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A Prototype Optical Tracking System Investigation and Development - page 108 / 170

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108 / 170

94

Hub Design

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Figure 7.7 Intersection of lines with a clock wise camera rotation of 90 degrees with respect to the world frame.

7.4.3

Minimisation

The function F (q) equals zero when q corresponds to the rotation between the camera co- ordinate frame and the world coordinate frame. To find a q that minimises F (q) a minimi- sation algorithm is required. There are many numerical optimisation methods including the method of steepest decent, conjugate gradient decent, Newton’s method, the Gauss- Newton method, and the Levenberg-Marquardt method. n overview of these methods is available in, for example, Methods for Non-Linear Least Squares Problems [91]. These

methods generally minimise a vector valued function in

n

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This is a problem as F (q) is

defined over the unit quaternions that form a 3D manifold in

4. Schmidt and Niemann

propose a method to apply optimisation techniques to a function of unit quaternions by de- veloping a mapping between three rotation parameters and the corresponding unit quater- nion [92] . In separate work, Ude [93] derives a method for applying the Gauss-Newton method using unit quaternions.

The method described here modifies the method of steepest descent to allow it to work with unit quaternions. The above algorithms were not used because they were discovered after this algorithm was implemented. The method of steepest descent was chosen due to its ease of implementation and because only first order derivatives of F (q) are used. Eval- uation of F (q) is computationally expensive meaning that calculation of first and second order derivatives is also expensive. In the future this minimisation could be replaced with a Gauss-Newton minimisation which has better convergence properties than the method of steepest descent when close to the solution [92]. However, as the Gauss-Newton method uses second order derivatives it should be examined to see whether the improvement in

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