Robot Modeling and Control. Mark W. Spong

Although we have derived the entries for in terms of the angle θ, it is not necessary that we do so. An alternative approach, and one that scales nicely to the three-dimensional case, is to build the rotation matrix by projecting the axes of frame o₁x₁y₁ onto the coordinate axes of frame o₀x₀y₀. Recalling that the dot product of two unit vectors gives the projection of one onto the other, we obtain

      which can be combined to obtain the rotation matrix

      Thus, the columns of specify the direction cosines of the coordinate axes of o₁x₁y₁ relative to the coordinate axes of o₀x₀y₀. For example, the first column (x₁ · x₀, x₁ · y₀) of specifies the direction of x₁ relative to the frame o₀x₀y₀. Note that the right-hand sides of these equations are defined in terms of geometric entities, and not in terms of their coordinates. Examining Figure 2.2 it can be seen that this method of defining the rotation matrix by projection gives the same result as we obtained in Equation (2.1).

      If we desired instead to describe the orientation of frame o₀x₀y₀ with respect to the frame o₁x₁y₁ (that is, if we desired to use the frame o₁x₁y₁ as the reference frame), we would construct a rotation matrix of the form

      Since the dot product is commutative, (that is, xi · yj = yj · xi), we see that

      In a geometric sense, the orientation of o₀x₀y₀ with respect to the frame o₁x₁y₁ is the inverse of the orientation of o₁x₁y₁ with respect to the frame o₀x₀y₀. Algebraically, using the fact that coordinate axes are mutually orthogonal, it can readily be seen that

      The above relationship implies that and it is easily shown that the column vectors of are of unit length and mutually orthogonal (Problem 2–4). Thus is an orthogonal matrix. It also follows from the above that (Problem 2–5) . If we restrict ourselves to right-handed coordinate frames, as defined in Appendix B, then (Problem 2–5).

      More generally, these properties extend to higher dimensions, which can be formalized as the so-called special orthogonal group of order n.

       Definition 2.1.

The special orthogonal group of order n, denoted SO(n), is the set of n × n real-valued matrices

      (2.2)

      Thus, for any the following properties hold

       

       The columns (and therefore the rows) of are mutually orthogonal

       Each column (and therefore each row) of is a unit vector

       

      The special case, SO(2), respectively, SO(3), is called the rotation group of order 2, respectively 3.

      To provide further geometric intuition for the notion of the inverse of a rotation matrix, note that in the two-dimensional case, the inverse of the rotation matrix corresponding to a rotation by angle θ can also be easily computed simply by constructing the rotation matrix for a rotation by the angle − θ:

      2.2.2 Rotations in Three Dimensions

      The projection technique described above scales nicely to the three-dimensional case. In three dimensions, each axis of the frame o₁x₁y₁z₁ is projected onto coordinate frame o₀x₀y₀z₀. The resulting rotation matrix R ∈ SO(3) is given by

      As was the case for rotation matrices in two dimensions, matrices in this form are orthogonal, with determinant equal to 1 and therefore elements of SO(3).

       Example 2.1.

Suppose the frame o₁x₁y₁z₁ is rotated through an angle θ about the z₀-axis, and we wish to find the resulting transformation matrix . By convention, the right hand rule (see Appendix B) defines the positive sense for the angle θ to be such that rotation by θ about the z-axis would advance a right-hand threaded screw along the positive z-axis.

From Figure 2.3 we see that

      and

      while all other dot products are zero. Thus, the rotation matrix has a particularly simple form in this case, namely

(2.3)

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