Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 65

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.