Page 161 - The Mechatronics Handbook

0066-frame-C09 Page 35 Friday, January 18, 2002 11:00 AM

z a
z b
ψ
y b
ψ

ψ O y a
FIGURE 9.31 An elementary rotation by angle φ about
x b
axis x. x a
The matrix elements of can be found by expanding the relation given above, using (λ), to give
C
S

( 1 – cos ψ)λ 1 + cos ψ ( 1 – cos ψ)λ 1 λ 2 + λ 3 sin ψ ( 1 – cos ψ)λ 1 λ 3 + λ 2 sin ψ
2
C = ( 1 – cos ψ)λ 2 λ 1 + λ 3 sin ψ ( 1 – cos ψ)λ 2 + cos ψ ( 1 – cos ψ)λ 2 λ 3 + λ 1 sin ψ (9.18)
2
( 1 – cos ψ)λ 3 λ 1 + λ 2 sin ψ ( 1 – cos ψ)λ 3 λ 2 + λ 1 sin ψ ( 1 – cos ψ)λ 3 + cos ψ
2

The value of this formulation is in identifying that there are formally defined principle axes, charac-
terized by the λ , and angles of rotation, ψ, that taken together define the body orientation. These
rotations describe classical angular variables formed by elementary (or principle) rotations, and it can
be shown that there are two cases of particular and practical interest, formed by two different axis rotation
sequences.
Elementary Rotations. Three elementary rotations are formed when the rotation axis (defined by the
eigenvector) coincides with one of the base vectors of a defined coordinate system. For example, letting
λ = [1, 0, 0] define an axis of rotation x, as in Fig. 9.31, with an elementary rotation of φ gives the
T
rotation matrix,

1 0 0
C x,φ = 0 cos φ sin φ
0 – sin φ cos φ

The two elementary rotations about the other two axes, y and z, are

cos θ 0 – sin θ cos ψ sin ψ 0
C y,θ = 0 1 0 and C z,ψ = – sin ψ cos ψ 0
sin θ 0 cos θ 0 0 1

These three elementary rotation matrices can be used in sequence to define a direction cosine matrix,
for example,

C = C z,ψ C C x,φ
y,θ
and the elementary rotations and the direction cosine matrix are all orthogonal; i.e.,

T T
C C = C C = E

E
where is the identity matrix. Consequently, the inverse of the rotation or coordinate transformation
matrix can be found by C −1 = C T .

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