Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

two pure‐rotation HTMs.

      (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

(3.212)

In Eq. (3.212),

      (3.213)

When Eqs. (3.212) and (3.193) are compared, it is seen that

(3.214)

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 so that OC = 20 cm. In the second position of the cube, the edge GF coincides with the second axis of so that OG = 15 cm. The reference frame that is fixed to the cube is . It is oriented in such a way that , , and .

Figure 3.4 Two positions of a cube.

      It is required to express the HTM between the two positions of the cube.

      The translation vector can be expressed in as follows:

      (3.215)

      On the other hand, is oriented with respect to so that

      (3.216)

Hence, in , the expression of the translation vector becomes

(3.217)

      Then, the column matrix representation of in is obtained as

      (3.218)

      As for the rotation of the cube, Figure 3.4 implies that

(3.219)

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.

(3.220)

      (3.221)

      (3.222)

      Hence,

      (3.223)

      Having found the rotational and translational displacement matrices, i.e. and , the HTM can then be constructed as follows:

      (3.224)

      In order to have a detailed expression, the rotational partition can be written as shown below.

      (3.225)

      Hence,

      (3.226)

As a verification of the expression of obtained above, the coordinates of the points A₂ and G₂