href="#ulink_e474a145-647a-5197-ac99-1925c3878b7a">2.29), then sθ > 0, and
(2.31)
(2.32)
If we choose the value for θ given by Equation (2.30), then sθ < 0, and
(2.33)
(2.34)
Thus, there are two solutions depending on the sign chosen for θ.
If r13 = r23 = 0, then the fact that
(2.35)
If r33 = 1, then cθ = 1 and sθ = 0, so that θ = 0. In this case, Equation (2.27) becomes
Thus, the sum ϕ + ψ can be determined as
(2.36)
Since only the sum ϕ + ψ can be determined in this case, there are infinitely many solutions. In this case, we may take ϕ = 0 by convention. If r33 = −1, then cθ = −1 and sθ = 0, so that θ = π. In this case Equation (2.27) becomes
(2.37)
The solution is thus
(2.38)
As before there are infinitely many solutions.
2.5.2 Roll, Pitch, Yaw Angles
A rotation matrix
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
There are several ways in which the matrix
(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
