2-D Magnetic Localization Newton–Raphson-Based Method Considering Measurement Uncertainties for Smart Surface Applications

Magnetic localization techniques present some interests for microrobotics applications using smart surfaces due to their inherent advantages, such as integration ability or low cost. However, at the microscale, these methods often encounter challenges to perform with high precision in real scenarios, because of the limitation of calculation algorithms (numerical stability, convergence, and approximation methods) and the impact of measurement uncertainties (misalignment, sensor performance, and so on). This article presents a new 2-D magnetic localization method based on a 3-D Hall effect sensor array system, in order to localize a cubic magnet. The proposed approach combines an efficient numerical resolution algorithm [Newton–Raphson (NR)] with precise analytical models of the magnetic field of the magnet to brave these measurement uncertainties and to achieve high-resolution 2-D magnetic localization. Through performance assessments from simulation calculations and experimentations, the operation and efficiency of the proposed method are demonstrated. Results indicate good performance of the NR method with micrometric accuracy, even when there are measurement uncertainties in real scenario. An experimental test realized with a prototype of 3-D Hall effect sensor array confirms the efficiency of the 2-D magnetic localization with an averaged 12.73- $\mu $ m localization error. A comparative analysis shows that this new magnetic localization method is well placed in the hierarchy of existing methods that estimate the 2-D position of a magnet by using a Hall effect sensor array system.