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the x-axis of F through an angle a, a rotation S
2
about the y-axis of S
1
(F) through an
angle b, and finally a rotation about the z-axis of S
2
(S
1
(F)) through an angle t.
Proof.
We shall again only consider the case where F is the standard frame and
p
is the origin. If R
1
, R
2
, and R
3
are the rotations about the standard coordinate axes
defined in Theorem 2.5.1.1, then
SR
SSRS
=
=
,
1
1
-
1
,
2
1
2
1
-
1
(
)
SSSRSS
=
,
3
2
1
3
2
1
so that S
3
S
2
S
1
= R
1
R
2
R
3
. Note that this composition is in the opposite order of the
composition of the maps in Theorem 2.5.1.1, but the matrix for S
3
S
2
S
1
will be similar
to the one shown in (2.37). Therefore we can set this matrix equal to the matrix for
R and solve for the angles just like in Theorem 2.5.1.1.
Definition.
Given a rotation R, the angles a, b, and t in Theorem 2.5.1.2 are called
the
X-Y-Z Euler angles of R
. The tuple [a,b,t,
p
] is called an
Euler angle representation
of the rotation R
(with respect to the frame F). [a,b,t] will denote the Euler angle
representation in the case where
p
is the origin.
The term Euler angles is also used in the case of any other choice of axes. For
example, if one were to rotate about the z-, y-, and z-axis, then one would get the
Z-Y-Z Euler angles
for a rotation, and so on. We shall only look at Euler angles in
the X-Y-Z case. The others are similar. The proofs of Theorems 2.5.1.1 and 2.5.1.2
show us how to compute the roll, pitch, yaw or Euler angles of a rotation. We
shall not pursue the subject here. These angles are often used to describe motions in
robotics.
Let us return to the main subject matter of this section, which is how to derive
equations for rigid motions. Since translations are trivial, we now work through an
example to show how one typically computes equations for geometrically defined
rotations. The idea is to express an arbitrary rotation in terms of rotations about the
x-, y-, and z-axis.
2.5.1.3. Example.
To show that any nonzero vector
v
can be rotated into one of the
coordinate axes by a composition of two rotations about coordinate axes.
Solution.
We sketch the construction in case we want to rotate
v
into the z-axis.
Rotating into the x- or y-axis would be done in a similar way. Let
w
by the orthogo-
nal projection of
v
onto the y-z plane. See Figure 2.28(a). A rotation R
1
about the x-
axis through an angle a, where a is the angle that
w
makes with the z-axis (or
e
3
),
will move
v
into a vector
v
¢ in the x-z plane. See Figure 2.28(b). A second rotation R
2
about the y-axis through the angle b, where b is the angle that
v
¢ makes with the z-
axis, will rotate
v
¢ into the z-axis. The composition R
2
R
1
then does what we want,
namely, move
v
into the z-axis.
2.5.1.4. Example.
To find the rotation R that rotates the plane
X
defined by
