In this paper a localization system of a passive 3-D coil is proposed and signal uncertainties due to the 3-D coil's arbitrary orientation are analyzed. The 3-D coil is excited by an alternating primary magnetic field. Geometrically distributed pick-up coils measure the 3-D coil's secondary field. By means of a simulated look-up table that assigns expected voltages from the pick-up coils to the positions of the 3-D coil, the position of the 3-D coil is deduced by a least-squares approach. A basic assumption is that the secondary field is invariant to the orientation of the 3-D coil. This allows a reduction of the computational effort for the look-up table generation and the table search during the localization phase since for each position the field distribution for only one orientation has to be calculated. However, the assumption of invariance to rotation is only valid for a dipole model. In this paper we investigate the localization error introduced by this assumption when using 3-D coils with a geometric extent in an inhomogeneous primary field. Optimized localization methods that decrease the statistical error are proposed. The theoretical results are verified with measurements conducted on a laboratory system.

Nearly every modern producing industry has a storage solution within their
logistics process. The most flexible way for picking items from a shelf is
still by human interaction. However, cost-intensive errors can occur when a
storekeeper picks an item from the wrong shelf. It is important to provide a
technology that gives the operator feedback in case of a wrong pick. Several
technologies already exist to support the operator. In pick-by-light systems
the shelf or container from which an item has to be picked is illuminated

In this paper we propose a magnetic-field-based approach for the localization
of an operator's hand in a shelf. The advantage of the magnetic approach is
that no direct line of sight is necessary and the installation costs are
relatively low. The estimation of the position and the orientation of a
magnetic source based on measured field values is a classic inverse problem.
The corresponding direct problem is the calculation of a field distribution
knowing the position, orientation and strength of the field sources, which
can be done using Maxwell's equations. The inverse problem, however, is often
ill-posed. That means it can have no solution or multiple solutions or that
the solution can be very sensitive to noisy data. In order to account for
that, regularization techniques are used

Several methods exist for the estimation of the position and the orientation
of a magnetic field source. The straightforward approach to estimate the
position of a field source is to compare a measured magnetic field
distribution at several points in space to a theoretical field distribution
that is calculated by means of a mathematical model for a certain position
and orientation of the field source. Analytical approaches as described in

All of the above methods need an active magnetic field source which they
mostly approximate by a simple dipole model. In contrast, we utilize a
passive field source, whose equivalent magnetic dipole moment is additionally
position dependent. Furthermore the model we utilize is more sophisticated
and takes the geometry of the current paths into account. The approach we
present in this paper is based on the GoalRef technology

For the adaption of that technology to our localization approach for
logistics, the new proposed system is extended with the ability to localize a
passive 3-D coil in three space dimensions in the proximity of a shelf.
Therefore a primary current loop is positioned around that shelf, which
generates a primary magnetic field. The field induces a current in a passive
resonantly tuned 3-D coil that is mounted on the hand of the operator. That
current flow in turn generates a secondary magnetic field that is measured by
pick-up coils around the shelf. From voltage measurements at these pick-up
coils the position of the 3-D coil is deduced by utilizing a look-up table
that is generated by the forward model presented in

The paper is structured as follows. In Sect.

In this section the experimental system setup and the basic localization method are explained.

Similarly to an LF RFID (low-frequency radio frequency identification)
system, our setup includes a reader that generates a continuous low-frequency
current in a primary exciter wire loop around a shelf as depicted in
Fig.

Shelf with the exciter wire, eight pick-up coils around the shelf and a 3-D coil in one compartment.

3-D coil with three passive resonantly tuned individual windings.

Different optimization algorithms as described in
Sect.

In our system the magnetic field strength of the localization object depends
on the object's position and orientation. Thus, a sophisticated mathematical
system model is needed to address that behavior. Such a model has been
presented in

If only a single coil was mounted on the localization object, a look-up table would need to include three entries for three dimensions of position and two dimensions of orientation. The third rotational degree of freedom is omitted due to rotation symmetry of a circular coil. Such a table with a fine enough grid to provide a good estimation accuracy to be generated in practice would require a high computational effort. Also, searching the whole table for the globally best solution with each localization step would take a long time, which makes real-time applications difficult. Another disadvantage of localizing only one passive coil in a primary magnetic field is that for any coil position there are orientations at which the coil is orthogonal to the primary field and hence cannot develop a secondary field. A localization would not be possible in such a case.

Since we are not interested in the orientation but only in the position of
the object, we propose an approach that makes the localization
computationally less expensive. Instead of using only one coil, we use a coil
system comprising three passive orthogonal coils. The basic assumption is
that the three coils provide an equivalent secondary magnetic moment

The reduced-complexity method described in Sect.

This section verifies analytically the assumption of the rotation-invariant secondary equivalent magnetic dipole field of a 3-D coil in a homogeneous primary magnetic field. In the following, complex values are underlined, the hat superscript stands for a peak value and the superscripts l and g on variables indicate the coordinate system in which they are expressed, where l stands for the local coordinate system of the moving 3-D coil and g for the fixed global coordinate system.

Let us assume a system comprising three orthogonal circular coils, all with
the same radius

In the practical case two assumptions made in Sect.

Simulated distributions of the induced voltage

In order to investigate the contribution of the primary field's inhomogeneity
to the voltage variance in the pick-up coil, the simulation model written in
Python which is described in

As a primary field source the rectangular exciting loop from the shelf
described in Sect.

Comparing Fig.

Simulated distributions of the induced voltage

The overall effect that takes both the inhomogeneity of the primary field as
well as the geometric extent of the 3-D coil into account is again
investigated using the Python simulation model. The calculations from
Sect.

Simulated distributions of the induced voltage

In order to be able to compare the contribution of the two individual effects
to the overall voltage variances, the variance coefficient

Simulated variance coefficient of the induced voltage

In Fig.

In this section six localization approaches are introduced, which are later
compared by means of simulations and measurements in
Sects.

The estimators presented in the following can be divided into two categories.
Stochastic estimators are built upon the statistical distribution of pick-up
coil voltages at each distance

Figure

Probability

The estimators described in this subsection are based on this distribution,
which is computationally very expensive to acquire. For each position (or
distance in the 1-D case) a large number

When choosing a fixed voltage

Having exact knowledge of the probability distributions, using an MLE is a common approach. However, if the distributions do not reflect the reality, the MLE will not provide optimal results.

Another possible estimator is the a posteriori mean (APM)

The prior knowledge term

This estimator is expected to have the smallest mean squared error.

The third estimator is the median of the a posteriori distribution from
Eq. (

This estimator is expected to have the least absolute error in the estimate.
Besides the mean and the median of the a posteriori distribution from
Eq. (

The precomputation of the distributions

In the precomputation phase only one instead of

For practical setups with more than one pick-up coil a suitable metric needs
to be selected for the comparison operation. A simple choice is the sum of
the squared differences, i.e., the square of the Euclidean distance. However,
as the induced voltage in the pick-up coils is a highly nonlinear function of
the 3-D coil's position, the Euclidean distance is often a suboptimal choice.
It tends to put the highest weight on the largest pick-up coil voltages,
which unfortunately are the ones with the largest coefficients of variation,
as shown in Sect.

In the following several possibilities for the forward solution

A simple approach to a forward solution for

A second possible forward solution approach is to calculate the induced
voltage

Accepting a higher computational effort, we can also calculate the induced
voltages for many different orientations of the 3-D coil for each position

In the following the localization errors of the six methods described in
Sect.

The results of the simulation are shown in Fig.

Simulated mean localization error at different positions

Simulated mean localization error at different positions

Especially in Fig.

The explanation of those effects can be found when looking at the individual
estimators' mapping functions of measured voltages

Except for the

Mapping of the measured voltage

Mapping of the measured voltage

In order to verify the theoretical distribution of the induced voltage, a
measurement has been conducted on the setup defined in
Sect.

Measured logarithmic variation coefficient

In a second analysis the estimators defined in Sect.

Measured mean localization error at different positions

The simulation and measurement results show that for the position estimation
in the 1-D case only positions close to the pick-up coil are sensitive to the
selection of a certain estimator. Further away from the pick-up coil noise is
the dominant influence on the localization error and equally influences all
estimators. Close to the pick-up coil the maximum likelihood estimator

Using more distributed pick-up coils and extending the 1-D localization to a 2-D or 3-D localization may have influence on the performance of the individual estimators. In contrast to the other estimators the stochastic estimators can include a priori knowledge, like not accessible positions due to the geometry of the shelf or due to the last known position combined with a movement model of the localization object. A weighting of the pick-up coil signals based on the signal-to-noise ratio may also be applied in the future.

The differences between the simulation and the measurements arise from the sensitive manual measurement setup, where the exact positioning of the 3-D coil is difficult. A big influence on the results may arise from different electrical and geometrical characteristics of the 3-D coils. In the simulations geometrically and electrically identical coils were assumed. However, in reality different coil areas, different qualities of the resonance as well as the exact resonance frequency itself have a huge influence on the amplitude of the secondary field strength of each of the three coils of the 3-D coil. On the one hand using high quality for the resonant circuit of the coil provides a higher current and hence a stronger secondary field. On the other hand a high-quality resonant circuit is very sensitive to the surrounding and may be detuned easily. A detuning of the resonance frequency leads to a drop in the field strength of that coil. It is assumed that this effect has a big influence on the localization accuracy and will therefore be investigated in the future.

In this paper the uncertainty of the field distribution from an arbitrarily oriented 3-D coil in a primary magnetic field is investigated analytically. The sensitivity of the field distribution against two effects, the inhomogeneity of the primary field and the geometrical extent of the 3-D coil, is presented. Three stochastic and three non-stochastic position estimators are proposed. Their theoretical absolute mean estimation errors are compared by simulations. Finally measurements on a laboratory setup for inductive localization are conducted to verify the field variation as well as the localization errors of the individual estimators. It is shown that from a certain distance between the field sensor and the 3-D coil, the effects of the field distribution on the localization error can be neglected. Future work will comprise analyses of the influence of the characteristics of the individual coils from the 3-D coil on the field distribution as well as the influence of the geometrical extent of the field sensors.

The corresponding data can be provided upon request to the author (rafael.psiuk@fau.de).

For an uncertainty analysis of the rotation effect on the induced voltages in
the system's pick-up coils, arbitrary orientations of the 3-D coil have to be
calculated. It is convenient to regard the two rotation angles

RP, AM and DC generated the main outcomes presented in this article. HB checked the structure and content of the article. HT and AH are supervising the corresponding field of research.

The authors declare that they have no conflict of interest.

The authors would like to thank Jörg Robert for his scientific advice, Ibrahim Ibrahim for fruitful discussions and Carina Schmitz for the support during the measurements.

This paper was edited by Rosario Morello and reviewed by two anonymous referees.