Mark W. Spong - Robot Modeling and Control

It is important to notice that the local coordinates of the corner of the block do not change as the block rotates, since they are defined in terms of the block’s own coordinate frame. Therefore, when the block’s frame is aligned with the reference frame o 0 x 0 y 0 z 0(that is, before the rotation is performed), the coordinates equals , since before the rotation is performed, the point pa is coincident with the corner of the block. Therefore, we can substitute into the previous equation to obtain

This equation shows how to use a rotation matrix to represent a rotational motion. In particular, if the point pb is obtained by rotating the point pa as defined by the rotation matrix , then the coordinates of pb with respect to the reference frame are given by

This same approach can be used to rotate vectors with respect to a coordinate frame, as the following example illustrates.

Example 2.3.

The vector v with coordinates v 0= (0, 1, 1) is rotated about y 0by as shown in Figure 2.7. The resulting vector v 1is given by

(2.10)

(2.11)

Thus, a third interpretation of a rotation matrix is as an operator acting on vectors in a fixed frame. In other words, instead of relating the coordinates of a fixed vector with respect to two different coordinate frames, Equation ( 2.10) can represent the coordinates in o 0 x 0 y 0 z 0of a vector v 1that is obtained from a vector v by a given rotation.

Figure 2.6 The block in (b) is obtained by rotating the block in (a) by π about z 0.

Figure 2.7 Rotating a vector about axis y 0.

As we have seen, rotation matrices can serve several roles. A rotation matrix, either or , can be interpreted in three distinct ways:

1 It represents a coordinate transformation relating the coordinates of a point p in two different frames.

2 It gives the orientation of a transformed coordinate frame with respect to a fixed coordinate frame.

3 It is an operator taking a vector and rotating it to give a new vector in the same coordinate frame.

The particular interpretation of a given rotation matrix should be made clear by the context.

A coordinate frame is defined by a set of basis vectors, for example, unit vectors along the three coordinate axes. This means that a rotation matrix, as a coordinate transformation, can also be viewed as defining a change of basis from one frame to another. The matrix representation of a general linear transformation is transformed from one frame to another using a so-called similarity transformation. For example, if A is the matrix representation of a given linear transformation in o 0 x 0 y 0 z 0and B is the representation of the same linear transformation in o 1 x 1 y 1 z 1then A and B are related as

(2.12)

where is the coordinate transformation between frames o 1 x 1 y 1 z 1and o 0 x 0 y 0 z 0. In particular, if A itself is a rotation, then so is B , and thus the use of similarity transformations allows us to express the same rotation easily with respect to different frames.

Example 2.4.

Henceforth, whenever convenient we use the shorthand notation c θ= cos θ , s θ= sin θ for trigonometric functions. Suppose frames o 0 x 0 y 0 z 0and o 1 x 1 y 1 z 1are related by the rotation

If A = R z, θrelative to the frame o 0 x 0 y 0 z 0, then, relative to frame o 1 x 1 y 1 z 1we have

In other words, B is a rotation about the z 0-axis but expressed relative to the frame o 1 x 1 y 1 z 1. This notion will be useful below and in later sections.

In this section we discuss the composition of rotations. It is important for subsequent chapters that the reader understand the material in this section thoroughly before moving on.