Position and orientation estimation of a permanent magnet using a small-scale sensor array

A maximum likelihood estimator for the determination of the position and orientation of a permanent magnet using an array of magnetometers is presented. To reduce the complexity and increase the robustness of the estimator, the likelihood function is concentrated and an iterative solution method for the resulting low-dimensional optimization problem is presented. The performance of the estimator is experimentally evaluated with a miniaturized sensor array that consists of 32 magnetometer triads. The results are compared to the Cramér-Rao bound for the estimation problem at hand. The comparisons show that the performance of the estimator is close to the Cramer-Rao bound; hence, the estimator is close to optimal. Further, the results illustrate that even with a matchbox-sized array and a small magnet with a dipole moment that has a magnitude of 7 2 · 10−3 Am2 the position and orientation of the magnet can, within a 80×80×80 mm volume, be estimated with a root mean square error of less than 10 mm and 15 deg, respectively.

[1]  Isaac Skog,et al.  Inertial sensor arrays — A literature review , 2016, 2016 European Navigation Conference (ENC).

[2]  Maysam Ghovanloo,et al.  Joint Magnetic Calibration and Localization Based on Expectation Maximization for Tongue Tracking , 2018, IEEE Transactions on Biomedical Engineering.

[3]  B. Lütkenhöner.  Dipole source localization by means of maximum likelihood estimation. II. Experimental evaluation. , 1998, Electroencephalography and clinical neurophysiology.

[4]  P. Dario,et al.  Capsule Endoscopy: From Current Achievements to Open Challenges , 2011, IEEE Reviews in Biomedical Engineering.

[5]  J. J. Abbott,et al.  Optimal Permanent-Magnet Geometries for Dipole Field Approximation , 2013, IEEE transactions on magnetics.

[6]  S. Ando,et al.  A Closed-Form Formula for Magnetic Dipole Localization by Measurement of Its Magnetic Field and Spatial Gradients , 2006, IEEE Transactions on Magnetics.

[7]  Sailes K. Sengijpta.  Fundamentals of Statistical Signal Processing: Estimation Theory , 1995 .

[8]  G. Golub,et al.  Separable nonlinear least squares: the variable projection method and its applications , 2003 .

[9]  Kaj Madsen,et al.  Methods for Non-Linear Least Squares Problems (2nd ed.) , 2004 .

[10]  B. Lutkenhoner.  Dipole source localization by means of maximum likelihood estimation. II. Experimental evaluation , 1998 .

[11]  P. Stoica,et al.  On the concentrated stochastic likelihood function in array signal processing , 1995 .

[12]  K. V. S. Hari,et al.  Classification of vehicles using magnetic dipole model , 2012, TENCON 2012 IEEE Region 10 Conference.

[13]  Fredrik Gustafsson,et al.  Silent Localization of Underwater Sensors Using Magnetometers , 2010, EURASIP J. Adv. Signal Process..

[14]  B. Lütkenhöner.  Dipole source localization by means of maximum likelihood estimation I. Theory and simulations. , 1998, Electroencephalography and clinical neurophysiology.

[15]  Fredrik Gustafsson,et al.  Single target tracking using vector magnetometers , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[16]  M. Perlmutter,et al.  High-performance, low cost inertial MEMS: A market in motion! , 2012, Proceedings of the 2012 IEEE/ION Position, Location and Navigation Symposium.

[17]  L. Scharf,et al.  Statistical Signal Processing: Detection, Estimation, and Time Series Analysis , 1991 .

[18]  Alessio De Angelis,et al.  Analysis of simultaneous 3D positioning and attitude estimation of a planar coil using inductive coupling , 2017, 2017 IEEE International Instrumentation and Measurement Technology Conference (I2MTC).

[19]  M Birsan,et al.  Recursive Bayesian Method for Magnetic Dipole Tracking With a Tensor Gradiometer , 2011, IEEE Transactions on Magnetics.

[20]  Joakim Jaldén,et al.  On-the-fly geometric calibration of inertial sensor arrays , 2017, 2017 International Conference on Indoor Positioning and Indoor Navigation (IPIN).