Page 164 - The Mechatronics Handbook

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set are referred to as Euler parameters, which are unit quaternions. There are many other possibilities,
but this four-parameter method is used in many areas, including spacecraft/flight dynamics, robotics,
and computational kinematics and dynamics. The term “quaternion” was coined by Hamilton in about
1840, but Euler himself had devised the use of Euler parameters 70 years before. Quaternions are discussed
by Goldstein [11], and their use in rigid body dynamics and attitude control dates back to the late 1950s
and early 1960s [13,24]. Application of quaternions is common in control applications in aerospace
applications [38] as well as in ocean vehicles [10]. More recently (past 20 years or so), these methods
have found their way into motion and control descriptions for robotics [34] and computational kine-
matics and dynamics [14,25,26]. An overview of quaternions and Euler parameters is given by
Wehage [37]. Quaternions and rotational sequences and their role in a wide variety of applications areas,
including sensing and graphics, are the subject of the book by Kuipers [19]. These are representative
references that may guide the reader to an application area of interest where related studies can be found.
In the following only a brief overview is given.
q
Quaternion. A quaternion is defined as the sum of a scalar, q 0 , and a vector, , or,
q = q 0 + q = q 0 + q 1 i + q 2 j + q 3 k ˆ
ˆ
ˆ
A specific algebra and calculus exists to handle these types of mathematical objects [7,19,37]. The
conjugate is defined as q = q 0 – q.
Euler Parameters. Euler parameters are normalized (unit) quaternions, and thus share the same
properties, algebra and calculus. A principal eigenvector of rotation has an eigenvalue of 1 and defines
the Euler axis of rotation (see Euler’s theorem discussion and [11]), with angle of rotation α. Let this
eigenvector be = [e 1 , e 2 , e 3 ] . Recall from Eq. (9.17), the direction cosine matrix is nowe T

(
C = ee T + Iee ) cosα − Se() sinα
–
T
e
S
where ( ) is a skew-symmetric matrix. The Euler parameters are defined as
cos ( α/2)
q 0
q = q 1 = e 1 sin ( α/2)
e 2 sin ( α/2)
q 2
e 3 sin ( α/2)
q 3
where

2
2
2
2
q 0 + q 1 + q 2 + q 3 = 1
Relating Quaternions and the Coordinate Transformation Matrix. The direction cosine matrix in
terms of Euler parameters is now
T
C q = q 0 –( 2 q q) E + 2qq T − 2q S q()
0
T
q
where = [q 1 , q 2 , q 3 ] , and E is the identity matrix. The direction cosine matrix is now written in
terms of quaternions
2 2 2 2
(
(
q 0 + q 1 – q 2 – q 3 2 q 1 q 2 + q 3 q 0 ) 2 q 1 q 3 – q 2 q 4 )
2
(
C q = 2 q 1 q 2 –( q 3 q 0 ) q 0 – q 1 + q 2 – q 3 2 2 q 2 q 3 + q 1 q 4 )
2
2
2 q 1 q 3 +( q 2 q 0 ) 2 q 1 q 2 + q 3 q 0 ) q 0 – q 1 – q 2 + q 3 2
(
2
2
2
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