Robot Modeling and Control. Mark W. Spong
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We can also use rotation matrices to represent rigid motions that correspond to pure rotation. For example, in Figure 2.6(a) one corner of the block is located at the point pa in space. Figure 2.6(b) shows the same block after it has been rotated about z0 by the angle π. The same corner of the block is now located at point pb in space. It is possible to derive the coordinates for pb given only the coordinates for pa and the rotation matrix that corresponds to the rotation about z0. To see how this can be accomplished, imagine that a coordinate frame is rigidly attached to the block in Figure 2.6(a), such that it is coincident with the frame o0x0y0z0. After the rotation by π, the block’s coordinate frame, which is rigidly attached to the block, is also rotated by π. If we denote this rotated frame by o1x1y1z1, we obtain
In the local coordinate frame o1x1y1z1, the point pb has the coordinate representation
It is important to notice that the local coordinates
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
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 v0 = (0, 1, 1) is rotated about y0 by
(2.10)
(2.11)
Thus, a third interpretation of a rotation matrix
Figure 2.6 The block in (b) is obtained by rotating the block in (a) by π about z0.
Figure 2.7 Rotating a vector about axis y0.
As we have seen, rotation matrices can serve several roles. A rotation matrix, either
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
Similarity Transformations
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