M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

More specifically, the function atan 2( η , ξ ) gives the outcome θ as follows depending on the values of the arguments η and ξ :

As mentioned in Chapter 1, the vectors are independent of the reference frames in which they are observed. However, their components are naturally dependent on the selected observation frames. In other words, they have different components and matrix representations in different reference frames. Their matrix representations in different reference frames are related to each other by means of the transformation matrices . The transformation matrix between two reference frames can be expressed in various ways. It can be expressed as a rotation matrix , or as a direction cosine matrix , or as a function of the Euler angles of a selected sequence, or as a function of the basis vectors of the relevant reference frames. All these expressions are studied in this chapter together with several examples. Moreover, the matrix representations of the position vectors of a point in differently oriented and/or located reference frames can be related by means of either affine or homogeneous transformations. So, the affine and homogeneous transformations together with the 4 × 4 homogeneous transformation matrices are also studied in this chapter.

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 3.1A 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.