Magnetic localization during manipulation by two robotized permanent magnets

Localization methods for magnetically actuated medical robots have long been a topic of research, as they are fundamental to closed-loop control and delivery of functionalities. However, magnetic localization has mainly been linked to robots under a single permanent magnet control. With the release of multi-magnet actuation systems for increased control and manipulability, new localization methods are needed to account for the added magnetic field sources. This paper presents a six degree of freedom localization method for magnetically actuated robots under two external permanent magnets control. The approach relies on the measurements of an accelerometer and gyroscope for the estimation of orientation in the Special Orthogonal Group SO (3), and the measurements of the actuating magnetic fields for the estimation of position. The observability analysis of the system is presented, and the relationship between the external permanent poses and conditioning of the system is explored. Additionally, a calibration procedure to determine the relative poses between the two external permanent magnets is presented where the path that the magnets travel is optimized for the best results. Lastly, the localization method was implemented in a magnetic soft continuum robot and achieved positional average errors of 3.5 mm in norm, and orientation errors of 2.5°, 1.5°, and 2.8° around x , y , and z , respectively.