Particle importance analysis: A fast localization method under indeterminate magnetic sources

For magnetically actuated robots driven by external magnetic sources (EMSs), rapid and stable localization of the internal permanent magnet (IPM) is essential for closed-loop control. Existing sensor-array-based localization methods require a fixed model with an offline-calibrated dipole moment to separate the magnetic field generated by the IPM from the measured total field. However, model accuracy varies with the distance to the sensor array, leading to inaccurate separation and unstable localization. Additionally, nonlinear optimization is typically used for localization with sensor arrays, which is computationally inefficient. We propose a sampling-based method that jointly estimates the dipole moments and the IPM position, thereby eliminating the need for magnetic field separation. Given the EMS positions and the measured total field, our particle importance analysis efficiently identifies the sampled particle that most closely approximates the dipole position and concurrently estimates the optimal dipole moments. Adaptive particle sampling enables high-accuracy localization using sparsely sampled particles, thereby improving computational efficiency. We validated the approach experimentally with EMSs actuated by manipulators and electromagnetic systems. With a 72-sensor array and the IPM located 20 cm above the sensor plane, the method achieved an average computation time of 1.2 ms and localization errors of 4.70 mm and 4.17°. Compared with an optimization-based method, our method reduced the maximum localization error by 59% and improved computational efficiency by a factor of 30. The method reliably localized in permanent magnet, electromagnetic, and hybrid magnetic fields. Finally, we applied the method to the simultaneous localization and actuation of a magnetic endoscope, validating its feasibility in a colon phantom.