Numerical and Experimental Estimation of Measurement Uncertainty in Magnetic Field Mapping

Abstract

This article proposes a general method to model and simulate the process of magnetic field mapping; aim of our work is the combination of the effect of the bench position uncertainty with the magnetic measurements’ uncertainty. The method is based on the study of positioner kinematics using a multi-body system approach. The geometrical errors of the manipulator, including the manufacturing tolerances and the assembling nonidealities are included in the model using the homogeneous transformation matrix (HTM), to numerically estimate the end-effector positioning uncertainty <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U_{p}$ </tex-math></inline-formula> . The positioning uncertainty is then combined with the magnetic measurement uncertainty using the magnetic field gradient as a sensitivity coefficient; in presence of strong field nonlinearities, the combination can be performed using Monte Carlo (MC) simulations in order to estimate how <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U_{p}$ </tex-math></inline-formula> propagates to the magnetic measurement uncertainty <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U_{f}$ </tex-math></inline-formula> at different positions. The method has been validated in the specific case of the compact field mapper (CFM), a bench featuring a Cartesian robot and a triaxial Hall sensor used to measure flux density maps in the accessible region of interest of the magnets of the Swiss Light Source, with an uncertainty below 0.5 %. The method allowed us to define, during the design phase, the characteristics of the positioning system (i.e., the mechanical positioning uncertainty <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U_{p}$ </tex-math></inline-formula> ) in order to obtain the desired magnetic measurement uncertainty <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U_{f}$ </tex-math></inline-formula> . Simulations and experiments in the case of a reference quadrupole are presented and discussed.