Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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      (3.171)

      It is called nonlinear if

      (3.172)

      In the case of a linear relationship, is expressed as follows in terms of an m × n matrix , which does not depend on :

      (3.173)

      Note that a linear relationship is also homogeneous, but a nonlinear relationship may or may not be homogeneous.

      1 (c) Affine Relationship

The relationship set up by is called affine, if is expressed as follows in terms of an m × n matrix and an m × 1 matrix , which are both independent of .

(3.174)

In Eq. (3.174), is defined as the bias term. It may or may not be zero, i.e. .

      Note that a general affine relationship with is both nonhomogeneous and nonlinear. However, a special affine relationship with happens to be both homogeneous and linear.

      3.9.3 Affine Coordinate Transformation Between Two Reference Frames

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)