Mark W. Spong - Robot Modeling and Control

To see this, suppose we have two frames o 0 x 0 y 0 z 0and o 1 x 1 y 1 z 1related by the rotational transformation . If R ∈ SO (3) represents a rotation relative to o 0 x 0 y 0 z 0, we know from Section 2.3 that the representation for R in the currentframe o 1 x 1 y 1 z 1is given by ( . Therefore, applying the composition law for rotations about the current axis yields

(2.20)

Thus, when a rotation is performed with respect to the world coordinate frame, the current rotation matrix is premultipliedby to obtain the desired rotation matrix.

Example 2.7.(Rotations about Fixed Axes)

Referring to Figure 2.9, suppose that a rotation matrix represents a rotation of angle ϕ about y 0followed by a rotation of angle θ about the fixed z 0. The second rotation about the fixed axis is given by , which is the basic rotation about the z -axis expressed relative to the frame o 1 x 1 y 1 z 1using a similarity transformation. Therefore, the composition rule for rotational transformations gives us

(2.21)

It is not necessary to remember the above derivation, only to note by comparing Equation ( 2.21) with Equation ( 2.18) that we obtain the same basic rotation matrices, but in the reverse order.

Figure 2.9 Composition of rotations about fixed axes.

We can summarize the rule of composition of rotational transformations by the following recipe. Given a fixed frame o 0 x 0 y 0 z 0and a current frame o 1 x 1 y 1 z 1, together with rotation matrix relating them, if a third frame o 2 x 2 y 2 z 2is obtained by a rotation performed relative to the current framethen postmultiply by to obtain

(2.22)

If the second rotation is to be performed relative to the fixed framethen it is both confusing and inappropriate to use the notation to represent this rotation. Therefore, if we represent the rotation by , we premultiply by to obtain

(2.23)

In each case represents the transformation between the frames o 0 x 0 y 0 z 0and o 2 x 2 y 2 z 2. The frame o 2 x 2 y 2 z 2that results from Equation ( 2.22) will be different from that resulting from Equation ( 2.23).

Using the above rule for composition of rotations, it is an easy matter to determine the result of multiple sequential rotational transformations.

Example 2.8.

Suppose R is defined by the following sequence of basic rotations in the order specified:

1 A rotation of θ about the current x-axis

2 A rotation of ϕ about the current z-axis

3 A rotation of α about the fixed z-axis

4 A rotation of β about the current y-axis

5 A rotation of δ about the fixed x-axis

In order to determine the cumulative effect of these rotations we simply begin with the first rotation R x, θand pre- or postmultiply as the case may be to obtain

(2.24)

The nine elements rij in a general rotational transformation are not independent quantities. Indeed, a rigid body possesses at most three rotational degrees of freedom, and thus at most three quantities are required to specify its orientation. This can be easily seen by examining the constraints that govern the matrices in SO (3):

(2.25)

(2.26)

Equation ( 2.25) follows from the fact that the columns of a rotation matrix are unit vectors, and Equation ( 2.26) follows from the fact that columns of a rotation matrix are mutually orthogonal. Together, these constraints define six independent equations with nine unknowns, which implies that there are three free variables.