Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 62

We can summarize the rule of composition of rotational transformations by the following recipe. Given a fixed frame o₀x₀y₀z₀ and a current frame o₁x₁y₁z₁, together with rotation matrix relating them, if a third frame o₂x₂y₂z₂ is obtained by a rotation performed relative to the current frame then postmultiply by to obtain

(2.22)

If the second rotation is to be performed relative to the fixed frame then it is both confusing and inappropriate to use the notation to represent this rotation. Therefore, if we represent the rotation by , we premultiply by to obtain

(2.23)

In each case represents the transformation between the frames o₀x₀y₀z₀ and o₂x₂y₂z₂. The frame o₂x₂y₂z₂ that results from Equation (2.22) will be different from that resulting from Equation (2.23).

Using the above rule for composition of rotations, it is an easy matter to determine the result of multiple sequential rotational transformations.

Example 2.8.

Suppose R is defined by the following sequence of basic rotations in the order specified:

1 A rotation of θ about the current x-axis

2 A rotation of ϕ about the current z-axis

3 A rotation of α about the fixed z-axis

4 A rotation of β about the current y-axis

5 A rotation of δ about the fixed x-axis

In order to determine the cumulative effect of these rotations we simply begin with the first rotation R_{x, θ} and pre- or postmultiply as the case may be to obtain

(2.24)