On-Orbit Determination with Radio Doppler Validation SSC16-P4-15 Ground Station Software Tools for Small

Nicholas A. Dallmann LA-UR-16-26025

Abstract & Motivation Analysis Flow Ephemeris Determination at the Ground Station

A brief overview of well established, complementary, and redundant • Block diagram below shows the operational (purple) and simulation • Coherent receivers will naturally methods for determining and validating ephemeris is given. (blue) analysis flow. measure the received carrier • Once satellites are on orbit, the first task is to quickly determine • The system model is the SGP4 orbit propagator and it is assumed to frequency. ephemeris sufficient to maintain reliable communications. represent ‘True’ dynamics. • Small satellite teams should be prepared with tools to accomplish • The error between expected and ‘True’ Doppler is simulated for periodic • The covariance of ephemeris this in the absence of external assistance. radio receptions throughout each . determined via Doppler error Los Alamos is in the process of adding these capabilities to its • The standard deviation, error, in measurements for both Doppler and measurement has been integrated ground station control software suite “GS Commander.” GPS (DOP) is known at each measurement investigated. • It is assumed that the standard deviation, error, in both Doppler and GPS measurements are known for each measurement and that they are Determining Which TLE Goes with Which Satellite normally distributed. Validating Ephemeris

• It is now common for many dozens of satellites to be released from ‘True’ ‘True’ Noise • The results below are from a simulated scenario of: SGP4 a single in fast succession. Ephemeris Doppler 1) On-orbit GPS measurements obtained periodically over 1 day Doppler Sim • JSpOC can quickly provide a list of candidate TLEs but it can be still Determined and subsequent on-orbit ephemeris determination SGP4 SGP4 Eigen challenging to determine which TLE is correct for which satellite. SGP4 Best Guess Ephemeris 2) Transmission of determined ephemeris and covariance estimates Orbit Decomp Ephemeris to a ground station • Given Doppler measurements of satellite transmissions, a simple Det Ephemeris SGP4 Error 3) Ground station determination of ephemeris and covariance analysis can quickly pair the satellites with correct TLEs. GPS Covariance ∑ Analysis ‘True’ Positions estimate via Doppler error measurements Measurement & Prop • The figure below shows 9 candidate TLEs each separated by 0.5 Times Noise 4) Analysis of the correlation between the Doppler and GPS results seconds of or a spread from leading to trailing of only ~30 km. for cross verification. Covariance and High Dimensional Ellipsoidal Geometry • The simulated Doppler measurements assume transmissions are in 5) Combining of the results to reduce covariance of propagated solution. unified s-band and are obtained with accuracy of 25 Hz (~1σ). • Classical differential correction (least squares) is used for the mixed Candidate TLEs measurement (Doppler & GPS) : • Los Alamos plans to incorporate automated analysis of this type = A A A = A into its small satellite ground station software to provide • Forward uncertainty propagationT −1 −1 fromT −1 the ephemerisT −1 covariance to redundancy in ephemeris determination and statistically sound 𝛿𝛿𝑥𝑥� 𝛴𝛴𝑦𝑦 𝛴𝛴𝑦𝑦 𝑏𝑏� 𝛴𝛴𝑥𝑥 𝛴𝛴𝑦𝑦 𝑏𝑏� Most correct the position covariance is defined as: verification of on-orbit ephemeris determination. TLE Available 𝛴𝛴𝑥𝑥 Satellite is = A A 𝛴𝛴𝑦𝑦 behind TLE • Eigenvector/value decomposition is used𝑇𝑇 to determine the size and 𝑦𝑦 𝑥𝑥 used during orientation of the multidimensional𝛴𝛴 𝛴𝛴error ellipsoid and projections. pass • Figure below is the projection of the 95% confidence position portion of the ellipsoid onto the Y-Z (JD2000) plane. The ellipsoid can be rotated and projected onto different planes for diverse investigations. • Forward uncertainty propagation can suffer from a poorly formed Jacobian [A]. However, it can be verified by performing a Monte Carlo analysis. The dots in the figure are full propagations with the ephemeris Satellite is set drawn from: ahead of TLE TLE used for = { : = + 7 } Doppler pre- used during 𝑥𝑥 pass correction 𝑋𝑋 𝑥𝑥 𝑥𝑥 𝑥𝑥̅ 𝛴𝛴 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 Doppler Error during pass Measurements

REFERENCES Mean Arg of Mean Inclination Eccentricity RAAN th Perigee Anomaly [1] David A. Vallado, Fundamentals of Astrodynamics and Applications 4 Ed. Hawthorne, CA: t0 i0 e0 B* Ω0 Microcosm Press, 2013. n0 ω0 M0 [2] M. Friendly, G. Monette, & J. Fox, “Elliptical Insights: Understanding Statistical Methods 6027.203 15.08409 0.2322E-4 28.4691° 2.798E-4 20.317° 339.9343° 156.2539° through Elliptical Geometry,” in Statistical Science, Vol. 28, No. 1, 2013, pp. 1-39. JD2000 {rev/day} {1/ER}