Coordinate system transformation

Chapter 3

Coordinate System
Transformations

3.1 Problem Statement
We have met the various coordinate systems that will be of primary interest
in the study of flight dynamics. Now we address the subject of how these
coordinate systems are related to one another. For instance, when we begin
to sum the external forces acting on the aircraft, we will have to relate all
these forces to a common reference frame. If we take some body-axis system
to be the common frame, then we need to be able to take the gravity vector
(weight) from the local horizontal reference frame, the thrust from some other
body-axis frame, and the aerodynamic forces from the wind-axis frame, and
represent all these forces in the body-axis frame.
Some of these relationships are easy because they are fixed: Two body-
axis systems, once defined, are always related by a single rotation around
their common y axis. Others are much more complicated because they vary
with time: The orientation of a given body-axis system with respect to the
wind axes determines certain aerodynamic forces and moments which change
that orientation.
First we must characterize the relationship between two coordinate sys-
tems at some frozen instant in time. The instantaneous relationship between
two coordinate systems will be addressed by determining a transformation
that will take the representation of an arbitrary vector in one system and
convert it to its representation in the other.

23

24 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS

Three approaches to finding the needed transformations will be presented.
The first is called Direction Cosines, a sort of brute-force approach to the
problem. Next we will discuss Euler angles, by far the most common ap-
proach but one with a potentially serious problem. Finally we will examine
Euler parameters, an elegant solution to the problem found with Euler angles.

3.2 Transformations
3.2.1 Definitions
Consider two reference frames, F1 and F2 , and a vector v whose components
are known in F1 , represented as {v}1 :
 
 v x1
 

{v}1 = v
 y1
 

v z1
We wish to determine the representation of the same vector in F2 , or
{v}2 :
 
 v x2
 

{v}2 = v
 y2
 

v z2
These are linear spaces, so the transformation of the vector from F1 to
F2 is simply a matrix multiplication which we will denote T2,1 , such that
{v}2 = T2,1 {v}1 . Transformations such as T2,1 are called similarity trans-
formations. The transformations involved in simple rotations of orthogonal
reference frames have many special properties that will be shown.
The order of subscripts of T2,1 is such that the left subscript goes with the
system of the vector on the left side of the equation and the right subscript
with the vector on the right. For the matrix multiplication to be conformal
T2,1 must be a 3x3 matrix:
 
t11 t12 t13
 
T2,1 =  t21 t22 t23 
t31 t32 t33

3.2. TRANSFORMATIONS 25

x1 x2

y1
z2 v

y2
z1

Figure 3.1: Two Coordinate Systems

(The tij probably need more subscripting to distinguish them from those
in other transformation matrices, but this gets cumbersome.)
The whole point of the three approaches to be presented is to figure out
how to evaluate the numbers tij . It is important to note that for a fixed
orientation between two coordinate systems, the numbers tij are the same
quantities no matter how they are evaluated.

3.2.2 Direction Cosines

Derivation

Consider our two reference frames F1 and F2 , and the vector v, shown in
figure 3.1. We claim to know the representation of v in F1 . The vector v
is the vector sum of the three components vx1 i1 , vy1 j1 , and vz1 k1 so we may
replace the vector by those three components, shown in figure 3.2
Now, the projection of v onto x2 is the same as the vector sum of the
projections of each of its components vx1 i1 , vy1 j1 , and vz1 k1 onto x2 , and
similarly for y2 and z2 . Define the angle generated in going from x2 to x1 as
θx2 x1 . The projection of vx1 i1 onto x2 therefore has magnitude vx1 cos θx2 x1
and direction i2 , as shown if figure 3.3.
To the vector vx1 cos θx2 x1 i2 must be added the other two projections,
vy1 cos θx2 y1 i2 and vz1 cos θx2 z1 i2 . Similar projections onto y2 and z2 result in:


x1 x2

y1
z2 v x 1i 1

v y 1j1 y2
z1

v z 1k1

Figure 3.2: Vector in Component Form

x1 x2

v x 1i 1

v x 1 cos θ x 2x 1 i 2
θ x 2x 1

Figure 3.3: One Component of the Vector


vx2 = vx1 cos θx2 x1 +vy1 cos θx2 y1 +vz1 cos θx2 z1
vy2 = vx1 cos θy2 x1 +vy1 cos θy2 y1 +vz1 cos θy2 z1
vz2 = vx1 cos θz2 x1 +vy1 cos θz2 y1 +vz1 cos θz2 z1
In vector-matrix notation, this may be written

    
 v x2
 
 
cos θx2 x1 cos θx2 y1 cos θx2 z1  vx1 

 
v =  cos θy2 x1 cos θy2 y1 cos θy2 z1  vy1
 y2
 
 
 

v z2 cos θz2 x1 cos θz2 y1 cos θz2 z1 v z1
Clearly this is {v}2 = T2,1 {v}1 , so we must have tij = cos θ(axis)2 (axis)1 in
which i or j is 1 if the corresponding (axis) is x, 2 if (axis) is y, and 3 if
(axis) is z.

 
cos θx2 x1 cos θx2 y1 cos θx2 z1
 
T2,1 =  cos θy2 x1 cos θy2 y1 cos θy2 z1  (3.1)
cos θz2 x1 cos θz2 y1 cos θz2 z1

Properties of the Direction Cosine Matrix
The transformation matrix (often called the direction cosine matrix, regard-
less of how it is derived or represented) does not depend on the vector v for
its existence. It is therefore the same no matter what we choose for v. If we
take

 
 1 
 
{v}1 = 0
 
 
0
Then {v}2 is just the ﬁrst column of the direction cosine matrix:

 
 cos θx2 x1
 

{v}2 = cos θy2 x1

 

cos θz2 x1


Since the length of v is 1, we have shown the well-known property of
direction cosines, that

cos2 θx2 x1 + cos2 θy2 x1 + cos2 θz2 x1 = 1

The same obviously holds true for any column of the direction cosine
matrix.
If we need to go the other way, {v}1 = T1,2 {v}2 , it is clear that T1,2 =
−1
T2,1 , and we might be tempted to invert the direction cosine matrix. On the
other hand, had we begun by assuming {v}2 was known, and ﬁguring out
how to get {v}1 = T1,2 {v}2 using similar arguments to the above, we would
have arrived at

    
 v x1
 
 cos θx1 x2 cos θx1 y2 cos θx1 z2  vx2
 

 
v =  cos θy1 x2 cos θy1 y2 cos θy1 z2  vy2
 y1
 
 
 

v z1 cos θz1 x2 cos θz1 y2 cos θz1 z2 v z2

Here we observe that cos θx1 x2 is the same as cos θx2 x1 , cos θx1 y2 is the
same as cos θy2 x1 , etc., since (even if we had deﬁned positive rotations of
these angles) the cosine is an even function. So the same transformation is

    
 v x1
 
 cos θx2 x1 cos θy2 x1 cos θz2 x1  vx2
 

 
v =  cos θx2 y1 cos θy2 y1 cos θz2 y1  vy2
 y1
 
 
 

v z1 cos θx2 z1 cos θy2 z1 cos θz2 z1 v z2

Obviously the columns of T1,2 are the rows of T2,1 (and have unit length
as well). This leads us to another nice property of these transformation ma-
trices, that the inverse of the direction cosine matrix is equal to its transpose,

−1 T
T1,2 = T2,1 = T2,1
−1
Since T2,1 T2,1 = T2,1 T2,1 = I3 , the 3 × 3 identity matrix, it must be true
T

that the scalar (dot) product of any row of T2,1 with any other row must
−1 T
be zero. It is easy to show that T1,2 T1,2 = T1,2 T1,2 = I3 , so the scalar (dot)
product of any column of T2,1 with any other column must be zero since the


columns of T2,1 are the rows of T1,2 . When the rows (or columns) of the
direction cosine matrix are viewed as vectors, this means the rows (or the
columns) form orthogonal bases for 3-dimensional space.
T
Also, with T2,1 T2,1 = I3 , if we take the determinant of each side and note
that T2,1 = |T2,1 |, we have
T

T2,1 T2,1 = |T2,1 | T2,1
T T

= |T2,1 | |T2,1 |
= |T2,1 |2 = 1

The only conclusion we can reach at this point is that |T2,1 | = ±1. Be-
cause the identity matrix is a transformation matrix with determinant +1,
and since all other transformations may be reached through continuous rota-
tions from the identity transformation, it seems unreasonable to think that
the sign of the determinant would be diﬀerent for some rotations and not
others. We will show in our discussion of Euler angles that the right answer
is indeed |T2,1 | = +1.
While we have 9 variables in T2,1 = {tij } , i, j = 1 . . . 3, there are 6
nonlinear constraining equations based on the orthogonality of the rows (or
columns) of the matrix. For the rows these are:

t2 + t2 + t2 = 1
11 12 13
t21 + t22 + t2 = 1
2 2
23
t31 + t32 + t2 = 1
2 2
33
t11 t21 +t12 t22 +t13 t23 = 0
t11 t31 +t12 t32 +t13 t33 = 0
t21 t31 +t22 t32 +t23 t33 = 0

The result of these constraints is that there are only 3 independent vari-
ables. In principle all nine may be derived given any three that do not violate
a constraint.

3.2.3 Euler angles
If there are only three independent variables in the direction cosine matrix,
then we should be able to express each of the tij in terms of some set of three


x1
θz x′

y1
θz

y′

z1 = z′

Figure 3.4: Rotation Through θz

(not necesarily unique) independent variables. One means of determining
these variables is by use of a famous theorem due to the Swiss mathematician
Leonhard Euler (1707-83). Briefly, this theorem holds that any arbitrarily
oriented reference frame may be placed in alignment with (made to have axes
parallel to) any other reference frame by three successive rotations about the
axes of the reference frame. The order of selection of axes in these rotations
is arbitrary, but the same axis may not be used twice in succesion. The
rotation sequences are usually denoted by three numbers, 1 for x, 2 for y,
and 3 for z. The twelve valid sequences are 123, 121, 131, 132, 213, 212,
231, 232, 312, 313, 321, and 323. The angles through which these rotations
are performed (defined as positive according to the right hand rule for right
handed coordinate systems) are called generically Euler angles.
The rotation sequence most often used in flight dynamics is the 321, or
z − y − x. Considering a rotation from F1 to F2 the first rotation (figure 3.4)
is about z1 through an angle θz which is positive according to the right hand
rule about the z1 axis. With two rotations to go, the resulting alignment in
general is oriented with neither F1 or F2 , but some intermediate reference
frame (the first of two) denoted F . Since the rotation was about z1 , z is
parallel to it but neither of the other two primed axes is.
The next rotation (figure 3.5) is through an angle θy about the axis y
of the first intermediate reference frame to the second intermediate reference
frame, F . Note that y = y , and neither y or z are necessarily axes of
either F1 or F2 .


x″

θy x′

θy y″ = y′
z″
z′

Figure 3.5: Rotation Through θy

The final rotation (figure 3.6) is about x through angle θx and the final
alignment is parallel to the axes of F2 .

Now assuming we know the angles θx , θy , and θz we need to relate them
to the elements of the direction cosine matrix T2,1 . We will do this by seeing
how the arbitrary vector v is represented in each of the intermediate and final
reference frames in terms of its representation in the prior reference frame.

Consider first the rotation about z1 (Figure 3.7). In terms of the direc-
tion cosines previously defined, the angles between the axes are as follows:
between z1 and z it is zero; between either z and any x or y it is 90 deg;
between x1 and x or y1 and y it is θz ; between x1 and y it is 90 deg +θz ;
and between y1 and x it is 90 deg −θz . We therefore may write


x″ = x 2

θx y″
z2 θx z″
y2

Figure 3.6: Rotation Through θx

x1 x′
θz

v x ′i′

vx i1 v

v y 1j 1 y1
θz
v y ′j′

y′

Figure 3.7: Rotation from F1 to F


    
 vx
 
 cos θx x1 cos θx y1 cos θx z1  vx1 
 
 
v = cos θy x1 cos θy y1 cos θy z1  vy1
 y
 
 
 

vz cos θz x1 cos θz y1 cos θz z1 v z1
  
cos θz cos (90 deg −θz ) cos 90 deg  vx1
 

 
= cos (90 deg +θz ) cos θz cos 90 deg  vy1

 

cos 90 deg cos 90 deg cos 0 v z1
  
cos θz sin θz 0  vx1 
 
 
= − sin θz cos θz 0  vy1

 

0 0 1 v z1

In short, {v} = TF ,1 {v}1 in which

 
cos θz sin θz 0
 
TF ,1 =  − sin θz cos θz 0  (3.2)
0 0 1

From the rotation about y we get {v} = TF ,F {v} in which

 
cos θy 0 − sin θy
 
TF ,F = 0 1 0  (3.3)
sin θy 0 cos θy

From the rotation about x , ﬁnally, {v}2 = T2,F {v} with

 
1 0 0
 
T2,F =  0 cos θx sin θx  (3.4)
0 − sin θx cos θx

We now cascade the relationships: {v}2 = T2,F {v} , {v} = TF ,F {v}
to arrive ﬁrst at {v}2 = T2,F TF ,F {v} and then at {v}2 = T2,F TF ,F TF ,1 {v}1 ,
or


   
1 0 0 cos θy 0 − sin θy cos θz sin θz 0
   
{v}2 =  0 cos θx sin θx   0 1 0   − sin θz cos θz 0  {v}1
0 − sin θx cos θx sin θy 0 cos θy 0 0 1

The transformation matrix we seek is the product of the three sequential
transformations (in the correct order!), or T2,1 = T2,F TF ,F TF ,1 . The details
are slightly tedious, but the result is

 
cos θy cos θz cos θy sin θz − sin θy
 
 sin θx sin θy cos θz sin θx sin θy sin θz 
 sin θx cos θy 
T2,1 =
 − cos θx sin θz + cos θx cos θz 

 
 cos θx sin θy cos θz cos θx sin θy sin θz 
cos θx cos θy
+ sin θx sin θz − sin θx cos θz

Since this is just a different way of representing the direction cosine ma-
trix, it must be true that cos θy cos θz = cos θx2 x1 , cos θy sin θz = cos θx2 y1 ,
etc. At this point we may observe that since T2,1 = T2,F TF ,F TF ,1 , we must
have

|T2,1 | = |T2,F TF ,F TF ,1 | = |T2,F | |TF ,F | |TF ,1 |

The determinant of each of the three intermediate transformations is eas-
ily verified to be +1, so that we have the expected result,

|T2,1 | = +1

It is very important to note that the Euler angles θx , θy , and θz have been
defined for a rotation from F1 to F2 using a 321 rotation sequence. Thus, a
321 rotation from F2 to F1 (T1,2 ) with suitably defined angles (say, φx , φy ,
and φz ) would have the same form as given for T2,1 , but the angles involved
would be physically different from those in T2,1 . So, for the rotation from F1
to F2 using a 321 rotation sequence:


 
cos φy cos φz cos φy sin φz − sin φy
 
 sin φx sin φy cos φz sin φx sin φy sin φz 
 sin φx cos φy 
T1,2 =
 − cos φx sin φz + cos φx cos φz 

 
 cos φx sin φy cos φz cos φx sin φy sin φz 
cos φx cos φy
+ sin φx sin φz − sin φx cos φz

−1 T
Alternatively we may note that T1,2 = T2,1 = T2,1 and may write the
matrix

 
sin θx sin θy cos θz cos θx sin θy cos θz
 cos θy cos θz 
 − cos θx sin θz + sin θx sin θz 
 
T1,2 =
 cos θ sin θ sin θx sin θy sin θz cos θx sin θy sin θz 

 y z 
 + cos θx cos θz − sin θx cos θz 
− sin θy sin θx cos θy cos θx cos θy

The two matrices are the same, only the definitions of the angles are
different. Clearly the relationships among the two sets of angles are non-
trivial.
In short, the definitions of Euler angles are unique to the rotation sequence
used and the decision as to which frame one goes from and which one goes
to in that sequence. We will normally define our Euler angles going in only
one direction using a 321 rotation sequence, and rely on relationships like
−1 T
T1,2 = T2,1 = T2,1 to obtain the other.

3.2.4 Euler parameters
The derivation of Euler parameters is at appendix A. They are based on the
observation that any two coordinate systems are instantaneously related by
a single rotation about some axis that has the same representation in each
system. The axis, called the eigenaxis, has direction cosines ξ, ζ, and χ; the
angle of rotation is η. Then, with the definition of the Euler parameters


.
q0 =cos (η/2)
.
q1 =ξ sin (η/2)
. (3.5)
q2 =ζ sin (η/2)
.
q3 =χ sin (η/2)

the transformation matrix becomes:

 
(q0 + q1 − q2 − q3 )
2 2 2 2
2 (q1 q2 + q0 q3 ) 2 (q1 q3 − q0 q2 )
 
T2,1 =  2 (q1 q2 − q0 q3 ) (q0 − q1 + q2 − q3 )
2 2 2 2
2 (q2 q3 + q0 q1 ) 
2 (q1 q3 + q0 q2 ) 2 (q2 q3 − q0 q1 ) (q0 − q1 − q2 + q3 )
2 2 2 2

(3.6)
Euler parameters have one great disadvantage relative to Euler angles:
Euler angles may in most cases be easily visualized. If one is given values of
θx , θy , and θz it is not hard to visualize the relative orientation of two coordi-
nate systems. A given set of Euler parameters, however, conveys almost no
information about how the systems are related. Thus even in applications
in which Euler parameters are preferred (such as in ﬂight simulation), the
results are very often converted to Euler angles for ease of interpretation and
visualization.
The direct approach to convert a set of Euler parameters to the corre-
sponding set of Euler angles is to equate corresponding elements of the two
representations of the transformation matrix. We may combine the (2,3) and
(3,3) entries to obtain

sin θx cos θy 2 (q2 q3 + q0 q1 )
= tan θx = 2
cos θx cos θy q0 − q1 − q2 + q 3
2 2 2

t23 2 (q2 q3 + q0 q1 )
θx = tan−1 = tan−1 , −π ≤ θx < π
t33 2
q0 − q 1 − q 2 + q3
2 2 2

From the (1,3) entry we have

− sin θy = 2 (q1 q3 − q0 q2 )


θy = − sin−1 (t13 ) = − sin−1 (2q1 q3 − 2q0 q2 ) , −π/2 ≤ θy ≤ π/2

Finally we use the (1,1) and (1,2) entries to yield

cos θy sin θz 2 (q1 q2 + q0 q3 )
= tan θz = 2
cos θy cos θz q 0 + q 1 − q2 − q 3
2 2 2

t12 2 (q1 q2 + q0 q3 )
θz = tan−1 = tan−1 , 0 ≤ θz ≤ 2π
t11 2
q0 + q 1 − q2 − q3
2 2 2

With the arctangent functions one has to be careful to not divide by
zero when evaluating the argument. Most software libraries have the two-
argument arctangent function which avoids this problem and helps keep track
of the quadrant. In summary,

θx = tan−1 2(q2 q3 +q0 q1 )
q0 −q1 −q2 +q3
2 2 2 2 , −π ≤ θx < π
θy = − sin−1 (2q1 q3 − 2q0 q2 ) , −π/2 ≤ θy ≤ π/2 (3.7)
θz = tan−1 2(q1 q2 +q0 q3 )
q0 +q1 −q2 −q3
2 2 2 2 , 0 ≤ θz ≤ 2π

Going the other way, from Euler angles to Euler parameters, has been
studied extensively. It may be done in a somewhat similar manner to equat-
ing elements of the transformation matrix. Another more elegant method
due to Junkins and Turner 1 yields a very nice result, here presented without
proof:

q0 = cos (θz /2) cos (θy /2) cos (θx /2) + sin (θz /2) sin (θy /2) sin (θx /2)
q1 = cos (θz /2) cos (θy /2) sin (θx /2) − sin (θz /2) sin (θy /2) cos (θx /2)
(3.8)
q2 = cos (θz /2) sin (θy /2) cos (θx /2) + sin (θz /2) cos (θy /2) sin (θx /2)
q3 = sin (θz /2) cos (θy /2) cos (θx /2) − cos (θz /2) sin (θy /2) sin (θx /2)
1
Junkins, J.L. and Turner, J.D., “Optional Continuous Torque Attitude Maneuvers,”
AIAA-AAS Astrodynamics Conference, Palo Alto, California, August 1978.


Just one final note on Euler parameters. Frequently in the literature
Euler parameters are referred to as quaternions. However, quaternions are
actually defined as half-scalar, half-vector entities in some coordinate system
˜
as Q = q0 + q1 i + q2 j + q3 k. The algebra of quaternions is useful for proving
theorems regarding Euler parameters, but will not be used in this course.

3.3 Transformations of Systems of Equations
Suppose we have a linear system of equations in some reference frame F1 :

{y}1 = A1 {x}1

We know the transformation to F2 (T2,1 ) and wish to represent the same
equations in that reference frame. Each side of {y}1 = A1 {x}1 is a vector in
F1 , so we transform both sides as T2,1 {y}1 = {y}2 = T2,1 A1 {x}1 . Then we
T
may insert the identity matrix in the form of T2,1 T2,1 in the right hand side
to yield {y}2 = T2,1 A1 T2,1 T2,1 {x}1 . The reason for doing this is to transform
T

the vector {x}1 to {x}2 = T2,1 {x}1 . Grouping terms we have the result,

{y}2 = T2,1 A1 T2,1 {x}2 = A2 {x}2
T

With the conclusion that the transformation of a matrix A1 from F1 to
F2 (or the equivalent operation performed by A1 in F2 ) is given by

T
A2 = T2,1 A1 T2,1 (3.9)

This is sometimes spoken of as the transformation of a matrix. Such
transformations may be very useful. For example, if we could find F2 and
T2,1 such that A2 is diagonal, then the system of equations {y}2 = A2 {x}2
is very easy to solve. The original variables can then be recovered by {y}1 =
T2,1 {y}2 , {x}1 = T2,1 {x}2 .
T T

3.4. CUSTOMS AND CONVENTIONS 39

3.4 Customs and Conventions
3.4.1 Names of Euler angles
Some transformation matrices occur frequently, and the 321 Euler angles
associated with them are given special symbols. These are summarized as
follows:

TF2 ,F1 θx θ y θz
TB,H φ θ ψ
(3.10)
TW,H µ γ χ
TB,W 0 α −β

3.4.2 Principal Values of Euler angles
The principal range of values of the Euler angles is ﬁxed largely by convention
and may vary according to the application. In this course the convention is

−π≤θx < π
−π/2≤θy ≤ π/2 (3.11)
0≤θz < 2π

These ranges are most frequently used in ﬂight dynamics, although occa-
sionaly one sees −π < θz ≤ π and 0 < θx ≤ 2π.

Coordinate system transformation

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Coordinate system transformation