Beacon-based geolocation using a low frequency electromagnetic field

1. A method of performing geolocation of an object, the method comprising:

detecting, with one or more receivers, magnetic field signals generated by one or more transmitters, with each transmitter comprising two or more coils that are mechanically stable with respect to one another, are positioned around a same point and are electrically variable with respect to each other, wherein each of the two or more coils generates a signal with a different frequency than others of the two or more coils and the coils do not include a common controller, the frequencies being in the ultra-low frequency (ULF) or very-low frequency (VLF) range;

decomposing the magnetic field signals from each of the coils of each of the two or more transmitters into individual magnetic field components for each coil of each transmitter;

calculating, from the individual magnetic field components, a set of invariant scalar values that are independent of orientation of one or more magnetic antennas associated with the one or more receivers;

using the set of invariant scalar values, determining a geographic position of the object by:

determining a line of bearing (LOB) to each of the one or more transmitters based on an analysis of the set of invariant scalar values, the set of invariant scalar values being derived from a total, n, of magnetic field signals generated by the two or more coils of each of the one or more transmitters, and wherein n corresponds to, for each transmitter, the number of co-located coils;

determining, for a given transmitter, i, an LOB dependent plane, {circumflex over (n)} i , based on the magnetic field signals generated by the first and second coils in accordance with the expression:

n

^

i

=

B

ω

1

i

×

B

ω

2

i



B

ω

_

1

i

×

B

ω

_

2

i



;

wherein B 1 i and B 2 i are the magnetic fields, respectively, of the two coils of the first coil and second coil of the transmitter i;

wherein a given individual coil (k) is characterized by a single a geometrical parameter comprising its orientation, {circumflex over (m)} k ; and

wherein a given pair of coils (k, l) is characterized by a transmission plane orientation in accordance with the expression:

u

^

=

m

^

k

×

m

^

l



m

^

k

×

m

^

l



;

wherein {circumflex over (m)} k and {circumflex over (m)} l are unitary vectors roughly perpendicular to respective areas of the pair of coils (k, l); and

outputting a global positioning system location of the object to a user over an output display based on the determined geographic position of the object.

2. The method of claim 1 , further comprising correlating LOB, {circumflex over (r)} i , for the transmitter i, with transmission plane orientation, û i , for the first and second coils of the transmitter i in accordance with the expressions:

{

u

)

i

·

n

^

i

=

3

⁢

(

r

^

i

·

u

^

i

)

2

-

2

4

-

3

⁢

(

r

^

i

·

u

^

i

)

2

(

u

)

i

×

n

^

i

)

·

r

)

i

=

0.

.

3. The method of claim 1 , wherein calculating the set of invariant scalar values further comprises calculating:

n

^

i

·

n

^

j

=

B

ω

1

i

×

B

ω

2

i



B

ω

_

1

i

×

B

ω

_

2

i



·

B

ω

1

j

×

B

ω

2

j



B

ω

_

1

j

×

B

ω

_

2

j



wherein {circumflex over (n)} i and {circumflex over (n)} j are the LOB dependent planes associated with pairs of coils of the transmitter i and another transmitter, j.

4. A geolocation system, comprising:

one or more receivers configured to detect magnetic field signals, the magnetic signal generated by one or more transmitters, each comprising two or more coils, that are electrically variable with respect to each other, and that have a mechanically stable orientation with respect to one another, wherein each of the two or more coils generates a signal with a different frequency than others of the two or more coils and the coils do not include a common controller, the frequencies being in the ultra-low frequency (ULF) or very-low frequency (VLF) range; and

a processing device configured to:

decompose the detected magnetic field signals into individual magnetic field components for each coil of each transmitter;

calculate, from the individual magnetic field components, a set of invariant scalar values that are independent of orientation of one or more magnetic antennas associated with the one or more receivers; and

using the set of invariant scalar values, determine a geographic position of an object by:

determining a line of bearing (LOB) to each of the two or more transmitters based on an analysis of the set of invariant scalar values, the set of invariant scalar values being derived from a total, n, of magnetic field signals generated by the two or more coils of each of the two or more transmitters, and wherein n corresponds to, for each transmitter, the number of coils;

determining, for a given transmitter, i, having a first coil and a second coil, an LOB dependent plane, {circumflex over (n)} h , based on the magnetic field signals generated by the first and second coils in accordance with the expression:

n

^

i

=

B

ω

1

i

×

B

ω

2

i



B

ω

_

1

i

×

B

ω

_

2

i



;

wherein B 1 i and B 2 i are the magnetic fields, respectively, of the first coil and second coil of the transmitter i;

wherein a given individual coil (k) is characterized by a single a geometrical parameter comprising its orientation, {circumflex over (m)} k ; and

wherein a given pair of coils (k, l) is characterized by a transmission plane orientation in accordance with the expression

u

^

=

m

^

k

×

m

^

l



m

^

k

×

m

^

l



;

wherein {circumflex over (m)} k and {circumflex over (m)} l are unitary vectors roughly perpendicular to respective areas of the pair of coils (k, l); and

outputting a global positioning system location of the object to a user over an output display based on the determined geographic position of the object.

5. The system of claim 4 , wherein the processing device is further configured to correlate LOB, {circumflex over (r)} i , for the transmitter i, with transmission plane orientation, û i , for the first and second coils of the transmitter i in accordance with the expressions:

{

u

)

i

·

n

^

i

=

3

⁢

(

r

^

i

·

u

^

i

)

2

-

2

4

-

3

⁢

(

r

^

i

·

u

^

i

)

2

(

u

)

i

×

n

^

i

)

·

r

)

i

=

0.

.

6. The system of claim 4 , wherein calculating the set of invariant scalar values further comprises calculating:

n

^

i

·

n

^

j

=

B

ω

1

i

×

B

ω

2

i



B

ω

_

1

i

×

B

ω

_

2

i



·

B

ω

1

j

×

B

ω

2

j



B

ω

_

1

j

×

B

ω

_

2

j



wherein {circumflex over (n)} i and {circumflex over (n)} j are the LOB dependent planes associated with pairs of coils of the transmitter i and another transmitter, j.