The S-band X and Y angles, SAMS, and TACAN range and bearing were incorporated into the ENTREE software for use by experimenters at LaRC for entry trajectory reconstruction purposes. Background discussions present the need for this added capability. Formulations for the various observables are presented. Both north-south and east-west antenna mounts were provided for in the S-band angle computations. Sub-vehicle terrain height variations are included in the SAMS model. Local magnetic variations were incorporated for the TACAN bearing computations. Observable formulations are discussed in detail along with the partial computations

Findlay, J. T.

Heck, M. L.

English

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Formulation of Additional Observables for ENTREE
(NASA-CR-165880) FORMULATION OF ADDIIICNIL 	 N82-2C171
OSSERVABLES FOR ENTREE (Analytical Mechanics
Associates, Inc.) 12 p HC A02/MF A01
CSCL 17G	 Unclas
G3/04 17722
John T. Findlay
Michael L. Heck
0 ^,
L y
0
AMA Report No. 80-16
September, 1980
19AS1-16087
ANALYTICAL MECHANICS ASSOCIATES, INC.
17 RESEARCH ROAD
HAMPTON, VIRGINIA 23666
Formulation of Additional Obse
4
John T. Findlay
Michael L. Heck
Analytical Mechanics Associates, Inc.
A DQTD A OFP
S band X and Y angles, SAMS, and TACAN range and bearing have been
incorporated into the ENTREE software for use by experimenters at La RC for
entry trajectory reconstruction purposes. Background discussions in Section I
present the need for this added capability. Formulations for the various obser-
vables are presented. Both north-south and east-west antenna mounts have been
provided for in the S band angle computations. Sub-vehicle terrain height vari-
ations are included in the SAMS model. Local magnetic variations have been
incorporated for the TACAN bearing computations. Observable formulations
are discussed in detail in Section II of this report. The partial computations
are discussed in Section III.
I. Background discussion
The principal entry trajectory reconstruction software (Ref. 1) for LaRC
experimenters provided for C-band range, azimuth and elevation observables
but was structured to accomodate additional modelling. Though not completely
formulated, Doppler, S band ranging, and altimeter data were included. Instan-
taneous formulations for Doppler and S-band ranging were modelled. AMA, Inc.,
under the subject contract, has rectified these models. (1) It is felt that the
documentations, References 1 and 2, are adequate for C-band data, Doppler
and S-band ranging. The purpose of this report is to provide a third source to
complete the documentation for altimeter modelling as well as describe the new
observable formulations required to process the entire spectrum of anticipated
data for the LaRC experiment. Thus, the remaining observable formulation in-
cluded herein pertains to:
OBSERVABLE
(a) SAMS
(b) S band X and Y angles
and (c) TACAN bearing (2)
Section 11 presents the observable modelling. SAMS, S-band angular obser-
vations and TACAN bearing computations are presented as sub sections Ila, Ilb,
and IIc, respectively. The partials of these observables with respect to the state
vector are presented as Section III in the same order.
(1)Analysis has shown that the instantaneous formulations are inadequate.
Reference 2 discusses the recommended modelling for the S-band ranging and
Doppler observables. Partials, however, are based on the instantaneous for-
mulations which are deemed sufficient.
(2 )The computation of the TACAN range measurement and the associated
partial vectors is identical to that of the previously documented (Ref. 1) C-band
range measurement.
11. Observable Modelling
(a) SAMS data
The - bservable formulation used to process the recently proposed SAMS
altimetric measurements computes the geocentric altitude of the spacecraft
above the reference ellipsoid as follows:
he
 = K( RS/c - RL
 - h  (0D , X ) ) + - b	 (1)
where
	 Rs/c is the geocentric radius vector of the spacecraft,
RL is the local radius of the Earth ellipsoid at the sub-
spacecraft point (geocentrically located,)
h 	 is the local terrain height, (3)
K	 is the instrument scale factor,
and	 hb	 is the instrument bias.
The local planet radius is computed as follows:
RE
RL =	 1 + ( RE )2
 - 1	 sing 'tc
	
(2)RP
in which	 RE is the Earth equatorial radius
RP
 is the Earth polar radius
and	 0 is the geocentric latitude
Though the expression in Eq. (1) is not a rigorous computation of the true
geodetic altitude above the Fischer ellipsoid it is considered sufficiently accurate
in its simpler form (— 2 ft).
(3)The terrain height is modelled as a bi-variate table with geodetic latitude
(0D) and longitude (X ) as the independent variables.
3
(b) S-band Angle Data
(1) X-Angle; For east-west antenna mounts, the X-angle is defined to
be the angle between the local vertical (up) direction and the projection of the
range vector onto the east-west local vertical plane, measured positive to the
east. For north-south mounts, the X-angle is the angle between local vertical
(up) and the projection of the range vector onto the local N-S vertical plane,
measured positive south. As shown in Figures 1 and 2, the X-angle can be
given as a function of the azimuth and elevaticn angles. In particular, for
east-west mounts:
1 Ys	 __	 1 sin Az 1X-angle
-W -tan	 X % tan-1  tan El Js
For north-south mounts:
_ ( -Zs 1 _	 1 - cos Az lX-ang a le
	 - \ X
	
tan	
tan El
s
Additionally, an X-angle bias term can be modelled, considered, or solved for
in ENTREE for each S-band station if so desired.
(3)
(4)
4
	 N
Anorth (Z )
s
down
Figure 2. East-West Mount X-ANGLE and Y-ANGLE Definition
Aeast (Y )
s
Figure 2. North-South Mount X-ANGLE and Y-ANGLE Definition
6
down
1"/ 1-Z%a,gac: r va Bast- west antenna mounts, the Y-angle is defined
to be the angle between the projection of the vector onto the local vertical
east-west plane and the position vector itself, measured positive to the
north. For north-south mounts, the Y-angle is measured between the pro-
jection of the position vector onto the north-south plane and the position vector
itself, positive to the east. Again, referring to Figures 1 and 2, the Y-angle
can be shown to be given as a function of the previously computed azimuth,
elevation, and X-angles.
For east-west mounts:
Z
	
Y-angle = tan-1 C	 s	 )	 (5a)
E-W	 Xs cos (X-Angle) + Ys sin (X-Angle)
E-W	 E-W
	
tan-I C	 cos Az	 J	 (5b)tan El cos (X-Angle) + sin Az sin (X-Angle)
E-W	 E-W
For north-south mounts:
	
Y-angle = tan (	 sin Az	 I	 (S)
N-S	 \ tan El cos (X-Angle) - cos Az sin (X-Angle)
N-5	 N-S
The Y-angle bias term is modelled in ENTREE as a constant, solve-for or
consider parameter for each S-band station if desired.
(c) TACAN bearing
TACAN bearing is very similar to C-band azimuth except the measured
angle is a "fly-to" angle for the benefit of pilots. In other words, if a vehicle
was due southwest of a tracking station, the C-band azimuth angle measurement
would be 2250 ( or -1350) from north, whereas the TACAN bearing angle
measurement would be +45 degrees with respect to north. In addition, the
=
7
reference north direction for TACAN is magnetic north (again in deference
to pilots flying with magnetic compasses) whereas it is true north for C-band
azimuth. Hence, TACAN bearing is given by
TAC-B = Az + ft - µ ( - 1f . 'R )	 (7)
where µ is the local true north to magnetic north variation (positivc to the
east).
8 -
tv l
	ah T	 0
c	 =	 0
	
S. V.	 0
1
0
0
0
0
0
0
M. Partials
A very simple definition
with respect to state, viz:
T
In Eqn. (8) S. V. represents the ENTREE internal state vector (Ref. 1)
141
which consists of:
u	 North component of the inertial velocity	 I 7
v	 East component of the inertial velocity
w	 Vertical component of the inertial velocity
N	 geocentric altitude of S/C above reference sphere
0c	geocentric latitude
X	 longitude
e0
e1	quaternions which define orientation of Shuttle
e2	body axes with respect to geocentric local vertical
Je3 
A very weak sensitivity of h e to latitude could be included but Viking exper-
ience showed no need for such treatment.
(b) X and Y-Angles
The X-Angle and Y•-Angle partials were coded into ENTREE making
use of the existing Az and El partials as follows:
a X-Angle _ a X-Angle a Az	 + a X-Angle a El
a S. V.	 s Az	 a s. V .	 a El	 a S. V.
a Y-Angle 
= 
aY-Angle a Az + a Y-Angle a El
aS. V.	 a Az	 a S. V.	 o E1	 ?^ S. V.
By using the X and Y-Angle computations shown earlier, the following
equations can be derived:
a X-Angle_ tan El cos Az
a Az	 tan2 El +sin2
 Az
aX-Angle	 - sin Az
a El	 sin  E1 + sin  Az cos 2 E1
a Y-Angle 
=	
D2 	 -sin Az	 cos Az	 a D	 1
a Az	 D2 + cost Az	 D	 D2	 a Az	 J
where
	 D = tan El cos (X-Angle) + sin Az sin (X-Angle)
and	 a D = -tan E1 sin (X-Angle) a X-Angle
a Az	 a Az
+ cos Az sin (X-Angle) + sin Az cos (X-Angle) a 'X-Angle)
a Az
a Y-Angle _ - cos Az	 a D
a E1	 D2 + cos2 Az	 d El
a D	 cos (X-Angle)	 a (X-tingle)where	 ^	 - tan El sin (X-Angle)	 +2
a E1	 cos El	 a E1
sin Az cos (X-Angle) a -Angle)
a El
(9)
(10)
(11)
(12)
(13)
(13a)
(13b)
(14)
(14a)
1
10
€"
The above equations apply for the east-west mounted antennas. For the north-
south mount antennas, substitute sin Az everywhere that cos Az appears, and
- cos Az everywhere that sin Az appears.
(c) TACAN Bearing
Since the TACAN bearing measurement differs from the azimuth measure-
mont by a constant amount, the partial of TACAN bearing with respect to the
state vector is identical to the azimuth partial documented in Ref. 1, 1. e. ,
a TAC-B	 a Az
a S. V.	 a S. V.	 (15)
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. Kelly, G. M. , "Recommended ENTREE S-band Range and Doppler
Models", Report No. 80-15, Analytical Mechanics Associates, Inc.,
September, 1980, NAS1-16087.
i
f
11


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