Mark W. Spong - Robot Modeling and Control

which can be combined to obtain the rotation matrix

Thus, the columns of specify the direction cosines of the coordinate axes of o 1 x 1 y 1relative to the coordinate axes of o 0 x 0 y 0. For example, the first column ( x 1· x 0, x 1· y 0) of specifies the direction of x 1relative to the frame o 0 x 0 y 0. 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 2.2it 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 o 0 x 0 y 0with respect to the frame o 1 x 1 y 1(that is, if we desired to use the frame o 1 x 1 y 1as 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 o 0 x 0 y 0with respect to the frame o 1 x 1 y 1is the inverse of the orientation of o 1 x 1 y 1with respect to the frame o 0 x 0 y 0. 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 :

The projection technique described above scales nicely to the three-dimensional case. In three dimensions, each axis of the frame o 1 x 1 y 1 z 1is projected onto coordinate frame o 0 x 0 y 0 z 0. 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 o 1 x 1 y 1 z 1is rotated through an angle θ about the z 0-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 2.3we 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)