Robot Modeling and Control. Mark W. Spong

rel="nofollow" href="#ulink_5ef0a5c0-6b64-58b2-9ead-d39f4927759d">2.19) we see that .

      2.4.2 Rotation with Respect to the Fixed Frame

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 x₀ followed by a rotation about y₀ (and not y₁!). We will refer to o₀x₀y₀z₀ as 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 order from 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.

      To see this, suppose we have two frames o₀x₀y₀z₀ and o₁x₁y₁z₁ related by the rotational transformation . If R ∈ SO(3) represents a rotation relative to o₀x₀y₀z₀, we know from Section 2.3 that the representation for R in the current frame o₁x₁y₁z₁ is 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 premultiplied by 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 y₀ followed by a rotation of angle θ about the fixed z₀. The second rotation about the fixed axis is given by , which is the basic rotation about the z-axis expressed relative to the frame o₁x₁y₁z₁ using 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.

      2.4.3 Rules for Composition of Rotations

We can summarize the rule of composition of rotational transformations by the following recipe. Given a fixed frame o₀x₀y₀z₀ and a current frame o₁x₁y₁z₁, together with rotation matrix relating them, if a third frame o₂x₂y₂z₂ is obtained by a rotation performed relative to the current frame then postmultiply by to obtain

(2.22)

      If the second rotation is to be performed relative to the fixed frame then 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 o₀x₀y₀z₀ and o₂x₂y₂z₂. The frame o₂x₂y₂z₂ that 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, θ} and pre- or postmultiply as the case may be to obtain

      (2.24)

      2.5 Parameterizations of Rotations

The nine elements rij in a general rotational transformation are not independent quantities.