Relationships between geographic and inertial coordinates of positio

Heffron, W. G.

Watson, S. B.

English

NASA Technical Reports Server

BELLCOMM, INC. 
COVER SHEET FOR TECHNICAL MEMORANDUM s 
- R e l a t i o n s h i p s  Between 
Geographic and I n e r t i a l  
Coord ina te s  of P o s i t i o n  
F I L ~ N G  CASE NOW- 310 
TM - 66-2012-1 
DATE- March 17, 1966 
FILING S\JBJECT(S) - Naviga t ion  
(ASSIGNED B Y  AUTHOR(S) - 
A6STRACT 
R e l a t i o n s h i p s  between geographic  ( l o n g i t u d e ,  g e o d e t i c  
l a t i t u d e ,  a l t i t u d e )  and i n e r t i a l  (x, y, z) c o o r d i n a t e s  o f  p o s i t i o n  
are examined and computa t iona l  techniques  g iven .  
between g e o d e t i c  l a t i t u d e ,  g e o c e n t r i c  l a t i t u d e  and g e o c e n t r i c  
d e c l i n a t i o n  a r e  shown: errors r e s u l t i n g  from misuse of these a n g l e s  
a r e  g iven .  
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SUBJECT: R e  1 a t  i on s h i p  s B e  t w e e n  G e og r ap h i  c DATE: March 15, 1966 
and I n e r t i a l  Coordinates  of  
P o s i t i o n  - Case 310 FROM: W .  G .  Hef f ron  
S .  B .  Watson 
TM-66-2012-1 
TECHNICAL MEMORANDUM 
T h i s  memorandum examines t h e  r e l a t i o n s h i p s  between 
geographic  c o o r d i n a t e s  ( a l t i t u d e ,  g e o d e t i c  ( o r  g e o c e n t r i c )  
l a t i t u d e  of  the  sub-vehic le  p o i n t ,  l o n g i t u d e )  and (x, y ,  z )  
components o f  i n e r t i a l  p o s i t i o n .  Formulas are developed f o r  
conve r s ion  from one system t o  t he  o t h e r .  S p e c i a l  emphasis is 
p l a c e d  on the  d i s t i n c t i o n  between g e o d e t i c  l a t i t u d e  (which 
a p p l i e s  only f o r  t he  sub-vehic le  p o i n t ) ,  t h e  g e o c e n t r i c  l a t -  
i t u d e s  of t h e  sub-vehic le  p o i n t  and t h e  g e o c e n t r i c  d e c l i n a t i o n  
( l a t i t u d e )  of t h e  v e h i c l e  i t s e l f .  It i s  shown t h a t  the  numeri- 
c a l  d i f f e r e n c e s  between these t h r e e  l a t i t u d e s  are  small .  Des- 
p i t e  t h e  small d i f f e r e n c e s ,  convers ion  from geographic  coor- 
d i n a t e s  t o  x, y ,  z coord ina te s  can lead t o  s e r i o u s  errcrs  if 
t h e  d i s t i n c t i o n s  between t h e  t h r e e  t y p e s  o f  l a t i t u d e  are not 
p r e s e r v e d .  T h i s  i s  shown by  a numer ica l  example. 
The q u a n t i t i e s  o f  i n t e r e s t  are:  
c o o r d i n a t e s  ',* o f  t h e  v e h i c l e .  
E a r t h ' s  North Pole a x i s  o f  t he  p r e s e n t  i n s t a n t ,  t he  
x a x i s  i s  towards t h e  f i r s t  p o i n t  o f  *T and the  y 
a x i s  completes a r igh t -handed  s e t .  See F igure  1. 
(x , y  , z ) The z a x i s  i s  the  
(w,z> c o o r d i n a t e s  of t h e  v e h i c l e  i n  t he  mer id ian  p l ane  
of t h e  v e h i c l e .  w =-, z i s  as above. 
.. BELLCOMM, I N C .  - 2 -  
R 
0 
x 
c o o r d i n a t e s  of t h e  sub-vehic le  p o i n t  i n  t h e  v e h i c l e  
m e r i d i a n  p l a n e .  The l i n e  between ( u , v )  and (w,z)  i s  
p e r p e n d i c u l a r  t o  t h e  r e f e r e n c e  e l l i p s i o d .  The 
v e h i c l e  i s  at  t h e  z e n i t h  r e l a t i v e  t o  t h e  ( u , v )  p o i n t .  
r a d i u s  t o  the  v e h i c l e  R = ( x 2  + y2 + z ) . 
g e o d e t i c  l a t i t u d e ,  which i s  t h e  ang le  between t h e  
e q u a t o r i a l  p l ane  and a l i n e  p a s s i n g  th rough  (u ,v )  
p e r p e n d i c u l a r  t o  the  e l l i p s o i d  ( t h e  z e n i t h  r a y ) .  
g e o c e n t r i c  l a t i t u d e  o f  t he  sub-veh ic l e  p o i n t ,  which 
is t he  a n g l e  between t h e  e q u a t o r i a l  p l a n e  and t h e  
r a d i u s  v e c t o r  t o  the  sub-vehic le  p o i n t  ( u , v ) .  
g e o c e n t r i c  d e c l i n a t i o n  of t he  v e h i c l e ,  which i s  t h e  
angle between t h e  e q u a t o r i a l  p l a n e  and t h e  r a d i u s  
v e c t o r  t o  t h e  v e h i c l e  p o i n t  ( x , y , z >  o r  (w ,z ) .  
l o n g i t u d e  ( e a r t h - f i x e d )  of the  veh ic l e ,wh ich  i s  t h e  
a n g l e  between t h e  Greenwich mer id i an  and the  v e h i c l e  
2 1 / 2  
mer id i an ,  p o s i t i v e  e a s t w a r d  from Greenwich. 
i n e r t i a l  p l a n e  and the  m e r i d i a n  of Greenwich. 
G H A T  Greenwich Hour Angle, t h e  a n g l e  between the  x-z 
H a l t i t u d e ,  which i s  t h e  d i s t a n c e  a long  t h e  z e n i t h  r a y  
from t h e  sub-vehic le  p o i n t  to t he  v e h i c l e .  
a,b major  and minor axes of t he  r e f e r e n c e  e l l i p s o i d .  
a t h e  ang le  0-6 
D a d i s t a n c e  which c l o s e l y  approximates  t he  a l t i t u d e .  
v i ewpo in t .  F i r s t  t h e  computation of ( x , y , z )  g iven  H ,  I$ and X 
( o r  Q o r  6 i n  p l a c e  of  t h e  standard c o o r d i n a t e  0 )  i s  c o n s i d e r e d ,  
and t h e n  t h e  i n v e r s e  problem i s  t r e a t e d .  The l a t t e r  i s  shown t o  
i n v o l v e  t r a n s c e n d e n t a l  f u n c t i o n s  i n  computing 0 and Q, s o  approx- 
imate methods are developed and numer i ca l  comparisons a r e  g i v e n .  
The r e l a t i o n s h i p s  a r e  deve loped  from a computa t iona l  
. . 
.. BELLCOMM, 1NC. - 3 -  
Computation of x ,y ,z  
The e q u a t i o n  of  the  r e f e r e n c e  e l l i p s e  (see F i g u r e  2 )  i s  1;) + (;)* = 1 
From t h i s  i t  can be deduced t h a t  
1. 
2. 
And s o ,  by d e f i n i t i o n  o f  t h e  ang le  $, t h e  r e l a t i o n s h i p  
t a n  4 =(+2 t a n  $ 
between $ and 4 is found t o  be 
3 . a .  
o r  
t a n  $ =[:I * t a n  4 3.b .  
These formulas  are independent  of t he  a l t i t u d e  of t he  
If 4 o r  $ i s  g i v e n ,  t h e  o t h e r  can be e a s i l y  found, and 
v e h i c l e ,  s i n c e  only  t h e  sub-veh ic l e  p o i n t  i s  i n v o l v e d .  
t h e  r a t i o  E formed. 
s o l v i n g  f o r  u ,  g i v e s  
S u b s t i t u t i o n  of  t h i s  i n  e q u a t i o n  1, and 
and t h e n  
v = u t a n  + = u k 2  tan 4 a 5 .  
r a y  
has 
It can a l s o  b e  deduced t h a t  t he  i n t e r c e p t  of t h e  z e n i t h  
w i t h  t he  e q u a t o r i a l  p l ane  o c c u r s  a t  u ( l  -M * >  s i n c e  t h i s  r a y  
s l o p e  t a n  4 and p a s s e s  through ( u , v ) .  
Using these r e l a t i o n s h i p s ,  t h e n  
w = u + H cos 4 
z = v + H s i n  4 
6.a. 
6 . b .  
and 
Z t a n  6 = w 7 .  
BELLCOMM, INC. - 4 -  * .  
b 
The e q u a t o r i a l  p l a n e  component w i s  r e s o l v e d  i n t o  x and 
y components u s i n g  the  l o n g i t u d e  X and the  Greenwich Hour Angle 
GHA. The s t a n d a r d  formula  f o r  GHA i s  (where t h e  xyz s y s t e m  i s  de- 
f i n e d  f o r  J anua ry  0 .5 ,  1 9 5 0 )  
where D = number of i n t e g r a l  days past the  g i v e n  epoch,  t i s  the  
time i n  seconds  p a s t  t h e  0 h r  a t  Greenwich, and w i s  g i v e n  by 
G H A ~  = 100.075543 t 0.g8564734600 t 2 . g 0 1 5 ~ 1 0 - ~ ~ ? ~ ~  + 
w =  360 deg/sec 9.  
86164.09972 t 4.49075 x 10  -8D 
and s o  
x = w COS ( A "  + CHAP) 
y = w s i n  ( A "  + GHAT) 
t y 2  + 
and f i n a l l y ,  
ComDutation of A .  cb .H 
2 2 1 / 2  R = ( x  = ( w 2  t z ) 
10. 
11. 
1 2 .  
Given x ,y , z ,  the l o n g i t u d e  i s  f i r s t  found u s i n g  t h e  
GHA fo rmula  and t h e  e q u a t i o n  
X O  = a r c t a n  O - GHATO 13.  
H and + are more d i f f i c u l t  t o  f i n d .  F i r s t  w i s  computed 
and t h e n  e q u a t i o n s  6a and 6b are  pu t  i n  t h e  form 
U = W - H C O S +  1 4 .  
v = z - H s i n  + 15 .  
If H and + are c o r r e c t ,  the  e q u a t i o n s  f o r  t he  e l l i p s e  
and f o r  t a n  + ( e q u a t i o n  1 a n d  2 )  w i l l  be  s a t i s f i e d ,  i . e .  
and 
z-Hsin+ 
tan ' =(d w-Hcos+ 
1 6 .  
1 7 .  
8 .  
. . 
. *  BELLCOMM, INC. - 5 -  
These are p u t  i n  t h e  following form 
2 f ( 4 , H )  = 0 = W 2' 
a 
2 /cos  2 4 + sin*+) -1 
H -  
b 2  2 \ a  
18.  
1 9 .  
Obviously t h e s e  equa t ions  are t r a n s k e n d e n t a l  and 
r e q u i r e  e i t h e r  use  o f  approximations o r  an i t e r a t i v e  procedure .  
The approximat ions  a r e  d i s c u s s e d  below and t h e n  an i t e r a t i v e  
t echn ique  ( f o r  which one i t e r a t i o n  appears  s u f f i c i e n t )  i s  g i v e n .  
The approximat ions  are d i s c u s s e d  fkst because  they  are good 
s t a r t i n g  p o i n t s  f o r  any i t e r a t i v e  procedure .  
Approximate H and I$ 
t h e  d i s t a n c e  D ( s e e  F igu re  2 )  i s  a n  e x c e l l e n t  approximation t o  H.  
F i g u r e  3 bears t h i s  o u t ,  showing t h e  d i f f e r e n c e  a l w a y s  to b e  less 
t h a n  36 m e t e r s .  Using t h e  F i s c h e r  r e f e r e n c e  e l l i p s o i d 3  f o r  which 
a = 6 378 166  m . ,  b = 6 356 784 m .  and f = 1/298.3, D i s  g i v e n  by  
Numerical computations as w e l l  as a n a l y s i s  show t h a t  
o r  
i I  
o r  
20.  
21.a. 
21 .b .  
. .  
and D i s  thus  a first approximation t o  H .  
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To develop  a n  approximation t o  I$, a s t a r t i n g  p o i n t  i s  
t o  rewri te  g(I$,H) i n  terms of t h e  small a n g l e  a = 4 - 6,and D .  
The a n g l e  6 i s  known from e q u a t i o n  7, and D i s  as j u s t  g i v e n .  
That a i s  small i s  shown i n  F igu re  4 ,  which g i v e s  t he  t r u e  a f o r  
s e v e r a l  l a t i t u d e s  and a l t i t u d e s .  
i s  e q u a l  t o  a f o r  H = 0 . )  
i s  more s e n s i t i v e  t o  a t h a n  t h e  f(4,H) f u n c t i o n .  
a n g l e  assumptions and making s e v e r a l  s i m p l i f l c a t i o n s  , t h e  approx- 
i m a t i o n  t o  a ,  c a l l e d  a + ,  i s  g iven  by 
(The d i f f e r e n c e  between 4 and JI 
The f u n c t i o n  g(I$,H) i s  used because  i t  
Using small 
o r  
and  t h e  approximat ion  t o  (I i s  
4 = 6 + a *  2 4 .  
Using a 27 s i g n i f i c a n t  b i t  d i g i t a l  computer f o r  c a l c u l a t i o n s ,  t h e  
e r r o r  i n  $I from t h i s  formula occur s  i n  t he  26th  and 27th b i t s ,  i . e . ,  
i s  less t h a n  2 ~ 1 0 - ~  degrees. 
s u f f i c i e n t l y  a c c u r a t e  formula f o r  $I. 
An I t e r a t i v e  Technique 
may be used .  One form i s  t h e  m a t r i x  e q u a t i o n  
Thus, t h i s  i s  f o r  most purposes  a 
If greater  p r e c i s i o n  i s  r e q u i r e d  a Newton-Raphson t e c h n i q u e  
.. BELLCOMM, INC. - 7 -  
where t h e  m a t r i x  A i s  t h e  i n v e r s e  of t h e  s e n s i t i v i t y  m a t r i x ,  i . e .  
26. 
Ana lys i s  shows t h a t  t he  d i a g o n a l  terms g r e a t l y  dominate 
t he  A m a t r i x ,  and tha t  a s u f f i c i e n t  s e t  of e q u a t i o n s  i s  
27. 
28. 
For a 27 s i g n i f i c a n t  b i t  d i g i t a l  computer t he  e r r o r  i n  H 
l i e s  i n  t he  2 4 t h  th rough 27th  b i t s  ( less  t h a n  0 .25  meters) and t h e  
e r r o r  i n  $ now l i e s  i n  t h e  26 th  and 27th b i t s  and i s  o f t e n  ze ro .  
M i s a p p l i c a t i o n  E r r o r s  
Unfo r tuna te ly  t h e  f a c t  t ha t  t he  d i f f e r e n c e  between 4, 
JI a n d 6  2 s  a lways  less t h a n  0.2' leads t o  a tendancy t o  ignDre the  
d i s t i n c t i o n  between them. T h i s  can lead t o  e r r o r s  i n  ( w , z )  and 
( x , y , z ) .  It i s  i m p o r t a n t  t h a t  t h e  a l t i t u d e  be  p r o j e c t e d  a long  t h e  
z e n i t h  r a y  f o r  c o r r e c t  computat ion o f  ( x , y , z ) .  F i g u r e  5 shows t h e  
c o r r e c t  a p p l i c a t i o n  and two p o s s i b l e  m i s a p p l i c a t i o n s .  A t  p o i n t  P ,  
t h e  c o r r e c t  g e o c e n t r i c  l a t i t u d e  of t he  sub-veh ic l e  p o i n t  has been 
computed b u t  t h e  a l t i t u d e  has been  p r o j e c t e d  a long  the  e x t e n s i o n  o f  
t h e  r a d i u s  t o  t h e  sub-vehic le  p o i n t .  For  an a l t i t u d e  of 1 0 0  NM, 
t h e  e r r o r  i s  619 meters, and f o r  an a l t i t u d e  of 200 ,000  NM t h e  e r r o r  
i s  1,238,848 meters ( abou t  2/3 t h e  r a d i u s  o f  the  moon). At p o i n t  Q 
t h e  g i v e n  " l a t i t u d e  = 45'" has been f a l s e l y  i n t e r p r e t e d  as geocen- 
t r i c  d e c l i n a t i o n  of  t h e  v e h i c l e  r a the r  t h a n  t h e  g e o d e t i c  l a t i t u d e  
of t h e  sub -veh ic l e  p o i n t  (this l a t t e r  b e i n g  t h e  s t a n d a r d  meaning of  
l a t i t u d e ) .  P o i n t  Q i s  i n  e r r o r  b y  21,448 meters, and t h i s  e r r o r  i s  
.. BELLCOMM, INC. - 8 -  
independent  of  a l t i t u d e .  None o f  these e r r o r s  should  be  
t o l e r a t e d  i n  most a p p l i c a t i o n s .  
Conclusions 
R e l a t i v e l y  s t r a i g h t f o r w a r d  convers ion  formulas  between 
geographic  and i n e r t i a l  components of p o s i t i o n  have been developed.  
The convers ion  from i n e r t i a l  t o  geographic  c o o r d i n a t e s  i n v o l v e s  
t r a n s c e n d e n t a l  equa t ions ,  b u t  t h e  accuracy o f  t h e  formulas  g iven  
i s  demonst ra ted .  The need f o r  d i s t i n c t i o n  between t h e  g e o d e t i c  
l a t i t u d e ,  t h e  g e o c e n t r i c  l a t i t u d e  o f  t h e  sub-vehic le  p o i n t  and 
t h e  g e o c e n t r i c  d e c l i n a t i o n  of t he  v e h i c l e  i s  demonstrated,  even 
though the  ang le s  d i f f e r  by l e s s  t h a n  0 . 2 O .  
W .  G .  Heffron 
WGH m r r  * 012-SBW-jvd S .  B.  Watson 
Attachment 
R e  f e r e n c e s  
6 .  BELLCOMM, INC. 
REFERENCES 
1. "The American P r a c t i c a l  Navigator" ,  N .  Bowditch e t  a l ,  
U.  S. Navy Hydrographic  O f f i c e ,  U . S .  Goverrment P r i n t i n g  
O f f i c e .  
2 .  " P r o j e c t  Apollo Coordina te  System S tanda rds" ,  NASA 
S tanda rd  SE 008-001-1, OMSF, NASA, June  1965 
3. "An A s t r o g e o d e t i c  World Datum from Geoida l  Heights  Based 
on t h e  F l a t t e n i n g  f-1/298.311, Jane F i s c h e r ,  J o u r n a l  of 
Geophys ica l  Research,  Vol. 6 5  No. 7 ,  page 2067. 
.. 
X 
. .-- - - . ... 
/ nER'D'An7 
) EARTH ROTATION 
Y 
4 F I R S T  POINT 
OF ARIES 
FIGURE I BASIC COORDINATE AXES 
h 
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0.  
u) 
(0 0 
hl 
0 
- a, 0 
0 
W 
0 
0 
c3 
0 
9 
0 
0 
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0 
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u) 
t- 
0 
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=t 
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0 
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Relationships between geographic and inertial coordinates of position

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