Radar-Optical Observation Mix

Transcription

1 Radar-Optical Observation Mix Felix R. Hoots" April 2010! ETG Systems Engineering Division April 19, 10 1

2 Background" Deep space satellites are those with period greater than or equal to 225 minutes! Synchronous! Semi-synchronous (either circular or highly eccentric)! Generally observed by optical sensors! A few radars are capable of observing deep space satellites but resources are limited! Optical data has very precise angles but no range information! Radar has very precise range but lesser quality angular measurements! April 19, 10 2

3 Central Question" What is the best mix of radar and optical data to optimize the quality of the orbit estimation?! The quality of the orbit is judged by how well it matches the actual trajectory as we predict to future times! Covariance is a way to measure the expected prediction errors! April 19, 10 3

4 Trajectory Model" The main source of error for deep space satellites is measurement noise, not modeling errors! Since a shortfall in modeling is not the issue, we can use a two body model for both the truth reference and the trajectory model! Assume circular motion in a plane this will accommodate the synchronous and semisynchronous (circular) cases! April 19, 10 4

5 Sensor Model" Since the satellites are synchronous or semisynchronous, we assume observations are always taken each time the sensor moves through the orbital plane! The observations are assumed to be evenly spaced in time! April 19, 10 5

6 Sensor Observations" Radar observations! Optical observations! April 19, 10 6

7 Least Squares Formulation" where! April 19, 10 7

8 Least Squares Solution" where! and we have summed over all observations! April 19, 10 8

9 Observation Distribution" Assume there are k radar observations and m optical observations! Assume the radar observations are equally spaced in time over some time interval T! Assume the optical observations are equally spaced in time over the same time interval T! We can develop an analytic expression for the matrices in the weighted least squares problem! After a lot of algebra...! April 19, 10 9

10 Analytic Formulation" where!!these terms are functions of the actual errors that occurred with each measurement! April 19, 10 10

11 Analytic Formulation (cont.)" where! April 19, 10 11

12 Covariance" The covariance can be used to predict the error at a future time, t! where! April 19, 10 12

13 Covariance (cont.)" where! April 19, 10 13

14 Covariance Properties" Several properties of the covariance are immediately obvious in this analytical formulation! Coordinate variances are directly proportional to the measurement variances! Coordinate variances are inversely proportional to the number of measurements, k+m! Coordinate variances are inversely proportional to the square of the fit span, T! The radial variance is a constant! The in-track variance grows quadratically with time! April 19, 10 14

15 Radar-Optical Mix" Assume the total number of observations, k+m, is fixed! We want to know how to best distribute the observations between k radar observations and m optical observations! Looking at the covariance, it is clear that the key to minimizing the prediction error is to make the denominator D as large as possible! April 19, 10 15

16 Covariance Minimization" We can reformulate D as! where!!p = satellite period!!i = number of revs in fit span! April 19, 10 16

17 Covariance Minimization (cont.)" Since the total number of observations and sensor statistics are fixed, the only choice is to use more observations with good statistics and less observations with poor statistics! Since the radar has both a good and a bad measurement component and the bad component is multiplied by the square of the number of revs in the fit span, it is not clear exactly how to optimize the mix! We will assume a value (.01 deg) for the worst component (radar angle) and vary the relative radar range and optical angle quality! April 19, 10 17

18 Decision Parameter" The radar and optical terms are (approximately)! Whichever term is bigger tells us to use that sensor exclusively! We create a decision parameter which is the radar term minus the optical term! Decision parameter > 0 use all radar! Decision parameter < 0 use all optical! April 19, 10 18

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29 Sample Application" April 19, 10 29

30 Conclusions" For circular satellite motion synchronized with the Earth motion, we are able to formulate the covariance analytically! The analytical expressions reveal several interesting properties of the covariance! The formulation allows determination of the choice of resources between radar and optical measurements! Reveals a dependence not only on sensor quality but also on fit span length! April 19, 10 30