Mathematical Optimization of a Magnetic Ruler Layout with Rotated Pole Boundaries

Abstract

Magnetic rulers for measuring systems are either based on incremental or absolute measuring methods. Incremental methods need to initialize a measurement cycle at a reference point. From there, the position is determined by counting increments of a periodic graduation. Absolute methods do not need reference points, since the position can be read directly from the ruler. In the state of the art approach the absolute position on the ruler is encoded using two tracks with different graduation. To use only one track for position encoding in absolute measuring a pattern of trapezoidal magnetic areas is considered instead of the common rectangular ones. We present a mixed integer programming model for an optimal placement of the trapezoidal magnetic areas to obtain the longest possible ruler under constraints conditioned by production techniques, physical limits as well as mathematical approximation of the magnetic field.