Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 76

As mentioned before in Section 3.9.1, the column matrices and consist of the coordinates of P in and . So, the following equation, which is the repetition of Eq. (3.167), constitutes an affine transformation between and , i.e. between the coordinates of P in and .

(3.175)

In Eq. (3.175), is the bias term of the affine transformation. It represents the translational displacement of with respect to . As for , it represents the rotational displacement of with respect to .

If the point P is observed in several different reference frames such as , , , …, , then the following affine transformation equations can be written between the successive reference frames.

(3.176)

(3.177)

(3.178)

(3.179)

As for the overall affine transformation equation, it can be written as

(3.180)

Upon successive substitutions, the preceding equations lead to the following combined equations.

(3.181)

(3.182)

(3.183)

Equations (3.180) and (3.183) show the necessity of using the following set of equations in order to obtain the combined rotation and translation matrices.

(3.184)

(3.185)