Page 162 - The Mechatronics Handbook

0066-frame-C09 Page 36 Friday, January 18, 2002 11:00 AM

z, z a

y a
y
φ
x
x a
(a)
z z
y a y a
y b
z a , z b
z a θ θ
x b
y y
φ φ
x x
x a x a
(b) (c)

FIGURE 9.32 The rotations defining the Euler angles (adapted from Goldstein [11]).

It can be shown that there exist two sequences that have independent rotation sequences, and these
lead to the well known Euler angle and Tait-Bryan or Cardan angle rotation descriptions [30].
Euler Angles. Euler angles are defined by a specific rotation sequence. Consider a right-handed axes
system defined by the base vectors, x, y, z, as shown in Fig. 9.32(a). The rotation sequence of interest
involves rotations about the axes in the following sequence: (1) φ about z, (2) θ about x a , then (3) ψ
about z b . This set of rotation sequences is defined by the elementary rotation matrices,

cos φ sin φ 0 1 0 0 cos ψ sin ψ 0
C z,φ = – sin φ cos φ 0 , C x a ,θ = 0 cos θ sin θ , C z b ,ψ = – sin ψ cos ψ 0
0 0 1 0 – sin θ cos θ 0 0 1

where the subscript on each denotes the axis and angle of rotation. Using these transformations relates
C
the quantity A in x, y, z to A b in x b , y b , z b , or
A b = C Euler A = C z b ,ψ C x a ,θ C z,φ A

where C Euler is given by

φ
cos ψcos – sin ψcos θsin φ cos ψsin φ +sin ψcos θcos φ sin ψsin θ
φ
φ
C Euler = – sin ψcos – cos ψcos θsin φ – sin ψsin + cos ψcos θcos φ cos ψsin θ (9.19)
sin θsin φ – sin θcos φ cos θ
Since C Euler is orthogonal, transforming between the two coordinate systems is relatively easy since the
inverse can be found simply by the transpose of Eq. (9.19).
In some applications, it is desirable to derive the angles given the direction cosine matrix. So, if the
(3,3) element of C Euler is given, then θ is easily found, but there can be difficulties in discerning small
angles. Also, if θ goes to zero, there is a singularity in solving for φ and ψ, so determining body orientation
becomes difficult. The problem also makes itself known when transforming angular velocities between
the coordinate systems. If the problem at hand avoids this case (i.e., θ never approaches zero), then Euler
angles are a viable solution. Many applications that cannot tolerate this problem adopt other represen-
tations, such as the Euler parameters to be discussed later.

©2002 CRC Press LLC