Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 48

Consider a vector and two different reference frames and , which are shown in Figure 3.1. Both and are assumed to be orthonormal, right‐handed, and equally scaled on their axes. They have different orientations described by the following basis vector triads.

(3.1)

In and , the observed vector is resolved differently as shown below.

(3.2)

In Eq. (3.2), the components of in and are obtained as follows for k ∈ {1, 2, 3}:

(3.3)

(3.4)

Figure 3.1 A vector observed in two different reference frames.

The components of that are obtained above can be stacked into the following column matrices, which are defined as the matrix representations of in and .

(3.5)

(3.6)

On the other hand, by recalling the definition of the basic column matrices from Chapter 1, the following equations can be written.

(3.7)

(3.8)

Hence, referring to Eq. (3.2), and can also be expressed as shown below.

(3.9)

(3.10)