ROTATION SEQUENCES AND EULER ANGLES
(c) 1993, 2009 M.Lampton
STELLAR SOFTWARE
The orientation of an object in three dimensions can be
described an a variety of ways. One such description as
follows: take an arbitrary point P that is not the origin, and
write its coordinates (x,y,z) in a frame of reference fixed in
the object. Also, write its coordinates (X,Y,Z) in a lab frame
of coordinates having the same center. The matrix M that
converts (x,y,z) into (X,Y,Z) is uniquely defined by the
orientation of the object's coordinate frame. Explicitly:
X | M11 M12 M13 | x
Y = | M21 M22 M23 | y
Z | M31 M32 M33 | z
This description is a set of nine numbers, namely the matrix
elements. They are not independent: the sums of the squares
along any column or row is exactly 1.0 because the length of
the (x,y,z) vector and the (X,Y,Z) vector have to come out
equal for any object vector. A matrix that satisfies these six
constraints is said to be unitary. Nine numbers and six
constraints means that there are just three degrees of freedom
for the three dimensional orientation.
Orientations can also be described using angles. For any
rotation, there exists some direction in space (two parameters)
and one angle around that axis that gives the rotation. In
Cartesian coordinates it is usually more convenient to break
this one eigenaxis rotation into three separate consecutive
rotations about the original or partly rotated axes.
How many such descriptions are there? The first motion can
be taken about the lab X, Y, or Z axes -- three possibilities.
Whichever is chosen, the second rotation cannot be around that
axis again, but now there are two new axes to choose from, plus
two unused of the original three axes. If the second angle is
one of the original three, it creates three new axes for a
total of eight, seven of which are possible third axes, giving
42 possible descriptions. If the second angle is one of the
two new axes produced by the first angle, it creates two new
axes for a total of seven, six of which are possible third
angle axes, giving 36 more descriptions. In all there are 78.
THE (X, y', z") ROTATION DESCRIPTION
In BEAM products, and in much of the optics industry, we use
the rotation-sequence described by successive rotations about
the X, y', and z" axes. This is just one way of doing
business, but it has the advantage that each coordinate appears
once so that if you have a simple motion about X, Y, or Z,
there is a one-angle description of it, and it is about the
obvious axis. Another advantage of (X,y',z") over say (X,Y,Z)
is that this is the way that actual mechanical goniometers
work: each goniometer cradle carries the base of the next
goniometer, so that the first motion is about a fixed lab
coordinate direction -- the second motion is about an axis that
depends on the first motion, and the third motion axis depends
on both the preceding motions. In all, 18 of the rotation
sequences are of the form (A, B', C") that can be created with
actual goniometers. Of these, (X, y', z") is the easiest to
remember.
Define these three angles in a positive right-handed sense:
Tilt "t" -- about lab +X axis
Pitch "p" -- about tilted +y axis
Roll "r" -- about tilted and pitched +z axis.
Abbreviate: ct = cos(t)
st = sin(t)
cp = cos(p)
sp = sin(p)
cr = cos(r)
sr = sin(r)
Then, any vector (xrot, yrot, zrot) seen in the frame of the
rotated coordinates can be converted to its lab frame
representation by the linear matrix operator M:
Xlab = | cp*cr -sr*cp sp | xrot
Ylab = | st*sp*cr+ct*sr ct*cr-sr*st*sp -cp*st | yrot
Zlab = | -sp*ct*cr+st*sr sp*sr*ct+st*cr ct*cp | zrot
This matrix M is unitary.
For the case of a flat surface whose orientation is defined by
the tilt-pitch-roll coordinates, its normal is the unit +z axis
in its local (rotated) frame. In lab coordinates this +z axis,
and hence the normal, is just the rightmost column of the
matrix M:
Xlab = sin(p)
Ylab = -cos(p)*sin(t)
Zlab = cos(p)*cos(t)
The inverse of any unitary matrix M is its transpose. The
conversion of a lab frame vector into that vector expressed in
the rotated coordinates is done with the transpose of M, called
M*:
xrot = | cp*cr st*sp*cr+ct*sr -sp*ct*cr+st*sr | Xlab
yrot = | -sr*cp ct*cr-sr*st*sp sp*sr*ct+st*cr | Ylab
zrot = | sp -cp*st ct*cp | Zlab
THE (X,Y,Z) ROTATION DESCRIPTION
Another of the many three-angle descriptions is the (X,Y,Z)
sequence: the first rotation is identical to tilt (right
handed about the lab X axis), the second rotation is about the
lab Y axis even though the optic is no longer aligned with the
lab frame, and the third rotation is about the lab Z axis,
ditto.
Call these angles ax, ay, and az, and abbreviate
cx = cos(ax)
sx = sin(ax)
cy = cos(ay)
sy = sin(ay)
cz = cos(az)
sz = sin(az)
Then, any point (xrot yrot zrot) in the frame of the rotated
coordinates can be converted to its lab frame representation by
the linear operator M(Z) * M(Y) * M(X) because successive
motions in the lab frame multiply onto the left hand side (the
lab side) of the building matrix. This product is:
Xlab = | cy*cz -cx*sz+sx*sy*cz cx*sy*cz+sx*sz | xrot
Ylab = | cy*sz cx*cz+sx*sy*sz -sx*cz+cx*sy*sz | yrot
Zlab = | -sy sx*cy cx*cy | zrot
There is a set of angles (ax, ay, az) that describes the same
orientation as the set of angles (tilt, pitch, roll). Of
course the angles are different, but the orientations are the
same. This fact gives us a way to convert angles in one scheme
to or from angles in another. We set up the rotation matrix
for each scheme. All nine of the matrix elements are equal
because the orientations are the same. So, solve either way
for the set of angles. This plan converts any rotation
description to any other.
Here's an example connecting (ax, ay, az) to (tilt, pitch,
roll). The third element is sin(pitch) in the tilt-pitch-roll
system. But at the same time the third element is also given by
cos(ax)*sin(ay)*cos(az)+sin(ax)*sin(az). So, we have a formula
for pitch, given ax ay az:
pitch = arcsin[ cos(ax)*sin(ay)*cos(az) + sin(ax)*sin(az) ]
Similarly, note that the ratio of the sixth matrix element to
the ninth matrix element is -tan(tilt). So:
-sin(ax)*cos(az) + cos(ax)*sin(ay)*sin(az)
tilt = -arctan[ ------------------------------------------ ]
cos(ax)*cos(ay)
Similarly, the ratio of the second matrix element to the first
matrix element is -tan(roll), and
-cos(ax)*sin(az) + sin(ax)*sin(ay)*cos(az)
roll = -arctan[ ------------------------------------------ ]
cos(ay)*cos(az)
By shopping for the simplest expression of variables you need
in one matrix, and using the equality of matrix elements, you
can write down any needed conversion formula.
This works in both directions. For example let's find the
angles ax ay and az that correspond to a given trio of tilt,
pitch, and roll. First, ax can be obtained by noting that
tan(ax) is the ratio of the eighth to ninth matrix element:
sin(p)*sin(r)*cos(t) + sin(t)*cos(r)
ax = arctan[ ------------------------------------ ]
cos(t)*cos(p)
The term sin(ay) appears by itself as the seventh matrix
element, so
ay = -arcsin[ -sin(p)*cos(t)*cos(r) + sin(t)*sin(r) ].
Finally, the tangent of az is the ratio of the fourth matrix
element of M to its first element:
sin(t)*sin(p)*cos(r) + cos(t)*sin(r)
az = arctan[ ------------------------------------ ]
cos(p)*cos(r)
THE (Y,X,Z) ROTATION DESCRIPTION
Yet another of the specifications is the (Y,X,Z) system. The
transformation M = R(Z) * R(X) * R(Y) works out to:
Xlab = | cy*cz-sx*sy*sz -cx*sz sy*cz+sx*cy*sz | xrot
Ylab = | cy*sz+sx*sy*cz cx*cz sy*sz-sx*cy*cz | yrot
Zlab = | -cx*sy sx cx*cy | zrot
Again, useful formulas going either way can be found by
identifying corresponding parts of this matrix and the (t,p,r)
matrix that performs the same transformation, from rotated to
lab coordinates.
THE (Z, x', z") ROTATION DESCRIPTION
This is Euler's original set of angles and is widely used in
dynamics although not much used in optics. Adopt the angle
names a = roll about Z, i = pitch about x', and p = roll about z".
Adopt the abbreviations
ca = cos(a)
sa = sin(a)
ci = cos(i)
si = sin(i)
cp = cos(p)
sp = sin(p)
Then the transformation M = R(Z) * R(x') * R(z") works out to:
Xlab = | ca*cp-sa*ci*sp -ca*sp-sa*ci*cp sa*si | xrot
Ylab = | sa*cp+ca*ci*sp ca*ci*cp-sa*sp -ca*si | yrot
Zlab = | si*sp si*cp ci | zrot
