Failure of Numerical modeling of 3-D Position Reconstruction from 3-Axial Planar Spiral Coil Sensor Sensitivity due to Existence of Quadratic Terms

A sensitivity profile of a planar spiral coil sensor (PSCS) is proposed and is used to generate the relation of 3-D position of object observed using three PSCSs, one in each x, y, and z axis to the sensors response. A numerical procedure using self consistent field-like method to reconstruct the real position of observed object from sensor sensitivity is presented and the results are discussed. Unfortunately, the procedure fails to approach the desired results due to the existence of quadratic terms.

💡 Research Summary

The paper proposes a mathematical model for the sensitivity of a planar spiral coil sensor (PSCS) and attempts to use three such sensors, each oriented in the xy, yz, and zx planes, to reconstruct the three‑dimensional position of a small spherical object. Assuming cylindrical symmetry, the authors describe the radial sensitivity with a Gaussian function (S₀ exp(‑ρ²/2σ²)) and the axial sensitivity with an exponential decay (S₀ exp(‑γh)). By combining these expressions they obtain a general sensitivity formula (Eq. 3) that depends on the radial distance ρ and the axial distance h between the object and a given sensor.

Three sensors are placed orthogonally, and the output of each sensor (S_ij) is expressed as a function of the object coordinates (x, y, z) via Eq. 5. To invert this relationship, the authors introduce auxiliary variables Sₓ, S_y, S_z and, after logarithmic manipulation, derive three equations (Eqs. 9‑11) linking the measured signals to the coordinates. Shifting the origin to (L/2, L/2, L/2) simplifies the system to Eqs. 12‑14, which are then merged into a single nonlinear equation (Eq. 15). This equation is intended to be solved iteratively using a self‑consistent field‑like algorithm: start with an initial guess (x₀, y₀, z₀), compute the next estimate via Eq. 15, and repeat until the changes Δx, Δy, Δz fall below a preset tolerance.

Numerical experiments are performed with L = 2 cm, σ = 0.5 cm, S₀ = 1 mV, and γ = 50 m⁻¹. Figure 3 shows the sensitivity profile, and Table 1 lists the sensor outputs for nine test positions spanning the unit cube. The data confirm the intuitive expectation that a displacement along a given axis is primarily sensed by the sensor whose plane is perpendicular to that axis (e.g., motion in x is reflected mainly in S_jk).

When the iterative inversion is applied, however, the reconstructed coordinates (Table 2) differ dramatically from the original test points. The algorithm often yields negative values or completely erroneous magnitudes. The authors attribute this failure to the presence of quadratic cross‑terms (x·y, y·z, z·x) in the mapping from (x, y, z) to (Sₓ, S_y, S_z). These terms cause the forward mapping to be non‑injective: different sign combinations of the coordinates produce identical sensor outputs, analogous to the loss of sign information in the cosine function over a restricted domain. Consequently, the inverse problem is ill‑posed and the self‑consistent iteration converges to spurious solutions.

The conclusion emphasizes that the direct implementation of the proposed self‑consistent method cannot uniquely recover the 3‑D position because of the quadratic terms. The paper suggests that either the sensitivity model must be reformulated to avoid such terms, or alternative reconstruction strategies—such as nonlinear optimization, Bayesian inference, or the addition of more sensors—should be explored to resolve the non‑uniqueness.

Overall, the work provides a clear description of a PSCS‑based positioning concept, identifies a fundamental mathematical obstacle, and points toward future directions for achieving reliable three‑dimensional localization with planar spiral coil sensors.