Without Differential method for determining the points of tangency with rotation surfaces when measured on CMMs

Progress in the advanced branches of technology has caused the need to manufacture parts of highly complex shapes that are often found through experimenting and only later is described with mathematical models. Most coordinate-measuring machines (CMM) now available have their capability based on implementation of the Cartesian coordinate system. This feature has become a bottleneck on the way to creating a new generation of CMM. A major breakthrough in this direction was the creation of a six-axis co-ordinate measuring machine. With six-axis probe control hexapods can control hard-to-reach surfaces of products and make measurements with output that is 2 to 5 times higher than the conventional CMM solutions. Absence of an integrated mathematical foundation remains the greatest problem, therefore multiple separate mathematical models are necessary for the control process. The “minimum volume of measured space” principle is going to become basis for science foundations of high-output control of complex surfaces to be created. According to the principle, top accuracy and output of control can be simultaneously achieved when the measured points cloud is located within encircling equidistant surfaces of minimum volume. Real-life realization of this approach rests on unique iterative methods of problem solving. The results of research will help industrial manufacturers adopt coordinate-measuring machines more intensively, also making it easier to adopt production technology considering information on tolerance in size, shape and surface layout. The article deals with without a differential method of finding points of contact with rotation surface when controlling on the CMM.

I. Introduction

PROGRESS in the advanced branches of technology has caused the need to manufacture parts of highly complex shapes that are often found through experimenting and only later is described with mathematical models, such as dual polynomial of high degree. Most coordinate-measuring machines (CMM) now available have their capability based on implementation of the Cartesian coordinate system [1]–[5]. This feature has become a bottleneck on the way to creating a new generation of CMM. Current potential of the conventional three-axis CMM is just not enough to measure complex dimensional surfaces.