Mark W. Spong - Robot Modeling and Control

In this section we derive three ways in which an arbitrary rotation can be represented using only three independent quantities: the Euler anglerepresentation, the roll-pitch-yawrepresentation, and the axis-anglerepresentation.

A common method of specifying a rotation matrix in terms of three independent quantities is to use the so-called Euler angles. Consider the fixed coordinate frame o 0 x 0 y 0 z 0and the rotated frame o 1 x 1 y 1 z 1shown in Figure 2.10. We can specify the orientation of the frame o 1 x 1 y 1 z 1relative to the frame o 0 x 0 y 0 z 0by three angles ( ϕ , θ , ψ ), known as Euler angles, and obtained by three successive rotations as follows. First rotate about the z -axis by the angle ϕ . Next rotate about the current y -axis by the angle θ . Finally rotate about the current z -axis by the angle ψ . In Figure 2.10, frame oaxayaza represents the new coordinate frame after the rotation by ϕ , frame obxbybzb represents the new coordinate frame after the rotation by θ , and frame o 1 x 1 y 1 z 1represents the final frame, after the rotation by ψ . Frames oaxayaza and obxbybzb are shown in the figure only to help visualize the rotations.

Figure 2.10 Euler angle representation.

In terms of the basic rotation matrices the resulting rotational transformation can be generated as the product

(2.27)

The matrix RZYZ in Equation ( 2.27) is called the ZYZ –Euler angle transformation.

The more important and more difficult problem is to determine for a particular R = ( rij ) the set of Euler angles ϕ , θ , and ψ , that satisfy

(2.28)

for a matrix R ∈ SO (3). This problem will be important later when we address the inverse kinematics problem for manipulators in Chapter 5.

To find a solution for this problem we break it down into two cases. First, suppose that not both of r 13, r 23are zero. Then from Equation ( 2.27) we deduce that s θ≠ 0, and hence that not both of r 31, r 32are zero. If not both r 13and r 23are zero, then r 33≠ ±1, and we have c θ= r 33, so

(2.29)

or

(2.30)

where the function Atan2 is the two-argument arctangent functiondefined in Appendix A.

If we choose the value for θ given by Equation ( 2.29), then s θ> 0, and

(2.31)

(2.32)

If we choose the value for θ given by Equation ( 2.30), then s θ< 0, and

(2.33)

(2.34)

Thus, there are two solutions depending on the sign chosen for θ .

If r 13= r 23= 0, then the fact that is orthogonal implies that r 33= ±1, and that r 31= r 32= 0. Thus, has the form

(2.35)

If r 33= 1, then c θ= 1 and s θ= 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 r 33= −1, then c θ= −1 and s θ= 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.

A rotation matrix can also be described as a product of successive rotations about the principal coordinate axes x 0, y 0, and z 0taken in a specific order. These rotations define the roll, pitch, and yawangles, which we shall also denote ϕ , θ , ψ , and which are shown in Figure 2.11.

Figure 2.11 Roll, pitch, and yaw angles.