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A Prototype Optical Tracking System Investigation and Development - page 36 / 170

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22

Theory and Design

nalysis

identities for multiplying i, j, and k are

i 2 = j 2 = k 2 = i j k =

1

(3.8)

q1q2

=

2 2 (w1 + x1i + y1j + z1k)(w2 + x2i + y j + z k)

=

w1w2 +

12 +y z

12 z y )i +

12 xz

12 + z x )j +

12 xy

12 y x )k

12 (w x + x1w2

12 xx

12 (w y + y1w2

12 yy

12 (w z + z1w2

12 zz

and from these identities, the remaining products such as ij = k can be derived.

Using Equation 3.6 and Equation 3.8, the quaternion product of q1, q2 is

(3.9)

The addition of quaternions q1, q2 is defined in the same manner as vectors, i.e.,

q1 + q2 = w1 + w2 + (x1 + x2)i + (y1 + y2)j + (z1 + z2)k.

(3.10)

Quaternion rotations can be constructed in terms of an axis and a rotation angle. nion q can be defined using a rotation around an axis (l, m, n) by an angle θ using

quater-

q = cos

θ

2

+ l s i n θ 2 i + m s i n θ 2 j + n s i n θ 2 k .

(3.11)

For Equation 3.11 to hold, the axis (l, m, n) must be normalised to length 1.

Similarly an axis and angle can be determined from the components of a quaternion q using

w = cos

θ

x2 + y2 + z2 = sin

2 θ

2

p

(3.12)

that was derived from the definition of a quaternion. This leads to an angle and axis of

θ = 2 cos

x l = 1

1 w2 w, ,

y m = 1

w2

,

z n = 1

w2

.

(3.13)

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