ªºrrc ªº occurs during the gap, and the nonzero values of the 11 n(0,4 ) °«»vvc «» 1 estimated maneuvering velocity mainly results in the °¬¼11 ¬¼ measurement error. ° II IIr °ªºIJ§·ªºIJªºª0 0 º 22 11 n(0,4 ) The analysis results of the observations indicate that ®«»II¨¸«» II«»«vv< » 1 °¬¼IJ22©¹¬¼I 11J¬0 ¼¬¼'YYPP our estimation method of the maneuvering mode is ° ªºrrc ªº efficient. It can preliminarily deduce the maneuvering ° 00 n(0,4 ) °«»vvc «» 0 parameters and explicate sensibly for the observations. ¯¬¼00 ¬¼ A further advantage is that the algorithm is little (7) affected by the observation gap which ranges from 2 hours to 24 hours in our testing data. In addition, the ªrrºªªºIJIIº where P 110 . Here, the perturbation method is straightforward. Even for the preliminary «vv»««» II » ¬ P ¼¬¬¼IJ110 ¼ orbits determined based on the coarse observations, a robust and relative accurate result can be also obtained. forces are not considered. The first equation in the maneuvering model of Eq. 7 is 3 PRECISE ESTIMATION OF ªºrc abbreviated to 1 -(,rv,,'YW )(n0,4). MANEUVERING PARAMETERS IN ONE «»vc 00P 1 TANGENTIAL MANEUVERING MODE ¬¼1 If the estimated maneuvering mode indicates that only Therefore, the parameter estimation problem can be one tangential impulse is exerted during the treated as a non-linear least square problem in Eq. 8: observation gap, we could reconstruct the maneuvering [,'YWˆ ˆ ] parameters more precisely. P 7 §·ªºrrcc§· ªº 11rv 1 rv arg min ¨¸'-( 00,,,YWPP)(4 1¨¸- 00,,,'YW ) 3.1 Method of parameter estimation in 'YW, «»vv «» P ¬¼11cc ¬¼ orbital maneuvering ©¹©¹ (8) Let W and 'Y denote the maneuvering time and P where the unknown parameter vector is velocity respectively. v and vc are the pre-maneuver 7 P P >@rv,,'YW , . Before the Gauss-Newton iterative velocity vector and the post-maneuver velocity vector 00 P at the time W , respectively. The space object algorithm being applied, we need compute the P linearization form of -(,rv,,'YW ) at the parameter maneuvers along the tangential direction, so we can 00 P vector : vvc 'YY< v Y v < derive PP PP, where PP2 , 2 A r v r v denotes the 2 -norm. 0 , 0 and 1 , 1 are the H Hr Hv HH () > ()00,(),('YW),(P )@ ªrr(1Q )ºªºc ª3V 1 º 00dU 3 ° «vv»«» « » ªºwwww---- (9) ¬ 00(1Q )¼¬¼c ¬3V Y 13 ¼ ,, , ° «»rv ° ¬¼www00'YWwP s.t. ®0(d'YQ1)d'Ymax (10) °WWd(1Q)dW The calculation of partial derivatives has no difficulty ° 01P ° but is complicated, so we don’t give the concrete forms ¯ due to the limited length of paper. where 'Y is the upper bound of the maneuvering The initial value (0) of the parameter vector max velocity, 1 is a 3-dimensional full-1 vector. The directly affects the convergence of the iterative 3 algorithm. It is important to choose an appropriate iterative process is an optimization problem, which can initial value based on the prior information of the be handled by the SOCP algorithm [9]. Therefore, the ˆ correlated orbits. In this paper, the observed values of maneuvering time WP and the maneuvering velocity rc vc 0 , 0 at the time W0 can be taken as the initial values 'Yˆ can be estimated by multiple iterations. r v of 0 , 0 . Moreover, let the observed orbit D and orbit E propagate in the time interval >WW, @ . We calculate 3.2 Detection performance of orbital 01 maneuver the intersection time WPc when the two orbits have the Using Eq. 9, wr / w and wwv / , we can easily minimum module difference of the position vectors 0 0 obtain the Cramer-Rao Lower Bound (CRLB) of the through cross propagation. Thus, WPc is chosen as the estimated parameters W and 'Y . Here, two common initial iterative value of W , and the corresponding P P gaps of the observations are used for the performance module value 'Yc of the two velocity vectors’ analysis, i.e.: one case is that the surveillance system difference at the intersection point is chosen as the observes two adjacent orbits of an object, whose time initial value of maneuvering velocity 'Y . interval is about an orbital period (about 2 hours for In order to obtain a more sensible and accurate result, LEO object); the other case is that the neighbouring the constraints for the unknown parameter vector ascending and descending arcs are observed, whose time interval is about 12 hours. In this paper, the should be applied in the iterative process. The simulation parameters are set as follows: the initial constrained iterative form for solving the non-linear least squares problem is: orbit element: Semi-major axis DN 7000 P, Eccentricity H 0.01, Inclination L 70q , Longitude of 7 §·ªºrc the ascending node : 170q , Argument of periapsis 1 H ªºˆ (QQ1) argmin¨¸«» -(()) (())()QQ (Q 1) q q ˆ(1Q ) vc ¬¼ ©¹¬¼1 Z 30 , Mean anomaly 0 30 ; the observation r errors are V U 10P and V Y 0.1PV/ . Subsequently, §·ªº1c <41 ¨¸-( (Q ))H ( ( QQQ ))ªº ( ) ˆ ( 1) we calculate the CRLBs in the simulation. For the non- 1 «»vc ¬¼ ©¹¬¼1 maneuvering object, the CRLB’s square root of the maneuvering velocity is 0.1m/s, which is similar to the observation error. For the maneuvering objects, the distribution of the CRLBs’ square root of the estimated maneuvering time and maneuvering velocity are shown in Fig. 3. ) ) s s e( e( m m i i
er t 80 er t 80
neuv 60 neuv 60
40 40 ated ma ated ma m m
ti 20 ti 20
of es 0 of es 0
2 20 2 20 1/ 1/
B 2 B 10 L 10 L 10 1 5 CR Maneuver velocity(m/s) Maneuver time(h) CR 0 Maneuver velocity(m/s) 0 Maneuver time(h)
(a) The CRLBs’ square root of the estimated maneuvering time ) ) s / /s (m (m y t ty i i oc oc
0.2 el 0.2 vel r er v e 0.15 0.15 uv aneuv ane 0.1 0.1 m
ed 0.05 0.05 t ated m m ma i t ti 0 0
20 es 20 f
2 o of es
10 2 2 10 10 1/ 1/ 1 5 B B Maneuver velocity(m/s) L L Maneuver time(h) Maneuver velocity(m/s) 0 0 Maneuver time(h) CR CR (b) The CRLBs’ square root of the estimated maneuvering velocity )LJXUH7KHGLVWULEXWLRQRIWKH&5/%V¶VTXDUHURRWRIWKHHVWLPDWHGPDQHXYHULQJWLPHDQGYHORFLW\
Fig. 3 indicates that the estimated precision fluctuates performance and parameter estimation precision of the with respect to the maneuvering time instead of proposed algorithm. The result is compared with changing monotonously. With the increasing CRLB. The parameters in the simulation are the same maneuvering velocity, the parameters will be estimated with aforementioned setting. with a higher precision. In addition, the estimation Firstly, we carry out the maneuver detection for the precision for a large gap WW of the observations is 10 non-maneuver object. The cumulative probability slightly higher than that for a small one. distribution of the estimated maneuver velocity is Furthermore, Monte Carlo simulations are carried out statistically shown in Tab. 3. to examine and analyze the maneuver detection 7DEOH7KHFXPXODWLYHSUREDELOLW\GLVWULEXWLRQRIWKHHVWLPDWHGPDQHXYHULQJYHORFLW\IRUWKHQRQPDQHXYHULQJREMHFW
Estimated maneuvering velocity (m/s) 10-3 10-2 0.1 0.3 0.6 Cumulative probability (2h) 0.4500 0.4550 0.6500 0.9450 1 Cumulative probability (12h) 0.5050 0.5250 0.7000 0.9950 1 Tab. 3 indicates that almost all the estimated the maneuver object. In this paper, when the estimation È maneuvering velocities are less than 3V Y , which result biases satisfy G ()WVP d 60 G ()'dYP0.3 / V, we from the observations error. In other words, there is assume that the orbital maneuver detection is correct. quite a small probability that a non-maneuver object is Only the correct maneuver detection is used to evaluate identified as a maneuver one. the accuracy of the parameter estimation. The result is shown in Fig. 4. Subsequently, the maneuver detection is carried out for y y t 1 1 li ilit 'v=1m/s obab 0.8 'v=5m/s r 0.8 p
' n 'v=1m/s on probabi v=10m/s i io t t c 0.6 'v=15m/s c 'v=5m/s e 0.6 e t
'v=20m/s e 'v=10m/s d r 'v=15m/s er det e
0.4 v 0.4 'v=20m/s aneuv aneu
m 0.2 m 0.2 t t c e rr orrec 0 o 0 C 0 0.5 1 1.5 2 C 0 2 4 6 8 10 12 Maneuver time(h) Maneuver time(h) (a) Probability of correct maneuver detection ) 40 ) 50 (s ' (s ' e v=1m/s e v=1m/s m m i 'v=5m/s i 'v=5m/s 40 er t 30 'v=10m/s er t 'v=10m/s 'v=15m/s 'v=15m/s neuv aneuv 'v=20m/s a 30 'v=20m/s
d m 20 e ed m t t a a 20 m m i i t t es 10 es 10 of of SE SE M M
R 0 R 0 0 0.5 1 1.5 2 0 2 4 6 8 10 12 Maneuver time(h) Maneuver time(h) (b) Root Mean Square Error (RMSE) of the estimated maneuvering time ) ) /s s 0.25 / 0.2 m ( 'v=1m/s (m 'v=1m/s ty y i t i c c
o 'v=5m/s 'v=5m/s l o
0.2 l e 'v=10m/s e 0.15 'v=10m/s v r e
'v=15m/s er v 'v=15m/s v 0.15 'v=20m/s 'v=20m/s neuv aneu
a 0.1 m
d 0.1 te ed m t a a m m
ti 0.05 i t s 0.05 f e es o of
SE 0 0 SE
M 0 0.5 1 1.5 2 0 2 4 6 8 10 12 M R Maneuver time(h) R Maneuver time(h) (c) Root Mean Square Error (RMSE) of the estimated maneuvering velocity )LJXUH7KHGHWHFWLRQSHUIRUPDQFHRIWKHSURSRVHGPDQHXYHUGHWHFWLRQDOJRULWKP
Fig. 4 suggests that the detection performance is Maneuver detection plays an important role in space closely related with the real maneuvering time and the surveillance. This paper presents a novel method for maneuvering velocity. Comparing the simulation addressing the maneuver detection problem under the results with the CRLBs, we can find that it will acquire hypothesis that only the tangential maneuvers exist. a high correct detection probability where the CRLBs We preliminarily estimate the maneuvering mode of the maneuvering time and velocity are low. When according to the changes of the observed semi-major the maneuvering velocity exceeds 5m/s, the correct axis and eccentricity. This method is very convenient maneuvering detection probability can achieve 1. With to handle the practical data and will be little affected by the increasing maneuvering velocity, the estimation the observation gap. Some typical observations are accuracy of the maneuvering time is improved. But the used to verify the robust and efficiency of our estimation accuracy of the maneuvering velocity is presented algorithm. If estimated maneuvering mode more sensitive to the real maneuvering time, which shows that only one tangential maneuver occurs during coincides with the trend of the CRLB. In addition, Fig. the gap, the reconstruction problem of the maneuvering 4 also shows that the larger gap interval of the parameters can be transformed into a non-linear least observations leads to a higher estimation accuracy than squares problem. To obtain a stable and precise result the smaller one, but the correct detection probability is of the least squares problem, some prior constrains are much lower. The reasons behind this outcome are that incorporated into the iterative processes. The SOCP it is much more difficult to get an accurate initial algorithm is utilized to calculate the optimal problem. iterative value in the case of the large gap interval than Finally, the performance of the proposed method is the small gap. The larger difference between the initial also analyzed in detail, and extensive simulations are iterative value and the real value will result in a higher carried out to validate the effectiveness of the probability of the non-convergence and local algorithm. However, the proposed algorithm is a convergence problem. This is a main reason why the general methodology, which can be also adapted and larger gap of the observations makes the orbital extended to many different situations. maneuver detection more difficult. 5 Reference 4 Conclusions [1] James, L. (2009). On keeping the space environment safe for civil and commercial users. %HIRUH WKH 6XEFRPPLWWHH RQ 6SDFH DQG $HURQDXWLFV +RXVH &RPPLWWHH RQ 6FLHQFH DQG 7HFKQRORJ\. [2] Schumacher, P.W. &Jr. (2009). US navel space surveillance upgrade program 1999-2003. )LIWK (XURSHDQ &RQIHUHQFH RQ 6SDFH 'HEULV 'DUPVWDGW*HUPDQ\(6$(62&. [3] Storch, T.R. (2005). 0DQHXYHU HVWLPDWLRQ PRGHO IRU UHODWLYH RUELW GHWHUPLQDWLRQ. Air Force Institute of Technology Wright-Patterson AFB OH School of Engineering And Management Air Force Institute of Technology, Ohio, Master thesis. [4] Patera, R.P. (2008). Space event detection method. -RXUQDO RI 6SDFHFUDIW DQG 5RFNHWV, 45(3), 554- 559. [5] Holzinger, M.J. & Scheeres, D.J. (2010). Object correlation and maneuver detection using optimal control performance metrics. 3URFHHGLQJV RI WKH $GYDQFHG 0DXL 2SWLFDO DQG 6SDFH 6XUYHLOODQFH 7HFKQRORJLHV&RQIHUHQFH, Hawaii. [6] Kelecy, T. & Jah, M. (2010). Detection and orbit determination of a satellite executing low thrust maneuvers. $FWD$VWURQDXWLFD, 66, 798-809. [7] Sidi, M.J. (1997). 6SDFHFUDIW G\QDPLFVDQG FRQWURO $ SUDFWLFDO HQJLQHHULQJ DSSURDFK. Cambridge University Press. [8] Der, G.J. (1997). An elegant state transition matrix. -RXUQDO RI WKH $VWURQDXWLFDO 6FLHQFHV, 45(4), 371-390. [9] Lobo, M., Vandenberghe, L., Boyd, S, et al. (1998). Applications of the second-order cone programming. /LQHDU$OJHEUDDQGLWV$SSLFDWLRQ, 284: 193-228.