Mark W. Spong - Robot Modeling and Control

Recall that the matrix in Equation ( 2.9) represents a rotational transformation between the frames o 0 x 0 y 0 z 0and o 1 x 1 y 1 z 1. Suppose we now add a third coordinate frame o 2 x 2 y 2 z 2related to the frames o 0 x 0 y 0 z 0and o 1 x 1 y 1 z 1by rotational transformations. A given point p can then be represented by coordinates specified with respect to any of these three frames: , , and The relationship among these representations of p is

(2.13)

(2.14)

(2.15)

where each is a rotation matrix. Substituting Equation ( 2.14) into Equation ( 2.13) gives

(2.16)

Note that and represent rotations relative to the frame o 0 x 0 y 0 z 0while represents a rotation relative to the frame o 1 x 1 y 1 z 1. Comparing Equations ( 2.15) and ( 2.16) we can immediately infer

(2.17)

Equation ( 2.17) is the composition law for rotational transformations. It states that, in order to transform the coordinates of a point p from its representation in the frame o 2 x 2 y 2 z 2to its representation in the frame o 0 x 0 y 0 z 0, we may first transform to its coordinates in the frame o 1 x 1 y 1 z 1using and then transform to using .

We may also interpret Equation ( 2.17) as follows. Suppose that initially all three of the coordinate frames coincide. We first rotate the frame o 1 x 1 y 1 z 1relative to o 0 x 0 y 0 z 0according to the transformation . Then, with the frames o 1 x 1 y 1 z 1and o 2 x 2 y 2 z 2coincident, we rotate o 2 x 2 y 2 z 2relative to o 1 x 1 y 1 z 1according to the transformation . The resulting frame, o 2 x 2 y 2 z 2has orientation with respect to o 0 x 0 y 0 z 0given by . We call the frame relative to which the rotation occurs the current frame.

Example 2.5.

Suppose a rotation matrix represents a rotation of angle ϕ about the current y -axis followed by a rotation of angle θ about the current z -axis as shown in Figure 2.8. Then the matrix is given by

(2.18)

Figure 2.8 Composition of rotations about current axes.

It is important to remember that the order in which a sequence of rotations is performed, and consequently the order in which the rotation matrices are multiplied together, is crucial. The reason is that rotation, unlike position, is not a vector quantity and so rotational transformations do not commute in general.

Example 2.6.

Suppose that the above rotations are performed in the reverse order, that is, first a rotation about the current z -axis followed by a rotation about the current y -axis. Then the resulting rotation matrix is given by

(2.19)

Comparing Equations ( 2.18) and ( 2.19) we see that .

Many times it is desired to perform a sequence of rotations, each about a given fixed coordinate frame, rather than about successive current frames. For example we may wish to perform a rotation about x 0followed by a rotation about y 0(and not y 1!). We will refer to o 0 x 0 y 0 z 0as the fixed frame. In this case the composition law given by Equation ( 2.17) is not valid. It turns out that the correct composition law in this case is simply to multiply the successive rotation matrices in the reverse orderfrom that given by Equation ( 2.17). Note that the rotations themselves are not performed in reverse order. Rather they are performed about the fixed frame instead of about the current frame.