Mark W. Spong - Robot Modeling and Control

We specify the order of rotation as x − y − z , in other words, first a yaw about x 0through an angle ψ , then pitch about the y 0by an angle θ , and finally roll about the z 0by an angle ϕ . 2Since the successive rotations are relative to the fixed frame, the resulting transformation matrix is given by

(2.39)

Of course, instead of yaw-pitch-roll relative to the fixed frames we could also interpret the above transformation as roll-pitch-yaw, in that order, each taken with respect to the current frame. The end result is the same matrix as in Equation ( 2.39).

The three angles ϕ , θ , and ψ can be obtained for a given rotation matrix using a method that is similar to that used to derive the Euler angles above.

Rotations are not always performed about the principal coordinate axes. We are often interested in a rotation about an arbitrary axis in space. This provides both a convenient way to describe rotations, and an alternative parameterization for rotation matrices. Let k = ( kx , ky , kz ), expressed in the frame o 0 x 0 y 0 z 0, be a unit vector defining an axis. We wish to derive the rotation matrix representing a rotation of θ about this axis.

There are several ways in which the matrix can be derived. One approach is to note that the rotational transformation will bring the world z -axis into alignment with the vector k . Therefore, a rotation about the axis k can be computed using a similarity transformation as

(2.40)

(2.41)

From Figure 2.12we see that

(2.42)

(2.43)

Note that the final two equations follow from the fact that k is a unit vector. Substituting Equations ( 2.42) and ( 2.43) into Equation ( 2.41), we obtain after some lengthy calculation (Problem 2–17)

(2.44)

where v θ= vers θ = 1 − c θ.

Figure 2.12 Rotation about an arbitrary axis.

In fact, any rotation matrix can be represented by a single rotation about a suitable axis in space by a suitable angle,

(2.45)

where k is a unit vector defining the axis of rotation, and θ is the angle of rotation about k . The pair ( k , θ ) is called the axis-angle representationof . Given an arbitrary rotation matrix with components rij , the equivalent angle θ and equivalent axis k are given by the expressions

and

(2.46)

These equations can be obtained by direct manipulation of the entries of the matrix given in Equation ( 2.44). The axis-angle representation is not unique since a rotation of − θ about − k is the same as a rotation of θ about k , that is,

(2.47)

If θ = 0 then is the identity matrix and the axis of rotation is undefined.

Example 2.9.

Suppose is generated by a rotation of 90° about z 0followed by a rotation of 30° about y 0followed by a rotation of 60° about x 0. Then

(2.48)

We see that and hence the equivalent angle is given by Equation (2.46) as

(2.49)

The equivalent axis is given from Equation ( 2.46) as

(2.50)

The above axis-angle representation characterizes a given rotation by four quantities, namely the three components of the equivalent axis k and the equivalent angle θ . However, since the equivalent axis k is given as a unit vector only two of its components are independent. The third is constrained by the condition that k is of unit length. Therefore, only three independent quantities are required in this representation of a rotation . We can represent the equivalent axis-angle by a single vector r as