Page 163 - The Mechatronics Handbook

0066-frame-C09 Page 37 Friday, January 18, 2002 11:00 AM

In classical rigid body dynamics, φ is called the precession angle, θ is the nutation angle, and ψ is the
spin angle. The relationship between the time derivative of the Euler angles, ϕ ˙ = [ φ, θ, ψ ˙ ] T , and the
˙ ˙
body angular velocity, ω = [ω x , ω y , ω z ] b , is given by [11]
T
ω = T ϕ()ϕ ˙ (9.20)
b
ϕ
where the transformation matrix, T (), is given by

sin θsin ψ cos ψ 0
T ϕ() = sin θcos ψ – sin ψ 0
cos θ 0 1

ϕ
Note here again that T ( ) will become singular at θ = ±π/2.
Tait-Bryan or Cardan Angles. The Tait-Bryan or Cardan angles are formed when the three rotation
sequences each occur about a different axis. This is the sequence preferred in flight and vehicle dynamics.
Specifically, these angles are formed by the sequence: (1) φ about z (yaw), (2) θ about y a (pitch), and
(3) φ about the final x b axis (roll), where a and b denote the second and third stage in a three-stage
sequence or axes (as used in the Euler angle description). These rotations define a transformation,

A b = C A = C x b ,ψ C y a ,θ C z,φ A

where

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

and the final coordinate transformation matrix for Tait-Bryan angles is

cos θcos φ cos θsin φ – sin θ
φ
φ
C Tait-Bryan = sin ψsin θcos – cos ψsin φ sin ψsin θsin + cos ψcos φ cos θsin ψ (9.21)
φ
φ
cos ψsin θcos + sin ψsin φ cos ψsin θsin – sin ψcos φ cos θcos ψ
A linearized form of C Trait-Bryan gives a form preferred to that derived for Euler angles, making it useful
in some forms of analysis and control. There remains the problem of a singularity, in this case when θ
approaches ±π /2.
For the Tait-Bryan angles, the transformation matrix relating to ω b is given by
ϕ ˙

– sin θ 0 1
T ϕ() = cos θsin ψ cos ψ 0

cos θcos ψ – sin ψ 0

which becomes singular at θ = 0, π.
Euler Parameters and Quaternions
The degenerate conditions in coordinate transformations for Euler and Tait-Bryan angles can be avoided
by using more than a minimal set of parameterizing variables (beyond the three angles). The most notable

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