Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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)

3.2 Transformation Matrices Between Reference Frames

      3.2.1 Definition and Usage of a Transformation Matrix

Consider two reference frames and . If they are differently oriented with respect to each other, a vector appears differently in and . In other words, . However, since and represent the same vector , they are nonetheless related so that

(3.11)

In Eq. (3.11), is defined as the transformation matrix between and . It can be considered as an operator that transforms the components of a vector from to . So, to be more specific, it may also be called a component transformation matrix.

      3.2.2 Basic Properties of a Transformation Matrix

      Equation (3.11) can also be written in the following two ways: first by inverting ; and then by interchanging the frame indicators a and b.

      (3.12)

(3.13)

Equations (3.11)–(3.13) imply the following equalities.

(3.14)

      (3.15)

      As also mentioned in Chapter 1, all the reference frames considered in this book are assumed to be orthonormal, right‐handed, and equally scaled on their axes. Therefore, the magnitude of a vector appears to be the same in every reference frame. This fact is expressed as follows:

(3.16)

After substituting Eq. (3.11), Eq. (3.16) can be manipulated as shown below.

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