To address the critical bottleneck of residual errors arising from intrinsic sensor imperfections and array-level misalignment in magnetic gradient tensor (MGT) systems, this paper aims to propose a systematic two-step calibration framework that overcomes the limitations of conventional methods, including underutilization of array measurement data and susceptibility to local optima in parameter estimation.
This paper constructs a sequential two-step calibration framework. In the first step, the firefly algorithm (FA) is used to independently solve for each sensor’s intrinsic error parameters (scale factors, biases and non-orthogonality angles). In the second step, with one sensor designated as the reference, FA is applied to minimize the vector discrepancies between the remaining sensors and the reference, thereby determining the inter-sensor misalignment matrices. The core innovation lies in integrating FA with a full-sensor data correction strategy that fully uses measurements from all array elements, overcoming the inherent limitation of conventional tensor-norm minimization methods, which rely on partial data and fail to uniquely determine array misalignment parameters.
Simulation and experimental validation on a cross-shaped MGT array demonstrate that, in simulation, the post-correction magnetic field strength error is reduced to below 3.69% of its initial value; FA reduces the maximum relative error in bias estimation from 41.10% (using particle swarm optimization) to 1.67%. For array misalignment calibration, the proposed full-data utilization strategy successfully recovers all misalignment parameters for all sensors, whereas the conventional tensor-norm minimization method yields estimation errors of up to several hundred percent for certain sensors. Experimental results confirm that the system measurement error is suppressed to 16.89% of the pre-calibration level, with axis-specific error suppression ratios ranging from 5.37% to 16.89%.
This paper introduces, for the first time, the integration of the FA into a two-step calibration framework for MGT systems, alongside a novel reference-sensor-based full-data utilization calibration strategy. In contrast to conventional tensor-norm minimization methods, this approach directly enforces vector consistency between all sensors and a reference sensor, enabling full utilization of the complete measurement data set and ensuring physical consistency of the entire array. The proposed framework is not limited to cross-shaped arrays but is extendable to arbitrary magnetometer array geometries, establishing a new calibration paradigm for high-precision MGT measurements.
