JP4149344B2 - Geomagnetic orientation sensor and method of using geomagnetic orientation sensor - Google Patents

Description

  The present invention relates to a geomagnetic azimuth sensor suitable for a small three-dimensional electronic magnetic compass that detects a magnetic azimuth regardless of tilt, and a method of using the geomagnetic azimuth sensor.

  Mobile devices such as mobile phones and PDAs (Personal Digital Assistants) use a magnetic compass that navigates directions along with position information from GPS (Global Positioning System). It has been. Since mobile devices are often used while being tilted, it is necessary to acquire information including the tilt angle. The magnetic direction can be detected by replacing the inclination angle with an inclination vector and removing the component along the inclination vector from the geomagnetic vector. The demand for miniaturization is large in portable devices, and the smallest semiconductor capacitance type tilt sensor is often used to determine the tilt angle.

  If the gravitational vector is obtained by the tilt sensor, the magnetic pole direction can be calculated from the geomagnetic vector. If the gravity vector of magnitude 1 is g and the geomagnetic vector is M, the size of the projection vector onto the gravity vector of the geomagnetic vector is the inner product g · M of both, so the projection vector is (g · M) g. Therefore, the azimuth vector of the magnetic pole is M- (g · M) g. When performing azimuth navigation, a map of the current location may be displayed on a plane perpendicular to g with the north magnetic pole direction M- (g · M) g as north.

  As a prior art, Patent Document 1 discloses an azimuth angle detection device that performs tilt correction from outputs of a triaxial geomagnetic sensor and a biaxial tilt sensor. As the triaxial geomagnetic sensor, a high-sensitivity Hall element using a compound semiconductor InSb, InAs, GaAs, or the like is used, and offset is canceled by using a chopper that switches the connection terminal of the Hall element.

  Patent Document 2 also discloses a method in which a triaxial magnetic sensor is configured by a biaxial fluxgate sensor and a uniaxial hall magnetic sensor, and tilt correction is performed by a biaxial tilt sensor.

JP 2003-65791 A JP 2002-196055 A

  In the prior art, a tilt sensor is required in addition to the magnetic sensor in order to correct the tilt of the magnetic compass, making it difficult to meet the demands for miniaturization and weight reduction. In addition, as mobile devices become lighter and more compact, they are often held with one hand. In such a state, only the tilt with the horizontal direction as the axis is required, so a magnetic compass with tilt correction that is optimal for each method of use is required. It is.

  Accordingly, an object of the present invention is to provide a geomagnetic azimuth sensor capable of detecting a magnetic azimuth regardless of inclination and a method of using the geomagnetic azimuth sensor.

That is, according to the present invention, in a geomagnetic azimuth sensor having a function of correcting a tilt angle by a three-axis (geo) magnetic sensor, a one-axis tilt correction is performed only by a three-axis geomagnetic vector and pre-given geomagnetic depression information. An axis in the inclined plane where the inclined plane of the geomagnetic azimuth sensor intersects the horizontal plane is an X axis, an axis perpendicular to the X axis in the inclined plane of the geomagnetic azimuth sensor is the Y axis, and the geomagnetic azimuth sensor When the vertical axis with respect to the inclined plane is Z-axis, the inclination angle formed by the inclined plane of the geomagnetic direction sensor and the horizontal plane is Φ, and the ground vertical vector orthogonal to the X axis is Select the angle α to be α = 90 ° + Φ, and coordinate the Y axis and Z axis components of the geomagnetic vector by the tilt angle around the X axis so that the determined ground vertical vector overlaps the Z axis Rotation change In other words, when the X component of the converted geomagnetic vector is Hx and the Y component is Hy, the azimuth angle θ with respect to the X-axis direction after the tilt angle correction is defined as θ = tan −1 (Hy / Hx) In the case where there are two different values of α satisfying the equation α = 90 ° + Φ as the ground vertical vector,
1) If the tilt angle Φ is within the range of the standard tilt angle ± 45 ° with respect to the predetermined standard tilt angle, select it,
2) When multiple measurements have been performed before this measurement, first select an inclination angle Φ that is closer to the ground vertical vector selected during the previous measurement, and detect it before correcting the inclination angle. The sign of the difference in the amount of change in each component in the current measurement with respect to each component in the previous measurement of the X and Z components of the geomagnetic vector is the same as in the previous measurement, and When the sign of the amount of change in the current measurement with respect to the value in the previous measurement of the azimuth angle θ after correction of the tilt angle is different from that in the previous measurement, another tilt angle Φ Change to
A geomagnetic azimuth sensor characterized in that the inclination angle Φ is selected in the priority order of the above 1) and 2) and the azimuth angle θ is determined.

Further, the present invention is characterized in that the geomagnetic direction sensor is a magnetic sensor in which three magnetic sensors having an angle of about 35 degrees with respect to the mounting substrate surface are arranged so as to be substantially equilateral triangles. It is a geomagnetic direction sensor.

Further, the present invention uses the magnetic azimuth sensor in which the impedance of the elongated magnetic body changes with respect to an external magnetic field by passing a high frequency current of 1 MHz or more through the elongated magnetic body having an easy magnetization axis perpendicular to the longitudinal direction. It is a geomagnetic direction sensor characterized by being a magnetic impedance sensor.

The present invention also provides a method of using a geomagnetic azimuth sensor that performs a uniaxial tilt correction using only the three-axis magnetic sensor and geomagnetic depression information by performing a calculation of a geomagnetic azimuth sensor that corrects the tilt angle using a three-axis magnetic sensor. The axis in the inclined plane where the inclined plane of the geomagnetic direction sensor intersects the horizontal plane is the X axis, the axis orthogonal to the X axis in the inclined plane of the geomagnetic direction sensor is the Y axis, and the tilt of the geomagnetic direction sensor is When the vertical axis with respect to the plane is the Z-axis, the angle formed between the inclined surface of the geomagnetic azimuth sensor and the horizontal plane is Φ, and the angle formed with the geomagnetic vector as the ground vertical vector orthogonal to the X-axis Select that α is α = 90 ° + Φ, and rotate and convert the Y axis and Z axis components of the geomagnetic vector by the tilt angle around the X axis so that the determined ground vertical vector overlaps the Z axis Shi If the X component of the converted geomagnetic vector is Hx and the Y component is Hy, the azimuth angle θ based on the X-axis direction after the tilt angle correction is detected as θ = tan −1 (Hy / Hx). A method of using a geomagnetic orientation sensor characterized in that
When there are two different values as α satisfying the formula α = 90 ° + Φ as the ground vertical vector,
1) If the tilt angle Φ is within the range of the standard tilt angle ± 45 ° with respect to the predetermined standard tilt angle, select it,
2) When multiple measurements have been performed before this measurement, first select an inclination angle Φ that is closer to the ground vertical vector selected during the previous measurement, and detect it before correcting the inclination angle. The sign of the difference in the amount of change in each component in the current measurement with respect to each component in the previous measurement of the X and Z components of the geomagnetic vector is the same as in the previous measurement, and When the sign of the amount of change in the current measurement with respect to the value in the previous measurement of the azimuth angle θ after correction of the tilt angle is different from that in the previous measurement, another tilt angle Φ Change to
A method of using a geomagnetic azimuth sensor, wherein the inclination angle Φ is selected in the priority order of the above 1) and 2), and the azimuth angle θ is determined.

  INDUSTRIAL APPLICABILITY According to the present invention, it is possible to provide a geomagnetic azimuth sensor and a method of using a geomagnetic azimuth sensor that can obtain a azimuth navigation system that is smaller and less expensive than the conventional system with a minimum shape and system suitable for the usage form of a portable device.

  A method of using the geomagnetic direction sensor and the geomagnetic direction sensor according to the embodiment of the present invention will be described below.

(Embodiment 1)
FIG. 2 is an explanatory diagram of a geomagnetic direction sensor according to Embodiment 1 of the present invention. A mounting substrate 10 in the case 1 is mounted with a display 11, a three-axis magnetic sensor 12, a signal processing circuit 13, and an azimuth and tilt angle calculation circuit 14.

  FIG. 3 is an explanatory diagram of a display of the geomagnetic direction sensor of the present invention. As shown in FIG. 3, the display 11 displays the magnetic compass display of the display 11 tilted so as to be horizontal to the ground in accordance with the tilt of the case. The magnetic compass always points to the north earth magnetic pole by geomagnetism.

  The triaxial magnetic sensor 12 is mounted with a geomagnetic sensor for detecting one directional component at a time as shown in FIG. The geomagnetic sensor chip 2 of each axis in FIG. 4 is constituted by a dielectric substrate 20 by an elongated magnetic body 21 and an extraction electrode 22, and is connected to the mounting substrate electrode 24 by solder 23. The longitudinal direction of the elongated magnetic body has an inclination of about 35 degrees with respect to the mounting substrate surface. Three such geomagnetic sensors are arranged so as to be substantially equilateral triangles as shown in the plan view of FIG. 4, and the detection axes thereof are orthogonal to each other. Further, a winding bobbin 3 for applying a magnetic bias in the vertical direction of the mounting substrate is attached. Such a configuration is desirable in that not only the height of the triaxial magnetic sensor can be reduced, but also a single bobbin is required, the shape can be reduced, and the cost can be reduced.

  A method for forming the magnetic film into an elongated rectangle will be described below. As a first method, a film is formed by sputtering, vapor deposition or plating after masking with a photoresist, and as a second method, a chemical after masking with a photoresist after the film forming method is performed. Etching or ion etching is selected. The magnetic body 21 formed in an elongated shape is subjected to heat treatment by applying a magnetic field in the width direction of the magnetic body in an inert atmosphere, thereby aligning the magnetization in the width direction of the magnetic body in a zero magnetic field. When a magnetic field is applied in the longitudinal direction of the magnetic body 21 after processing, the magnetization is aligned with a magnetic field of several Gauss in the longitudinal direction, and accordingly, a high-frequency impedance of 1 MHz or more once increases and becomes saturated as shown in FIG. After that, the magnetic sensor has a characteristic of decreasing in the longitudinal direction and directivity in the longitudinal direction.

  The geomagnetic vector is detected by the three-axis magnetic sensor 12 and the signal processing circuit 13. As shown in the circuit diagram of FIG. 6, the signal processing circuit 13 includes an oscillation circuit 31, a signal detection circuit 32, and an amplification circuit 33. The oscillation circuit 31 generates a high-frequency voltage of 1 MHz or more by a multivibrator, a crystal resonator, etc., and a signal sensor circuit 32 changes a divided voltage of a magnetic sensor and a resistor to which an appropriate magnetic bias is applied by a winding bobbin 3. The voltage amplitude is detected by peak hold.

  Further, the signal is amplified to an appropriate signal level by the amplifier circuit 33. At this time, the magnetic bias of the winding bobbin 3 is alternately applied to the same amount of positive and negative to detect a difference in magnetic sensor signal output when the magnetic bias is positive or negative, or one side of the magnetic bias when the magnetic bias is positive or negative It is desirable to use a method of suppressing the offset fluctuation of the magnetic sensor output by a method of inverting and smoothing the sensor signal output.

  The azimuth and tilt angle calculation circuit 14 temporarily rotates the signal processing circuit 13 using a primary conversion matrix to convert the signal processing circuit 13 into a coordinate system having axes in the mounting substrate plane and in the substrate plane vertical. As shown in FIG. 11, the detection axes S1, S2, and S3 of the three-axis magnetic sensor are respectively displayed as vector components (1, 0, 0), (0, 1, 0), (0, 0, 1). When the magnetic bias Hb is added in the (1, 1, 1) direction in the case of replacement, a magnetic field is equally applied to each of the three-axis magnetic sensors.

When the direction vector of the magnetic bias Hb is normalized, it becomes [1 / (3) ^1/2 , 1 / (3) ^1/2 , 1 / (3) ^1/2 ], and each detection axis S1 of the 3-axis magnetic sensor , S2 and S3 are approximately 55 °. When the magnetic bias direction is set to be perpendicular to the mounting substrate, the angle formed by the mounting substrate and the magnetic sensor detection axis is about 35 °, which is obtained by subtracting the angle formed by 90 ° from Hb from FIG.

  FIG. 13 is an explanatory diagram in which Hb is set in the direction perpendicular to the paper surface and the magnetic sensor is projected onto the mounting substrate S1, S2, and S3. As shown in FIG. 13, Hb is set in the direction perpendicular to the paper surface, and projections of S1, S2, and S3 onto the mounting substrate are 120 ° intervals.

In order to set the detection axis in the mounting board plane and in the vertical direction, the two axes in the mounting board surface are (1, 0, 0) and (0, 1, 0), and the vertical axis on the mounting board is ( (0, 0, 1), it is rotated by -tan ⁻¹ [(2) ^1/2 ] ° (≈−55 °) about the (1, 0, 0) axis, and (0, 0, 1) A coordinate transformation matrix that is rotated by −45 ° about the axis is expressed as follows.

  Assuming that the outputs of the magnetic sensors S1, S2, and S3 are V1, V2, and V3, the output after the axis conversion is expressed as Vx, Vy, and Vz.

  FIG. 14 is an explanatory diagram of the relationship between the magnetic sensors S1, S2, and S3 and the detection axes of the post-axis conversion outputs Vx, Vy, and Vz. In this manner, the relative relationship between the outputs of the magnetic sensors S1, S2, and S3 and the post-axis conversion outputs Vx, Vy, and Vz can be expressed using graphics.

  FIG. 1 is a diagram showing the principle of the present invention. Consider a portable device as shown in FIG. The X axis is taken as a vertical coordinate system (width direction of the portable device), the Y axis is taken as the longitudinal direction of the portable device, and the Z axis is taken as the coordinate system centered on the portable device in the thickness direction of the portable device. The ground is inclined by an angle about the X axis as seen from the portable device. Let M be the magnetic vector detected by the 3-axis magnetic sensor. The ground viewed from the portable device rotates around the X axis.

When using a portable device with one hand, it can be used with its body laid down as shown in Fig. 7 (b) or standing up as shown in Fig. 7 (a). Only the direction along the direction. This axis is set as the X axis. If a vector perpendicular to the ground and having a size of 1 is G,
M · G = | M | cos (θ) (1)
However, θ = 90 ° + the angle of depression.

If the dip angle is given in advance, the direction of G can be specified without using an inclination sensor if M is measured. From FIG. 7, since the direction of G is always perpendicular to the X axis, it exists in the YZ plane. In order to use the magnetic vector as the reference vector,
m = M / | M | (2)
far. If the angle formed by the Y axis of the G vector is φ,
G = [0, cos (φ), sin (φ)] (3)
Furthermore, if the angle formed by the projection vector of the m vector onto the YZ plane with the Y axis is α and the angle between the X axis and the m vector is β,
m = [cos (β), sin (β) cos (α), sin (β) sin (α)] (4)

Equation (3) uses equation (4),
sin (β) [cos (φ) cos (α) + sin (φ) sin (α)]
= Sin (β) cos (φ−α) = cos (θ) (5)

  From the relationship of equation (5), α and β can be obtained from the geomagnetic vector M and equation (2), and θ can be obtained from the geomagnetic dip angle to obtain G and φ. The calculation procedure is shown below. Here, arc tan is an inverse function of tan, and arc cos is an inverse function of cos.

i)
Geomagnetic vector M = (Mx, My, Mz)
α = arc tan (Mz / My)
β = arc tan [(My ² ) ^1/2 + (Mz ² ) ^1/2 / Mx]
(Two solutions of arc tan are discriminated from the codes of Mx and My.)

ii)
θ = Angle + 90 °

iii)
φ = α ± arc cos {cos (θ) / sin (β)}
G = [0, cos (φ), sin (φ)]

  FIG. 8 is a rewrite of FIG. 1 from the XY plane, but the projection of the vector group having the angle formed by θ from the geomagnetic vector onto the XY plane rides on an elliptical locus and rides on the Y axis. The trajectory is a candidate for G. It can be seen from the figure that there are generally two solutions. Also, there may be one solution as a multiple solution when two solutions overlap. It is necessary to select the correct solution in a certain procedure.

  The tilt angle of the portable device can be obtained from FIG. 1 by tilt angle = 90 ° −φ. FIG. 9 is a diagram showing a solution of the equation (iii) with respect to the orientation of the portable device and the orientation angle limited to 0 to 90 ° when the geomagnetic vector is (cos 30 °, 0, −sin 30 °). FIG. 10 is a diagram showing a solution of the equation (iii) with respect to the orientation of the portable device and the attitude angle limited to 0 to 90 ° when the magnetic vector is (cos 30 °, 0, −sin 30 °).

As described with reference to FIG. 1, the inclination and magnetic orientation of case 1 can be calculated from a three-dimensional geomagnetic vector. First, as described in the item of action for the geomagnetic vector,
m = M / | M |
The angles α and β are obtained from m = [cos (β), sin (β) cos (α), sin (β) sin (α)].

Next, get information on geomagnetic dip,
sin (β) [cos (φ) cos (α) + sin (φ) sin (α)]
= Sin (β) cos (φ−α) = cos (θ)
To obtain the inclination angle.

Since there are two solutions of the above equation, the inclination angle Φ is selected in the following priority order, and the azimuth angle θ is determined.
1) If there is a solution with a tilt angle Φ within a standard tilt angle ± 45 ° with respect to a predetermined standard tilt angle , select it.
2) When multiple measurements have been performed before the current measurement, first select an inclination angle Φ that is closer to the ground vertical vector selected during the previous measurement, and detect it before correcting the inclination angle. The sign of the difference between the immediately preceding component and the currently read component of the X and Z components of the geomagnetic vector is the same as in the previous measurement, and the azimuth angle θ after tilt angle correction When the sign of the amount of change per hour is different from that in the previous measurement, it is changed to another inclination angle Φ .

The Y-axis and Z-axis components of the 3-axis geomagnetic vector are rotated by the tilt angle around the X-axis so that the determined ground vertical vector overlaps the Z-axis, and the X component of the converted geomagnetic vector is converted to the Hx and Y components. Is Hy, it is possible to detect the azimuth angle θ with reference to the X-axis direction after tilt angle correction as θ = tan ⁻¹ (Hy / Hx).

Geomagnetic dip information needs to be provided in advance at the time of the above calculations.

Explanatory drawing of the principle of the geomagnetic direction sensor of embodiment of this invention. Explanatory drawing of the geomagnetic direction sensor by Embodiment 1 of this invention. Explanatory drawing of the display of the geomagnetic direction sensor of this invention. The figure which has arrange | positioned three magnetic sensors so that it may become a substantially equilateral triangle, and mutually made the detection axis orthogonal. The figure which shows the high frequency impedance characteristic of 1 MHz or more. The circuit diagram which consists of an oscillation circuit, a signal detection circuit, and an amplifier circuit. The figure which shows the portable apparatus carrying the geomagnetic direction sensor of this invention. The figure which rewrote FIG. 1 from XY plane. The figure which shows the solution of (iii) type | formula with respect to the orientation angle | corner restrict | limited to the azimuth | direction of a portable apparatus and 0-90 degrees, when a geomagnetic vector is set to (cos30 degrees, 0, -sin30 degrees). The figure which shows the solution of (iii) type | formula with respect to the orientation angle | corner restrict | limited to the azimuth | direction of a portable apparatus and 0-90 degrees, when a geomagnetic vector is set to (cos30 degrees, 0, -sin30 degrees). Explanatory drawing of each detection axis S1, S2, S3 of a three-axis magnetic sensor. Explanatory drawing of the angle | corner which an attachment board | substrate and a magnetic sensor detection axis | shaft make. Explanatory drawing which projected the magnetic sensor to the attachment board | substrate of S1, S2, S3, setting Hb to the paper surface perpendicular | vertical direction. Explanatory drawing of the relationship between magnetic sensor S1, S2, S3 and the detection axis | shaft of the outputs Vx, Vy, and Vz after axis conversion.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Case 2 Geomagnetic sensor chip 3 Winding bobbin 10 Board | substrate 11 Display 12 Triaxial magnetic sensor 13 Signal processing circuit 14 Inclination angle calculation circuit 20 Dielectric board | substrate 21 Magnetic body 22 Electrode 23 Solder 24 Substrate electrode 31 Oscillation circuit 32 Signal detection circuit 33 Amplifier circuit

Claims (4)

In the geomagnetic azimuth sensor having a function of correcting the tilt angle by the three-axis magnetic sensor, the tilt correction of one axis is performed only by the three-axis geomagnetic vector and the geomagnetic depression information given in advance .
The axis in the inclined plane where the inclined plane of the geomagnetic orientation sensor intersects the horizontal plane is the X axis, the axis perpendicular to the X axis in the inclined plane of the geomagnetic bearing sensor is the Y axis, and the axis perpendicular to the inclined plane of the geomagnetic bearing sensor is In the case of the Z axis, an inclination angle that is an angle formed between the inclined surface of the geomagnetic orientation sensor and the horizontal plane is Φ, and an angle α formed with a geomagnetic vector as a ground vertical vector orthogonal to the X axis is α = Select the one that is 90 ° + Φ, and convert the Y-axis and Z-axis components of the geomagnetic vector by the tilt angle around the X-axis so that the determined ground vertical vector overlaps the Z-axis. When the X component of the geomagnetic vector is Hx and the Y component is Hy, the azimuth angle θ with respect to the X axis direction after the inclination angle correction is
θ = tan ⁻¹ (Hy / Hx)
It is a geomagnetic direction sensor characterized by detecting as
When there are two different values as α satisfying the formula α = 90 ° + Φ as the ground vertical vector,
1) If the tilt angle Φ is within the range of the standard tilt angle ± 45 ° with respect to the predetermined standard tilt angle, select it,
2) When multiple measurements have been performed before this measurement, first select an inclination angle Φ that is closer to the ground vertical vector selected during the previous measurement, and detect it before correcting the inclination angle. The sign of the difference in the amount of change in each component in the current measurement with respect to each component in the previous measurement of the X and Z components of the geomagnetic vector is the same as in the previous measurement, and When the sign of the amount of change in the current measurement with respect to the value in the previous measurement of the azimuth angle θ after correction of the tilt angle is different from that in the previous measurement, another tilt angle Φ Change to
A geomagnetic azimuth sensor characterized in that the inclination angle Φ is selected in the priority order of the above 1) and 2) and the azimuth angle θ is determined. The geomagnetic direction sensor according to claim 1 is a magnetic sensor in which three magnetic sensors having an angle of about 35 degrees with respect to the mounting substrate surface are arranged so as to be substantially equilateral triangles. Direction sensor. The geomagnetic azimuth sensor according to claim 1 or 2 , wherein a high-frequency current of 1 MHz or more is applied to an elongated magnetic body having an easy magnetization axis perpendicular to the longitudinal direction, and the impedance of the elongated magnetic body changes with respect to an external magnetic field. A geomagnetic orientation sensor characterized by being a magneto-impedance sensor utilizing the above. A method of using a geomagnetic azimuth sensor that performs a calculation of a geomagnetic azimuth sensor that corrects an inclination angle by a triaxial magnetic sensor and performs uniaxial inclination correction only by the triaxial magnetic sensor and geomagnetic dip angle information. X axis the axis of the inclined surface and the inclined surface and the horizontal plane intersect the azimuth sensor, Y-axis the axis perpendicular to the X-axis within the tilt plane of the geomagnetic direction sensor, an axis perpendicular to the inclined surface of the geomagnetic direction sensor Z Assuming that the axis is an axis , the inclination angle formed by the inclined plane of the geomagnetic direction sensor and the horizontal plane is Φ, and the angle α formed by the geomagnetic vector as the ground vertical vector orthogonal to the X axis is α = 90 ° + [Phi become ones chosen, Y-axis of the geomagnetic vector as determined ground vertical vector is overlapped with the Z-axis, Z-axis component of the angle of inclination only coordinate rotation transformation about the X-axis geomagnetic converted Hx the X component of the vector, when the Y component was Hy, the azimuth angle θ relative to the X-axis direction after inclination angle correction,
θ = tan ⁻¹ (Hy / Hx)
Detecting a use of the geomagnetic direction sensor, wherein a,
When there are two different values as α satisfying the formula α = 90 ° + Φ as the ground vertical vector,
1) If the tilt angle Φ is within the range of the standard tilt angle ± 45 ° with respect to the predetermined standard tilt angle, select it,
2) When multiple measurements have been performed before this measurement, first select an inclination angle Φ that is closer to the ground vertical vector selected during the previous measurement, and detect it before correcting the inclination angle. The sign of the difference in the amount of change in each component in the current measurement with respect to each component in the previous measurement of the X and Z components of the geomagnetic vector is the same as in the previous measurement, and When the sign of the amount of change in the current measurement with respect to the value in the previous measurement of the azimuth angle θ after correction of the tilt angle is different from that in the previous measurement, another tilt angle Φ Change to
A method of using a geomagnetic azimuth sensor, wherein the inclination angle Φ is selected in the priority order of the above 1) and 2), and the azimuth angle θ is determined.

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