AAS/AIAA Astrodynamics Specialists Conference

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1 Paper AAS 5-34 COVARIANCE ANALYSIS FOR DEEP- SPACE SATELLITES WITH RADAR AND OPTICAL TRACKING DATA James G. Miller The MITRE Corporation AAS/AIAA Astrodynamics Specialists Conference Lake Tahoe, CA, Agst 7-, 25 AAS Pblications Office, P.O. Box 283, San Diego, CA 9298

2 AAS 5-34 COVARIANCE ANALYSIS FOR DEEP-SPACE SATELLITES WITH RADAR AND OPTICAL TRACKING DATA INTRODUCTION James G. Miller * Coariance analysis for the special pertrbations orbit determination problem for deep-space satellites is considered to determine the relatie merit of radar and optical tracking data. Deep-space radars proide ery accrate range measrements, bt less accrate angle measrements. Optical sensors proide ery accrate angle measrements, bt make no range measrements since they are passie systems. The relationship of the size of the spatial part of the coariance to the relatie track density of radar and optical tracks in the orbit determination fit interal is illstrated for arios satellite orbits, inclding circlar semisynchronos, highly eccentric semi-synchronos, and geosynchronos. The US Strategic Command Space Sreillance Netork (SSN) is tasked to track satellites to maintain a special pertrbations (SP) satellite catalog at the Space Control Center (SCC) at Cheyenne Montain Air Force Station and at the Alternate Space Control Center (ASCC) at Dahlgren, VA. There is sfficient track capacity in the SSN to proide many tracks per day on near-earth satellites ith period less than 225 mintes. Hoeer, there is not enogh capacity in the SSN to track eery deep-space satellite ith period greater than or eqal to 225 mintes at least once per day. The SSN has grondbased optical sensors dedicated to tracking deep-space satellites, bt sch sensors are limited to night time ieing ith clear eather. The Space Based Visible (SBV) sensor onboard the Midcorse Space Experiment (MSX) satellite is the only optical sensor that does not hae either of these limitations. The recent pgrade of the Grond-based Electro-Optical Deep-Space Sreillance (GEODSS) sites ith Charged-Copled Deices (CCDs) has more than dobled the track capacity of the GEODSS sites. It is no feasible to obtain an optical track on each deep-space satellite eery other day. The sitation is mch orse for obtaining radar tracks on deep-space satellites. There are only three deep-space radars in the SSN, namely ALTAIR at the Regan Test Site on the Kajalein atoll, the Millstone Hill radar in Massachsetts, and the Globs II radar in Noray. The track capacity of each of these radars is ery limited de to the necessity to integrate many plses oer an extended period of time in order to bild p the signal-to-noise ratio to obtain a detection. Becase a radar is an actie system that transmits a radio freqency ae and receies the reflected ae from the satellite, the sensitiity of a radar is inersely proportional to the forth poer of the range to the satellite. These radars make ery accrate range and range rate measrements, bt less accrate angle measrements. Becase an optical sensor is a passie system that receies reflected light from the sn, the sensitiity of an optical sensor is inersely proportional to the sqare of the range to the satellite. The ability of an optical sensor to detect a satellite also depends on the solar phase angle. Optical sensors proide ery accrate angle measrements at fast integration times, bt proide no range measrements. * The MITRE Corporation, 55 Academy Park Loop, Colorado Springs, CO , USA.

3 The ne SP Sensor Tasking Prototype that tasks the SSN to maintain the SP satellite catalog has the ability to manage reqirements for radar and optical tracks independently. Coariance analysis for the SP orbit determination problem for deep-space satellites is sed to determine the relatie merit of radar and optical tracking data. This analysis ill be sed to proide track reqirements for the SP Sensor Tasking Prototype for deep-space satellites. COVARIANCE ANALYSIS The SP differential correction to a state ector is obtained by soling the linear eqation. AX = B by eighted least sqares, here the matrix A is obtained as the prodct of the partial deriaties of the measrements ith respect to the state ariables times the state transition matrix, the colmn ector X is the differential correction (DC) to the state ector, and the colmn ector B is the difference beteen the obsered sensor measrements and the predicted measrements obtained by propagating the state ector to the time of the obserations. The matrix D is the diagonal matrix hose entries are the reciprocals of the standard deiations of the sensor measrement errors. The eighted least sqares problem is obtain by mltiplying both sides of eqation Eq. () by D, hich yields 2. (DA)X = DB. The optimal least sqares soltion of Eq. (2) satisfies the normal eqations 3. (A T WA)X = A T WB, here W = D T D and the sperscript T indicates the transpose of the matrix. The coariance matrix C is the inerse of A T WA and is gien by 4. C = (A T WA) -. Coariance analysis assmes the only errors are sensor measrement errors characterized by their standard deiations, and no model errors. The Air Force Space Command (AFSPC) Astrodynamics Standards Look Angle Modle (LAMOD) is sed to generate perfect simlated obserations in an Orbit Determination Interal (ODI) by propagating an SP state ector. The left-hand side of Eq. (3) does not depend on the actal ale of the sensor measrements, bt only on the time of the obserations and the standard deiation of the measrements. The perfect simlated obserations are sed in the AFSPC Special Pertrbations Differential Correction (SPDC) softare to obtain a ale for the coariance matrix C. The coariance matrix is then propagated forard from epoch an ODI length of time to see ho the errors gro. The least sqares coariance matrix tends to be an optimistic estimate of the actal state ector errors, bt this analysis is more concerned ith the relatie merit of radar erss optical tacking data, and not the absolte errors from the SPDC process ith real obserations. For this analysis, radar tracks consist of six obserations and optical tracks consist of eight obserations, hich is typical of hat the deep-space SSN sensors proide. A GEODSS telescope at Socorro and the Millstone radar are sed as the sensors for this analysis. The force model sed is a 24 by 24 graitational potential, and solar and lnar pertrbations.

4 ANALYSIS RESULTS Since the amont of reflected sn light is dependent on the solar phase angle, optical sensors tend to collect obserations arond the minimm phase angle. Hoeer, it is better for the orbit determination problem to hae the tracks distribted arond the orbit rather than at one point of the orbit. Figre shos the extreme case here an optical sensor proides a track each day at the minimm solar phase angle for an ODI of 2 days on a circlar, semi-synchronos satellite, and there is no radar data. The components,, and are the radial, in-track, and cross-track errors, respectiely, obtained from the sqare root of the diagonal elements in the spatial part of the of the coariance matrix. The figre shos the propagated coariance oer one ODI from epoch, not the coariance oer the ODI itself, een thogh there is no eidence of error groth oer time. The in-track errors are ery large for this extreme sitation, and they hae the same period as the period of the satellite, namely 2 hors. Figre 2 shos the same satellite here the tracks from day to day hae some ariability arond the minimm solar phase angle. The in-track error is still the largest error for optical only tracking data, bt it is no mch smaller. The cross-track error is the smallest error ith optical only data. These figres illstrates the point that optical sensors shold ary the tracks from day to day abot the minimm solar phase angle, and not take all the tracks right at the minimm phase angle. Circlar, Semi-Synchronos Satellite Hors Since Epoch Figre 2-Day ODI, 2 Optical Tracks at Minimm Solar Phase Angle Circlar, Semi-Synchronos Satellite

5 Hors Since Epoch Figre 2 2-Day ODI, 2 Optical Tracks Figre 3 shos the same satellite ith one radar track each day and no optical data. For radar only data, the largest error is the cross-track error, and the smallest is the radial error. Oer all, radar

6 only data proides a more accrate state ector than optical only data. Circlar, Semi-Synchronos Satellite Hors Since Epoch Figre 3 2-Day ODI, 2 Radar Tracks The best reslts are obtained ith a combination of optical and radar data. Hoeer, it is nrealistic to get ery mch radar data on deep-space satellites. A realistic goal is to get an optical track eery other day and as fe as necessary radar tracks to complement the optical data and drie don the

7 in-track error. Figre 4 shos the same satellite ith an optical track eery other day and jst to radar tracks. This combination of optical and radar tracks has less error than either the optical only data or radar only data. Circlar, Semi-Synchronos Satellite Hors Since Epoch Figre 4 2-Day ODI, Optical Tracks, 2 Radar Tracks Figre 5 shos the same satellite ith an optical track eery other day and jst one radar track. A comparison of Figre 2 and Figre 5 shos the importance of getting jst one radar track, bt to radar tracks ithin the ODI is preferable.

8 Circlar, Semi-Synchronos Satellite Hors Since Epoch Figre 5 2-Day ODI, Optical Tracks, Radar Track Figre 6 shos a highly eccentric, semi-synchronos satellite ith an optical track eery day and no radar data. The ODI is also 2 days. Again, the in-track error is the largest and the cross-track error is the smallest for optical only tracking data. Highly Eccentric, Semi-Synchronos Satellite

9 Hors Since Epoch Figre 6 2-Day ODI, 2 Optical Tracks Figre 7 shos the same satellite ith one radar track each day and no optical data. No, the intrack error is the largest and the radial error is the smallest for radar only data. Highly Eccentric, Semi-Synchronos Satellite

10 Hors Since Epoch Figre 7 2-Day ODI, 2 Radar Tracks Figre 8 shos the same satellite ith an optical track eery other day and jst to radar tracks. This combination of optical and radar tracks has less error than either the optical only data or radar only data. Highly Eccentric, Semi-Synchronos Satellite

11 Hors Since Epoch Figre 8 2-Day ODI, Optical Tracks, 2 Radar Tracks Figre 9 shos the same satellite ith an optical track eery other day and jst one radar track. A comparison of Figre 6 and Figre 9 shos the importance of getting jst one radar track, bt to radar tracks ithin the ODI is preferable. Highly Eccentric, Semi-Synchronos Satellite

12 Hors Since Epoch Figre 9 2-Day ODI, Optical Tracks, Radar Track Figre shos a geosynchronos satellite ith an optical track eery day and no radar data. The ODI is 28 days. Again, the in-track error is the largest and the cross-track error is the smallest for optical only tracking data. Geosynchronos Satellite

13 Hors Since Epoch Figre 28-Day ODI, 28 Optical Tracks Figre shos the same satellite ith one radar track each day and no optical data. Like the circlar, semi-synchronos satellite, the cross-track error is the largest and the radial error is the smallest for radar only data on a geosynchronos satellite. Geosynchronos Satellite

14 Hors Since Epoch Figre 28-Day ODI, 28 Radar Tracks Figre 2 shos the same satellite ith an optical track eery other day and jst to radar tracks. This combination of optical and radar tracks has less error than either the optical only data or radar only data. Geosynchronos Satellite

15 Hors Since Epoch Figre 2 28-Day ODI, 4 Optical Tracks, 2 Radar Tracks Figre 3 shos the same satellite ith an optical track eery other day and jst one radar track. A comparison of Figre and Figre 3 shos the importance of getting jst one radar track, bt to radar tracks ithin the ODI is preferable. Geosynchronos Satellite

16 Hors Since Epoch Figre 3 28-Day ODI, 4 Optical Tracks, Radar Track CONCLUSIONS In general, radar only data proides a more accrate SP state ector than optical only data for deep-space satellites. Hoeer, the best case is a combination of optical and radar data, namely an optical track eery other day and to radar tracks ithin the ODI. This coariance analysis ill be sed to establish separate optical and radar tracking reqirements for deep-space satellites in the ne SP Sensor Tasking Prototype, hich ill optimally allocate the ery limited deep-space radar tracks. REFERENCES. James G. Miller, A Ne Sensor Resorce Allocation Algorithm for the Space Sreillance Netork in Spport of the Special Pertrbations Satellite Catalog, AAS 3-669, AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Montana, Agst 23.