Contribution of nonlinear optimization to the determination of measurement uncertainties

Much work has already been realized concerning the association of surfaces to points digitized with a Coordinate Measuring Machine (CMM). The developed procedures are usually based on the minimization of the distance between the measured points and the geometric element. They depend on the criterion used for the optimization. The function to minimize is always non-linear. To reduce the computing time, it is therefore transformed usually into linear variational equations, which are solved by an iterative process. Such algorithms require however to define initial parameters close to the final solution. The method presented here is based on a nonlinear least squares optimization. It does not require any coordinate transformation and is not sensitive to the selection of initial intrinsic parameters. It applies to all the classical surfaces, i.e. lines, planes, circles, cylinders, cones and spheres. In accordance with quality standards, the uncertainties of the estimated parameters are also defined. These values are derived from the residue between the measured points and the optimized associated surface.