M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(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)

Referring to Eqs. (3.184)and (3.185), it is seen that the result of a combination of several affine transformations necessitates carrying out a considerable number of addition and multiplication operations involving 3 × 1 and 3 × 3 matrices. However, a large number of matrix operations is not desirable of course especially from the viewpoint of computational efficiency.

On the other hand, if the transformations are expressed homogeneously, the number of necessary matrix operations reduces considerably to such an extent that only a minimal number of multiplications are required without any additions. However, this reduction in the number of operations necessitates the introduction of 4 × 1 and 4 × 4 augmented matrices in return. Even so, the advantage of the reduction in the number of operations emphatically overcomes the disadvantage of the increased dimension of the matrices.

The affine transformation expressed by Eq. (3.175)can be converted into a homogeneous transformation as explained below.

Equation (3.175)can be combined with the trivial equation 1 = 1 in order to set up the following system of equations.

(3.186)

The preceding system of equations can be written as the following single matrix equation.

(3.187)

Equation (3.187)suggests the following definitions.

is defined as the augmented position matrix of P in . It is a 4 × 1 matrix formed as

(3.188)

is defined as the augmented position matrix of P in . It is a 4 × 1 matrix formed as

(3.189)

is defined as the homogeneous transformation matrix (HTM) between and . It is a 4 × 4 matrix formed as