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. Note to Practitioners—This paper was motivated by the challenge of tracking flexible surgical tools, such as heart catheters and endoscopes, during medical procedures. Many tracking algorithms become more complex when they estimate all six degrees of spatial information, including the tool’s rotation around its own axis. For tasks controlled through the tool centerline and pointing direction, the feedback state comprises position and direction. In this paper, we suggest a simplified localization framework that focuses on the tool’s 3D position and its pointing direction. By removing axial rotation from the estimated state, the estimator becomes simpler and more stable in the tested configurations. This approach can also help engineers use smaller or fewer sensors while preserving accuracy in the evaluated sensor layouts. Our results show that a two-stage calculation process, which separates the location search from the direction search, achieves accurate and numerically stable estimator performance. Currently, the system assumes a relatively clean magnetic environment. In future research, we will address how to compensate for interference caused by nearby metal objects or operating room equipment. Beyond surgery, tracking focused on the axis may also be useful for continuum robots used in confined industrial spaces where knowing the tool’s pointing direction is the primary requirement.