Information-Theoretic Geometry Optimization and Physics-Aware Learning for Calibration-Free Magnetic Localization

Figure 1: Schematic representation of the 5-DOF localization system based on the magnetic dipole model. The state of the permanent magnet is defined by its center position \mathbf{p} and orientation vector \mathbf{n} . The displacement vector \mathbf{d}_{i} originates from the magnet towards the i -th sensor located at \mathbf{r}_{i} in the dual-layer sensor array. The staggered height configuration of the sensors (indicated by green and grey colors) is designed to improve the gradient information in the vertical direction.

Figure 2: Overall architecture of the proposed Phy-GAANet. (A) The overall network pipeline. The model accepts a physics-augmented input tensor ( 4\times 4\times 9 ), processes it through the MagPhyResNet backbone, and outputs position ( \mathbf{p} ) and orientation ( \mathbf{n} ) estimates via decoupled MLP heads. (B) Physics-augmented input stage. To handle the unique characteristics of magnetic signals, three feature maps are concatenated: Log Transform (for compressing the high dynamic range), Saturation Mask (indicating clipped sensor readings), and Sensor Geometry (providing spatial coor

Figure 4: The custom-designed hardware validation platform. The system features a dual-layer 4\times 4 sensor array positioned at Z=20 mm and Z=180 mm, a calibration board located at Z=40 mm, a set of calibration pillars with varying heights, and an MCU-based acquisition unit. This configuration enables selective activation of 16-sensor subsets for a controlled comparison between planar and staggered spatial layouts under identical environmental conditions.

Figure 7: Performance comparison at the critical boundary heights Z=50 mm and Z=150 mm. Under the Staggered Split Array topology, both heights correspond to near-field regimes where partial sensor saturation occurs and the dipole model begins to break down. Panels (a) and (b) compare the mean position and orientation errors, respectively, across representative baseline methods and the proposed Phy-GAANet.

Figure 3: Comparative evaluation of Fisher Information-based observability metrics for three distinct sensor array configurations over a 200,000-sample validation set with \sigma=10\,\mu\text{T} noise. (a) and (b) show the theoretical lower bounds for position and orientation RMSE (CRLB); lower values indicate higher achievable precision. (c) displays the condition number \kappa , representing numerical stability. (d) illustrates the minimum eigenvalue \lambda_{\min} , quantifying the information density in the least observable direction. While the “Single-Split” array (green) attempts to intr

Figure 8: Top row: comparison of three rigid external sensor layouts, including the planar baseline, the staggered split baseline, and the shell-constrained free optimum. Bottom row: corresponding normalized performance profiles under a shared logarithmic y-axis, where larger values indicate better performance. The four metrics are mean log-determinant, median minimum eigenvalue, position precision, and orientation precision, with the latter two defined relative to the planar baseline through the inverse CRLB.