Gyromagnetic geopositioning system
A geopositioning system. The geopositioning system includes an accelerometer including three sensing axes, a gyroscope including three sensing axes, and a magnetometer including three sensing axes, and a processing circuit. The processing circuit is configured to calculate a location of the geopositioning system as a latitude, longitude, and altitude with respect to the Earth.
1. A geopositioning system comprising:
an accelerometer comprising three sensing axes;
a gyroscope comprising three sensing axes;
a magnetometer comprising three sensing axes; and
a processing circuit comprising a non-transitory storage medium, the non-transitory storage medium having instructions stored therein, which, when executed by the processing circuit, cause the processing circuit to calculate a location of the geopositioning system as a latitude, longitude, and altitude with respect to the Earth based on a combined output of the accelerometer, the gyroscope, and the magnetometer,
wherein the calculating of the location comprises:
determining a plane of a local horizon from measurements performed by the accelerometer;
determining a direction of a spin axis of the Earth from measurements performed by the gyroscope; and
calculating the latitude from the angle between the local horizon and the spin axis.
2. The geopositioning system of claim 1 , wherein the calculating of the latitude comprises:
forming a dot product between
a normal vector to the plane of the local horizon, and
a normalized vector in the direction of the spin axis of the Earth;
calculating an inverse cosine of the dot product; and
subtracting 90 degrees from the inverse cosine.
3. The geopositioning system of claim 2 , wherein the calculating of the location further comprises:
calculating the longitude from
the spin axis; and
measurements obtained by the magnetometer.
4. The geopositioning system of claim 3 , wherein the calculating of the longitude comprises numerically solving for a value of l for which
a
→
n
·
B
→
n
a
→
n
·
B
→
n
=
-
2
·
[
g
1
0
sL
+
(
g
1
1
cl
+
h
1
1
sl
)
cL
]
[
g
1
0
cL
-
(
g
1
1
cl
+
h
1
1
sl
)
sL
]
2
+
[
g
1
1
sl
+
h
1
1
cl
]
2
+
4
·
[
g
1
0
sL
+
(
g
1
1
cl
+
h
1
1
sl
)
cL
]
2
wherein
{right arrow over (a)} n is an acceleration vector measured by the accelerometer;
{right arrow over (B)} n is a magnetic field vector measured by the magnetometer;
g 1 0 , g 1 1 , and h 1 1 are three coefficients of a magnetic field model for the Earth;
cL and sL are the sine and cosine respectively of the latitude;
l is the longitude; and
cl, and sl are the sine and cosine respectively of the longitude.
5. The geopositioning system of claim 4 , wherein:
g 1 0 =−29438.5 nanoTesla (nT);
g 1 1 =−1501.1 nT; and
h 1 1 =4796.2 nT.
6. The geopositioning system of claim 1 , wherein the calculating of the altitude comprises calculating the altitude from measurements obtained by the accelerometer.
7. The geopositioning system of claim 6 , wherein the calculating of the altitude from measurements obtained by the accelerometer comprises:
calculating a magnitude of an acceleration vector measured by the accelerometer; and
determining an altitude at which a magnitude predicted by a gravitational model of the Earth equals the magnitude of the acceleration vector.
8. The geopositioning system of claim 7 , wherein the gravitational model of the Earth represents the Earth's gravitational field as being proportional to
1
r
2
,
where r is the radius from the center of the Earth.
9. The geopositioning system of claim 1 , wherein the calculating of the altitude comprises calculating the altitude from measurements obtained by the magnetometer.
10. The geopositioning system of claim 9 , wherein the calculating of the altitude from measurements obtained by the magnetometer comprises:
calculating a magnitude of a magnetic field vector measured by the magnetometer; and
determining an altitude at which a magnitude predicted by a magnetic model of the Earth equals the magnitude of the magnetic field vector.
11. The geopositioning system of claim 10 , wherein the magnetic model represents the Earth's magnetic field as being proportional to
1
r
3
,
where r is the radius from the center of the Earth.
12. The geopositioning system of claim 1 , wherein the calculating of the altitude comprises calculating the altitude from:
measurements obtained by the accelerometer; and
measurements obtained by the magnetometer.
13. The geopositioning system of claim 12 , wherein the calculating of the altitude comprises calculating the altitude as a weighted average of:
an altitude calculated from measurements obtained by the accelerometer; and
an altitude calculated from measurements obtained by the magnetometer.
14. The geopositioning system of claim 13 , wherein weights of the weighted average are selected to provide, in the weighted average, a measurement error less than the measurement error in an equally-weighted average of
the altitude calculated from measurements obtained by the accelerometer; and
the altitude calculated from measurements obtained by the magnetometer.
15. A geopositioning system comprising:
an accelerometer comprising three sensing axes;
a gyroscope comprising three sensing axes;
a magnetometer comprising three sensing axes;
a gimbaled support; and
a processing circuit comprising a non-transitory storage medium, the accelerometer, the gyroscope, and the magnetometer being secured to the gimbaled support,
wherein the non-transitory storage medium has instructions stored therein, which, when executed by the processing circuit, cause the processing circuit to control the gimbaled support and to calculate a location of the geopositioning system as a latitude, longitude, and altitude with respect to the Earth based on a combined of the accelerometer, the gyroscope, and the magnetometer, and
wherein the calculating of the location comprises:
determining a plane of a local horizon from measurements performed by the accelerometer;
determining a direction of a spin axis of the Earth from measurements performed by the gyroscope; and
calculating the latitude from the angle between the local horizon and the spin axis.
16. The geopositioning system of claim 15 , wherein the two sensing axes of the accelerometer define a plane, and the gyroscope is secured to the gimbaled support with the sensing axes of the gyroscope in the plane.
17. The geopositioning system of claim 15 , wherein the processing circuit is configured to rotate the gimbaled support to an attitude at which each of the two sensing axes of the accelerometer measures no acceleration, and at which the sensing axis of the gyroscope measures no rotation.
18. The geopositioning system of claim 17 , wherein the processing circuit is further configured to calculate the longitude by numerically solving for a value of l for which
a
→
n
·
B
→
n
a
→
n
·
B
→
n
=
-
2
·
[
g
1
0
sL
+
(
g
1
1
cl
+
h
1
1
sl
)
cL
]
[
g
1
0
cL
-
(
g
1
1
cl
+
h
1
1
sl
)
sL
]
2
+
[
g
1
1
sl
+
h
1
1
cl
]
2
+
4
·
[
g
1
0
sL
+
(
g
1
1
cl
+
h
1
1
sl
)
cL
]
2
wherein
{right arrow over (a)} n is an acceleration vector measured by the accelerometer;
{right arrow over (B)} n is a magnetic field vector measured by the magnetometer;
g 1 0 , g 1 1 , and h 1 1 are three first Gauss coefficients of a magnetic field model for the Earth;
cL and sL are the sine and cosine respectively of the latitude;
l is the longitude; and
cl, and sl are the sine and cosine respectively of the longitude.
19. The geopositioning system of claim 17 , wherein the calculating of the altitude comprises calculating the altitude as a weighted average of:
an altitude calculated from measurements obtained by the accelerometer; and
an altitude calculated from measurements obtained by the magnetometer.
20. A method for determining a location of a geopositioning system, the geopositioning system comprising:
an accelerometer comprising three sensing axes;
a gyroscope comprising three sensing axes; and
a magnetometer comprising three sensing axes,
the method comprising calculating the location of the geopositioning system as a latitude, longitude, and altitude with respect to the Earth based on a combined output of the accelerometer, the gyroscope, and the magnetometer,
wherein the calculating of the location comprises:
determining a plane of a local horizon from measurements performed by the accelerometer;
determining a direction of a spin axis of the Earth from measurements performed by the gyroscope; and
calculating the latitude from the angle between the local horizon and the spin axis.