Page 160 - The Mechatronics Handbook

0066-frame-C09 Page 34 Friday, January 18, 2002 11:00 AM

Matrix Formulation and Coordinate Transformations
A vector in three-dimensional space characterized by the right-handed reference frame x a , y a , z a ,
ˆ
ˆ
ˆ
A = A x i a + A y j a + A z k a, can be represented as an ordered triplet,
A x
A = A y = A x A y A z a T

A z a
where the elements of the column vector represent the vector projections on the unit axes. Let A a denote
the column vector relative to the axes x a , y a , z a . It can be shown that the vector A can be expressed in
another right-handed reference frame x b , y b , z b , by the transformation relation

A b = C A a (9.15)
ab
where C ab is a 3 × 3 matrix,

cx a x b cx a y b cx a z b
C (9.16)
ab = cy a x b cy a y b cy a z b
cz a x b cz a y b cz a z b

The elements of this matrix are the cosines of the angles between the respective axes. For example, cz a y b
is the cosine of the angle between z a and y b . This is the rotational transformation matrix and it must be
orthogonal, or

C ab = C ab = C ba
–
T
1
and for right-handed systems, let C ab = +1.
Angle Representations of Rotation
The six degrees of freedom needed to describe general motion of a rigid body are characterized by three
degrees of freedom each for translation and for rotation. The focus here is on methods for describing
rotation.
Euler’s theorem (11) confirms that only three parameters are needed to characterize rotation. Two
parameters define an axis of rotation and another defines an angle about that axis. These parameters
define three positional degrees of freedom for a rigid body. The three rotational parameters help construct
a rotation matrix, . The following discussion describes how the rotation matrix, or direction cosine
C
matrix, can be formulated.
General Rotation. Unit vectors for a system a, u ˆ a, are said to be carried into b, as = C u ˆ a. It can
u ˆ b
ba
be shown that a direction cosine matrix can be formulated by [30]
T
T
C = λλ + ( E – λλ )cos ψ – S λ()sin ψ (19.17)
λ
λ
T
where is the identity matrix, and represents a unit vector, = [λ 1 , λ 2 , λ 3 ] , which is parallel to the
E
axis of rotation, and ψ is the angle of rotation about that axis [30]. In this relation, ( ) is a skew-S λ
symmetric matrix, which is defined by the form
0 – λ 3 λ 2
S λ() = 0 –
λ 3 λ 1
– λ 2 λ 1 0

©2002 CRC Press LLC