NAMIC Wiki - User contributions [en]

5DOF Electromagnetic Tracker Notes

2022-10-13T13:06:12Z

Traneus: 2. changed "plane of the array" to "plane of the single coil".

5DOF (five-degree-of-freedom) electromagnetic trackers track a single dipole coil receiver against a spread-out array of transmitter coils. Because the single coil is symmetrical about its axis, orientation roll about the single coil's axis cannot be tracked. Orientation latitude and longitude can be tracked, as can X Y Z position.

The single receiver coil does not need to be characterized, as long as the coil is small enough to be a dipole, as the coil's gain is tracked as part of the position-and-orientation algorithm.

The single coil can be used as a transmitter, and the array as receivers, though it can be difficult to get enough signal.

There are many choices for the spread-out array:

1. Six or more coils pointed in various directions. Any particular arrangement needs to be analyzed and simulated for trackability.

2. Twelve coils in a field-and-gradient array. Think of the single coil as a transmitter. The twelve-coil array then measures the field and gradient from the single coil. By electromagnetic reciprocity, single coil as receiver works the same. There is a direct analytic solution for position using this array. One problem is that the gradient matrix is singular when the array is in the equatorial plane of the single coil. See the following paper for discussion and workarounds:

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

3. Array of spiral coils on a printed-circuit board. This has the advantage of precisely-known locations of coil turns.

As mentioned above, the 5DOF tracker has an unmeasurable degree of freedom, single-coil roll. If the tracking algorithm is calculated in the coordinate system of the single-coil receiver, the unmeasurable degree of freedom is explicitly present, and its effects can be evaluated when converting from receiver coordinates to transmitter coordinates. If the tracking algorithm is calculated directly in the transmitter coordinates, the unmeasurable degree of freedom is hidden, and its effects can come as a surprise (as discussed in 2. above).

Open Source Electromagnetic Trackers

2018-01-14T22:13:58Z

Traneus: /* References */ added Plotkin etal characterization procedure

__NOTOC__

[[Image:Dry0030.png]]
[[Image:Dry_elphel_model_1_rcvr_coils.jpg]]

In the photo, each of the three receiver coils is ten millimeters long, and is a
[[http://www.sonion.com Sonion]] T 20 AG telecoil usually used in hearing aids.

==Key Personnel==
* [[User:Traneus | Peter Traneus Anderson]]
* [[User:Tkapur | Tina Kapur]]
* [[User:SPujol | Sonia Pujol]]

==Goals of the Project==
To teach the process of developing electromagnetic trackers for research, to foster an open community of researchers developing electromagnetic trackers, to develop open-source software and open-source hardware for working research electromagnetic trackers interfacing to Slicer through [[OpenIGTLink|OpenIGTLink]].

==Current Progress==

[[2014_Summer_Project_Week:Open_source_electromagnetic_trackers_usingOpenIGTLink | 2014 Summer Project Week]]

[[6DOF_Electromagnetic_Tracker_Construction_HOWTO|6DOF Electromagnetic Tracker Construction HOWTO]]

[[5DOF_Electromagnetic_Tracker_Notes|5DOF Electromagnetic Tracker Notes]]

Pete's current efforts are towards developing low-cost coil-characterization methods which make sense electromagnetically, aiming for a published paper. Pete has yet to encounter a paper which covers this topic.

==References==
* https://github.com/traneus/emtrackers

* http://www.hackster.io/plume/plume http://rose.eu.org/2014/tag/plume a dead 6DOF EM tracker project.

* Project started at [[2011_Summer_project_Week_Open_Source_Electromagnetic_Trackers_using_OpenIGTLink| 2011 Summer Project Week]]

*Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic systems, Vol. AES-15, No. 4, September 1979, pages 709-718. Iterative solution for 6DOF tracker. Includes sensitivity matrix of magnetic couplings partial derivatives with respect to position and orientation changes.

*Frederick H. Raab, "Quasi-Static Magnetic-Field Technique for Determining Position and Orientation", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-19, No. 4, October 1981, pages 235-243. Direct solution for 6DOF tracker.

*Tobias Schroeder, "An accurate magnetic field solution for medical electromagnetic tracking coils at close range", Journal of Applied Physics 117, 224504 (2015). Current-sheet model for cubical coils.

*C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

*C. L. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level," Proceedings of the IRE (now part of the IEEE), Vol. 35, pp. 335-348, June, 1946. The original Dolph-Chebyshev Fourier-transform window article. Dolph-Chebyshev window can give 140 dB rejection in the stopband.

*Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1) 84-91, doi:10.1109/TASSP.1981.1163506, "U.S. Government work not subject to U.S. copyright", in particular Figure 10 window for -L/2 < t < L/2: w(t) = (1/L) (10/32 + 15/32 cos(2pi t/L) + 6/32 cos(4pi t/L) + 1/32 cos(6pi t/L)) has first sidelobe at -61 dB and 42 dB/octave sidelobe rolloff.

*Eugene Paperno, "Suppression of magnetic noise in the fundamental-mode orthogonal fluxgate", Elsevier, Sensors and Actuators A 116 (2004) 405-409. Picotesla noise in 20 mm long 1 mm diameter fluxgate magnetometer. To get low noise, the drive flux swings between saturation in one direction and zero flux. The usual noisy fluxgate drive flux swings between saturation in one direction and saturation in the other direction, to ease measurement down to DC.

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

*James M. Chappell, Samuel P. Drake, Cameron L. Seidel, Lachlan J. Gunn, Azhar Iqbal, Andrew Allison, Derek Abbott, "Geometric Algebra for Electrical and Electronic Engineers", Proceedings of the IEEE, Vol. 102, No. 9, September 2014, pages 1340 to 1363. Clifford algebra formulation of electromagnetics using vectors, bivectors, trivector.

*Anton Plotkin, Vladimir Kucher, Yoram Horen, and Eugene Paperno, "A New Calibration Procedure for Magnetic Tracking Systems", IEEE Transactions on Magnetics, Volume 44, Number 11, November 2008, Pages 4525 to 4528. In-system coil characterization using just receiver positions on the plane closest to the transmitter, which makes electromagnetic sense.

==Citations==

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-12-08T02:28:09Z

Traneus: Finished discussion

This paper covers the electromagnetics, and includes sensitivity matrices as part of its iterative method of tracking position and orientation:

* Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic systems, Vol. AES-15, No. 4, September 1979, pages 709-718.

"range" is used here as the distance between transmitter coil trio and receiver coil trio.

Raab etal use (range, alpha, beta) three-dimensional polar coordinates for position, and spherical coordinates (azimuth, elevation, roll) for orientation. This separation of range from position and orientation angles, is electromagnetically sensible:

* The dipole magnetic field is inversely proportional to the cube of range, so d_signal/signal = 3 * d_range/range.

* At a constant range, the field strength on the axis of the dipole transmitter, is twice the field strength on the equatorial plane of the dipole transmitter. This factor of two, is what permits the separation of transverse position changes (changes in position angles without change in range) from changes in orientation angles. From the sensitivity matrix R in Raab etal, we can derive that the difficulty of separating position and orientation angles leads to: d_signal/signal = 0.3 * d_angle_in_radians. See [[EM_Tracker_HFluxPerI_Derivation | EM_Tracker_HFluxPerI_Derivation]] for details.

To illustrate the calculation, we consider a specific example, for a desired 95% accuracy of 1 millimeter at a maximum distance of 30 centimeters, assigning half the error budget to noise:

* Rmax = maximum range needed = 30 centimeters = 0.3 meters

* Pnoise = 0.15 millimeters_RMS = 0.5 millimeters_95%_probability

Calculate the required signal-to-noise ratio, SNR, at maximum range Rmax:

* d_angle_in_radians = Pnoise / Rmax = 0.0005 radians = 0.5 milliradians

* d_signal/signal = 0.3 * d_angle_in_radians = 0.00015 radians = 0.15 milliradians

* SNR_Rmax = 1/(d_signal/signal) = 6700 = 6.7e+03 = 76 dB = 1/(0.015%)

The best audio analog-to-digital converters, ADCs, have an SNR over a 40-kHz bandwidth of 120 dB:

* SNR_adc = 120 dB = 1.0e+06

* BW_adc = 40 kHz = 40,000 Hz

Assume that the narrowband filters after the ADC have bandwidth of 200 Hz, and calculate the SNR over the narrow bandwidth:

* BW_narrow = 200 Hz

* BW_narrow / BW_adc = 1/200

* SNR_narrow = SNR_adc * sqrt(BW_adc/BW_narrow) = 1.0e+06 * sqrt(200) = 1.41e+07 = 143 dB

Assume the receiver noise is the same as the narrowband noise:

* SNR_receiver = 0.5 * SNR_narrow = 7.1e+06 = 137 dB

SNR_receiver is much more than SNR_Rmax, so we can meet our spec at maximum range. We have spare SNR to use for dynamic range:

* Dynamic_range = SNR_receiver / SNR_Rmax = 1055 = 137 dB - 76 dB = 61 dB

As the receiver moves closer to the transmitter, the signal increases by the inverse of the cube of range, so we can calculate the minimum range, Rmin, our tracker can operate at:

Rmin = Rmax / cuberoot(Dynamic_range)= 30 centimeters / 10.2 = 3 centimeters

Mechanical interference will probably prevent reaching Rmin in this example. Also, non-dipole coil-finite-size effects will reduce accuracy at small ranges.

At large ranges, in the absence of magnetic-field distorters, the dominant error is due to electric-field coupling. See [[6DOF_Electromagnetic_Tracker_Electric_Field]]. At really large ranges, the radiation field starts to matter. See Raab's expired U.S. Patent 4,346,384 for discussion.

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-12-08T02:12:47Z

Traneus: added more

This paper covers the electromagnetics, and includes sensitivity matrices as part of its iterative method of tracking position and orientation:

* Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic systems, Vol. AES-15, No. 4, September 1979, pages 709-718.

"range" is used here as the distance between transmitter coil trio and receiver coil trio.

Raab etal use (range, alpha, beta) three-dimensional polar coordinates for position, and spherical coordinates (azimuth, elevation, roll) for orientation. This separation of range from position and orientation angles, is electromagnetically sensible:

* The dipole magnetic field is inversely proportional to the cube of range, so d_signal/signal = 3 * d_range/range.

* At a constant range, the field strength on the axis of the dipole transmitter, is twice the field strength on the equatorial plane of the dipole transmitter. This factor of two, is what permits the separation of transverse position changes (changes in position angles without change in range) from changes in orientation angles. From the sensitivity matrix R in Raab etal, we can derive that the difficulty of separating position and orientation angles leads to: d_signal/signal = 0.3 * d_angle_in_radians. See [[EM_Tracker_HFluxPerI_Derivation | EM_Tracker_HFluxPerI_Derivation]] for details.

To illustrate the calculation, we consider a specific example, for a desired 95% accuracy of 1 millimeter at a maximum distance of 30 centimeters, assigning half the error budget to noise:

* Rmax = maximum range needed = 30 centimeters = 0.3 meters

* Pnoise = 0.15 millimeters_RMS = 0.5 millimeters_95%_probability

Calculate the required signal-to-noise ratio, SNR, at maximum range Rmax:

* d_angle_in_radians = Pnoise / Rmax = 0.0005 radians = 0.5 milliradians

* d_signal/signal = 0.3 * d_angle_in_radians = 0.00015 radians = 0.15 milliradians

* SNR_Rmax = 1/(d_signal/signal) = 6700 = 6.7e+03 = 76 dB = 1/(0.015%)

The best audio analog-to-digital converters, ADCs, have an SNR over a 40-kHz bandwidth of 120 dB:

* SNR_adc = 120 dB = 1.0e+06

* BW_adc = 40 kHz = 40,000 Hz

Assume that the narrowband filters after the ADC have bandwidth of 200 Hz, and calculate the SNR over the narrow bandwidth:

* BW_narrow = 200 Hz

* BW_narrow / BW_adc = 1/200

* SNR_narrow = SNR_adc * sqrt(BW_adc/BW_narrow) = 1.0e+06 * sqrt(200) = 1.41e+07 = 143 dB

Assume the receiver noise is the same as the narrowband noise:

* SNR_receiver = 0.5 * SNR_narrow = 7.1e+06 = 137 dB

More to come...

Note that analog gain switches to increase dynamic range, are not used, as the ratios of the gain states cannot be controlled precisely enough.

At large ranges, in the absence of magnetic-field distorters, the dominant error is due to electric-field coupling. See [[6DOF_Electromagnetic_Tracker_Electric_Field]].

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-12-08T01:57:56Z

Traneus: Added more

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-12-08T01:49:04Z

Traneus: Adding more discussion

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-12-08T01:44:44Z

Traneus: Completing the discussion

EM Tracker HFluxPerI Derivation

2017-11-19T01:09:28Z

Traneus:

All of the following can be found at (or derived from) https://en.wikipedia.org/wiki/Magnetic_dipole and in classical-electromagnetics textbooks.

We assume that, at the working frequency or frequencies, the wavelength is large compared to the distance between transmitter coil trio and receiver coil trio. This is the quasi-static approximation, which permits us to ignore radiation fields.

We assume that each coil is so small that its shape does not matter, only its size times its number of turns. This is the dipole approximation. Each coil has an effective-area vector, Aeff_vect (measured in square meters) which completely describes a dipole coil's quasi-static magnetic properties.

Consider a single transmitter coil, with effective-area vector Aeff_tmtr_vect. We pass a current Itmtr (measured in amperes) through the transmitter coil, which causes the coil to emit a vector magnetic field Hvect (measured in amperes per meter) which varies depending upon where we observe the magnetic field.

Let Rvect (measured in meters) be the vector from the transmitter coil to the position of the magnetic-field observer. The magnetic field at the observation point is Hvect(Rvect).

We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:

Rvect = Rmag Runit_vect

The vector magnetic field Hvect (measured in amperes per meter) at the observation point is then:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)

This can be written more compactly (being a little free with the notation) as:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:

HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)

Substituting for Hvect(Rvect) gives:

Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

HFluxPerI is a purely geometrical property of the coils and their relationship in space.

If we replace Rvect with -Rvect, Rmag is unchanged, Runit_vect is replaced by -Runit_vect, and HFluxPerI is unchanged. This is the hemisphere ambiguity.

If we swap Aeff_rcvr_vect and Aeff_tmtr_vect, HFluxPerI is unchanged. This is electromagnetic reciprocity.

The induced voltage Vrcvr (measured in volts) across the receiver coil is:

Vrcvr(t) = -d/dt(Uo HFluxPerI Itmtr(t))

Uo = pi 4e-07 volts*seconds/(amperes*meters) is the magnetic permeability of free space, usually called mu-nought.

If Itmtr(t) is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:

Itmtr(t) = Itmtr_peak sin(2 pi F t)

Vrcvr(t) = Vrcvr_peak cos(2 pi F t)

Vrcvr_peak = -Uo HFluxPerI Itmtr_peak 2 pi F

The minus sign comes from the electromagnetics of induced voltages.

So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Next, we expand this to three receiver coils and three transmitter coils. Scalar HFluxPerI becomes a 3x3 matrix. Vector Aeff_rcvr_vect becomes the product of two 3x3 matrices:

Aeff_rcvr_mat Rcvr_rotation_mat

Vector Aeff_tmtr_vect becomes the product of two 3x3 matrices:

Tmtr_rotation_mat Aeff_tmtr_mat

Assume we are working in a coordinate system where Runit_vect = (1,0,0). This leaves a free choice of rotations about Runit_vec, called "position roll" in Raab's 1981 paper and in the Raab, Blood, Steiner, Jones 1979 paper. Position roll drops out when converting the two rotation matrices to receiver position and orientation.

The parenthetical expression containing Runit_vect becomes a 3x3 matrix Ro. In a coordinate system where Runit_vect = (1,0,0), Ro is:

<tt>
(+2, +0, +0)
 
(+0, -1, +0) = Ro
 
(+0, +0, -1)
</tt>

For comparison, the identity matrix I is:

<tt>
(+1, +0, +0)
 
(+0, +1, +0) = I
 
(+0, +0, +1)
</tt>

Putting it all together, we get:

HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat

Multiplying both sides by 4 pi Rmag^3 gives:

HFluxPerI_mat 4 pi Rmag^3 = Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat

Multiplying both sides by the inverses of the Aeff matrices gives:

Aeff_Rcvr_mat_inverse HFluxPerI_mat 4 pi Rmag^3 Aeff_tmtr_mag_inverse = Rcvr_rotation_mat Ro Tmtr_rotation_mat 

HFluxPerI is measured by the tracker electronics.

Aeff_rcvr_mat is known from the receiver coil characterization process.

Aeff_tmtr_mat is known from the transmitter coil characterization process.

Calculating the sum of the squares of the nine elements in HFluxPerI, permits calculating Rmag independently of the rotation matrices. This needs more detail here, see the Raab and Raab Blood Steiner Jones papers.

The left side is thus known. The tracker position-and-orientation algorithm does the equivalent of calculating the two rotation matrices from the known left side and the known Ro. Note that if Ro were the identity matrix, the two rotation matrices could not be calculated separately.

EM Tracker HFluxPerI Derivation

2017-11-19T01:05:47Z

Traneus:

All of the following can be found at (or derived from) https://en.wikipedia.org/wiki/Magnetic_dipole and in classical-electromagnetics textbooks.

We assume that, at the working frequency or frequencies, the wavelength is large compared to the distance between transmitter coil trio and receiver coil trio. This is the quasi-static approximation, which permits us to ignore radiation fields.

We assume that each coil is so small that its shape does not matter, only its size times its number of turns. This is the dipole approximation. Each coil has an effective-area vector, Aeff_vect (measured in square meters) which completely describes a dipole coil's quasi-static magnetic properties.

Consider a single transmitter coil, with effective-area vector Aeff_tmtr_vect. We pass a current Itmtr (measured in amperes) through the transmitter coil, which causes the coil to emit a vector magnetic field Hvect (measured in amperes per meter) which varies depending upon where we observe the magnetic field.

Let Rvect (measured in meters) be the vector from the transmitter coil to the position of the magnetic-field observer. The magnetic field at the observation point is Hvect(Rvect).

We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:

Rvect = Rmag Runit_vect

The vector magnetic field Hvect (measured in amperes per meter) at the observation point is then:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)

This can be written more compactly (being a little free with the notation) as:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:

HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)

Substituting for Hvect(Rvect) gives:

Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

HFluxPerI is a purely geometrical property of the coils and their relationship in space.

If we replace Rvect with -Rvect, Rmag is unchanged, Runit_vect is replaced by -Runit_vect, and HFluxPerI is unchanged. This is the hemisphere ambiguity.

If we swap Aeff_rcvr_vect and Aeff_tmtr_vect, HFluxPerI is unchanged. This is electromagnetic reciprocity.

The induced voltage Vrcvr (measured in volts) across the receiver coil is:

Vrcvr(t) = -d/dt(Uo HFluxPerI Itmtr(t))

Uo = pi 4e-07 volts*seconds/(amperes*meters) is the magnetic permeability of free space, usually called mu-nought.

If Itmtr(t) is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:

Itmtr(t) = Itmtr_peak sin(2 pi F t)

Vrcvr(t) = Vrcvr_peak cos(2 pi F t)

Vrcvr_peak = -Uo HFluxPerI Itmtr_peak 2 pi F

The minus sign comes from the electromagnetics of induced voltages.

So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Next, we expand this to three receiver coils and three transmitter coils. Scalar HFluxPerI becomes a 3x3 matrix. Vector Aeff_rcvr_vect becomes the product of two 3x3 matrices:

Aeff_rcvr_mat Rcvr_rotation_mat

Vector Aeff_tmtr_vect becomes the product of two 3x3 matrices:

Tmtr_rotation_mat Aeff_tmtr_mat

Assume we are working in a coordinate system where Runit_vect = (1,0,0). This leaves a free choice of rotations about Runit_vec, called "position roll" in Raab's 1981 paper and in the Raab, Blood, Steiner, Jones 1979 paper. Position roll drops out when converting the two rotation matrices to receiver position and orientation.

The parenthetical expression containing Runit_vect becomes a 3x3 matrix Ro. In a coordinate system where Runit_vect = (1,0,0), Ro is:

<tt>
(+2, +0, +0)
 
(+0, -1, +0) = Ro
 
(+0, +0, -1)
</tt>

For comparison, the identity matrix I is:

<tt>
(+1, +0, +0)
 
(+0, +1, +0) = I
 
(+0, +0, +1)
</tt>

Putting it all together, we get:

HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat

Multiplying both sides by 4 pi Rmag^3 gives:

HFluxPerI_mat 4 pi Rmag^3 = Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat

Multiplying both sides by the inverses of the Aeff matrices gives:

Aeff_Rcvr_mat_inverse HFluxPerI_mat 4 pi Rmag^3 Aeff_tmtr_mag_inverse = Rcvr_rotation_mat Ro Tmtr_rotation_mat 

HFluxPerI is measured by the tracker electronics.

Aeff_rcvr_mat is known from the receiver coil characterization process.

Aeff_tmtr_mat is known from the transmitter coil characterization process.

Calculating the sum of the squares of the nine elements in HFluxPerI, permits calculating Rmag independently of the rotation matrices. This needs more detail here, see the Raab and raab etal papers.

The left side is thus known. The tracker position-and-orientation algorithm does the equivalent of calculating the two rotation matrices from the known left side and the known Ro. Note that if Ro were the identity matrix, the two rotation matrices could not be calculated separately.

EM Tracker HFluxPerI Derivation

2017-11-19T00:48:45Z

Traneus:

All of the following can be found at (or derived from) https://en.wikipedia.org/wiki/Magnetic_dipole and in classical-electromagnetics textbooks.

We assume that, at the working frequency or frequencies, the wavelength is large compared to the distance between transmitter coil trio and receiver coil trio. This is the quasi-static approximation, which permits us to ignore radiation fields.

We assume that each coil is so small that its shape does not matter, only its size times its number of turns. This is the dipole approximation. Each coil has an effective-area vector, Aeff_vect (measured in square meters) which completely describes a dipole coil's quasi-static magnetic properties.

Consider a single transmitter coil, with effective-area vector Aeff_tmtr_vect. We pass a current Itmtr (measured in amperes) through the transmitter coil, which causes the coil to emit a vector magnetic field Hvect (measured in amperes per meter) which varies depending upon where we observe the magnetic field.

Let Rvect (measured in meters) be the vector from the transmitter coil to the position of the magnetic-field observer. The magnetic field at the observation point is Hvect(Rvect).

We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:

Rvect = Rmag Runit_vect

The vector magnetic field Hvect (measured in amperes per meter) at the observation point is then:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)

This can be written more compactly (being a little free with the notation) as:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:

HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)

Substituting for Hvect(Rvect) gives:

Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

HFluxPerI is a purely geometrical property of the coils and their relationship in space.

If we replace Rvect with -Rvect, Rmag is unchanged, Runit_vect is replaced by -Runit_vect, and HFluxPerI is unchanged. This is the hemisphere ambiguity.

If we swap Aeff_rcvr_vect and Aeff_tmtr_vect, HFluxPerI is unchanged. This is electromagnetic reciprocity.

The induced voltage Vrcvr (measured in volts) across the receiver coil is:

Vrcvr(t) = -d/dt(Uo HFluxPerI Itmtr(t))

Uo = pi 4e-07 volts*seconds/(amperes*meters) is the magnetic permeability of free space, usually called mu-nought.

If Itmtr(t) is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:

Itmtr(t) = Itmtr_peak sin(2 pi F t)

Vrcvr(t) = Vrcvr_peak cos(2 pi F t)

Vrcvr_peak = -Uo HFluxPerI Itmtr_peak 2 pi F

The minus sign comes from the electromagnetics of induced voltages.

So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Next, we expand this to three receiver coils and three transmitter coils. Scalar HFluxPerI becomes a 3x3 matrix. Vector Aeff_rcvr_vect becomes the product of two 3x3 matrices:

Aeff_rcvr_mat Rcvr_rotation_mat

Vector Aeff_tmtr_vect becomes the product of two 3x3 matrices:

Tmtr_rotation_mat Aeff_tmtr_mat

Assume we are working in a coordinate system where Runit_vect = (1,0,0). This leaves a free choice of rotations about Runit_vec, called "position roll" in Raab's 1981 paper and in the Raab, Blood, Steiner, Jones 1979 paper. Position roll drops out when converting the two rotation matrices to receiver position and orientation.

The parenthetical expression containing Runit_vect becomes a 3x3 matrix Ro. In a coordinate system where Runit_vect = (1,0,0), Ro is:

<tt>
(+2, +0, +0)
 
(+0, -1, +0) = Ro
 
(+0, +0, -1)
</tt>

Putting it all together, we get:

HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat

EM Tracker HFluxPerI Derivation

2017-11-19T00:47:45Z

Traneus:

All of the following can be found at (or derived from) https://en.wikipedia.org/wiki/Magnetic_dipole and in classical-electromagnetics textbooks.

We assume that, at the working frequency or frequencies, the wavelength is large compared to the distance between transmitter coil trio and receiver coil trio. This is the quasi-static approximation, which permits us to ignore radiation fields.

We assume that each coil is so small that its shape does not matter, only its size times its number of turns. This is the dipole approximation. Each coil has an effective-area vector, Aeff_vect (measured in square meters) which completely describes a dipole coil's quasi-static magnetic properties.

Consider a single transmitter coil, with effective-area vector Aeff_tmtr_vect. We pass a current Itmtr (measured in amperes) through the transmitter coil, which causes the coil to emit a vector magnetic field Hvect (measured in amperes per meter) which varies depending upon where we observe the magnetic field.

Let Rvect (measured in meters) be the vector from the transmitter coil to the position of the magnetic-field observer. The magnetic field at the observation point is Hvect(Rvect).

We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:

Rvect = Rmag Runit_vect

The vector magnetic field Hvect (measured in amperes per meter) at the observation point is then:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)

This can be written more compactly (being a little free with the notation) as:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:

HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)

Substituting for Hvect(Rvect) gives:

Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

HFluxPerI is a purely geometrical property of the coils and their relationship in space.

If we replace Rvect with -Rvect, Rmag is unchanged, Runit_vect is replaced by -Runit_vect, and HFluxPerI is unchanged. This is the hemisphere ambiguity.

If we swap Aeff_rcvr_vect and Aeff_tmtr_vect, HFluxPerI is unchanged. This is electromagnetic reciprocity.

The induced voltage Vrcvr (measured in volts) across the receiver coil is:

Vrcvr(t) = -d/dt(Uo HFluxPerI Itmtr(t))

Uo = pi 4e-07 volts*seconds/(amperes*meters) is the magnetic permeability of free space, usually called mu-nought.

If Itmtr(t) is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:

Itmtr(t) = Itmtr_peak sin(2 pi F t)

Vrcvr(t) = Vrcvr_peak cos(2 pi F t)

Vrcvr_peak = -Uo HFluxPerI Itmtr_peak 2 pi F

The minus sign comes from the electromagnetics of induced voltages.

So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Next, we expand this to three receiver coils and three transmitter coils. Scalar HFluxPerI becomes a 3x3 matrix. Vector Aeff_rcvr_vect becomes the product of two 3x3 matrices:

Aeff_rcvr_mat Rcvr_rotation_mat

Vector Aeff_tmtr_vect becomes the product of two 3x3 matrices:

Tmtr_rotation_mat Aeff_tmtr_mat

Assume we are working in a coordinate system where Runit_vect = (1,0,0). This leaves a free choice of rotations about Runit_vec, called "position roll" in Raab's 1981 paper and in the Raab, Blood, Steiner, Jones 1979 paper. Position roll drops out when converting the two rotation matrices to receiver position and orientation.

The parenthetical expression containing Runit_vect becomes a 3x3 matrix Ro. In a coordinate system where Runit_vect = (1,0,0), Ro is:

<tt>
(+2, +0, +0)
 
(+0, -1, +0) = Ro
 
(+0, +0, -1)
</tt>

Putting it all together, we get:

HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat_Aeff_tmtr_mat

EM Tracker HFluxPerI Derivation

2017-11-19T00:28:57Z

Traneus: Expanded discussion to coil trios

All of the following can be found at (or derived from) https://en.wikipedia.org/wiki/Magnetic_dipole and in classical-electromagnetics textbooks.

We assume that, at the working frequency or frequencies, the wavelength is large compared to the distance between transmitter coil trio and receiver coil trio. This is the quasi-static approximation, which permits us to ignore radiation fields.

We assume that each coil is so small that its shape does not matter, only its size times its number of turns. This is the dipole approximation. Each coil has an effective-area vector, Aeff_vect (measured in square meters) which completely describes a dipole coil's quasi-static magnetic properties.

Consider a single transmitter coil, with effective-area vector Aeff_tmtr_vect. We pass a current Itmtr (measured in amperes) through the transmitter coil, which causes the coil to emit a vector magnetic field Hvect (measured in amperes per meter) which varies depending upon where we observe the magnetic field.

Let Rvect (measured in meters) be the vector from the transmitter coil to the position of the magnetic-field observer. The magnetic field at the observation point is Hvect(Rvect).

We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:

Rvect = Rmag Runit_vect

The vector magnetic field Hvect (measured in amperes per meter) at the observation point is then:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)

This can be written more compactly (being a little free with the notation) as:

Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:

HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)

Substituting for Hvect(Rvect) gives:

Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

HFluxPerI is a purely geometrical property of the coils and their relationship in space.

If we replace Rvect with -Rvect, Rmag is unchanged, Runit_vect is replaced by -Runit_vect, and HFluxPerI is unchanged. This is the hemisphere ambiguity.

If we swap Aeff_rcvr_vect and Aeff_tmtr_vect, HFluxPerI is unchanged. This is electromagnetic reciprocity.

The induced voltage Vrcvr (measured in volts) across the receiver coil is:

Vrcvr(t) = -d/dt(Uo HFluxPerI Itmtr(t))

Uo = pi 4e-07 volts*seconds/(amperes*meters) is the magnetic permeability of free space, usually called mu-nought.

If Itmtr(t) is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:

Itmtr(t) = Itmtr_peak sin(2 pi F t)

Vrcvr(t) = Vrcvr_peak cos(2 pi F t)

Vrcvr_peak = -Uo HFluxPerI Itmtr_peak 2 pi F

The minus sign comes from the electromagnetics of induced voltages.

So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:

HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect

Next, we expand this to three receiver coils and three transmitter coils. Scalar HFluxPerI becomes a 3x3 matrix. Vector Aeff_rcvr_vect becomes the product of two 3x3 matrices:

Aeff_rcvr_mat Rcvr_rotation_mat

Vector Aeff_tmtr_vect becomes the product of two 3x3 matrices:

Tmtr_rotation_mat Aeff_tmtr_mat

The parenthetical expression containing Runit_vect becomes a 3x3 matrix Ro. Assume we are working in a coordinate system where Runit_vect = (1,0,0). Then
Ro is:

<tt>
(+2, +0, +0)
 
(+0, -1, +0) = Ro
 
(+0, +0, -1)
</tt>

Next...

EM Tracker HFluxPerI Derivation

2017-11-19T00:11:03Z

Traneus:

EM Tracker HFluxPerI Derivation

2017-11-19T00:05:05Z

Traneus: Bolded equations for better readability, and corrected some errors.

EM Tracker HFluxPerI Derivation

2017-11-18T23:56:01Z

Traneus:

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-11-12T22:36:28Z

Traneus:

6DOF Electromagnetic Tracker Construction HOWTO

2017-11-12T22:27:40Z

Traneus: Added patent reference and improved equation formatting

A basic 6DOF (six degrees of freedom: three of position and three of orientation) electromagnetic tracker can contain the parts shown in this block diagram, though there are many variations. [[File:6DOF_tracker_block_diagram.png]]

* https://web.archive.org/web/20151002101401/http://home.comcast.net/~traneus/dry_emtrackertricoil.htm is an example of a breadboard 6DOF tracker.

* Transmitter contains three colocated orthogonal coils. The coils are approximated as magnetic dipoles.

* Receiver contains three colocated orthogonal coils. the coils are approximated as dipoles.

* [[Media:EM_tracker_classic_coil_trio.jpg]] Photo of classic coil trio used as transmitter or receiver. The three coils are wound on a black plastic cube about one centimeter on a side.

* [[http://rose.eu.org/2014/tag/plume Scroll down to see photo of hand-building a transmitter coil trio.]]

* [[Media:Dry_elphel_model_1_rcvr_coils.jpg]] Photo of crude handmade receiver coil trio using [[http://www.sonion.com Sonion]] T 20 AG telecoils. Each coil is ten millimeters long.

* Three transmitter coils times three receiver coils gives nine coil-coupling measurements, expressable as a 3x3 signal matrix, HFluxPerIMeasured.

* Each component of HFLuxPerIMeasured is the magnetic flux through one receiver coil (due to magnetic field H from transmitter coil), divided by the current I in one transmitter coil. HFLuxPerIMeasured has units of meters, and is a geometrical property of the coils' sizes, shapes, number of turns, ferromagnetic core (if any), positions, and orientations. [[EM_Tracker_HFluxPerI_Derivation | HFluxPerI coupling between two dipole coils]].

* Algorithm software converts HFluxPerIMeasured to estimated receiver position and orientation, using direct-solution algorithm in Raab's 1981 paper or iterative solution in Raab etal's 1979 paper.

* Frederick H. Raab, "Quasi-Static Magnetic-Field Technique for Determining Position and Orientation", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-19, No. 4, October 1981, pages 235-243, describes closed-form algorithm for concentric-dipole coil trios. Position is calculated first, directly in cartesian coordinates. Orientation is then calculated.

* Frederick H. Raab, "Remote Object Position Locater", expired U.S. Patent 4,054,881. Describes frequency-multiplexed hardware.

* Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-15, No. 5, September 1979, pages 709-718, describes iterative algorithm for concentric-dipole coil trios, using small-angle approximation for changes in position and in orientation. Includes sensitivity matrix of magnetic couplings partial derivatives with respect to changes in position and orientation.

* Berthold K. P. Horn, "Closed-form solution of absolute orientation using unit quaternions", Journal of the Optical Society of America A, volume 4, April, 1987, pages 629-642, has algorithm for converting from orthonormal rotation matrices to quaternions. Note error: r[2][1] on page 641 is incorrect, while r[2][1] on page 643 is correct.

* [[File:Dry0097.c]] is a simulator program containing an implementation of Raab's algorithm.

* The software which calculates position and orientation from HFluxPerI measurements, is an example of realtime embedded computational electromagnetics.

* Needed HFluxPerI measurement accuracy can be calculated by a sensitivity analysis. As in Raab, Blood, Steiner, Jones, treating position in spherical coordinates gives one distance (called range, the distance between transmitter and receiver) and five angles.

* The electromagnetics results in the signal-to-noise ratio in the five angles being 3.4 times worse than the HFluxPerIMeasured signal-to-noise ratio, due to interactions between position errors and orientation errors.

* The electromagnetics results in the signal-to-noise ratio in range being 3 times better than the HFluxPerIMeasured signal-to-noise ratio, due to the inverse-cube law of dipole-dipole field coupling.

* [[6DOF_Electromagnetic_Tracker_Signal_to_Noise_Requirements_Calculation]] details calculating signal-to-noise ratio (SNR) from accuracy requirements.

* There is an inherent hemisphere ambiguity, since receiver at position = (Xo,Yo,Zo) and receiver at position = (-Xo,-Yo,-Zo) show identical HFluxPerIMeasured for identical orientations. This ambiguity can be resolved by using additional transmitter or receiver coils spaced away from the colocated transmitter or receiver coils.

* The receiver is normally kept on one side of the transmitter, to avoid the hemisphere ambiguity.

* The transmitter field on the unused side of the transmitter, can be eliminated by using a magnetic mirror: Reference expired U.S. patent 5,640,170, which references many older expired EM-tracker patents.

* Accuracy is poor for lined-up pose: receiver positioned on a transmitter-coil axis, with receiver oriented to make receiver-coil axes parallel with transmitter-coil axes. Some of the first-order partial derivatives go to zero in these cases, causing the position-and-orientation solution to separate into four separate partial solutions.

* Poses with poor tracking accuracy, should be good for coil characterization. Coil-characterization poses should be chosen based on electromagnetic theory, rather than on mechanical measurements. Use receiver positions close to transmitter, based on boundary-condition principles of EM theory. Receiver positions on both sides of transmitter (even if only one side is used in operation), are necessary to distinguish transmitter gain from transmitter nonconcentricity. [[EM_Tracker_Coil_Characterization]] has further discussion.

* The poor-accuracy poses can be reduced, by replacing the three-orthogonal-coil receiver with a receiver comprising four colocated coils (the transmitter remains three orthogonal coils). The four receiver coils point in the directions of the vertices of a regular tetrahedron. When the four-coil receiver is positioned on a transmitter-coil axis, with one receiver coil oriented parallel to a transmitter-coil axis, the remaining three receiver coils' axes cannot be parallel to transmitter-coil axes.

* Three transmitter coils times three receiver coils, gives nine coil-coupling measurements. These can be performed sequentially, partially sequentially and partially simultaneously, or all simultaneously. Sequential measurements simplify the electronics, but impair dynamic accuracy: When the receiver is moving, sequential measurements result in inconsistent datasets, leading to position and orientation dynamic errors.

* Receiver coil signals can be measured simultaneously or sequentially. Simultaneous measurements improve signal-to-noise ratio.

* Many designs used one operating frequency, driving the transmitter coils sequentially. Use of one frequency simplifies handling frequency-dependent effects.

* Multiple-frequency designs drive the three transmitter coils simultaneously, with sinewaves at three distinct frequencies. This improves signal-to-noise ratio by lengthening measurement time.

* Operating frequencies are typically 30 Hz to 15000 Hz. 1000 Hz, 1300 Hz, and 1600 Hz are a good starting point. Higher frequencies give higher induced voltages, lower frequencies reduce error-causing eddy-current effects.

* The transmitter coils are usually series tuned with capacitors.

* The transmitter-coil currents must be measured. The currents vary slowly due to coil heating, so currents can be measured periodically.

* Some designs use DC pulses to drive the transmitter coils, instead of AC frequencies. This simplifies driver design, but makes receiver signal recovery more difficult. Pulse-driven transmitter coils must be driven sequentially.

* Data-acquisition electronics measures the currents in the three transmitter coils, and measures the voltages induced in the three receiver coils.

* 24-bit audio ADCs have enough dynamic range to avoid the need for gain-switching.

* Avoid gain-switching, as the ratios of the gain states are not precisely-enough known.

* A six-ADC electronics can measure three transmitter-coil currents and three receiver-coil voltages continually and simultaneously.

* Add three more ADCs for each additional receiver coil trio.

* A four-ADC electronics can use one channel to measure the currents periodically over time (The currents change slowly as the transmitter coils warm up.), and three channels to measure the three voltages continually and simultaneously.

* A two-ADC system can measure currents sequentially with one ADC and voltages sequentially with the other ADC.

* A single-ADC electronics can measure the currents and voltages sequentially.

* C. L. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level," Proc. IRE, Vol. 35, pp. 335-348, June, 1946. The original Dolph-Chebyshev window article. This window is capable of 140 dB rejection of out-of-band signals.

* Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1) 84-91, February 1981, doi:10.1109/TASSP.1981.1163506, "U.S. Government work not subject to U.S. copyright". The window in Figure 10 of this paper exhibits sidelobe peak, four DFT bins from the central peak, 91 dB down from the central peak (The window and its first through fifth derivatives are all continuous for all t, giving 42 dB/octave rolloff of the sidelobes) and is (for symmetrical limits |t|<=L/2, and zero for all t outside the limits):

<tt>
w(t) = (1/L)(10/32 + 15/32 cos(2pi t/L) + 6/32 cos(4pi t/L) + 1/32 cos(6pi t/L))
</tt>
 

* Expired U.S. Patent 4,109,199 describes the use of a calibration coil in the receiver to calibrate the gains of the electronics.

* More elaborate algorithms provide higher accuracy at the expense of much more computation. by modeling the non-dipole and/or non-concentric parts of the coils. Expired U.S. Patent 5,307,072 is an early example.

* C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006, reports low-cost accuracy-testing methods using a known-flat nonmagnetic surface (such as a granite surface plate).

* A 6DOF tracker using four-coil printed-circuit transmitter and receiver (optimized for academic originality) is discussed in: Peter Traneus Anderson, "A Source of Accurately Calculable Quasi-Static Magnetic Fields", dissertation presented to the Faculty of the Graduate College of the University of Vermont, October 2001, stored here as three files: [[Media:AndersonPeterDissertation.pdf]] is the main body. [[Media:AndersonPeterDissertationReadme.pdf]] contains copyright license, additional comments, and four figures that are blank in the main body. [[Media:AndersonPeterDissertationFig14r1.jpg]] is the color original photo of two of the figures. Expired U.S. Patent 1,172,017 discloses a direct-conversion radio receiver. Peter intended to include this reference as reference 16 in his dissertation, but was unable to find the patent number before Google Patents existed, so made do with existing indirect reference 16.

* A 6DOF tracker using two transmitter coils (instead of three) can be built; Frederick Raab calls this [[6DOF_Two_State_Electromagnetic_Trackers|two-state excitation]] in his 1981 paper referenced above. Two-state trackers are severely limited, as they cannot track near the axis of the missing transmitter coil and cannot track near the plane of the two existing transmitter coils.

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-11-12T21:48:10Z

Traneus:

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-11-12T00:23:20Z

Traneus: Added ((1,0,0)(0,-1/2,0)(0,0,-1/2) matrix.

5DOF Electromagnetic Tracker Notes

2017-04-06T22:19:44Z

Traneus:

5DOF (five-degree-of-freedom) electromagnetic trackers track a single dipole coil receiver against a spread-out array of transmitter coils. Because the single coil is symmetrical about its axis, orientation roll about the single coil's axis cannot be tracked. Orientation latitude and longitude can be tracked, as can X Y Z position.

The single receiver coil does not need to be characterized, as long as the coil is small enough to be a dipole, as the coil's gain is tracked as part of the position-and-orientation algorithm.

The single coil can be used as a transmitter, and the array as receivers, though it can be difficult to get enough signal.

There are many choices for the spread-out array:

1. Six or more coils pointed in various directions. Any particular arrangement needs to be analyzed and simulated for trackability.

2. Twelve coils in a field-and-gradient array. Think of the single coil as a transmitter. The twelve-coil array then measures the field and gradient from the single coil. By electromagnetic reciprocity, single coil as receiver works the same. There is a direct analytic solution for position using this array. One problem is that the gradient matrix is singular when the array is in the equatorial plane of the array. See the following paper for discussion and workarounds:

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

3. Array of spiral coils on a printed-circuit board. This has the advantage of precisely-known locations of coil turns.

As mentioned above, the 5DOF tracker has an unmeasurable degree of freedom, single-coil roll. If the tracking algorithm is calculated in the coordinate system of the single-coil receiver, the unmeasurable degree of freedom is explicitly present, and its effects can be evaluated when converting from receiver coordinates to transmitter coordinates. If the tracking algorithm is calculated directly in the transmitter coordinates, the unmeasurable degree of freedom is hidden, and its effects can come as a surprise (as discussed in 2. above).

5DOF Electromagnetic Tracker Notes

2017-04-02T20:59:37Z

Traneus:

5DOF (five-degree-of-freedom) electromagnetic trackers track a single dipole coil receiver against a spread-out array of transmitter coils. Because the single coil is symmetrical about its axis, orientation roll about the single coil's axis cannot be tracked. Orientation latitude and longitude can be tracked, as can X Y Z position.

The single receiver coil does not need to be characterized, as long as the coil is small enough to be a dipole, as the coil's gain is tracked as part of the position-and-orientation algorithm.

The single coil can be used as a transmitter, and the array as receivers, though it can be difficult to get enough signal.

There are many choices for the spread-out array:

1. Six or more coils pointed in various directions. Any particular arrangement needs to be analyzed and simulated for trackability.

2. Twelve coils in a field-and-gradient array. Think of the single coil as a transmitter. The twelve-coil array then measures the field and gradient from the single coil. By electromagnetic reciprocity, single coil as receiver works the same. There is a direct analytic solution for position using this array. One problem is that the gradient matrix is singular when the array is in the equatorial plane of the array. See the following paper for discussion and workarounds:

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

3. Array of spiral coils on a printed-circuit board. This has the advantage of precisely-known locations of coil turns.

4. Array of three or more 6DOF three-orthogonal-coil transmitters. The positions and orientations of the transmitters are tracked using a 6DOF three-orthogonal-coil receiver, so the positions and orientations of the transmitters can be permitted to change while tracking a single-coil receiver.

As mentioned above, the 5DOF tracker has an unmeasurable degree of freedom, single-coil roll. If the tracking algorithm is calculated in the coordinate system of the single-coil receiver, the unmeasurable degree of freedom is explicitly present, and its effects can be evaluated when converting from receiver coordinates to transmitter coordinates. If the tracking algorithm is calculated directly in the transmitter coordinates, the unmeasurable degree of freedom is hidden, and its effects can come as a surprise (as discussed in 2. above).

5DOF Electromagnetic Tracker Notes

2017-04-02T20:48:27Z

Traneus:

5DOF (five-degree-of-freedom) electromagnetic trackers track a single dipole coil receiver against a spread-out array of transmitter coils. Because the single coil is symmetrical about its axis, orientation roll about the single coil's axis cannot be tracked. Orientation latitude and longitude can be tracked, as can X Y Z position.

The single receiver coil does not need to be characterized, as long as the coil is small enough to be a dipole, as the coil's gain is tracked as part of the position-and-orientation algorithm.

The single coil can be used as a transmitter, and the array as receivers, though it can be difficult to get enough signal.

There are many choices for the spread-out array:

1. Eight or more coils pointed in various directions.

2. Twelve coils in a field-and-gradient array. Think of the single coil as a transmitter. The twelve-coil array then measures the field and gradient from the single coil. By electromagnetic duality, single coil as receiver works the same. There is a direct analytic solution for position using this array. One problem is that the gradient matrix is singular when the array is in the equatorial plane of the array. See the following paper for discussion and workarounds:

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

3. Array of spiral coils on a printed-circuit board. This has the advantage of precisely-known locations of coil turns.

4. Array of three or more 6DOF three-orthogonal-coil transmitters. The positions and orientations of the transmitters are tracked using a 6DOF three-orthogonal-coil receiver, so the positions and orientations of the transmitters can be permitted to change while tracking a single-coil receiver.

5DOF Electromagnetic Tracker Notes

2017-04-02T20:44:25Z

Traneus:

5DOF Electromagnetic Tracker Notes

2017-04-02T20:41:17Z

Traneus:

5DOF Electromagnetic Tracker Notes

2017-04-02T20:35:05Z

Traneus: Notes on differences between 5DOF and 6DOF trackers

5DOF (five-degree-of-freedom) electromagnetic trackers track a single dipole coil against a spread-out array of coils. Because the single coil is symmetrical about its axis, orientation roll about the single coil's axis cannot be tracked. Orientation latitude and longitude can be tracked, as can X Y Z position.

The single does not need to be characterized, as long as the coil is small enough to be a dipole, as the coil's gain is tracked as part of the position-and-orientation algorithm.

Open Source Electromagnetic Trackers

2017-04-02T20:28:55Z

Traneus: /* Current Progress */

__NOTOC__

[[Image:Dry0030.png]]
[[Image:Dry_elphel_model_1_rcvr_coils.jpg]]

In the photo, each of the three receiver coils is ten millimeters long, and is a
[[http://www.sonion.com Sonion]] T 20 AG telecoil usually used in hearing aids.

==Key Personnel==
* [[User:Traneus | Peter Traneus Anderson]]
* [[User:Tkapur | Tina Kapur]]
* [[User:SPujol | Sonia Pujol]]

==Goals of the Project==
To teach the process of developing electromagnetic trackers for research, to foster an open community of researchers developing electromagnetic trackers, to develop open-source software and open-source hardware for working research electromagnetic trackers interfacing to Slicer through [[OpenIGTLink|OpenIGTLink]].

==Current Progress==

[[2014_Summer_Project_Week:Open_source_electromagnetic_trackers_usingOpenIGTLink | 2014 Summer Project Week]]

[[6DOF_Electromagnetic_Tracker_Construction_HOWTO|6DOF Electromagnetic Tracker Construction HOWTO]]

[[5DOF_Electromagnetic_Tracker_Notes|5DOF Electromagnetic Tracker Notes]]

Pete's current efforts are towards developing low-cost coil-characterization methods which make sense electromagnetically, aiming for a published paper. Pete has yet to encounter a paper which covers this topic.

==References==
* https://github.com/traneus/emtrackers

* http://www.hackster.io/plume/plume http://rose.eu.org/2014/tag/plume another 6DOF EM tracker project.

* Project started at [[2011_Summer_project_Week_Open_Source_Electromagnetic_Trackers_using_OpenIGTLink| 2011 Summer Project Week]]

*Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic systems, Vol. AES-15, No. 4, September 1979, pages 709-718. Iterative solution for 6DOF tracker. Includes sensitivity matrix of magnetic couplings partial derivatives with respect to position and orientation changes.

*Frederick H. Raab, "Quasi-Static Magnetic-Field Technique for Determining Position and Orientation", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-19, No. 4, October 1981, pages 235-243. Direct solution for 6DOF tracker.

*Tobias Schroeder, "An accurate magnetic field solution for medical electromagnetic tracking coils at close range", Journal of Applied Physics 117, 224504 (2015). Current-sheet model for cubical coils.

*C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

*C. L. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level," Proceedings of the IRE (now part of the IEEE), Vol. 35, pp. 335-348, June, 1946. The original Dolph-Chebyshev Fourier-transform window article. Dolph-Chebyshev window can give 140 dB rejection in the stopband.

*Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1) 84-91, doi:10.1109/TASSP.1981.1163506, "U.S. Government work not subject to U.S. copyright", in particular Figure 10 window for -L/2 < t < L/2: w(t) = (1/L) (10/32 + 15/32 cos(2pi t/L) + 6/32 cos(4pi t/L) + 1/32 cos(6pi t/L)) has first sidelobe at -61 dB and 42 dB/octave sidelobe rolloff.

*Eugene Paperno, "Suppression of magnetic noise in the fundamental-mode orthogonal fluxgate", Elsevier, Sensors and Actuators A 116 (2004) 405-409. Picotesla noise in 20 mm long 1 mm diameter fluxgate magnetometer. To get low noise, the drive flux swings between saturation in one direction and zero flux. The usual noisy fluxgate drive flux swings between saturation in one direction and saturation in the other direction, to ease measurement down to DC.

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

*James M. Chappell, Samuel P. Drake, Cameron L. Seidel, Lachlan J. Gunn, Azhar Iqbal, Andrew Allison, Derek Abbott, "Geometric Algebra for Electrical and Electronic Engineers", Proceedings of the IEEE, Vol. 102, No. 9, September 2014, pages 1340 to 1363. Clifford algebra formulation of electromagnetics using vectors, bivectors, trivector.

==Citations==

EM Tracker Coil Characterization

2017-01-18T19:06:01Z

Traneus: Added detail

EM Tracker Coil Characterization

When we test a tracker for accuracy, we generally check using point sets that make mechanical sense. For example, we may use a 3D robot to move the receiver all over the working volume (also called the motion box).

In 6DOF trackers, the coils must be precisely characterized for their electromagnetic properties: gains, non-orthogonalities, non-concentricities, and finite-size effects. Since characterization is measuring electromagnetic properties, the point set chosen should make electromagnetic sense rather than mechanical sense.

The coils can only be manufactured to so much precision. Characterization measures the actual properties of each coil, to improve the tracker accuracy.

One important property of electromagnetics, is the boundary-condition property: If we know the magnetic field everywhere on the boundary of the working volume, we can calculate the field inside the working volume.

For coil characterization, the boundary-condition property suggests characterizing coils using only the the plane of the working volume closest to the transmitter. Farther-away data points provide no characterization-dependence not already in the closest points, so farther-away points can confuse characterization fitters.

Pete believes (though has not verified) that the scribble-test data-collection method in this paper would serve for coil characterization:

* C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

The scribble data-collection method involves mounting the receiver on a small flat slider, and slowly sliding the receiver around (and rotating the receiver) on a known-flat surface (such as a granite surface plate). We know that the points mechanically are all on a plane (though we do not know exactly what plane), and that the receiver orientations must all be the same within rotations about the axis perpendicular to the plane.

For coil characterization, the transmitter is mounted under the center of the known-flat surface, as close as the minimum range (minimum transmitter to receiver distance) of the tracker permits.

EM Tracker Coil Characterization

2017-01-18T19:04:04Z

Traneus: Added detail

EM Tracker Coil Characterization

2017-01-18T18:22:28Z

Traneus:

2017 Winter Project Week/Open Source Electromagnetic Trackers

2017-01-13T15:24:15Z

Traneus:

__NOTOC__
<gallery>
Image:PW-Winter2017.png|link=2017_Winter_Project_Week#Projects|[[2017_Winter_Project_Week#Projects|Projects List]]

</gallery>

==Key Investigators==

* Peter Traneus Anderson, retired

==Project Description==
{| class="wikitable"
! style="text-align: left; width:27%" | Objective
! style="text-align: left; width:27%" | Approach and Plan
! style="text-align: left; width:27%" | Progress and Next Steps
|- style="vertical-align:top;"
|

* Add further references and discussion of two-state 6DOF tracker.
* Add further discussion of coil characterization.
|

* Add pages to Open Source Electromagnetic Trackers on this Wiki.
|

* Added page with discussion and key references for two-state 6DOF tracker.
* Added page discussing the effects of electric-field coupling in 6DOF tracker and method of preventing e-field coupling.
* Added page showing example calculation of signal/noise needed to achieve given accuracy. Next step is to finish the discussion and example calculation.
|}

==Background and References==

6DOF Electromagnetic Tracker Electric Field

2017-01-13T15:10:08Z

Traneus:

6DOF trackers exhibiting submillimeter error at range of half a meter, can be accuracy-limited by electric-field coupling from transmitter to receiver.

Magnetic charges do not exist, so the magnetic field exhibits dipole-dipole magnetic (inductive)coupling from transmitter to receiver, so the receiver signal is inversely proportional to the cube of range for ranges short compared to a wavelength (near-field regime).

Electric charges do exist, so the lowest-order electric (capacitive) coupling is monopole-monopole, giving receiver signal inversely proportional to range.

Electric_signal/magnetic_signal is thus proportional to the square of range, increasing at large ranges.

Thus, the angular error due to electric coupling is proportional to the square of range, increasing at larger ranges. The electric-coupling-caused transverse position error in millimeters is an angle times range so is proportional to the cube of range.

Electric coupling can be blocked by an electrically-slightly-conducting grounded shield layer surrounding the receiver.

6DOF Electromagnetic Tracker Electric Field

2017-01-13T15:04:04Z

Traneus:

Magnetic charges do not exist, so the magnetic field exhibits dipole-dipole magnetic (inductive)coupling from transmitter to receiver, so the receiver signal is inversely proportional to the cube of range for ranges short compared to a wavelength (near-field regime).

Electric charges do exist, so the lowest-order electric (capacitive) coupling is monopole-monopole, giving receiver signal inversely proportional to range.

Thus, the angular error due to electric coupling is proportional to the square of range, increasing at larger ranges. The transverse position error in millimeters is thus proportional to the cube of range.

6DOF Electromagnetic Tracker Electric Field

2017-01-13T14:59:40Z

Traneus: Created page with "The magnetic field exhibits dipole-dipole coupling from transmitter to receiver."

The magnetic field exhibits dipole-dipole coupling from transmitter to receiver.

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-01-13T14:58:46Z

Traneus: added E-field discussion link

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-01-12T18:25:43Z

Traneus:

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-01-12T18:23:24Z

Traneus:

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-01-12T18:11:44Z

Traneus:

6DOF Electromagnetic Tracker Signal to Noise Requirements Calculation

2017-01-12T18:03:56Z

Traneus: Created page with "We start with the position-accuracy requirement, as that is usually more stringent than the orientation-accuracy requirement."

We start with the position-accuracy requirement, as that is usually more stringent than the orientation-accuracy requirement.

6DOF Electromagnetic Tracker Construction HOWTO

2017-01-12T18:01:25Z

Traneus:

A basic 6DOF (six degrees of freedom: three of position and three of orientation) electromagnetic tracker can contain the parts shown in this block diagram, though there are many variations. [[File:6DOF_tracker_block_diagram.png]]

* https://web.archive.org/web/20151002101401/http://home.comcast.net/~traneus/dry_emtrackertricoil.htm is an example of a breadboard 6DOF tracker.

* Transmitter contains three colocated orthogonal coils. The coils are approximated as magnetic dipoles.

* Receiver contains three colocated orthogonal coils. the coils are approximated as dipoles.

* [[Media:EM_tracker_classic_coil_trio.jpg]] Photo of classic coil trio used as transmitter or receiver. The three coils are wound on a black plastic cube about one centimeter on a side.

* [[http://rose.eu.org/2014/tag/plume Scroll down to see photo of hand-building a transmitter coil trio.]]

* [[Media:Dry_elphel_model_1_rcvr_coils.jpg]] Photo of crude handmade receiver coil trio using [[http://www.sonion.com Sonion]] T 20 AG telecoils. Each coil is ten millimeters long.

* Three transmitter coils times three receiver coils gives nine coil-coupling measurements, expressable as a 3x3 signal matrix, HFluxPerIMeasured.

* Each component of HFLuxPerIMeasured is the magnetic flux through one receiver coil (due to magnetic field H from transmitter coil), divided by the current I in one transmitter coil. HFLuxPerIMeasured has units of meters, and is a geometrical property of the coils' sizes, shapes, number of turns, ferromagnetic core (if any), positions, and orientations. [[EM_Tracker_HFluxPerI_Derivation | HFluxPerI coupling between two dipole coils]].

* Algorithm software converts HFluxPerIMeasured to estimated receiver position and orientation, using direct-solution algorithm in Raab's 1981 paper or iterative solution in Raab etal's 1979 paper.

* Frederick H. Raab, "Quasi-Static Magnetic-Field Technique for Determining Position and Orientation", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-19, No. 4, October 1981, pages 235-243, describes closed-form algorithm for concentric-dipole coil trios. Position is calculated first, directly in cartesian coordinates. Orientation is then calculated.

* Frederick H. Raab, "Remote Object Position Locater", expired U.S. Patent 4,054,881. Describes frequency-multiplexed hardware.

* Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-15, No. 5, September 1979, pages 709-718, describes iterative algorithm for concentric-dipole coil trios, using small-angle approximation for changes in position and in orientation. Includes sensitivity matrix of magnetic couplings partial derivatives with respect to changes in position and orientation.

* Berthold K. P. Horn, "Closed-form solution of absolute orientation using unit quaternions", Journal of the Optical Society of America A, volume 4, April, 1987, pages 629-642, has algorithm for converting from orthonormal rotation matrices to quaternions. Note error: r[2][1] on page 641 is incorrect, while r[2][1] on page 643 is correct.

* [[File:Dry0097.c]] is a simulator program containing an implementation of Raab's algorithm.

* The software which calculates position and orientation from HFluxPerI measurements, is an example of realtime embedded computational electromagnetics.

* Needed HFluxPerI measurement accuracy can be calculated by a sensitivity analysis. As in Raab, Blood, Steiner, Jones, treating position in spherical coordinates gives one distance (called range, the distance between transmitter and receiver) and five angles.

* The electromagnetics results in the signal-to-noise ratio in the five angles being 3.4 times worse than the HFluxPerIMeasured signal-to-noise ratio, due to interactions between position errors and orientation errors.

* The electromagnetics results in the signal-to-noise ratio in range being 3 times better than the HFluxPerIMeasured signal-to-noise ratio, due to the inverse-cube law of dipole-dipole field coupling.

* [[6DOF_Electromagnetic_Tracker_Signal_to_Noise_Requirements_Calculation]] details calculating signal-to-noise ratio (SNR) from accuracy requirements.

* There is an inherent hemisphere ambiguity, since receiver at position = (Xo,Yo,Zo) and receiver at position = (-Xo,-Yo,-Zo) show identical HFluxPerIMeasured for identical orientations. This ambiguity can be resolved by using additional transmitter or receiver coils spaced away from the colocated transmitter or receiver coils.

* The receiver is normally kept on one side of the transmitter, to avoid the hemisphere ambiguity.

* The transmitter field on the unused side of the transmitter, can be eliminated by using a magnetic mirror: Reference expired U.S. patent 5,640,170, which references many older expired EM-tracker patents.

* Accuracy is poor for lined-up pose: receiver positioned on a transmitter-coil axis, with receiver oriented to make receiver-coil axes parallel with transmitter-coil axes. Some of the first-order partial derivatives go to zero in these cases, causing the position-and-orientation solution to separate into four separate partial solutions.

* Poses with poor tracking accuracy, should be good for coil characterization. Coil-characterization poses should be chosen based on electromagnetic theory, rather than on mechanical measurements. Use receiver positions close to transmitter, based on boundary-condition principles of EM theory. Receiver positions on both sides of transmitter (even if only one side is used in operation), are necessary to distinguish transmitter gain from transmitter nonconcentricity. [[EM_Tracker_Coil_Characterization]] has further discussion.

* The poor-accuracy poses can be reduced, by replacing the three-orthogonal-coil receiver with a receiver comprising four colocated coils (the transmitter remains three orthogonal coils). The four receiver coils point in the directions of the vertices of a regular tetrahedron. When the four-coil receiver is positioned on a transmitter-coil axis, with one receiver coil oriented parallel to a transmitter-coil axis, the remaining three receiver coils' axes cannot be parallel to transmitter-coil axes.

* Three transmitter coils times three receiver coils, gives nine coil-coupling measurements. These can be performed sequentially, partially sequentially and partially simultaneously, or all simultaneously. Sequential measurements simplify the electronics, but impair dynamic accuracy: When the receiver is moving, sequential measurements result in inconsistent datasets, leading to position and orientation dynamic errors.

* Receiver coil signals can be measured simultaneously or sequentially. Simultaneous measurements improve signal-to-noise ratio.

* Many designs used one operating frequency, driving the transmitter coils sequentially. Use of one frequency simplifies handling frequency-dependent effects.

* Multiple-frequency designs drive the three transmitter coils simultaneously, with sinewaves at three distinct frequencies. This improves signal-to-noise ratio by lengthening measurement time.

* Operating frequencies are typically 30 Hz to 15000 Hz. 1000 Hz, 1300 Hz, and 1600 Hz are a good starting point. Higher frequencies give higher induced voltages, lower frequencies reduce error-causing eddy-current effects.

* The transmitter coils are usually series tuned with capacitors.

* The transmitter-coil currents must be measured. The currents vary slowly due to coil heating, so currents can be measured periodically.

* Some designs use DC pulses to drive the transmitter coils, instead of AC frequencies. This simplifies driver design, but makes receiver signal recovery more difficult. Pulse-driven transmitter coils must be driven sequentially.

* Data-acquisition electronics measures the currents in the three transmitter coils, and measures the voltages induced in the three receiver coils.

* 24-bit audio ADCs have enough dynamic range to avoid the need for gain-switching.

* Avoid gain-switching, as the ratios of the gain states are not precisely-enough known.

* A six-ADC electronics can measure three transmitter-coil currents and three receiver-coil voltages continually and simultaneously.

* Add three more ADCs for each additional receiver coil trio.

* A four-ADC electronics can use one channel to measure the currents periodically over time (The currents change slowly as the transmitter coils warm up.), and three channels to measure the three voltages continually and simultaneously.

* A two-ADC system can measure currents sequentially with one ADC and voltages sequentially with the other ADC.

* A single-ADC electronics can measure the currents and voltages sequentially.

* C. L. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level," Proc. IRE, Vol. 35, pp. 335-348, June, 1946. The original Dolph-Chebyshev window article. This window is capable of 140 dB rejection of out-of-band signals.

* Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1) 84-91, February 1981, doi:10.1109/TASSP.1981.1163506, "U.S. Government work not subject to U.S. copyright". The window in Figure 10 of this paper is (for symmetrical limits |t|<=L/2): w(t) = (1/L)(10/32 + 15/32 cos(2pi t/L) + 6/32 cos(4pi t/L) + 1/32 cos(6pi t/L)), and is zero for all t outside the L/2 limits. The sidelobe peak four DFT bins from the central peak is 91 dB down from the central peak. This window function and its first through fifth derivatives are all continuous for all t, giving 42 dB/octave rolloff of sidelobes.

* Expired U.S. Patent 4,109,199 describes the use of a calibration coil in the receiver to calibrate the gains of the electronics.

* More elaborate algorithms provide higher accuracy at the expense of much more computation. by modeling the non-dipole and/or non-concentric parts of the coils. Expired U.S. Patent 5,307,072 is an early example.

* C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006, reports low-cost accuracy-testing methods using a known-flat nonmagnetic surface (such as a granite surface plate).

* A 6DOF tracker using four-coil printed-circuit transmitter and receiver (optimized for academic originality) is discussed in: Peter Traneus Anderson, "A Source of Accurately Calculable Quasi-Static Magnetic Fields", dissertation presented to the Faculty of the Graduate College of the University of Vermont, October 2001, stored here as three files: [[Media:AndersonPeterDissertation.pdf]] is the main body. [[Media:AndersonPeterDissertationReadme.pdf]] contains copyright license, additional comments, and four figures that are blank in the main body. [[Media:AndersonPeterDissertationFig14r1.jpg]] is the color original photo of two of the figures.

* A 6DOF tracker using two transmitter coils (instead of three) can be built; Frederick Raab calls this [[6DOF_Two_State_Electromagnetic_Trackers|two-state excitation]] in his 1981 paper referenced above.

Open Source Electromagnetic Trackers

2017-01-12T17:54:33Z

Traneus:

__NOTOC__

[[Image:Dry0030.png]]
[[Image:Dry_elphel_model_1_rcvr_coils.jpg]]

In the photo, each of the three receiver coils is ten millimeters long, and is a
[[http://www.sonion.com Sonion]] T 20 AG telecoil usually used in hearing aids.

==Key Personnel==
* [[User:Traneus | Peter Traneus Anderson]]
* [[User:Tkapur | Tina Kapur]]
* [[User:SPujol | Sonia Pujol]]

==Goals of the Project==
To teach the process of developing electromagnetic trackers for research, to foster an open community of researchers developing electromagnetic trackers, to develop open-source software and open-source hardware for working research electromagnetic trackers interfacing to Slicer through [[OpenIGTLink|OpenIGTLink]].

==Current Progress==

[[2014_Summer_Project_Week:Open_source_electromagnetic_trackers_usingOpenIGTLink | 2014 Summer Project Week]]

[[6DOF_Electromagnetic_Tracker_Construction_HOWTO|6DOF Electromagnetic Tracker Construction HOWTO]]

Pete's current efforts are towards developing low-cost coil-characterization methods which make sense electromagnetically, aiming for a published paper. Pete has yet to encounter a paper which covers this topic.

==References==
* https://github.com/traneus/emtrackers

* http://www.hackster.io/plume/plume http://rose.eu.org/2014/tag/plume another 6DOF EM tracker project.

* Project started at [[2011_Summer_project_Week_Open_Source_Electromagnetic_Trackers_using_OpenIGTLink| 2011 Summer Project Week]]

*Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic systems, Vol. AES-15, No. 4, September 1979, pages 709-718. Iterative solution for 6DOF tracker. Includes sensitivity matrix of magnetic couplings partial derivatives with respect to position and orientation changes.

*Frederick H. Raab, "Quasi-Static Magnetic-Field Technique for Determining Position and Orientation", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-19, No. 4, October 1981, pages 235-243. Direct solution for 6DOF tracker.

*Tobias Schroeder, "An accurate magnetic field solution for medical electromagnetic tracking coils at close range", Journal of Applied Physics 117, 224504 (2015). Current-sheet model for cubical coils.

*C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

*C. L. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level," Proceedings of the IRE (now part of the IEEE), Vol. 35, pp. 335-348, June, 1946. The original Dolph-Chebyshev Fourier-transform window article. Dolph-Chebyshev window can give 140 dB rejection in the stopband.

*Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1) 84-91, doi:10.1109/TASSP.1981.1163506, "U.S. Government work not subject to U.S. copyright", in particular Figure 10 window for -L/2 < t < L/2: w(t) = (1/L) (10/32 + 15/32 cos(2pi t/L) + 6/32 cos(4pi t/L) + 1/32 cos(6pi t/L)) has first sidelobe at -61 dB and 42 dB/octave sidelobe rolloff.

*Eugene Paperno, "Suppression of magnetic noise in the fundamental-mode orthogonal fluxgate", Elsevier, Sensors and Actuators A 116 (2004) 405-409. Picotesla noise in 20 mm long 1 mm diameter fluxgate magnetometer. To get low noise, the drive flux swings between saturation in one direction and zero flux. The usual noisy fluxgate drive flux swings between saturation in one direction and saturation in the other direction, to ease measurement down to DC.

*[[http://scitation.aip.org/content/aip/journal/jap/115/17/10.1063/1.4861675 | Nara etal, "Moore-Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole"]][[http://dx.doi.org/10.1063/1.4861675 | J. Appl. Phys. 115, 17E504 (2014)]] on field-and-gradient single-coil 5DOF tracking closed-form algorithm.

*James M. Chappell, Samuel P. Drake, Cameron L. Seidel, Lachlan J. Gunn, Azhar Iqbal, Andrew Allison, Derek Abbott, "Geometric Algebra for Electrical and Electronic Engineers", Proceedings of the IEEE, Vol. 102, No. 9, September 2014, pages 1340 to 1363. Clifford algebra formulation of electromagnetics using vectors, bivectors, trivector.

==Citations==

EM Tracker Coil Characterization

2017-01-11T17:31:31Z

Traneus: Improved some language.

EM Tracker Coil Characterization

When we test a tracker for accuracy, we generally check using point sets that make mechanical sense. For example, we may use a 3D robot to move the receiver all over the working volume (also called the motion box).

In 6DOF trackers, the coils must be precisely characterized for their electromagnetic properties: gains, non-orthogonalities, non-concentricities, and finite-size effects. Since characterization is measuring electromagnetic properties, the point set chosen should make electromagnetic sense rather than mechanical sense.

The coils can only be manufactured to so much precision. Characterization measures the actual properties of each coil, to improve the tracker accuracy.

One important property of electromagnetics, is the boundary-condition property: If we know the magnetic field everywhere on the boundary of the working volume, we can calculate the field inside the working volume.

For coil characterization, the boundary-condition property suggests characterizing coils using only the the plane of the working volume closest to the transmitter. This could be done with a 2D robot or with a 3D robot. If a 3D robot is used, we can then use the remainder of the 3D robot points to check mechanical accuracy.

Pete believes (though has not verified) that the 2D robot can be replaced by the scribble-test data-collection method in this paper:

* C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

The scribble data-collection method involves mounting the receiver on a small flat slider, and slowly sliding the receiver around (and rotating the receiver) on a known-flat surface (such as a granite surface plate). We know that the points mechanically are all on a plane (though we do not know exactly what plane), and that the receiver orientations must all be the same within rotations about the axis perpendicular to the plane.

EM Tracker Coil Characterization

2017-01-09T21:21:49Z

Traneus: Further discussion added

EM Tracker Coil Characterization

When we test a tracker for accuracy, we generally check using point sets that make mechanical sense. For example, we may use a 3D robot to move the receiver all over the working volume (also called the motion box).

In 6DOF trackers, the coils must be precisely characterized for their electromagnetic properties: gains, non-orthogonalities, non-concentricities, and finite-size effects. Since characterization is measuring electromagnetic properties, the point set chosen should make electromagnetic sense rather than mechanical sense.

The coils can only be manufactured to so much precision. Characterization measures the actual properties of each coil, to improve the tracker accuracy.

The important electromagnetic property, is the boundary-condition property: If we know the magnetic field everywhere on the boundary of the working volume, we can calculate the field inside the working volume.

For coil characterization, the boundary-condition property suggests characterizing coils using only the the plane of the working volume closest to the transmitter. This could be done with a 2D robot or with a 3D robot. If a 3D robot is used, we can then use the remainder of the 3D robot points to check mechanical accuracy.

Pete believes (though has not verified) that the 2D robot can be replaced by the scribble-test data-collection method in this paper:

* C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

The scribble data-collection method involves mounting the receiver on a small flat slider, and slowly sliding the receiver around (and rotating the receiver) on a known-flat surface (such as a granite surface plate). We know that the points mechanically are all on a plane (though we do not know exactly what plane), and that the receiver orientations must all be the same, except for rotations about the axis perpendicular to the plane.

EM Tracker Coil Characterization

2017-01-09T21:17:23Z

Traneus:

EM Tracker Coil Characterization

When we test a tracker for accuracy, we generally check using point sets that make mechanical sense. For example, we may use a 3D robot to move the receiver all over the working volume (also called the motion box).

In 6DOF trackers, the coils must be precisely characterized for their electromagnetic properties: gains, non-orthogonalities, non-concentricities, and finite-size effects. Since characterization is measuring electromagnetic properties, the point set chosen should make electromagnetic sense rather than mechanical sense.

The coils can only be manufactured to so much precision. Characterization measures the actual properties of each coil, to improve the tracker accuracy.

The important electromagnetic property, is the boundary-condition property: If we know the magnetic field everywhere on the boundary of the working volume, we can calculate the field inside the working volume.

For coil characterization, the boundary-condition property suggests characterizing coils using only the the plane of the working volume closest to the transmitter. This could be done with a 2D robot or with a 3D robot. If a 3D robot is used, we can then use the remainder of the 3D robot points to check mechanical accuracy.

Pete believes (though has not verified) that the 2D robot can be replaced by the scribble-test data-collection method in this paper:

C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006. Reports low-cost accuracy-measuring techniques and results for various trackers.

EM Tracker Coil Characterization

2017-01-09T21:14:46Z

Traneus:

EM Tracker Coil Characterization

2017-01-09T21:08:51Z

Traneus:

EM Tracker Coil Characterization

2017-01-09T21:04:47Z

Traneus:

EM Tracker Coil Characterization

2017-01-09T21:01:52Z

Traneus: Coil Characterization is EM problem, not mechanical problem

6DOF Electromagnetic Tracker Construction HOWTO

2017-01-09T20:58:35Z

Traneus:

A basic 6DOF (six degrees of freedom: three of position and three of orientation) electromagnetic tracker can contain the parts shown in this block diagram, though there are many variations. [[File:6DOF_tracker_block_diagram.png]]

* https://web.archive.org/web/20151002101401/http://home.comcast.net/~traneus/dry_emtrackertricoil.htm is an example of a breadboard 6DOF tracker.

* Transmitter contains three colocated orthogonal coils. The coils are approximated as magnetic dipoles.

* Receiver contains three colocated orthogonal coils. the coils are approximated as dipoles.

* [[Media:EM_tracker_classic_coil_trio.jpg]] Photo of classic coil trio used as transmitter or receiver. The three coils are wound on a black plastic cube about one centimeter on a side.

* [[http://rose.eu.org/2014/tag/plume Scroll down to see photo of hand-building a transmitter coil trio.]]

* [[Media:Dry_elphel_model_1_rcvr_coils.jpg]] Photo of crude handmade receiver coil trio using [[http://www.sonion.com Sonion]] T 20 AG telecoils. Each coil is ten millimeters long.

* Three transmitter coils times three receiver coils gives nine coil-coupling measurements, expressable as a 3x3 signal matrix, HFluxPerIMeasured.

* Each component of HFLuxPerIMeasured is the magnetic flux through one receiver coil (due to magnetic field H from transmitter coil), divided by the current I in one transmitter coil. HFLuxPerIMeasured has units of meters, and is a geometrical property of the coils' sizes, shapes, number of turns, ferromagnetic core (if any), positions, and orientations. [[EM_Tracker_HFluxPerI_Derivation | HFluxPerI coupling between two dipole coils]].

* Algorithm software converts HFluxPerIMeasured to estimated receiver position and orientation, using direct-solution algorithm in Raab's 1981 paper or iterative solution in Raab etal's 1979 paper.

* Frederick H. Raab, "Quasi-Static Magnetic-Field Technique for Determining Position and Orientation", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-19, No. 4, October 1981, pages 235-243, describes closed-form algorithm for concentric-dipole coil trios. Position is calculated first, directly in cartesian coordinates. Orientation is then calculated.

* Frederick H. Raab, "Remote Object Position Locater", expired U.S. Patent 4,054,881. Describes frequency-multiplexed hardware.

* Frederick H. Raab, Ernest B. Blood, Terry O. Steiner, Herbert R. Jones, "Magnetic Position and Orientation Tracking System", IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-15, No. 5, September 1979, pages 709-718, describes iterative algorithm for concentric-dipole coil trios, using small-angle approximation for changes in position and in orientation. Useful for sensitivity analyses.

* Berthold K. P. Horn, "Closed-form solution of absolute orientation using unit quaternions", Journal of the Optical Society of America A, volume 4, April, 1987, pages 629-642, has algorithm for converting from orthonormal rotation matrices to quaternions. Note error: r[2][1] on page 641 is incorrect, while r[2][1] on page 643 is correct.

* [[File:Dry0097.c]] is a simulator program containing an implementation of Raab's algorithm.

* The software which calculates position and orientation from HFluxPerI measurements, is an example of realtime embedded computational electromagnetics.

* Needed HFluxPerI measurement accuracy can be calculated by a sensitivity analysis. As in Raab, Blood, Steiner, Jones, treating position in spherical coordinates gives one distance (called range, the distance between transmitter and receiver) and five angles.

* The electromagnetics results in the signal-to-noise ratio in the five angles being 3.4 times worse than the HFluxPerIMeasured signal-to-noise ratio, due to interactions between position errors and orientation errors.

* The electromagnetics results in the signal-to-noise ratio in range being 3 times better than the HFluxPerIMeasured signal-to-noise ratio, due to the inverse-cube law of dipole-dipole field coupling.

* There is an inherent hemisphere ambiguity, since receiver at position = (Xo,Yo,Zo) and receiver at position = (-Xo,-Yo,-Zo) show identical HFluxPerIMeasured for identical orientations. This ambiguity can be resolved by using additional transmitter or receiver coils spaced away from the colocated transmitter or receiver coils.

* The receiver is normally kept on one side of the transmitter, to avoid the hemisphere ambiguity.

* The transmitter field on the unused side of the transmitter, can be eliminated by using a magnetic mirror: Reference expired U.S. patent 5,640,170, which references many older expired EM-tracker patents.

* Accuracy is poor for lined-up pose: receiver positioned on a transmitter-coil axis, with receiver oriented to make receiver-coil axes parallel with transmitter-coil axes. Some of the first-order partial derivatives go to zero in these cases, causing the position-and-orientation solution to separate into four separate partial solutions.

* Poses with poor tracking accuracy, should be good for coil characterization. Coil-characterization poses should be chosen based on electromagnetic theory, rather than on mechanical measurements. Use receiver positions close to transmitter, based on boundary-condition principles of EM theory. Receiver positions on both sides of transmitter (even if only one side is used in operation), are necessary to distinguish transmitter gain from transmitter nonconcentricity. [[EM_Tracker_Coil_Characterization]] has further discussion.

* The poor-accuracy poses can be reduced, by replacing the three-orthogonal-coil receiver with a receiver comprising four colocated coils (the transmitter remains three orthogonal coils). The four receiver coils point in the directions of the vertices of a regular tetrahedron. When the four-coil receiver is positioned on a transmitter-coil axis, with one receiver coil oriented parallel to a transmitter-coil axis, the remaining three receiver coils' axes cannot be parallel to transmitter-coil axes.

* Three transmitter coils times three receiver coils, gives nine coil-coupling measurements. These can be performed sequentially, partially sequentially and partially simultaneously, or all simultaneously. Sequential measurements simplify the electronics, but impair dynamic accuracy: When the receiver is moving, sequential measurements result in inconsistent datasets, leading to position and orientation dynamic errors.

* Receiver coil signals can be measured simultaneously or sequentially. Simultaneous measurements improve signal-to-noise ratio.

* Many designs used one operating frequency, driving the transmitter coils sequentially. Use of one frequency simplifies handling frequency-dependent effects.

* Multiple-frequency designs drive the three transmitter coils simultaneously, with sinewaves at three distinct frequencies. This improves signal-to-noise ratio by lengthening measurement time.

* Operating frequencies are typically 30 Hz to 15000 Hz. 1000 Hz, 1300 Hz, and 1600 Hz are a good starting point. Higher frequencies give higher induced voltages, lower frequencies reduce error-causing eddy-current effects.

* The transmitter coils are usually series tuned with capacitors.

* The transmitter-coil currents must be measured. The currents vary slowly due to coil heating, so currents can be measured periodically.

* Some designs use DC pulses to drive the transmitter coils, instead of AC frequencies. This simplifies driver design, but makes receiver signal recovery more difficult. Pulse-driven transmitter coils must be driven sequentially.

* Data-acquisition electronics measures the currents in the three transmitter coils, and measures the voltages induced in the three receiver coils.

* 24-bit audio ADCs have enough dynamic range to avoid the need for gain-switching.

* Avoid gain-switching, as the ratios of the gain states are not precisely-enough known.

* A six-ADC electronics can measure three transmitter-coil currents and three receiver-coil voltages continually and simultaneously.

* Add three more ADCs for each additional receiver coil trio.

* A four-ADC electronics can use one channel to measure the currents periodically over time (The currents change slowly as the transmitter coils warm up.), and three channels to measure the three voltages continually and simultaneously.

* A two-ADC system can measure currents sequentially with one ADC and voltages sequentially with the other ADC.

* A single-ADC electronics can measure the currents and voltages sequentially.

* C. L. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level," Proc. IRE, Vol. 35, pp. 335-348, June, 1946. The original Dolph-Chebyshev window article. This window is capable of 140 dB rejection of out-of-band signals.

* Albert H. Nuttall, "Some Windows with Very Good Sidelobe Behavior", IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1) 84-91, February 1981, doi:10.1109/TASSP.1981.1163506, "U.S. Government work not subject to U.S. copyright". The window in Figure 10 of this paper is (for symmetrical limits |t|<=L/2): w(t) = (1/L)(10/32 + 15/32 cos(2pi t/L) + 6/32 cos(4pi t/L) + 1/32 cos(6pi t/L)), and is zero for all t outside the L/2 limits. The sidelobe peak four DFT bins from the central peak is 91 dB down from the central peak. This window function and its first through fifth derivatives are all continuous for all t, giving 42 dB/octave rolloff of sidelobes.

* Expired U.S. Patent 4,109,199 describes the use of a calibration coil in the receiver to calibrate the gains of the electronics.

* More elaborate algorithms provide higher accuracy at the expense of much more computation. by modeling the non-dipole and/or non-concentric parts of the coils. Expired U.S. Patent 5,307,072 is an early example.

* C.A. Nafis, V. Jensen, L. Beauregard, P.T. Anderson, "Method for estimating dynamic EM tracking accuracy of Surgical Navigation tools", SPIE Medical Imaging Proceedings, 2006, reports low-cost accuracy-testing methods using a known-flat nonmagnetic surface (such as a granite surface plate).

* A 6DOF tracker using four-coil printed-circuit transmitter and receiver (optimized for academic originality) is discussed in: Peter Traneus Anderson, "A Source of Accurately Calculable Quasi-Static Magnetic Fields", dissertation presented to the Faculty of the Graduate College of the University of Vermont, October 2001, stored here as three files: [[Media:AndersonPeterDissertation.pdf]] is the main body. [[Media:AndersonPeterDissertationReadme.pdf]] contains copyright license, additional comments, and four figures that are blank in the main body. [[Media:AndersonPeterDissertationFig14r1.jpg]] is the color original photo of two of the figures.

* A 6DOF tracker using two transmitter coils (instead of three) can be built; Frederick Raab calls this [[6DOF_Two_State_Electromagnetic_Trackers|two-state excitation]] in his 1981 paper referenced above.

2017 Winter Project Week/Open Source Electromagnetic Trackers

2017-01-01T23:20:29Z

Traneus: /* Project Description */

__NOTOC__
<gallery>
Image:PW-Winter2017.png|link=2017_Winter_Project_Week#Projects|[[2017_Winter_Project_Week#Projects|Projects List]]

</gallery>

==Key Investigators==

* Peter Traneus Anderson, retired

==Project Description==
{| class="wikitable"
! style="text-align: left; width:27%" | Objective
! style="text-align: left; width:27%" | Approach and Plan
! style="text-align: left; width:27%" | Progress and Next Steps
|- style="vertical-align:top;"
|

* Add further references and discussion of two-state 6DOF tracker.
* Add further discussion of coil characterization.
|

* Add pages to Open Source Electromagnetic Trackers on this Wiki.
|

*
|}

==Background and References==

2017 Winter Project Week/Open Source Electromagnetic Trackers

2017-01-01T23:16:59Z

Traneus: /* Key Investigators */