Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 33

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)

(2.39)

(2.40)

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)