Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 69

We have now seen how to represent both positions and orientations. We combine these two concepts in this section to define a rigid motion and, in the next section, we derive an efficient matrix representation for rigid motions using the notion of homogeneous transformation.

Definition 2.2.

A rigid motion is an ordered pair (d, R) where and R ∈ SO(3). The group of all rigid motions is known as the special Euclidean group and is denoted by SE(3). We see then that .

A rigid motion is a pure translation together with a pure rotation.³ Let be the rotation matrix that specifies the orientation of frame o₁x₁y₁z₁ with respect to o₀x₀y₀z₀, and be the vector from the origin of frame o₀x₀y₀z₀ to the origin of frame o₁x₁y₁z₁. Suppose the point is rigidly attached to coordinate frame o₁x₁y₁z₁, with local coordinates . We can express the coordinates of with respect to frame o₀x₀y₀z₀ using

(2.58)

Now consider three coordinate frames o₀x₀y₀z₀, o₁x₁y₁z₁, and o₂x₂y₂z₂. Let d₁ be the vector from the origin of o₀x₀y₀z₀ to the origin of o₁x₁y₁z₁ and d₂ be the vector from the origin of o₁x₁y₁z₁ to the origin of o₂x₂y₂z₂. If the point p is attached to frame o₂x₂y₂z₂ with local coordinates , we can compute its coordinates relative to frame o₀x₀y₀z₀ using

(2.59)

and

(2.60)

The composition of these two equations defines a third rigid motion, which we can describe by substituting the expression for from Equation (2.59) into Equation (2.60)

(2.61)

Since the relationship between and is also a rigid motion, we can equally describe it as

(2.62)

Comparing Equations (2.61) and (2.62) we have the relationships

(2.63)

(2.64)

Equation (2.63) shows that the orientation transformations can simply be multiplied together and Equation (2.64) shows that the vector from the origin o₀ to the origin o₂ has coordinates given by the sum of (the vector from o₀ to o₁ expressed with respect to o₀x₀y₀z₀) and (the vector from o₁ to o₂, expressed in the orientation of the coordinate frame o₀x₀y₀z₀).