Graphics Reference
In-Depth Information
Fig. 9.6
Three types of gimbal joints
lock, on the other hand, is a weakness associated with Euler rotations when certain
combinations of angles are used.
To understand this phenomenon, consider a simple gimbal which is a pivoted
support that permits rotation about an axis, as shown in Fig.
9.6
(a). If two gimbals
are combined, as shown in Fig.
9.6
(b), the inner cradle remains level with some ref-
erence plane as the assembly rolls and pitches. Such a combination has two degrees
of rotational freedom. By adding a third gimbal so that the entire structure is free to
rotate about a vertical axis, an extra degree of rotational freedom is introduced and
is often used for mounting a camera on a tripod, as shown in Fig.
9.6
(c).
A mechanical gimbal joint with three degrees of freedom is represented math-
ematically by a composite Euler rotation transform. For example, say we choose
R
90
°
,y
R
90
°
,x
R
90
°
,z
to rotate our unit cube as shown in Fig.
9.7
(a). The cube's
faces containing vertices 1, 5, 7, 3 and 0, 2, 6, 4 are first rotated about the perpen-
dicular
z
-axis, as shown in Fig.
9.7
(b). The second transform rotates the cube's
faces containing vertices 0, 4, 5, 1 and 2, 3, 7, 6 about the perpendicular
x
-axis,
as shown in Fig.
9.7
(c). If we now attempt to rotate the cube about the
y
-axis,
asshowninFig.
9.7
(d), the cube's faces containing 0, 2, 6, 4 and 1, 5, 7, 3 are
rotated again. Effectively we have lost the ability to rotate a cube about one of its
axes, and such a condition is called
gimbal lock
. There is little we can do about this,
apart from use another composite transform, but it, too, will have a similar restric-
tion. For example, Appendix A shows that
R
90
°
,x
R
90
°
,z
R
90
°
,y
,
R
90
°
,y
R
90
°
,z
R
90
°
,x
,
R
90
°
,z
R
90
°
,x
R
90
°
,y
and
R
90
°
,z
R
90
°
,y
R
90
°
,x
all possess a similar affliction. Fortu-
nately, there are other ways of rotating an object, which we will explore later.
