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A Prototype Optical Tracking System Investigation and Development - page 110 / 170

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the quaternion qx is defined as

qx = cos

θ

+ 1 × sin

θ

2

2

i + 0 × sin

θ j + 0 × sin 2

θ 2

k = cos

θ 2

  • +

    sin

θ 2

i.

(7.7)

Similarly, qy, qz can be calculated by substituting the Y, and Z axes for (l, m, n), i.e., (0, 1, 0), (0, 0, 1). Here θ represents a very small angle. This leads to a new definition of the gradient

F (q) =

l i m θ 0

F (qxq) F (q)

θ

F (qyq) F (q)

θ

F (qzq) F (q)

θ

.

(7.8)

Each element is the difference between the function’s value moving in one of the three directions and the function at q. The gradient vector is normalised before it is used in the following steps.

new update step must now be defined that uses the new gradient vector. The update step must produce a new quaternion based on the gradient vector and the step size. The quaternion must represent a move in the direction given by the gradient vector. Each e l e m e n t o f t h e g r a d i e n t v e c t o r F 1 ( q ) , F 2 ( q ) , F 3 ( q ) r e p r e s e n t s a p r o p o r t i o n t o r o t a t e a r o the X, Y, and Z axes. These can be expressed as angles of rotation around the principal axes given by u n d

θ x = λ F 1 ( q ) ,

θ y = λ F 2 ( q ) , θ z = λ F 3 ( q ) .

(7.9)

Using the angle-axis construction again, these rotations can be expressed as three new

quaternions q

x, q y, and q zwhere

q

x

= cos

θx

2

  • +

    1 × sin

θx

i = cos

1 λF (q)

+ 1 × sin

1 λF (q)

2

2

2

i,

(7.10)

etc. Now a quaternion corresponding to the overall rotation defined by these three rota- tions is defined as

q

=q

xq

y

q

z

.

(7.11)

The steepest descent algorithm is altered so that it has the form

qn+1

= qnq .

(7.12)

This states that the next quaternion is equal to the previous quaternion multiplied by the rotation quaternion q that rotates q in the direction of steepest descent. The step size λ is

reduced whenever moving in the direction of the gradient does not produce a lower value of F . Once the step size is below a threshold, the algorithm stops and the current estimate

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