# A Multi-axis Compact Positioner with a 6-coil Platen Moving Over a Superimposed Halbach Magnet Matrix

## Abstract

A multi-axis compact positioner is designed and implemented in this thesis. The single-moving-part positioner is designed to move in the magnetic field generated by a superimposed concentrated-field permanent magnet matrix. The compact positioner is primarily for the stepping and scanning applications that require 3-DOF planar motions. In which, the travel ranges in two orthogonal directions are on the order of 100 mm. The moving platen, which has the size of 185.4 mm x 157.9 mm and weighs 0.64 kg, mainly comprises of a plastic frame and six copper coils. It is actuated in the horizontal plane by flowing six independent electric currents into the coils. The platen is supported against gravity by three air bearings.

Force calculation is based on the Lorentz force law. With a current-carrying rectangular coil placed in the magnetic field of the supper-imposed Hallbach magnet matrix, the force acting on the coil is calculated by volume integration. The distances between the longer sides and between the shorter sides of the rectangular coil are designed to fit a half pitch and one pitch of the Hallbach magnet array, respectively. Therefore, the volume integration is simplified considerably. The force-current relation for the entire platen with six coils is derived.

Three Hall-effect sensors are attached to the moving platen to measure the magnetic flux densities at the center points of the sensors. The position of the moving platen is determined by the field solution of the magnet matrix and the magnetic flux densities sensed by the Hall-effect sensors. A new discrete PID-like controller is proposed and tested. For the step responses with the step sizes within 1000 micrometers, the overshoots and the steady state errors are negligible. The achieved velocity in x is 10.50 cm/s and in y is 16.25 cm/s, respectively. The achieved acceleration in x is 43.75 cm/s^2 and in y is 95.59 cm/s^2, respectively. The achieved travel ranges are 15.24 cm in x, 20.32 cm in y, and 0.21 rad in the rotational motions about the vertical axis. The positioning resolution in x and y is 8 micrometers with the rms positioning error of 6 micrometers. The positioning resolution in rotation about the vertical axis is 130 microrad.