Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 74

Figure 3.3 shows a point P, which is observed in two different reference frames and . The reference frame has a general (translating and rotating) displacement with respect to . This general displacement is represented by the translation vector and the rotation operator rot(a, b), which functions to rotate into for k ∈ {1, 2, 3}. The position vectors of P appear as and respectively, in and . The components of in and in are the coordinates of P in and . In other words, the column matrices and consist of the coordinates of P in and . On the other hand, as explained in Section 3.5, the transformation matrix between and can be expressed in terms of the matrix representations of the rotation operator as .

Figure 3.3 A point observed in two different reference frames.

As seen in Figure 3.3, the position vectors of P are related to each other as follows:

(3.165)

Equation (3.165), which is a vector equation, can be written as the following matrix equation in one of the involved reference frames, say .

(3.166)

However, it is more convenient to express in rather than in . By doing so, Eq. (3.166) can be written again as follows:

(3.167)