Robot Modeling and Control. Mark W. Spong

      As before there are infinitely many solutions.

      2.5.2 Roll, Pitch, Yaw Angles

A rotation matrix can also be described as a product of successive rotations about the principal coordinate axes x₀, y₀, and z₀ taken in a specific order. These rotations define the roll, pitch, and yaw angles, which we shall also denote ϕ, θ, ψ, and which are shown in Figure 2.11.

Figure 2.11 Roll, pitch, and yaw angles.

      We specify the order of rotation as x − y − z, in other words, first a yaw about x₀ through an angle ψ, then pitch about the y₀ by an angle θ, and finally roll about the z₀ by an angle ϕ.² Since 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.

      2.5.3 Axis-Angle Representation

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 o₀x₀y₀z₀, 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 2.12 we 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 representation of . 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 z₀ followed by a rotation of 30° about y₀ followed by a rotation of 60° about x₀. 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 (