2/6/2001
Coordinate Systems
ASEN 3200 Notes
George H. Born
Several Coordinate Systems are used extensively in Orbit Mechanics. They will be
reviewed here.
1. Earth Centered Interval (ECI) and Earth Centered Fixed (ECF)
The ECI frame has its origin at the center of mass of the Earth but has a fixed
inertial direction along the intersection of the Earth equatorial plane and the ecliptic
plane. Although this frame is referred to as inertial it is actually only pseudo inertial
because the center of mass of the Earth accelerates due to perturbations from the
Moon and other planets. The ECF frame has the same origin but is fixed in the Earth
with its X  - axis through the Greenwich meridian (zero longitude).
Z, Z 
 g  Greenwich Siderial Time
Y

 g =  g   (t  t 0 )
0
ECI
Y
(1)
 g  Greenwich hour angle
0
ECF
X
(Greenwich Siderial Time)
at the epoch, t 0 .
g
X
The transformation matrix between these frames is
X 
Cg
Y    S
 
 g
 Z  ECI  0
 Sg
Cg
0
0
0

1
 X 
Y   .
 
 Z   ECF
(2)
1
2. Another useful frame is the P,Q,W frame
where P and Q are unit vectors in the orbit plane with P directed to perigee,
W along the angular momentum vector and Q completing a right hand triad,
i.e., P  Q  W
Z
Q
To go from XYZ (ECI) to P,Q,W
P
We first rotate about Z through  ,
k
W
the right ascention of the ascending
j
i
node. Next rotate about the new
Y

I

X-axis through I, the inclination,
then rotate about the resulting Z-axis
Equator
through  , the argument of perigee.
X
S/C Orbit Plane
Here P, Q, W are the components of a vector expressed in the PQW frame, i.e.,
r PQW
= PP  QQ  WW
.
(3)
Remember P , Q , W are unit vectors. These three rotations will yield the desired
transformation matrix as the product of three matrices, i.e.,
 C
P
 Q  =  S
 
 
 0
W 
S
C
0
0
0

1
0
1
0 C
I

0  S I
or,
 C C  S C I S 
P
 Q  =  S C  C C S
 I 
  
 
S I S

W 
0
SI 

C I 
 C
 S
 
 0
C S   S C I C
 S S   C C I C
 S I C
S
C
0
0
0

1
X 
Y 
 
 Z 
S S I 
C S I 

C I 
X 
Y 
 
 Z 
(4)
(5)
2
Hence,

 C C  S C I S 
=  S C  C C I S 

S I S

ECI
PQW
C S   S C I C
 S S   C C I C
 S I C
S S I 
C S I  .

C I 
Then,
P
Q  =
 
W 
 PQW
ECI
X 
 Y  and
 
 Z 
X 
Y  =
 
 Z 

ECI
PQW
P
T  
 Q  ,
W 
(6)
and
 PQW
ECI
=
T
  ECI
PQW  .
If we have unit vectors P , Q , W expressed in the ECI frame (as we obtain using Gibbs
Method for example), then
P  X P i  YP j  Z P k ,
Q  X Q i  YQ j  Z Q k ,
W  X W i  YW j  Z W k ,
and
P 
XP
 

Q  =  X Q
W 
 X W
YP
YQ
YW
ZP 
ZQ 

Z W 
i 
 .
 j
k 
(7)
Where P , Q , W and i , j, k are unit vectors along the P, Q, W and X, Y, Z axes
respectively and the X P ,YP , Z P etc. are direction cosines.
Thus, we have
 ECI
PQW
directly, i.e.,
 PQW
XP
X
 Q
 X W
ECI
=
YP
YQ
YW
ZP 
ZQ  .

Z W 
(8)
3
Also, note that P lies along the eccentricity vector and W lies along the angular
momentum vector. Consequently, if the position and velocity vectors are available in the
ECI frame. We may obtain P , Q , W directly as follows
e
1  ECI 
r 
 r h 
,

r 
e
,
e
(10)
h  r  ECI r ,
(11)
P
W 
h
,
h
(12)
Q  W  P since P , Q , W form a right hand triad.
and
(9)
(13)
3. Another frame often used is the RIC frame with unit vectors R, I and C .
In this frame R lies along the instantaneous radius vector, I lies in the orbit plane
normal to R and in the direction of motion of the spacecraft. C is normal to the orbit
plane and lies along the angular momentum vector. Hence, C  W of the PQW frame.
R and I align with P and Q when the spacecraft is at perigee. Although R and
I rotate with the spacecraft radius vector, at each instant in time the frame is
considered fixed, so we do not differentiate these unit vectors when transforming
velocity to this frame i.e., velocity magnitude in this frame has the same value as in
the inertial, ECI, frame. This frame is useful for displaying the difference between
two orbits in the radial, in-track, and cross-track directions. To obtain the
transformation between the ECI and RIC frames, assume we are given the position
and velocity vectors in the ECI frame. Then
r  X i Y j  Z k ,
(14)
r  Xi  Yj  Zk ,
(15)
ECI
h  r  ECI r ,
(16)
and
4

r
X
Y
Z
R   i  j k ,
r
r
r
r
C
(17)
h
r  ECI r

,
h
r  ECI r
(18)
I C R.
(19)
Or in matrix notation
Rx
R 

 
 I  = Ix
C x
C 
Ry
Iy
Cy
Rz 

Iz 
C z 
Rx

= Ix
C x
Ry
Iy
Cy
Rz 

Iz ,
C z 
i 
 .
 j
k 
(20)
Define
ECI
 RIC
where the elements of
ECI
 RIC
are the direction cosines between the ECI and RIC frames
and are given by Eqns (17), (18), and (19).
Hence, the coordinates in one frame can be computed from coordinate in the other, i.e.,
R
I  =
 
C 
ECI
 RIC
X 
Y 
 
 Z 
and
X 
Y  =
 
 Z 
ECI
  RIC
R
T 
 I .
C 
(21)
If we wish to examine the difference between two orbits in the R, I and C directions, we
do the following. Designate one orbit as the reference orbit. Call the position and velocity
in this orbit r  and
ECI
unit vectors R , I , C and
r  . Then use equation (17), (18), (19) and (20) to compute the
ECI
 RIC
using r  and
ECI
r  . Next compute position and velocity
difference in the ECI frame,
X 
 Y   r   r  r  ,
ECI
ECI
 
 Z 
X 
 

 ECI 
 Y   r ECI  r  r
 Z 
 


ECI
.
(22)
Then use equation (21) to compute differences in the radial, in-track and cross-track
directions in position and velocity i.e.,
5
 R 
X 
 I    ECI  Y  ,
RIC 
 

C 
 Z 
 R 
X 
 
ECI 

 I    RIC  Y  .
C 
 Z 
 
 
(23)
6