Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

Kinematics of General Spatial Mechanical Systems

Скачать книгу

φ₃ take place about two axes that have become parallel, either codirectionally if images

or oppositely if images

. Therefore, only the resultant rotation by the angle images

can be recognized but the angles φ₁ and φ₃ become obscure and they cannot be distinguished from each other.

3.9 Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices

3.9.1 Position of a Point Expressed in Different Reference Frames

Figure 3.3 shows a point P, which is observed in two different reference frames images and images . The reference frame images has a general (translating and rotating) displacement with respect to images . This general displacement is represented by the translation vector images and the rotation operator rot(a, b), which functions to rotate images into images for k ∈ {1, 2, 3}. The position vectors of P appear as images and images respectively, in images and images . The components of images in images and images in images are the coordinates of P in images and images . In other words, the column matrices images and images consist of the coordinates of P in images and images . On the other hand, as explained in Section 3.5, the transformation matrix between images and images can be expressed in terms of the matrix representations of the rotation operator as images .

Vector diagram of a point observed in two different reference frames.

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

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

(3.166) equation

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

(3.167) equation

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

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

(3.168) equation

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

1 (a) Homogeneous Versus Nonhomogeneous Relationships

The relationship set up by images is called homogeneous if

(3.169) equation

It is called nonhomogeneous if

(3.170) equation

1 (b) Linear Versus Nonlinear Relationships

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