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Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
Читать онлайн.Название Kinematics of General Spatial Mechanical Systems
Год выпуска 0
isbn 9781119195764
Автор произведения M. Kemal Ozgoren
Жанр Математика
Издательство John Wiley & Sons Limited
(2.28)
Referring to Section for the discussion about the basic column matrix , it is to be noted that, just like
, the basic rotation matrix
is also an entity that is not associated with any reference frame. This is because
represents the rotation operator
in its own frame
, whatever
is. In other words,
(2.29)
By using Eqs., can be expressed in three equivalent ways as shown in the following equations.
(2.31)
(2.32)
Upon inserting the expressions of the basic column matrices into Eq. (2.30), the basic rotation matrices can be expressed element by element as shown below.
(2.33)
(2.34)
(2.35)
2.5 Successive Rotations
Suppose a vector is first rotated into a vector
and then
is rotated into another vector
. These two successive rotations can be described as indicated below.
(2.36)
On the other hand, according to Euler's theorem, the rotation of into
can also be achieved directly in one step. That is,
(2.37)
The following matrix equations can be written for the rotational steps described above as observed in a reference frame .
(2.38)
Equations (2.39) and (2.40) show that the overall rotation matrix is obtained as the following multiplicative combination of the intermediate rotation matrices
and
.
(2.41)
As a general notational feature, the rotation matrix between and
can be denoted by two alternative but equivalent symbols, which are shown below.
(2.42)
Although and
are mathematically equivalent, their verbal descriptions are not the same.
is called a rotation matrix that describes the rotation of
into
, whereas
is called an orientation matrix that describes the relative orientation of
with respect to
.
In a case of m successive rotational steps, the following equations can be written by using the alternative notations described above.
(2.43)
(2.44)
(2.45)