Based on comparisons between instantaneous and light time delay (LTD) formulations for S band range and Doppler observables, expanded models for instantaneous S band observables are recommended for inclusion in the ENTREE program. In a worst case situation using Guam tracking for a representative shuttle entry trajectory, the proposed ENTREE model differs from the LTD model by approximately + or - 0.2 ft in range and + or - 0.02 Hz in Doppler. In contrast, the maximum differences between the existing ENTREE model and the LTD model are approximately (-80, +60) ft in range and (-4, -1) Hz in Doppler

Kelly, G. M.

English

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1	 ,
Recommended ENTREE 3 - band Range and Doppler Models
(NASA-CR-165884) RECOMMENDED ENTREE S-BAND
	 N82-21169
AND DOPPLER MODELS (Analytical. Mechanics
Associates, Inc.) 14 p HC A02/MF A01
C SCL 17G	 Dnclas
G3/04 17720
^ O
r
G. Mel Kelly
AMA Report No. 80-15
September, 1980
NASI-16087
ANALYTICAL MECHANICS ASSOCIATES. INC.
17 RESEARCH ROAD
HAMPMN. VIRGINIA 29888
i
iRecommended ENTREE S-band Range and Doppler Models
G. Mel Kelly
Analytical Mechanics Associates, Inc.
Based on comparisons between instantaneous and light-time-delay (LTD)
formulations for S- band range and Doppler observables, the following expanded
models for instantaneous S- band observables are recommended for inclusion in
the ENTREE program (Ref. 1):
R p	 p
.2	 2
2K	 T
where
Rs , f2 = S band range and Doppler, respectively
	
P , p	 - Instantaneous slant range and slant range rate, respectively
	
c	 = Speed of light
	
K	 = Doppler conversion factor
	
P , p	 •	 = Higher order derivatives of instantaneous slant range
	
Tc	= Doppler count time
In a worst case situation using Guam tracking for a representative Shuttle
entry trajectory, the proposed ENTREE model differs from the LTD model by
approximately f 0.2 ft in range and :k 0.02 Hz in Doppler. in contrast, the max-
imum differences between the existing ENTREE model and the LTD model are
approxmately ( -80, + 60) ft in range and ( -4, -1) Hz in Doppler.
3i
i
INTRODUCTION
Instantaneous formulations for S-band range and Doppler observables
are reasonable apprwdmations when applied to near-Earth spacecraft. Never-
theless, the modeling errors resulting from instantaneous assumptions can be
orders cf magnitude larger than the accuracies of the measurements. In order
to provide the best mathematical models for use in ENTREE , an effort was
undertaken (pursuant to Task I of this contract) for the purposes of: (1) Determining
limiting accuracies of existing formulations in ENTREE; (2) Recommending
improvements which do not dramatically alter the current software (e.g., an
iterative light time solution algorithm is not considered viable for inclusion in
ENTREE).
The S- band range and Doppler formulations given in Ref. 2 were adopted
as the real world for this study. Comparisons were performed against instan-
taneous models with various levels of improvements. Simulated tracking data
from Guam for a representative Shuttle entry trajectory (or`ained from Dick
Powell, VAB/SSD, in Feb., 1980) were utilized. The Guam tracking pass is a
worst case illustration of the limitations of the instantaneous formulations because
the slant range and slant range rate are very large near the ends of the data arc.
Ii
MATHEMATICAL MODELS
Light-Time Delay (LTD) Model
Very rigorous formulations for S-band Doppler and range obser-
vables are given in Sections VIII and IX, respectively, of Ref. 2. The deriva-
tions take into account: (1) Relativistic effects; (2) Station, SIC, Earth motion
over the signal transit time (light time); (3) Differing time references (ET, UTC,
UT1) and their interrelationships. Corrections which must be applied to the
computed observables to account for the effects of: (1) Offset of tracking point
from station location; (2) Earth's troposphere; (3) Earth's ionosphere are
discussed in Section XII of Ref. 2.
The observation equations can be summarized as:
Rs = 2 (%+Td +...)
f2 = c ( u +rd +...)
where
Tu , Td = converged light travel times for the up and down-
legs, respectively
u , rd = rate of change of up, down-leg slant ranges,
respectively (Note: u , rd are not instantaneous
slant ranges)
Existing ENTREE Model
The S-band model currently implemented uses the simplest instan-
taneous formulations:
$ = P
f = 2K2	 c P
as
1
L
pExpanded Instantaneous Model
The inherent inaccuracies of the simplest instantaneous formulations
are due to neglect of: (1) one-way light time in range, Doppler computations;
(2) Count (averaging) time in Doppler computations. Improvements can be gained
by using the simple notion that both range and Doppler are measurements of where
the S/C was Td seconds earlier, not where it is at station receive time. As a
result, the error in range is approximately equal to 
rd times the slant range
rate. Therefore, a better approximation for S-band range is:
s=P - Td p
or	
s P IT p
Neglecting the count time, a better approximation for Doppler may be obtained by
taking the time derivative of s and converting to Rz. The resulting equation is
.2
2K
if the speed of light were infinite, Doppler would still not be .range rate because
it is averaged over the count; time, Tc. Taylor series expansions about the
mid-point of the count interval yield
T	 T2	 T3
PL PM - P  2 pM 8 `M 24
Tc .. T c 2	 c_
PR QM +PM 2 +PM 8 +PM 24 +
The average range late over the count interval is then
T 2
P	 = pR_ PL	 +P• cAvg	 Tcf M 24
2
where L, R, M represent the left, right end-points and the mid-point of the
count interval, respectively. Addition of this count time correction term
gives
.2	 2
••	 T
2	 c	 c	 c	 24
SOFTWARE DESCRIPTION
Comparative data between the LTD and instantaneous formulations were
generated using the author's program SESIMS (Shuttle Entry Simulated Observa-
blep). SESIMS has been utilized a number of times since March, 1980 to gener-
ate simulated instantaneous observations for ENTREE checkout. Its primary
purpose, however, is to perform model comparisons.
The complexity of both the LTD and instantaneous formulations can be
controlled on input. The LTD model can be simplified by "turning off' in any
combination: (1) Relativistic effects on light time solution; (2) Higher order
speed of light terms in Doppler calculation; (3) Earth motion over the light time.
This model can also be regulated by the input values assigned to the speed of light
and the Doppler count time. The default instantaneous model is the existing
ENTREE model. It can be expanded by "turning on" in any combination the
correction terms containing: (1) p ; (2) p ; (3) p .
RESULTS
Figs. 1-4 show the range, Doppler differences between the LTD and
instantaneous models for various levels of "expansion" in the instantaneous
formulations. A Doppler count time of i s and the most complete LTD model
were used throughout. The "stray" points in several of the plots are due to
acceleration discontinuities (reaction jet firings) and have no bearing on the
overall comparisons.
Fig. 1 illustrates the limitations of the existing ENTREE models.
Fig. 2 shows the improvement obtained by adding the p correction term, which
results in a complete model for range, but not for Doppler. The range comparisons
3
are excellent; the only noticeable change in Doppler is in the signature,
although some of the bias has been removed. Fig. 3 contains comparative
data based on the addition of b and p correction terms. (The range com-
parisons in Figs. 3a, 4a are identical to those shown in Fig. 2a, since no
further expansion beyond p correction is required). The Doppler differences
in Fig. 3b are unbiased, which represents an improvement. The spread, how-
ever, is approximately f 1.5 Hz and is due to neglect of the i s Doppler count
time. The differences will vary directly with the square of the count time and
could be as much as t 150 Hz for a 10 8
 count interval. The importance of the
count time correction term (which requires p) is thus established. Fig. 4b
shows the excellent Doppler comparisons resulting from the incorporation of
the P term.
A numerical example showing the magnitude of the range, Doppler cor-
rection terms at Os into the tracking are is given in Table I.
ALTERNATE METHODS FOR IMPROVING INSTANTANEOUS MODELS
Time-shifting the range observable by the one-way light time yields the
same level of improvement as the addition of the correction term(- C ). How-
ever, time-shifts of the Doppler observable are equivalent to the p correction
term only. If the Doppler count time were zero, then a time-shift with the
addition of the p correction term would produce comparisons as good as those
shown in Fig. 4b. Unfortunately, as shown in Fig. 3b, Doppler count time may
be a very significant error source, so time-shifting is no help for Doppler.
A range-difference formulation for Doppler would eliminate the need for
incorporating 0, p computations in ENTREE. By definition, the count time
problem would disappear. Time shifts to accomodate the light time effects could
probably be used when evaluating range at the end points of the count interval. A
range-difference Doppler formulation is costly, however, since effectively twice
as many observations are computed. The number of file positioning steps for
state interpolation also goes up dramatically, although the positioning may not
be too costly in ENTREE becruse of the indexed sequential PQR file.
4
E-_	 _
RECOMMENDATIO:^tS FOR FURTHER STUDY
The improvcmients in the instantaneous formulations for S-band range
and Doppler observables discussed in this report are considered an important
first step in upgrading ENTREE modeling accuracy. Additional refinements
will necessarily include: (1) Algorithms for correcting observables for the
effects of Earth's troposphereand perhaps antenna offsets; (2) Provision for
time system differences at S/C and tracker (Station clock measures UTC,
station rotates in UT1 and S/C moves in ET); (3) Thorough review of models
for all other data types (TACAN, C-Band, Altimeter) to determine the adequacy
of existing formulations.
5
(a)
N2
-3
CL0
0
0
-6
lY / I NSTRNTRNEOUS DIFFERENCES
EXISTING ENTREE MODEL
0	 50	 100	 1 0
	 200	 250
TIME (SEC)
(b)
Fig. 1
i
6
rTIME DELAY / INSTANTANEOUS DIFFERENCES
P CORRECTION TO ENTREE MODEL
.,
v 00
z
cn
*++ 4-6*
' * *^ ++ +
+
++ +++*0.4*
+
+
I
(a)
0
N
-3
0
v
L"')
-f;
0	 vo	 1 G0	 I So	 AGO
MIME [SECI
(b)
Fig. 2
7
}	 +
1}
t	 + t	 +}+
'^YA'IF+	 + ^fi^► '	 ++Tom +
:.*+ *+ +*AyJ
4.
'TT +,+	 'T + }
++
+
^y
. j
u-
V
^^Z3
V7
-.J
3
TIME DELAY / INSTANTANEOUS DIFFERENCES
. .
p , p CORRECTIONS TO ENTREE MODEL
(a)
1L
N
O'U
U-)
n
ice^	 2C^	 C
TIME !SEC'
(n )
Fig. 3
6
. C r,
r^
T
(1tJ
L _ .
U
U
i
1.7
rr
i+ J
L
r r r r ^ r ^ 	 'i c ^^
„+++ +^ +	 i
TIME DELAY / INSTANTANEOUS DIFFERENCES
.. ...
p , p , p CORRECTIONS TO ENTREE MODEL
r
LL
C
CD
r^
C7
r
t	 w }
++
*+ + QI,Ah* 	 + t 1M►+ + }t	 *
}
^ t *+ }1F
+.60.
 	 +t +f	 t•F	 +
+ }	 t
tt ++++*
t
^WF*
+
1
(a)
7'TME ' L ' -
(b)
Fig. 4
9
f^
TERMS IN EXPANDED INSTANTANEOUS MODEL
(ELAPSED ARC 71ME - Od)
P	 = 2765477.4 ft
p	 - -20904.582 ft/sec
P	 = 49.541 ft/sect
p	 = 1.168 ft/sec3
c	 = 9.8357106 E8 ft/sec
2K	
_ 4.6637701 Hz/(ft/sec)C
J-	 = 2.812 E-3 sec
c
9-i	
_ +58.8 ft	 (Range correction term)
c
2K (p)	 _ -97494.165 Hz
.2
-
2c( A-) -2.072 Hz
_
2c (
-0.650 Hz	 (Doppler correction terms)c
T 22K (24p") +0.227 Hz
Table I
10
REFERENCES:
1. Waligora, S. R. and Mittelman, J. P., "Entry Trajectory Estimation
(ENTREE) Program System Description and User's Guide, CSC/SD-
79/6145, Computer Sciences Corporation, November, 1979.
2. Moyer, T. D., "Mathematical Formulation of the Double Precision
Orbit Determination Program (DPODP)' I , JPL TR 32-1527, May 15, 1971.
11


Recommended ENTREE S-band and Doppler models

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