The iterative calculation of the Cartesian geometry of general non-Cartesian structures

This paper describes an iterative scheme for the calculation of the Cartesian coordinates of the nodes of any rigid structure from a knowledge of internodal distances and angles. Since internodal distances can be determined accurately by laser interferometry, it is possible to derive coordinates to very high precision. The techniques constitute analytical vector and matrix methods which have been programmed in a general way to provide generic software capable of finding iteratively the Cartesian geometry of complex structures such as non-Cartesian machines. It has proved possible to include certain constraints such that specified nodes should not move in certain Cartesian directions and specified connecting lines should remain horizontal or vertical, etc. The method requires an initial estimate of the Cartesian coordinates of the nodes and then adjusts the coordinates until calculated internodal distances and angles are within tolerance. No reprogramming is necessary to analyse a new structure. Full details of the mathematical derivations are included, as are details and results of the application of the method to the Stewart platform.