Robot Modeling and Control. Mark W. Spong

by o₁x₁y₁z₁, we obtain

      In the local coordinate frame o₁x₁y₁z₁, the point pb has the coordinate representation . To obtain its coordinates with respect to frame o₀x₀y₀z₀, we merely apply the coordinate transformation Equation (2.9), giving

      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 o₀x₀y₀z₀ (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 v⁰ = (0, 1, 1) is rotated about y0 by as shown in Figure 2.7. The resulting vector v1 is 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 o₀x₀y₀z₀ of a vector v₁ that 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 z₀.

Figure 2.7 Rotating a vector about axis y₀.

      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.

      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 transformation. For example, if A is the matrix representation of a given linear transformation in o₀x₀y₀z₀ and B is the representation of the same linear transformation in o₁x₁y₁z₁ then A and B are related as

      (2.12)

      where is the coordinate transformation between frames o₁x₁y₁z₁ and o₀x₀y₀z₀. 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 o₀x₀y₀z₀ and o₁x₁y₁z₁ are related by the rotation

      If A = R_{z, θ} relative to the frame o₀x₀y₀z₀, then, relative to frame o₁x₁y₁z₁ we have

      In