Robot Modeling and Control. Mark W. Spong

Figure 2.3 Rotation about z₀ by an angle θ.

      The rotation matrix given in Equation (2.3) is called a basic rotation matrix (about the z-axis). In this case we find it useful to use the more descriptive notation instead of to denote the matrix. It is easy to verify that the basic rotation matrix has the properties

(2.4)

      (2.5)

      which together imply

(2.6)

Similarly, the basic rotation matrices representing rotations about the x and y-axes are given as (Problem 2–8)

(2.7)

(2.8)

      which also satisfy properties analogous to Equations (2.4)–(2.6).

       Example 2.2.

Consider the frames o₀x₀y₀z₀ and o₁x₁y₁z₁ shown in Figure 2.4.

      Projecting the unit vectors x₁, y₁, z₁ onto x₀, y₀, z₀ gives the coordinates of x₁, y₁, z₁ in the o₀x₀y₀z₀ frame as

The rotation matrix specifying the orientation of o₁x₁y₁z₁ relative to o₀x₀y₀z₀ has these as its column vectors, that is,

Figure 2.4 Defining the relative orientation of two frames.

      2.3 Rotational Transformations

Figure 2.5 shows a rigid object S to which a coordinate frame o₁x₁y₁z₁ is attached. Given the coordinates of the point p (in other words, given the coordinates of p with respect to the frame o₁x₁y₁z₁), we wish to determine the coordinates of p relative to a fixed reference frame o₀x₀y₀z₀. The coordinates satisfy the equation

Figure 2.5 Coordinate frame attached to a rigid body.

      In a similar way, we can obtain an expression for the coordinates by projecting the point p onto the coordinate axes of the frame o₀x₀y₀z₀, giving

      Combining these two equations we obtain

But the matrix in this final equation is merely the rotation matrix , which leads to

(2.9)

      Thus, the rotation matrix can be used not only to represent the orientation of coordinate frame o₁x₁y₁z₁ with respect to frame o₀x₀y₀z₀, but also to transform the coordinates of a point from one frame to another. If a given point is expressed relative to o₁x₁y₁z₁ by coordinates , then represents the same point expressed relative to the frame o₀x₀y₀z₀.

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