Free-form surface fitting for precision coordinate metrology

Free-form surfaces are increasingly used in optical and mechanical devices due to their superior optical and aerodynamic properties. The form quality plays an essential role in the characteristics of a free-form component. In order to assess the form error, it is necessary to fit the measurement data with a nominal template or analytical function. This thesis focuses on investigating and developing appropriate fitting (matching) algorithms for different kinds of free-form surfaces.

A new algorithm called the Structured Region Signature (SRS) is proposed to provide a rough matching between the data and template. SRS is a global generalised feature which represents the surface shape by a one dimensional function. The candidate location which occupies the most similar signature with the measurement data is considered to be a correct matching position.

The fitted result is then refined to improve its accuracy and robustness. The widely used Iterative Closest Point technique suffers from a slow convergence rate and the local minimum problem. In this thesis the nominal template is reconstructed into a continuous representation using NURBS or radial basis functions if provided as a CAD model or a discrete-point set. The Levenberg-Marquardt algorithm is then applied to calculate the final result. The solution of the traditional algebraic fitting may be biased. The orthogonal distance fitting techniques can effectively overcome this problem. If the template function is explicit, the projection points can be updated simultaneously with the motion and shape parameters; whereas a nested approach is adopted to update the projection points and motion parameters alternately when the template is in a parametric form.

A proper error metric should be employed according to the distribution of the measurement noise, so that the solution can be guaranteed robust and unbiased. Simulation and experimental results are presented to validate the developed algorithms and techniques.