Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

Figure 3.3 A point observed in two different reference frames.

      As seen in Figure 3.3, the position vectors of P are related to each other as follows:

(3.165)

Equation (3.165), which is a vector equation, can be written as the following matrix equation in one of the involved reference frames, say .

(3.166)

However, it is more convenient to express in rather than in . By doing so, Eq. (3.166) can be written again as follows:

(3.167)

      3.9.2 Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships

      Consider two column matrices and . Suppose they are related to each other by means of a function so that

(3.168)

Depending on the mathematical features of the function , the relationship described by Eq. (3.168) is characterized by various designations, which are explained below.

      The relationship set up by is called homogeneous if

      (3.169)

      It is called nonhomogeneous if

      (3.170)

      The relationship set up by is called linear if, for a scalar k and for all ,

      (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.

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

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