Coordinate Transformations
ASEN 3200
2/22/04
George H. Born
In engineering it is often necessary to express vectors in different coordinate frames.
This requires the rotation matrix, which relates coordinates and basis (unit) vectors in one frame
to those in another frame.
Consider for example two frames with a common origin but separated by a positive
rotation* about their common Z-axis through the positive angle θ.
Z z
K̂ kˆ
ĵ
Ĵ
Î
A
Y
î
X
y
θ
x
B
Figure 1 - Definitions of the A and B frames and a positive rotation, θ, about the Z-axis
We wish to express a vector given in the A frame coordinates in terms of B frame
coordinates, i.e.,
 r B  T B  r A .
A
(1)
A simple way to do this is to express the unit vectors of the B frame in terms of those in the A
frame. Since we are rotating about the Z-axis let's look at a view down the Z-axis toward the
origin:
*
a positive rotation means that it obeys the right hand rule, i.e., place the thumb of the right hand in the positive
direction of the rotation axis and the fingers will indicate the positive direction of rotation.
1
ĵ
K̂, kˆ
θ
Ĵ
î
Î
θ
In order to do this we first write the projection of the Î and Ĵ vectors (or X and Y coordinates of a
vector) on the î and ĵ vectors (or x and y axes). Note, for example, that the projection of Î on î is
obtained by drawing a line from Î perpendicular to î. Hence,
iˆ  cos Iˆ  sin  Jˆ
ˆj   sin  Iˆ  cos Jˆ
(2)
kˆ  Kˆ .
In matrix form:
 iˆ   cos
 ˆ 
 j     sin 
 kˆ   0
  
sin 
cos
0
0   Iˆ 
 
0   Jˆ  .
1   Kˆ 
 
(3)
To transform a vector we replace the unit vectors with the coordinates of that vector, i.e., the
ˆ K
ˆ frame, r  XIˆ  YJˆ  ZKˆ , transforms to the ˆi, ˆj, kˆ frame as
coordinates of a vector in the ˆI, J,
 x   cos
 y     sin 
  
 z   0
2
sin 
cos
0
0  X 
0  Y  .
1   Z 
(4)
Similarly a transformation involving a positive rotation through the angle β about the Yaxis would be as follows (from a view down the Y-axis):
K̂
β
k̂
β
Jˆ , ˆj
β
Î
î
iˆ  cos  Iˆ  sin  Kˆ
ˆj  Jˆ
kˆ  sin  Iˆ  cos  Kˆ
or
0  sin    Iˆ 
 
1
0   Jˆ  .
0 cos    Kˆ 
 
 iˆ  cos 
 ˆ 
 j   0
 kˆ   sin 
  
(5)
Finally, a rotation about the X-axis through the positive angle α is given by (draw in the
projections for yourself)
k̂
α
K̂
ĵ
α
Î, î
Ĵ
 iˆ  1
0
 ˆ 
 j   0 cos 
 kˆ  0  sin 
  
0   Iˆ 
 
sin    Jˆ  .
cos    Kˆ 
 
(6)
If we wish to transform from the B frame to the A frame we could simply write (using Eq. 3)
3
 Iˆ   cos
  
 Jˆ     sin 
 ˆ  0
 K  
sin 
cos
0
0
0 
1 
1
 iˆ 
 ˆ
 j .
 kˆ 
 
(7)
But transformation matrices are orthogonal so that their inverse is equal to their transpose; hence,
 Iˆ  cos
  
 Jˆ    sin 
 ˆ  0
 K  
 sin 
cos
0
0
0 
1 
1
 iˆ 
 ˆ
 j .
 kˆ 
 
(8)
If the transformation from one frame to another requires several rotations, the final
rotation matrix is the product of the individual rotation (transformation) matrices.
Examples
ˆ W
ˆ K
ˆ to P,
ˆ Q,
ˆ
Transformation from ˆI, J,
ˆ K
ˆ to the PQW frame with
The transformation from the ECI frame with unit vectors ˆI, J,
ˆ W
ˆ Q,
ˆ is described below. Note that P̂ is in the orbit plane in the direction of
unit vectors P,
perigee, Q̂ is (in the orbit plane) perpendicular to P̂ and in the direction of motion and Ŵ lie
along the angular momentum vector. If this were a retrograde orbit Q̂ and Ŵ would be in the
opposite direction to that depicted here.
K̂
Ŵ
orbit
Q̂
i
ω
Î
P̂
i
Ω
line of
nodes
4
Ĵ
ˆ W
ˆ K
ˆ frame to P,
ˆ Q,
ˆ involves three rotations:
The transformation from the ˆI, J,
1. A rotation about the K̂ axis through the RAAN (Ω) to align the X-axis of this
intermediate frame (which we will call B ) with the line of nodes.
2. A rotation about the B frame X-axis through the inclination, i, to place the X and
Y axis of this intermediate frame (called C ) in the orbit plane. Note that the Zaxis of the C frame is along the angular momentum vector and will require no
further transformation.
3. Finally a rotation about the C frame Z-axis to align its X-axis with the P̂ vector,
and Y-axis with the Q̂ vector.
Hence, the transformation equation given by
 r PQW
ˆ ˆ ˆ  T PQW
ˆ ˆ ˆ T C T B  r  A
C
B
A
 T PQW
ˆ ˆ ˆ  r A
A
(9)
where T B is a rotation about Z through Ω. From Eq. (3)
A
T B
A
 cos  sin  0 
   sin  cos  0 .
 0
0
1 
(10)
The second rotation is about the X-axis through i and is given by Eq. (6)
T C
B
0
0 
1

 0 cos i sin i  .
0  sin i cos i 
(11)
Finally we have another rotation about the Z-axis through the argument of perigee, ω, and from
Eq. (3)
T 
C
ˆˆ ˆ
PQW
 cos  sin  0
   sin  cos  0 .
 0
0
1 
(12)
Multiplying these three matrices in the order given by Eq. (9) yields the final result which is the
transpose of the matrix given by Eq. (2.6-14) in Bate, et. al. You should carry out this
multiplication for yourself.
5
ˆ E,
ˆ Z
ˆ to ˆI, J,
ˆ K
ˆ
Transformation from S,
ˆ E,
ˆ Z
ˆ frame to the ˆI, J,
ˆ K
ˆ (ECI) frame is
The transformation from the topocentric S,
developed next.
K̂
s/c
Ẑ 
R

Ê
Ŝ
Ĵ
Î
ˆ E,
ˆ Z
ˆ has its origin at a tracking station and Ŝ directed south, Ê directed east and Ẑ
The S,
directed up along the local vertical. Note that Ẑ is perpendicular to a tangent to the oblate earth
at the location of the tracking station. Hence, the angle between Ẑ and the equatorial plane is
the local geodetic latitude,  . Also, because of the oblateness of the earth the Ẑ axis is not
parallel with the station position vector, R . Because in all mathematical operational vectors may
ˆ E,
ˆ Z
ˆ frame to the origin of the
be treated as free vectors we may move the origin of the S,
ˆI, J,
ˆ K
ˆ frame in the center of the Earth. This will aid our visualization of the necessary
transformations.
K̂
Ẑ

Ê

A
Ŝ
Î
θ
6
Ĵ
We wish to determine
  A  T A   SEZ
(13)
  SEZ  S Sˆ   E Eˆ  Z Zˆ .
(14)
SEZ
where
ˆ E,
ˆ Z
ˆ frame to the ˆI, J,
ˆ K
ˆ frame involves two rotations:
The transformation from the S,
1. The first rotation is about the E (or Y) axis through the angle 360° - (90° - φ) =
270° + φ (remember we must rotate in the positive direction). This aligns the Ẑ
ˆ K
ˆ frame and will be called the D frame.
axis with the K̂ axis of the ˆI, J,
2. The second rotation is a positive rotation about the new Ẑ axis through the angle
360° - θ = -θ to align the Ŝ and Ê axis with the Î and Ĵ (X and Y) axis of the A
or ECI frame. Here theta is the angle between the X-axis and the meridian of the
tracking station and is called the Greenwich sidereal time (Bate, et al.).
Hence, the transformation equation is given by
  A  T A T D   SEZ
D
where T D
SEZ
SEZ
(15)
is a rotation about Y through the angle 270° +  and is given by Eq. (5)
T D
SEZ
cos  270    0  sin  270    



0
1
0

 sin  270    0 cos  270    
 sin  0 cos  
  0
1
0  .
  cos  0 sin  
7
(16)
The second rotation is about the Z-axis through the angle -θ, and is given by Eq. (3)
T A
D
The final result, T  A T D
D
SEZ
 cos    sin    0 


   sin    cos    0 

0
0
1 
cos  sin  0 
  sin  cos 0  .
 0
0
1 
(17)
is given by
T SEZ
A
sin  cos
  sin  sin 
  cos 
 sin 
cos
0
cos  cos 
cos  sin  
sin  
which agrees with Eq. (2.6-12) of Bate, et al.
Reference: Bate, R., D. Mueller and J. White, Fundamentals of Astrodynamics, Dover, 1971.
8
(18)