M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

Note that Eq. (3.203)is actually valid for any pivot point whatsoever. Therefore, the HTM of a pure rotational displacement does not actually need a subscript and thus it may be denoted even in the following simplest form.

(3.204)

Note also that Eqs. (3.203)and (3.204)verify the well‐known fact that a rotation operator is indifferent to the location of the pivot point.

1 (e) HTM of a Pure Translation

A pure translational displacement of with respect to or a pure translational displacement of with respect to can be expressed by one of the following HTM expressions, depending on the selected one of and , in which the translation vector is observed.

(3.205)

(3.206)

1 (f) Observation in a Third Different Reference Frame

In general, the point P and the reference frames and may be observed in a different reference frame . In such a case, considering that the vectors and are conveniently resolved in and , Eq. (3.175)can be written as follows:

(3.207)

The above affine relationship can be expressed in the following homogeneous form.

(3.208)

In Eq. (3.208), the coefficient matrix on the left‐hand side is the HTM of a pure rotation from to and the coefficient matrix on the right‐hand side is the HTM that expresses the overall rotation from to together with the translation from A to B as observed in . Thus, Eq. (3.208)can be written compactly as

(3.209)

In case of a pure rotation with B = A , Eq. (3.209)takes the following form that involves 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.

Figure 3.4shows 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 .