Bundle adjustment of multi-position measurements using the Mahalanobis distance

Precision three-dimensional metrology frequently involves the measurement of common points by several three-dimensional measurement systems. These can be laser tracking interferometers, electronic theodolites, etc. A new analysis technique has been developed that fully takes into account the uncertainty ellipsoid for each measurement point. This technique solves for the positions and orientations of the measurement systems while determining the optimal fit to each physical point. No a priori knowledge of the location and orientation of the measurement systems is required. An initial estimate for the location and orientation of the measurement systems is derived from the measurement data by assuming equal uncertainties for each data point. Then a merit function is minimized to determine the optimized location and orientation of the measurement systems and the weighted mean for the position estimates of the physical points. This merit function is based on the Mahalanobis distance from multivariate statistics and takes into account the particular shape and orientation of the three-dimensional uncertainty ellipsoid of each data point from each measurement system. This technique can utilize data from differing types of three-dimensional measurement systems including distance only and angle only measurements, evaluate the “strength” of a measurement configuration and accommodate missing data points from some of the measurement systems.