(3.210)
In case of a pure translation with b = a, Eq. (3.209) takes the following form.
(3.211)
As another point of concern, note that Eq. (3.209) can also be written as
In Eq. (3.212),
(3.213)
When Eqs. (3.212) and (3.193) are compared, it is seen that
Equation (3.214) shows how an HTM can be adapted to the selected observation frame.
3.9.6 Example 3.2
Figure 3.4 shows the initial and final positions of a cube. The length of each edge of the cube is L = 10 cm. In the first position of the cube, the edge BC coincides with the first axis of the base frame
Figure 3.4 Two positions of a cube.
It is required to express the HTM
The translation vector can be expressed in
(3.215)
On the other hand,
(3.216)
Hence, in
Then, the column matrix representation of
(3.218)
As for the rotation of the cube, Figure 3.4 implies that
Note that Description (3.219) describes an IFB rotation sequence. Therefore, referring to Section 3.7, the relevant transformation matrices can be obtained as shown below.
(3.221)
(3.222)
Hence,
(3.223)
Having found the rotational and translational displacement matrices, i.e.
(3.224)
In order to have a detailed expression, the rotational partition
(3.225)
Hence,
(3.226)
As a verification of the expression of