According to Euler's rotation theorem, any rotation may be described using three angles.
If the rotations are written in terms of rotation
matrices ,
, and
, then a general rotation
can be written as
(1)
|
The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles, depending on the axes about which the rotations
are carried out. Write the matrix as
(2)
|
The so-called "-convention,"
illustrated above, is the most common definition. In this convention, the rotation
given by Euler angles
,
where
1. the first rotation is by an angle about the z-axis using
,
2. the second rotation is by an angle about the former x-axis
(now
)
using
,
and
3. the third rotation is by an angle about the former z-axis
(now
)
using
.
Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use , Tuma (1974) says
is used in aeronautical engineering in the analysis
of space vehicles (but claims that
is used in the analysis of gyroscopic motion),
while Bate et al. (1971) use
. Goldstein remarks that continental authors
usually use
,
and warns that left-handed coordinate systems are also in occasional use (Osgood
1937, Margenau and Murphy 1956-64). Varshalovich (1988, pp. 21-23) uses the
notation
or
to denote the
Euler angles, and gives three different angle conventions, none of which corresponds
to the
-convention.
Here, the notation
is used, a convention that could be used in versions of the Wolfram
Language prior to 6 as RotationMatrix3D[phi, theta, psi]
(which could be run after loading Geometry`Rotations`) and RotateShape[g,
phi, theta, psi] (which could be run after loading Geometry`Shapes`).
In the
-convention,
the component rotations are then given by
(3)
| |||
(4)
| |||
(5)
|
so
(6)
| |||
(7)
| |||
(8)
| |||
(9)
| |||
(10)
| |||
(11)
| |||
(12)
| |||
(13)
| |||
(14)
|
To obtain the components of the angular velocity in the body axes, note that for
a matrix
(15)
|
it is true that
(16)
| |||
(17)
|
Now,
corresponds to rotation about the
axis, so look at the
component of
,
(18)
| |||
(19)
|
The line of nodes corresponds to a rotation by about the
-axis, so look at the
component of
,
(20)
| |||
(21)
| |||
(22)
|
Similarly, to find rotation by about the remaining axis, look at the
component of
,
(23)
| |||
(24)
| |||
(25)
|
Combining the pieces gives
(26)
|
For more details, see Goldstein (1980, p. 176) and Landau and Lifschitz (1976, p. 111).
The -convention
Euler angles are given in terms of the Cayley-Klein
parameters by
(27)
| |||
(28)
| |||
(29)
|
In the "-convention,"
(30)
| |||
(31)
|
Therefore,
(32)
| |||
(33)
| |||
(34)
| |||
(35)
|
giving rotation matrices
(36)
| |||
(37)
| |||
(38)
|
and
is given by
(39)
| |||
(40)
| |||
(41)
| |||
(42)
| |||
(43)
| |||
(44)
| |||
(45)
| |||
(46)
| |||
(47)
|
In the "
(pitch-roll-yaw) convention,"
is pitch,
is roll, and
is yaw.
(48)
| |||
(49)
| |||
(50)
|
and
is given by
(51)
| |||
(52)
| |||
(53)
| |||
(54)
| |||
(55)
| |||
(56)
| |||
(57)
| |||
(58)
| |||
(59)
|
A set of parameters sometimes used instead of angles are the Euler parameters ,
,
and
, defined by
(60)
| |||
(61)
|
Using Euler parameters (which are quaternions), an arbitrary rotation matrix can be described by
(62)
| |||
(63)
| |||
(64)
| |||
(65)
| |||
(66)
| |||
(67)
| |||
(68)
| |||
(69)
| |||
(70)
|
(Goldstein 1980, p. 153).
If the coordinates of two sets of points
and
are known, one rotated with respect to the other, then
the Euler rotation matrix can be obtained in a straightforward manner using least
squares fitting. Write the points as arrays of vectors, so
(71)
|
Writing the arrays of vectors as matrices gives
(72)
|
(73)
|
and solving for
gives
(74)
|
However, we want the angles ,
, and
, not their combinations contained in the matrix
. Therefore, write the
matrix
(75)
|
as a vector
(76)
|
Now set up the matrices
(77)
|
Using nonlinear least squares fitting then gives solutions which converge to .