exist more observational equations than unknowns, and a least squares
adjustment leading to a 2n+7 by 2n *7 system of normal equations can be
performed. Although the size of the normal equations increcses linearly with
the number of measured points, their solution presents no difficulties, even on
a small computer. This is because they possess a patterned coefficient matrix
that can be exploited to collapse the system to one of order 7x7, involving
only the parameters of the comparator. Once these have been determined, an
independent, four station, least squares trilateration can be performed to
establish the coordinates of each point. Details of the data reduction are to be
found in the reference cited earlier. Suffice it to say here, that the internal
contradiction resulting from the redundancy of the measuring process can be
exploited to effect an accurate calibration of the parameters of the comparator
for the particular plate being measured. Hence, the designation of the
instrument as a self-calibrating multilaterative comparator.
THE COMPUTER PROGRAM
If developed fifteen years ago, the Multilaterative Comparator would have
been little more than an interesting academic curiosity, for no one could have
tolerated its computational requirements. Accordingly, the digital computer, so
commonplace today, is to be regarded as an integral part of the comparator
system. For this reason, a fully documented FORTRAN program is provided with
the comparator. The program is designed so that a minimum of modification is
required to adapt it to almost any computer. One version of the program is
designed specifically to run on a minimal computer configuration such as an
IBM 1130 with card input. Another version is designed for medium to large scale
computers. Both versions feature automatic editing and rigorous error propagation.
Typical running time on an IBM 7094 for the reduction of a plate containing 50
measured images is well under one minute.
In addition to producing the final coordinates of the measured points, the
program generates the four measuring residuals for each point and the rms closure
of trilateration.
RESULTS
The comparator has undergone extensive testing over the past eighteen
months. Ten have already been placed in operation and are to be followed by
another ten by the end of the year. In comparisons employing plates with well
defined images measured both on a conventional, two screw comparator and on
the multilaterative comparator, an rms agreement of final coordinates ranging
between +2 and +3 microns is usually obtained after allowance has been made
for a translation and rotation of the one coordinate system to conform to the
other. If one were to assume that the discrepancies between the results are
equally attributable to both comparators, one would conclude that the rms
accuracy of the coordinates produced by the multilaterative comparator generally
range between 41.4 and +2. 1 microns, a range compatible with typical setting
accuracies. Thus external evidence indicates that the multilaterative comparator
is capable of producing accuracies at least comparable with those produced by
conventional, one micron comparators.
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