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A Prototype Optical Tracking System Investigation and Development - page 109 / 170

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7.4 Pose estimation using quaternions

95

the speed of convergence is negated by the extra processing required to calculate second

order derivatives.

Consider a function F (x) that defines an error residual where x is a vector in

n. The stan-

dard steepest descent algorithm begins with an estimate x0 of the minimum of F and the gradient of the function at x0 is calculated. The gradient is a vector of partial derivatives with respect to each of the degrees of freedom of x as given by

F (x) =

∂x ∂F (x) ∂xn . . . ∂F (x)

.

(7.4)

The negative of the gradient corresponds to the direction of steepest descent. The steepest descent algorithm moves from the initial estimate in the direction of steepest descent by a step size λ to produce a new value for x. This update step is expressed as

xn

1

= xn

λF (xn).

(7.5)

Moving in the direction of steepest descent generally yields a better estimate of the mini- mum of F (x). In the case where this is not a better estimate of the minimum, the step size λ is reduced and xn 1 is calculated again. This continues until the step size is below a predetermined value and the resulting x is assumed to be the minimum of F (x). Unfortu- nately, the method may converge to a local minimum rather than the global minimum and this is a good reason for considering another minimisation method in the future.

To minimise F (q) the steepest descent method must be modified to work in the unit quater-

nion domain.

ll unit quaternions lie on the unit sphere S3 in

4

.

new definition of the

gradient is required that depends on a parameterisation of S3. The update step in Equation 7.5 must also be modified to yield a new unit quaternion that corresponds to movement from the previous quaternion in the direction of steepest descent.

First the revised definition of the gradient is defined. The gradient is a vector of small movements in each of the degrees of freedom, i.e., in the original gradient definition these small movements are given by the partial derivatives of F (x) with respect to each degree of freedom. In the new definition the degrees of freedom are defined as small rotations around the three principal axes X, Y, Z expressed as quaternions. The function is evaluated at q and at the three quaternions qxq, qyq, qzq. These quaternions are small rotations around the X, Y, and Z axes measured from q. Using the angle-axis construction , i.e.,

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 ,

(7.6)

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