A Real-Time Magnetic Adhesion Force Estimation Method for Wall-Climbing Robots Equipped with Halbach Permanent Magnet Arrays

This paper presents a real-time magnetic adhesion force estimation framework for wall-climbing robots equipped with Halbach permanent magnet arrays (PMAs) and air-gap–adjustable mechanisms. Accurately computing the magnetic adhesion force between a PMA and a large ferromagnetic surface is challenging due to the nonlinear magnetization behavior of soft magnetic materials and the strongly coupled, highly nonuniform magnetic fields generated by Halbach arrays. Conventional analytical models fail to capture these effects, while finite element methods (FEM) incur prohibitive computational cost for real-time applications. To address this, we propose an analytical magnetic-force estimation model based on the magnetostatic MoI (Method of Images), which replaces the unknown magnetization inside the steel plate with an equivalent image magnet distribution that satisfies boundary conditions at the air–steel interface. The method avoids solving complex magnetization in soft magnetic media and enables a unified force computation for arbitrarily oriented magnet elements. Additionally, complex Halbach PMA geometries are approximated through cuboid-element segmentation into cuboid magnet array, allowing efficient force evaluation. Comparative studies demonstrate that the proposed method achieves accuracy comparable to FEM while reducing computation time by several orders of magnitude. Experimental validation using a linear Halbach array and a large steel plate proved that the framework can reliably estimate magnetic adhesion force across varying air-gap distances, meeting the real-time requirements of air-gap–adjustable wall-climbing robots.