Decoupled 5-DoF Magnetic Localization: Position and Principal Axis Estimation via Gradient Invariants

Magnetic localization tracks slender robotic instruments through tissue without line-of-sight, making it attractive for catheter guidance in confined anatomy. In many tasks, navigation uses the distal position and the pointing direction of a functional axis as the feedback state. Existing magnetic localization methods often improve observability by adding sensing hardware, active excitation, auxiliary modalities, or full pose optimization on ℝ³ × SO(3). When feedback uses position and functional axis direction, full rotation recovery creates a task mismatch that can increase computation and sensitivity to initialization, local minima, and sensor geometry. To address this mismatch, we propose a decoupled magnetic localization framework with two stages that uses compact magnetometer array measurements to estimate position and principal axis direction. First, the local gradient tensor is extracted from differential magnetic measurements, and eigenvalue spectra invariant to rotation are used for position estimation. Second, to resolve the principal axis, we introduce a Spherical Successive Linearization (SSL) algorithm that iteratively optimizes proximal convex surrogates on the unit sphere. A spectral heuristic initialization and a stable closed-form proximal update reduce the online principal axis search to a unit sphere problem and provide convergence to a first-order stationary point under a standard proximal gain condition. Furthermore, we establish theoretical observability limits that map sensor geometry to spatial estimation uncertainty using the Fisher information matrix (FIM). Experimental validation across representative sensor layouts and trajectories demonstrates millimeter-level position accuracy and stable principal axis recovery, supporting estimator use in downstream robotic navigation pipelines.