An Overlap Comparison Using a Batch Filter with the Statistical Orbit
Determination I Project
Rodney L. Anderson
George H. Born
11/30/2004
In this study, a comparison of solutions for tracking data from the term project
obtained by using two different overlapping sets of range and range rate data was
completed. The original data file contained 385 points over a time period of 18,340
seconds, or approximately 5 hours. This data was divided into two parts with some
overlapping data points. The first data set covered the time period from the initial time to
12,600 seconds, which resulted in a data file with 288 points. The second data set began
at 5,400 seconds and continued until the end of the data, giving a data file with 297
points. In both cases, several passes through the batch filter were used to generate a best
estimate for the state and the covariance at the time that the corresponding data set
started. The trajectory was then integrated forward and the covariance was mapped to the
T
desired time using the equation Pk  Φt k , t o  Po Φt k , t o  . A comparison was desired
during the overlap time period between 1.5 and 3.5 hours, so plots were generated with
 one standard deviation (1 magnitudes from Pk, and the differences in the solution
overlaps for this period. They are given in the RIC (radial, in-track, cross-track)
coordinate system in Figure 1. The equations for converting the state and covariance
matrix from the ECI to RIC frame are given in section 4.16.1 of the class text.
Figure 1 Overlap comparison using Range and Range Rate
It can be seen from these plots that the actual differences in the radial and in-track
directions generally fell within the values of  obtained from the early and the late cases.
Recall that for the trivariate normal case the probability of the errors falling within the 1
error ellipsoid is 0.200 (see section 4.16 of class text). In the cross-track direction, the
actual differences were larger than the values of  obtained for the late data case, but they
exhibited the same trend as the  values with each case having a minimum at nearly the
same point.
In addition to generating solutions using both the range and range rate data, the batch
algorithm was also used with each data type separately. The overlap comparison using
just range data, shown in Figure 2, was very similar to the results obtained when both
data types were used. In both of these cases, the RMS value of the range residual for the
final solution was less than 0.01 m.
Figure 2 Overlap comparison using Range
It was found that for the range rate overlap comparison, plotted in Figure 3, the overlap
differences were much larger than the differences in the range or range plus range rate
cases. The actual RMS values for the range and range rate residuals are given in Table 1.
These results demonstrate that in general range is a stronger data type than range rate.
This can be explained by noting that range measurements, which for this study are
accurate to 0.01 m at 20 sec intervals, essentially provides the information equivalent to
range rate with an accuracy of 0.01/20 or 0.5 x 10-3 m/sec. Hence, range observations
provide not only a measure of the range, but also a measure of range rate. Each range
rate measurement is accompanied by an unknown value of range. This situation could be
mitigated if the range rate were generated by continuous count Doppler. In this case it
could be processing as accumulated or integrated range. This would result in only one
unknown epoch value of range and would increase the strength of the range rate data.
Figure 3 Overlap comparison using Range Rate
Note that in all cases the cross-track standard deviation and overlap differences exhibit a
once per rev sinusoidal behavior. In this case the maximum differences occur when the
satellite is at its highest latitude, and the minimum is at the equatorial crossing. To first
order, only a difference in inclination or right ascension of the ascending node will cause
a cross-track deviation and these will be 90 out of phase. A difference in node will
cause a maximum deviation at the equatorial crossing while a difference in inclination
will have a maximum effect on cross track deviation at maximum latitude. Therefore, it
is concluded that this difference is primarily caused by differences in the inclination of
the two orbits.
In general the formal or data noise covariance matrix would not be this indicative of
actual errors in the orbit determination solution because errors would arise from many
sources which are unmodeled or inaccurately modeled. However, in this case the
mathematical model contains all quantities affecting the satellite’s motion and there are
no measurement errors except random noise. Consequently, the formal covariance
matrix should be an accurate measure of estimation error which it is.
Table 1 RMS Values for the Range and Range Rate Residuals
Observation Types
Early Data Set
Late Data Set
Used
ρ RMS (m/s)  RMS (m)
ρ RMS (m/s)
 RMS (m)
Range & Range Rate
0.00965737
0.00101025
0.0097145
0.001012809
Range Only
0.00965711
0.00101032
0.0097144
0.001012848
Range Rate Only
0.18664792
0.00098957
0.3139124
0.000989415