Magnetic data processing device

BACKGROUND OF THE INVENTION

00011. Technical Field of the Invention

0002The present invention relates to a magnetic data processing device, magnetic data processing method, and magnetic data processing program, and more particularly to a technology for correcting an offset of two-dimensional and three-dimensional magnetic sensors.

00032. Description of the Related Art

0004A conventional magnetic sensor mounted on a moving body such as a mobile phone or a vehicle detects the direction of an Earth's magnetic field or geomagnetism. The magnetic sensor includes a set of magnetic sensor modules to detect scalar components of the magnetic field vector in mutually orthogonal directions. Magnetic data output from the magnetic sensor consists of a combination of outputs of the magnetic sensor modules, and thus the magnetic data is vector data that is a linear combination of mutually orthogonal unit vectors (fundamental vectors). The direction and magnitude of the magnetic data corresponds to the direction and magnitude of a magnetic field detected by the magnetic sensor. When specifying the direction or magnitude of an Earth's magnetic field based on outputs of the magnetic sensor, it is necessary to perform a process for correcting the outputs of the magnetic sensor in order to negate measurement errors caused by magnetization of the moving body or inherent temperature characteristics of the magnetic sensor. A control value of this correction process is referred to as an offset and a process for deriving the offset is referred to as calibration (see, for example, International Patent Publication No. 2004-003476). The offset is also vector data and defined as magnetic data output from the magnetic sensor when the intensity of external magnetic fields is zero. Such measurement errors are negated by subtracting the offset from the magnetic data output from the magnetic sensor. In a two-dimensional (2D) magnetic sensor, the offset corresponds to a position vector of the center of a circle on which a magnetic data set is distributed. However, practically, the distribution of a magnetic data set output from the 2D magnetic sensor does not form a perfect circle. The reasons are that outputs of magnetic sensor modules inherently have measurement errors following Gaussian distribution, a magnetic field measured by the 2D magnetic sensor varies during a period in which a data set of statistical population is stored to calculate the offset since in practice there is no completely uniform magnetic field, and calculation errors occur during AD conversion.

0005The 2D magnetic sensor outputs a data set of statistical population required to derive the offset while a moving body including the 2D magnetic sensor mounted thereon rotates such that the 2D magnetic sensor rotates about a rotation axis parallel to the direction perpendicular to orthogonal sensitive directions of its magnetic sensor modules. To move, in this manner, a moving body such as a mobile phone or a vehicle which is movable three-dimensionally, it is necessary for the user to intentionally operate the moving body so that it moves in such a manner. Accordingly, an offset derivation algorithm for a magnetic data processing device for deriving the offset of a 2D magnetic sensor is designed under the assumption that the user is explicitly informed of start of calibration and the user operates the moving body appropriately. However, it is troublesome and complicated for the user to perform the operation for calibration. In the conventional calibration method, it is determined, through binary decision, whether or not a reliable data set of statistical population has been stored, and, when the user has not correctly performed the operation for calibration, the calibration fails without storing a reliable data set of statistical population. This requires the user to repeat the operation for storing a reliable data set of statistical population.

0006A conventional 3-dimensional (3D) magnetic sensor mounted on a moving body such as a mobile phone or a vehicle detects the direction of an Earth's magnetic field. The 3D magnetic sensor generally includes 3 magnetic sensor modules to detect scalar components of the magnetic field vector in 3 orthogonal directions. Magnetic data output from the 3D magnetic sensor consists of a combination of outputs of the 3 magnetic sensor modules, and thus the magnetic data is 3D vector data that is a linear combination of mutually orthogonal unit vectors (fundamental vectors). The direction and magnitude of the magnetic data corresponds to the direction and magnitude of a magnetic field detected by the 3D magnetic sensor. When specifying the direction or magnitude of an Earth's magnetic field based on outputs of the 3D magnetic sensor, it is necessary to perform a process for correcting the outputs of the 3D magnetic sensor in order to negate measurement errors caused by magnetization of the moving body or inherent temperature characteristics of the magnetic sensor. A control value of this correction process is referred to as an offset. The offset is vector data indicating a magnetic field caused by the magnetization components of the moving body detected by the 3D magnetic sensor. Such measurement errors are negated by subtracting the offset from the magnetic data output from the 3D magnetic sensor. It is possible to calculate the offset by obtaining the center of a spherical surface on which a magnetic data set is distributed.

0007However, practically, the distribution of magnetic data does not form a perfect sphere. The reasons are that outputs of the 3D magnetic sensor inherently have measurement errors following Gaussian distribution, a magnetic field measured by the 3D magnetic sensor varies during a period in which magnetic data required to calculate the offset is stored since in practice there is no completely uniform magnetic field, and calculation errors occur until digital values are obtained from the outputs of the 3D magnetic sensor.

0008A conventional method for deriving a magnetic sensor offset stores a large number of magnetic data and derives the offset through a statistical process of the stored magnetic data. Thus, in the conventional method, a magnetic data set required to accurately update the offset is not stored unless the user intentionally changes the attitude or posture of the moving body, and a long time is required to update the offset after the need to update the offset occurs. Usually, magnetic data distributed only in two dimensions is stored since a significant three-dimensional change in the attitude of the magnetic sensor mounted on the vehicle is rare. Thus, it is undesirable to wait until a magnetic data set evenly distributed on a spherical surface is stored in order to accurately update the offset of the magnetic sensor mounted on the vehicle.

0009International Patent Publication No. 2005-061990 has disclosed an algorithm which can correct the offset even if the distribution of a magnetic data set is two-dimensional. However, it is not easy to implement a program according to the algorithm described in International Patent Publication No. 2005-061990 since it is complicated.

SUMMARY OF THE INVENTION

0010It is an object of the present invention to improve the usability of a two-dimensional (2D) magnetic sensor and three-dimensional (3D) magnetic sensor.

0011It is another object of the present invention to provide a magnetic data processing device, magnetic data processing method, and magnetic data processing program, and a magnetic measurement apparatus, which can correct an offset using a stored magnetic data set through a simple process, regardless of distribution of the data set of statistical population.

0012In a first aspect of the invention, a magnetic data processing device for accomplishing the above objects comprises an input part that sequentially inputs magnetic data outputted from a two-dimensional (2D) magnetic sensor, the magnetic data being 2D vector data that is a linear combination of a set of first fundamental vectors; a storage part that stores a plurality of the inputted magnetic data as a data set of statistical population in order to update an old offset of the magnetic data with a new offset; and an offset derivation part that derives the new offset based on the old offset and the data set of statistical population under a constraint condition that the new offset be obtained as the sum of the old offset and a correction vector, where the correction vector is a linear combination of a set of second fundamental vectors defined in principle axis directions of distribution of the data set of statistical population, and respective coefficients of the linear combination of the second fundamental vectors representing the correction vector are obtained by weighting respective coefficients of a temporary position vector of a temporary offset relative to the old offset according to a ratio of principle values of the distribution of the data set of statistical population, the temporary offset being derived from the data set of statistical population without using the old offset, the temporary position vector being a linear combination of the second fundamental vectors.

0013In a second aspect of the invention, a magnetic data processing device for accomplishing the above objects comprises an input part that sequentially inputs magnetic data outputted from a three-dimensional (3D) magnetic sensor, the magnetic data being 3D vector data that is a linear combination of a set of first fundamental vectors; a storage part that stores a plurality of the inputted magnetic data as a data set of statistical population in order to update an old offset of the magnetic data with a new offset; and an offset derivation part that derives the new offset based on the old offset and the data set of statistical population under a constraint condition that the new offset be obtained as the sum of the old offset and a correction vector, where the correction vector is a linear combination of a set of second fundamental vectors defined in principle axis directions of distribution of the data set of statistical population, and respective coefficients of the linear combination of the second fundamental vectors representing the correction vector are obtained by weighting respective coefficients of a position vector of a temporary offset relative to the old offset according to ratios of principle values of the distribution of the data set of statistical population, the temporary offset being derived from the data set of statistical population without using the old offset, the position vector of the temporary offset being a linear combination of the second fundamental vectors.

BRIEF DESCRIPTION OF THE DRAWINGS

0014<figref idref="DRAWINGS">FIG. 1</figref> is a schematic diagram of the embodiments of the first aspect of the present invention.

0015<figref idref="DRAWINGS">FIG. 2</figref> is a schematic diagram of the embodiments of the first aspect of the present invention.

0016<figref idref="DRAWINGS">FIG. 3</figref> is a block diagram of the embodiments of the first aspect of the present invention.

0017<figref idref="DRAWINGS">FIG. 4</figref> is a block diagram of the embodiments of the first aspect of the present invention.

0018<figref idref="DRAWINGS">FIG. 5</figref> is a flow chart of the first embodiment of the first aspect of the present invention.

0019<figref idref="DRAWINGS">FIG. 6</figref> is a schematic diagram of the first embodiment of the first aspect of the present invention.

0020<figref idref="DRAWINGS">FIG. 7</figref> is a schematic diagram of the first embodiment of the first aspect of the present invention.

0021<figref idref="DRAWINGS">FIG. 8</figref> is a flow chart of the second embodiment of the first aspect of the present invention.

0022<figref idref="DRAWINGS">FIG. 9</figref> is a graph associated with the second embodiment of the first aspect of the present invention.

0023<figref idref="DRAWINGS">FIG. 10</figref> is a graph associated with the second embodiment of the first aspect of the present invention.

0024<figref idref="DRAWINGS">FIG. 11</figref> is a schematic diagram of the embodiments of the second aspect of the present invention.

0025<figref idref="DRAWINGS">FIG. 12</figref> is a schematic diagram of the embodiments of the second aspect of the present invention.

0026<figref idref="DRAWINGS">FIG. 13</figref> is a block diagram of the embodiments of the second aspect of the present invention.

0027<figref idref="DRAWINGS">FIG. 14</figref> is a block diagram of the embodiments of the second aspect of the present invention.

0028<figref idref="DRAWINGS">FIG. 15</figref> is a flow chart of the first embodiment of the second aspect of the present invention.

0029<figref idref="DRAWINGS">FIG. 16</figref> is a schematic diagram of the first embodiment of the second aspect of the present invention.

0030<figref idref="DRAWINGS">FIG. 17</figref> is a schematic diagram of the first embodiment of the second aspect of the present invention.

0031<figref idref="DRAWINGS">FIG. 18</figref> is a flow chart of the second embodiment of the second aspect of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

0032A principle and algorithm used in the first aspect of the invention is described below in detail with reference to <figref idref="DRAWINGS">FIG. 1</figref>. Key points of this algorithm are that a magnetic data set distributed in a principle axis direction with larger dispersion is estimated to be more significant elements of statistical population for use in updating the offset, and a magnetic data set distributed in a principle axis direction with smaller dispersion is estimated to be less significant elements of statistical population for use in updating the offset. Each of the old offset c₀, the new offset c, and the temporary offset, which corresponds to a position vector of the end point of “g” relative to the point of origin “0”, is 2D position vector data that is a linear combination of a fundamental vector set of magnetic data. That is, each of the offsets is vector data represented in the xy coordinate system. The new offset c is derived based on the old offset c₀and a magnetic data set that is stored to update the old offset c₀with the new offset c.

0033The data set of statistical population, which is a magnetic data set stored to update the old offset with the new offset c, may include a magnetic data set that has been stored in a predetermined period of time and may include a magnetic data set including a predetermined number of magnetic data and may also include a magnetic data set including any number of magnetic data that has been stored at a certain time (for example, at the time when an offset update request is made). The old offset c₀may be derived using the same method as the new offset c and may also be predetermined.

0034Although the temporary offset is defined to be derived from the data set of statistical population without using the old offset c₀, this definition is introduced to define the constraint condition under which the new offset c is derived and the temporary offset is actually not data which must be derived. If the temporary offset is actually derived from the data set of statistical population without using the old offset c₀, the temporary offset is a position vector of the center of a circumference near which the data set of statistical population is distributed. However, if the data set of statistical population is unevenly distributed near a narrow central-angle arc which is a part of the circumference derived from the data set of statistical population, an error included in each element of the data set of statistical population greatly affects the derivation result of the circumference and thus there is a possibility that a temporary offset distant from the true offset is derived. For example, let us consider that a data set of statistical population is unevenly distributed near a narrow central-angle arc and mutually orthogonal eigenvectors of the data set of statistical population are u₁and u₂as shown in <figref idref="DRAWINGS">FIG. 1</figref>. In this case, since the variance of the data set of statistical population is small in the direction of the eigenvector u₂of the distribution corresponding to the smaller principle value, there is a high probability that a temporary offset derived from the data set of statistical population is distant from the true offset in the direction of the eigenvector u₂. On the other hand, in this case, since the variance of the data set of statistical population is large in the direction of the eigenvector u₁of the distribution, there is a high probability that the temporary offset derived from the data set of statistical population is near the true offset in the direction of the eigenvector u₁.

0035Since the variances in the principle axis directions of the distribution can be expressed using the principle values λ₁and λ₂of the distribution as indicators of the distribution, this device estimates elements of statistical population distributed in the directions corresponding respectively to the principle values according to the ratios of the principle values λ₁and λ₂. Specifically, first, a correction vector f, which is a position vector of the new offset c relative to the old offset c₀, and a temporary position vector g, which is a position vector of the temporary offset relative to the old offset c₀, can be defined in a coordinate system having coordinate axes α and β coincident with the principle axis directions of the distribution. That is, each of the correction vector f and the temporary position vector g can be defined to be a linear combination of fundamental vectors of the principle axis directions of the distribution. This corresponds to conversion into principle axis values. If the components f_αand f_βof the correction vector f are derived by weighting the components g_αand g_βof the position vector g of the temporary offset relative to the old offset c₀according to the measures of the corresponding principle values u₁and u₂of the distribution, it is possible to derive the correction vector f by increasing the reliability of the elements of statistical population in a direction with large dispersion and decreasing the reliability of the elements of statistical population in a direction with small dispersion. However, such definitions of the correction vector f and the temporary position vector g are also introduced to define the constraint condition under which the new offset c is derived and each of the correction vector f and the temporary position vector g is actually not data which needs to be derived.

0036By deriving the new offset c under a constraint condition that the new offset c be obtained as the sum of the old offset c₀and the correction vector f defined as described above, it is possible, regardless of the distribution of the data set of statistical population, to derive the new offset while estimating that a magnetic data set distributed in a principle axis direction with larger dispersion is more significant elements of statistical population and a magnetic data set distributed in a principle axis direction with smaller dispersion is less significant elements of statistical population. One example technique for deriving the new offset in this manner is to formulate the distribution as an optimization problem. No matter which operation for calibration the user performs, this device can derive the most probable new offset which can be derived from an actually performed operation and thus does not require the user to perform a predetermined operation.

0037In the inventive magnetic data processing device, the constraint condition may be that a weighting factor of the temporary position vector for the coefficient of the second fundamental vector in the principle axis direction corresponding to a smaller one of the principle values be zero if a ratio of the smaller principle value to a larger one of the principle values is equal to or less than a predetermined threshold.

0038This device discretely estimates the distribution of the data set of statistical population and discretely weights the coefficients of the position vector of the temporary offset relative to the old offset c₀according to the discrete estimation result. Specifically, the weight of the coefficient of the temporary position vector g in the direction in which the value of the distribution is relatively small is “0”. That is, no data set of statistical population is estimated in the direction in which the value of the distribution is smaller than a threshold.

0039In the inventive magnetic data processing device, the constraint condition may be that the coefficients of the correction vector be values obtained by weighting the coefficients of the temporary position vector by weighting factors continuously corresponding to the ratios of the principle values of the distribution of the data set of statistical population.

0040This device can increase the substantial use efficiency of the data set of statistical population since the weighting factors have a continuous association with the distribution. In addition, the device can simplify the offset update process since it is possible to derive the new offset without changing the process according to the distribution of the data set of statistical population.

0041In the inventive magnetic data processing device, the respective weighting factors for the coefficients of the temporary position vector may be normalized by setting, to one, the weighting factor for the coefficient of the second fundamental vector in the principle axis direction corresponding to the larger principle value.

0042In the case where estimation indicators of the distribution of the data set of statistical population other than the principle values of the distribution are introduced, it is not necessary to normalize the weighting factors by setting the larger weighting factor to one. For example, the maximum weighting factor may be set to less than one according to the ratio of the maximum distance between magnetic data in a principle axis direction (i.e., the principle direction) corresponding to the larger principle value to the radius of a circle that has been derived as a circumference, having a portion near which a data set of statistical population is distributed, based on the data set of statistical population.

0043In the inventive magnetic data processing device for accomplishing the above object, the offset derivation part may derive “c” that minimizes the following objective function f(c) under the constraint condition: <br />f(c)=(Xc−j)^T(Xc−j),

0044where “X” and “j” are as follows when the magnetic data is represented by q_i=(q_ix,q_iy) (i=0,1,2, . . .):

0045<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mrow><mi>X</mi><mo>=</mo><mrow><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr></mtable><mo>]</mo></mrow><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>j</mi></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁡</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>1</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></math></maths><img file="USRE44582E_D0001.tif" />

0046In this specification, all vectors are column vectors and row vectors are expressed as transposed matrices of column vectors. A symbol “T” is attached to each transposed matrix at the right upper or shoulder side thereof [i.e., in the form of ( )^T].

0047Since this device derives a new offset as an optimization problem of the distribution with a constraint condition, it is possible to derive the new offset, for example, by solving simple simultaneous linear equations as described later in the embodiments. That is, no matter what the distribution of the data set of statistical population is, it is possible for this device to derive, through a simple process, the most probable new offset which can be derived from the data set of statistical population.

0048The function of each of the plurality of parts included in the device of the present invention is realized by a hardware resource, the function of which is specified by its construction, a hardware resource, the function of which is specified by a program, or a combination of these resources. The function of each of the plurality of part a is not necessarily realized by a physically independent hardware resource. The present invention can be specified not only by a device but also by a program, a recording medium on which the program is recorded, and a method. The operations of the method described in the claims are not necessarily performed in the order as described in the claims and may be performed in any other order or at the same time, provided there is no technical impediment.

0049Embodiments of the first aspect of the present invention will be described in the following order.

A. FIRST EMBODIMENT

0050[1. General Description]

00511-1. Hardware Structure

00521-2. Software Structure

0053[2. Procedure]

00542-1. Overall Flow

00552-2. Buffer Update

00562-3. Estimation of Distribution

00572-4. Derivation of New Offset through Optimization Problem

00582-5. Derivation of New Offset when the Distribution is Two-Dimensional

00592-6. Derivation of New Offset when the Distribution is Substantially One-Dimensional

00602-7. Summary

0061B. Second Embodiment

0062Overview

0063Estimation of Distribution

0064Derivation of New Offset

0065C. Other Embodiments

0066[General Description]

00671. Hardware Structure

0068<figref idref="DRAWINGS">FIG. 2</figref> is a schematic diagram of an external appearance of an automobile <b>2</b> that is an example of a moving body to which the present invention is applied. The automobile <b>2</b> includes a 2-dimensional (2D) magnetic sensor <b>4</b>. The 2D magnetic sensor <b>4</b> detects the direction and intensity of a magnetic field by detecting respective intensities of the magnetic field in two orthogonal directions (x, y). The 2D magnetic sensor <b>4</b>, which constitutes a part of a navigation system mounted on the automobile <b>2</b>, is used to specify the travel direction of the automobile <b>2</b>.

0069<figref idref="DRAWINGS">FIG. 3</figref> is a block diagram of a magnetic measurement device which includes a 2D magnetic sensor <b>4</b> and a magnetic data processing device <b>1</b>. The 2D magnetic sensor <b>4</b> includes x and y-axis sensors <b>30</b> and <b>32</b> that detect x and y direction components of a magnetic field vector due to terrestrial magnetism. Each of the x and y-axis sensors <b>30</b> and <b>32</b> includes a magnetic resistance element, a hall sensor, or the like, which may be any type of 1-dimensional magnetic sensor provided that it has directivity. The x and y-axis sensors <b>30</b> and <b>32</b> are fixed so that their sensitive directions are perpendicular to each other. Outputs of the x and y-axis sensors <b>30</b> and <b>32</b> are time-divided and input to a magnetic sensor interface (I/F) <b>22</b>. The magnetic sensor interface <b>22</b> analog-to-digital converts inputs from the x and y-axis sensors <b>30</b> and <b>32</b> after amplifying the inputs. Digital magnetic data output from the magnetic sensor interface <b>22</b> is input to the magnetic data processing device <b>1</b> through a bus <b>5</b>.

0070The magnetic data processing device <b>1</b> is a computer including a CPU <b>40</b>, a ROM <b>42</b>, and a RAM <b>44</b>. The CPU <b>40</b> controls overall operations of, for example, the navigation system. The ROM <b>42</b> is a nonvolatile storage medium which stores a magnetic data processing program or a variety of programs used to implement functions of the navigation system, which are executed by the CPU <b>40</b>. The RAM <b>44</b> is a volatile storage medium which temporarily stores data to be processed by the CPU <b>40</b>. The magnetic data processing device <b>1</b> and the 2D magnetic sensor <b>4</b> may be constructed as a one-chip magnetic measurement device.

00711-2. Software Structure

0072<figref idref="DRAWINGS">FIG. 4</figref> is a block diagram of a magnetic data processing program <b>90</b>. The magnetic data processing program <b>90</b> is stored in the ROM <b>42</b> to provide orientation data to a locator <b>98</b>. The orientation data is 2D vector data representing the orientation of the Earth's magnetic field.

0073The magnetic data processing program <b>90</b> is constructed as a group of modules such as a buffer management module <b>92</b>, an offset derivation module <b>94</b>, and an orientation derivation module <b>96</b>.

0074The buffer management module <b>92</b> is a program part that receives a plurality of magnetic data sequentially output from the magnetic sensor <b>4</b> and stores the received magnetic data in a buffer in order to use the magnetic data in offset update. The buffer management module <b>92</b> allows the CPU <b>40</b>, the RAM <b>44</b>, and the ROM <b>42</b> to function as an input part and storage part. This buffer may be embodied not only in hardware but also in software. A magnetic data set stored in this buffer will now be referred to as a data set of statistical population.

0075The offset derivation module <b>94</b> is a program part that derives a new offset based on a data set of statistical population held by the buffer management module <b>92</b> and an old offset held by the offset derivation module <b>94</b> and updates the old offset with the new offset. The offset derivation module <b>94</b> allows the CPU <b>40</b>, the RAM <b>44</b>, and the ROM <b>42</b> to function as an offset derivation part. Since updating the old offset with the new offset causes the new offset to become an old offset, the “old offset” will be referred to simply as an “offset” in context in which it causes no misunderstanding. Actually, an offset used for orientation data correction is set in one variable and the new offset is derived as a different variable from that variable. When the new offset is derived, it is set in the variable used for orientation data correction. Therefore, the variable used for orientation data correction is that in which the old offset is stored.

0076The orientation derivation module <b>96</b> is a program part that corrects the magnetic data sequentially output from the magnetic sensor using the offset held by the offset derivation module <b>94</b> to create orientation data.

0077Specifically, the orientation derivation module <b>96</b> outputs, as orientation data, data including 2 components obtained by subtracting the components of the offset from the components of the magnetic data which is 2D vector data.

0078The locator <b>98</b> is a known program that specifies the current position of the automobile <b>2</b> through autonomous navigation. Specifically, the locator <b>98</b> specifies the travel direction of the automobile <b>2</b> based on the orientation data and specifies the position of the automobile <b>2</b> relative to a base point based on both the travel direction and the travel distance. The orientation data may be used only to display north, south, east and west by characters or arrows on the screen and may also be used for heading-map processing of the map displayed on the screen.

0079[2. Procedure]

00802-1. Overall Flow

0081<figref idref="DRAWINGS">FIG. 5</figref> is a flow chart illustrating a new offset derivation procedure. The CPU <b>40</b> performs the procedure of <figref idref="DRAWINGS">FIG. 5</figref> by executing the buffer management module and the offset derivation module <b>94</b> when an offset update request has been made. The offset update request may be made at predetermined time intervals and may also be made by an explicit instruction of the driver.

00822-2. Buffer Update

0083At step S<b>100</b>, all magnetic data stored in the buffer, in which a magnetic data set (data set of statistical population) used for deriving a new offset is stored, is deleted. As a result, in this process, a data set of statistical population used for deriving the old offset is deleted.

0084At step S<b>102</b>, magnetic data used for deriving a new offset is input and stored in the buffer. When a plurality of magnetic data is sequentially input from the magnetic sensor <b>4</b> with almost no change in the travel direction of the automobile <b>2</b>, the distance between two sequentially input magnetic data (or values) is small. Storing a plurality of near magnetic data in a buffer with a limited capacity wastes memory resources and causes unnecessary buffer update processes. In addition, if a new offset is derived based on a set of near magnetic data, there is a possibility that an inaccurate new offset is derived based on an unevenly distributed data set of statistical population. Whether or not it is necessary to update the buffer may be determined in the following manner. For example, if the distance between the last input magnetic data and magnetic data stored in the buffer immediately before the last input magnetic data is less than a given threshold, it is determined that it is unnecessary to update the buffer and the last input magnetic data is discarded without being stored in the buffer.

0085At step S<b>104</b>, it is determined whether or not a specified number of magnetic data required to derive an accurate new offset has been stored in the buffer. That is, the number of elements of the data set of statistical population is predetermined. Setting a small number of elements of the data set of statistical population improves response to the offset update request. The processes of steps S<b>102</b> and S<b>104</b> are repeated until the specified number of magnetic data is stored in the buffer.

00862-3. Estimation of Distribution

0087Once the specified number of magnetic data is stored in the buffer, the distribution of the data set of statistical population is estimated (S<b>106</b>). The distribution is estimated based on principle values of the distribution. When the magnetic data set is expressed by the following Equation (1), the principle values of the distribution are eigenvalues of a symmetric matrix A defined by Equations (2), (3), and (4) using the sum of vectors starting from a center (average) of the data set of statistical population and ending with the respective magnetic data.

0088<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>=</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>ix</mi></msub><mo>,</mo><msub><mi>q</mi><mi>iy</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi></mrow><mo>)</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>A</mi><mo>=</mo><mrow><msup><mi>X</mi><mi>T</mi></msup><mo>⁢</mo><mi>X</mi></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><mrow><mi>X</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mover><mi>q</mi><mi>_</mi></mover><mo>=</mo><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><msub><mi>q</mi><mi>i</mi></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0002.tif" />

0089As the matrix A may also be rewritten as Equation (5), the matrix A corresponds to N times a variance-covariance matrix.

0090<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>A</mi><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0003.tif" />

0091Let λ₁and λ₂be the eigenvalues of the matrix A in increasing order. Let u₁and u₂be mutually orthogonal eigenvectors that correspond to λ₁and λ₂and have been normalized to size 1. In this embodiment, it is assumed that the matrix A is nonsingular and the ranges of λ₁and λ₂are λ₁>0 and λ₂>0. When the smaller eigenvalue λ₂of the matrix A is zero, i.e., when the rank of the matrix A is one or less, there is no need to consider it since the number of elements of the data set of statistical population is one or the distribution is a perfectly straight line. Each of the eigenvalues must be zero or a positive real number since the matrix A is a positive semi-definite matrix from its definition.

0092The distribution of the data set of statistical population is estimated based on the ratio λ₂/λ₁of the smaller eigenvalue to the larger eigenvalue.

0093At step S<b>106</b>, it is determined whether or not the distribution of the data set of statistical population is sufficiently two-dimensional. Specifically, the determination is affirmative when the following condition (6) is satisfied and negative when it is not satisfied. <br />λ₂/λ₁>t   (6)

0094The condition (6) is satisfied when the data set of statistical population is distributed over a wide range along a specific circumference.

0095If the determination of step S<b>106</b> is negative, the distribution of the data set of statistical population is substantially one-dimensional. When the data set of statistical population is unevenly distributed over a narrow central-angle arc of the specific circumference, the distribution of the data set of statistical population is substantially one-dimensional.

00962-4. Derivation of New Offset through Optimization Problem

0097An optimization problem for deriving a new offset will now be described.

0098When the data set of statistical population includes 3 magnetic data not present on the same line, a circumference on which the data set of statistical population is distributed is uniquely specified without using a statistical technique. A position vector c=(c_x, c_y) of the center of this circumference is obtained by solving simultaneous equations (7). Although three equality constraints exist for two variables, the equations (7) must have a solution since one of the three equality constraints is redundant.

0099<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr></mtable><mo>]</mo></mrow><mo>⁢</mo><mi>c</mi></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁡</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>1</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><mrow><mi>R</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><msubsup><mi>q</mi><mi>i</mi><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mi>i</mi></msub></mrow></mrow></mrow></mrow><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mrow></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0004.tif" />

0100When the number of elements of the data set of statistical population is 4 or more, “j” is defined by the following equation (9).

0101<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>j</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁡</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>1</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0005.tif" />

0102Here, if simultaneous linear equations (10) for “c” have a solution, the solution is the center of a circumference on which the data set of statistical population is distributed. <br />Xc=j   (10)

0103However, if an inherent measurement error of the 2D magnetic sensor <b>4</b> is considered, it is practically impossible for Equations (10) to have a solution. A vector “e” defined by the following equation (11) is introduced to obtain a plausible solution through a statistical technique. <br />e=Xc−j   (11)

0104“c” which minimizes ∥e∥₂²(that is, e^Te) can be considered to be plausible as the center of a circumference closest to the distribution of the data set of statistical population. A problem for finding the value “c” minimizing ∥e∥₂²is an optimization problem for minimizing an objective function of the following equation (12) when the matrix A is nonsingular. <br />f(c)=(Xc−j)^T(Xc−j)   (12)

01052-5. Derivation of New Offset when the Distribution is Two-Dimensional

0106When the distribution of the data set of statistical population is two-dimensional as shown in HG. <b>6</b>, the data set of statistical population is sufficiently reliable in its entirety and therefore a new offset is derived by obtaining “c” which minimizes the objective function f(c) of Equation (12) without any constraint condition (S<b>108</b>). The value “c” which minimizes the objective function f(c) without any constraint condition can be written as Equation (13) when X^TX assumed in this embodiment is nonsingular. <br />c=(X^TX)⁻¹X^Tj   (13)

0107When “c” satisfying Equation (13) has been derived, a new offset, which can be obtained by correcting the old offset in two directions, is derived based on the data set of statistical population without using the old offset.

0108When the data set of statistical population is sufficiently two-dimensional, it is unnecessary to use the old offset when deriving the new offset. An algorithm for deriving a new offset based on the data set of statistical population without using the old offset may be an algorithm that, as in this embodiment, uses one of a variety of statistical techniques which have been suggested and may also be an algorithm that uses no statistical technique.

01092-6. Derivation of New Offset when the Distribution is Substantially One-Dimensional

0110As shown in <figref idref="DRAWINGS">FIG. 7</figref>, when the data set of statistical population is distributed over a narrow central-angle arc of the specific circumference and thus the distribution of the data set of statistical population is substantially one-dimensional (i.e., linear), a new offset is derived by restricting directions in which the old offset is corrected to one principle direction of the distribution (S<b>110</b>). When the data set of statistical population is distributed near a specific line, the distribution of the data set of statistical population in the direction of the line is sufficiently reliable while the distribution of the data set of statistical population in other directions is unreliable. In this case, the old offset is not corrected in directions other than the direction of the line, thereby preventing the offset from being updated based on unreliable information.

0111When the data set of statistical population is distributed near a specific line, the direction of the line is coincident with the direction (i.e., principle direction) of an eigenvector u₁corresponding to the larger eigenvalue λ₁and the direction of an eigenvector u₂corresponding to the smaller eigenvalue λ₂is perpendicular to the line. Accordingly, in order to derive a new offset only in the direction of the line, a new offset c which minimizes the objective function of Equation (12) is found under a constraint condition expressed by the following equation (14). <br />u₂^T(c−c₀)=0   (14)

0112The equation for solving the optimization problem of Equation (12) under the constraint condition of Equation (14) can be modified to its equivalent simultaneous equations using the method of Lagrange multipliers. When an unknown constant multiplier ρ is introduced and “x” is defined by the following equation (15), simultaneous linear equations (16) of “x” are the above-mentioned simultaneous equations.

0113<maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>x</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>ρ</mi></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>B</mi><mn>3</mn></msub><mo>⁢</mo><mi>x</mi></mrow><mo>=</mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>B</mi><mn>3</mn></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>A</mi></mtd><mtd><msub><mi>u</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle></mrow></mtd><mtd><mrow><mo>(</mo><mn>17</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>b</mi><mn>3</mn></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><msup><mi>X</mi><mi>T</mi></msup><mo>⁢</mo><mi>j</mi></mrow></mtd></mtr><mtr><mtd><mrow><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>18</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0006.tif" />

0114As can be understood from the above description, if the distribution of the data set of statistical population is substantially one-dimensional, the process for deriving the new offset at step S<b>110</b> is to solve the simultaneous linear equations (16). The solution “x” must be uniquely specified since the rank of the matrix B₃must be 3.

01152-7. Summary

0116The processes of steps S<b>108</b> and S<b>110</b> will now be described using spatial concepts with reference to <figref idref="DRAWINGS">FIGS. 6 and 7</figref>. If it is assumed that the data set of statistical population is completely reliable, the new offset c is defined by the following equation (19) by considering the new offset c as the sum of the old offset c₀and a position vector (i.e., temporary position vector) g of the center of a circumference, derived from only the data set of statistical population, relative to the old offset c₀. <br />c=c₀+g   (19)

0117The position vector “c” of the new offset can be considered to be the sum of the old offset “c₀” and a correction vector “f” which is a linear combination of fundamental vectors in the same directions as the eigenvectors u₁and u₂of the distribution. Therefore, the correction vector “f”, which corresponds to a vector corrected from the position vector “g” according to the respective degrees of reliability of the components of the position vector “g” corresponding to the center of the data set of statistical population, can be obtained by weighting coefficients g_αand g_βof the position vector “g” according to the respective degrees of reliability of the data set of statistical population in the corresponding principle axis directions.

0118In the process of step S<b>108</b> which is performed when the distribution of the data set of statistical population is sufficiently two-dimensional as shown in <figref idref="DRAWINGS">FIG. 6</figref>, “c” which minimizes the objective function of Equation (12) is obtained without any constraint condition as described above.

0119However, it can be considered that this process is performed under a constraint condition that the new offset “c” be obtained as the sum of the old offset c₀and the correction vector “f” that is obtained by weighting both the components g_αand g_βof the temporary position vector “g” in two principle axis directions of the distribution by a weighting factor of “1”. In <figref idref="DRAWINGS">FIG. 6</figref>, the correction vector “f” is not shown since it is coincident with the position vector “g”.

0120In the process of step S<b>110</b> which is performed when the distribution of the data set of statistical population is substantially one-dimensional as shown in <figref idref="DRAWINGS">FIG. 7</figref>, the following constraint condition is imposed when deriving a new offset based on the old offset c₀and the data set of statistical population. The constraint condition is that the new offset c be obtained as the sum of the old offset c₀and a correction vector “f” that is obtained by weighting a coefficient g_αof the temporary position vector “g” in a principle axis direction (or a principle direction) of the distribution corresponding to the larger principle value of the distribution (i.e., corresponding to the larger eigenvalue λ₁) by a weighting factor of “1” of the temporary position vector “g” and weighting a coefficient g_βin a principle axis direction of the distribution corresponding to the smaller principle value of the distribution (i.e., corresponding to the smaller eigenvalue λ₂) by a weighting factor of “0” of the position vector “g”.

0121The key point of the algorithm in this embodiment is that “c” which minimizes the objective function of Equation (12) is obtained under a constraint condition that weighting factors for both the components of the temporary position vector “g” are set to “1” if the ratio between principle values of the distribution of the data set of statistical population is greater than a predetermined threshold “t” and a weighting factor for the component of the temporary position vector “g” in the principle direction of the distribution is set to “1” and a weighting factor for the component of the temporary position vector “g” in a principle axis direction of the distribution with a smaller level of distribution is set to “0” if the ratio between principle values of the distribution of the data set of statistical population is less than or equal to the predetermined threshold “t”.

B. SECOND EMBODIMENT

0122Overview

0123In the first embodiment, the distribution of the data set of statistical population is estimated discretely and the new offset “c” is derived using different techniques when the distribution is sufficiently two-dimensional and when the distribution is substantially one-dimensional. In the second embodiment, a description will be given of a simple, highly accurate algorithm that can eliminate the need to perform different processes according to estimations of the distribution as in the first embodiment and can also derive a more probable new offset by efficiently using the data set of statistical population.

0124<figref idref="DRAWINGS">FIG. 8</figref> is a flow chart illustrating a new offset derivation process. In the same manner as in the first embodiment, the CPU <b>40</b> performs the procedure of <figref idref="DRAWINGS">FIG. 8</figref> by executing the offset derivation module <b>94</b> when an offset update request has been made. The process of step S<b>200</b> is the same as that of the process of step S<b>100</b> described above in the first embodiment. The process of step S<b>202</b> is the same as that of the process of step S<b>102</b> described above in the first embodiment. The process of step S<b>204</b> is the same as that of the process of step S<b>104</b> described above in the first embodiment.

0125Estimation of Distribution

0126At step S<b>206</b>, a distribution indicator of the data set of statistical population is derived. Specifically, the distribution of the data set of statistical population is estimated as continuous values by deriving, as a distribution indicator, m₂defined by the following equation (20).

0127<maths id="MATH-US-00007" num="00007"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mrow><mn>1</mn><mo>-</mo><msup><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><msub><mi>k</mi><mn>2</mn></msub></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>20</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0007.tif" />

0128Here, “k₂” is a predetermined positive constant number.

0129The value of k₂determines the association between principle values and the degrees of reliability of the corresponding principle axis directions of the data set of statistical population. Here, “m₂” must satisfy the following condition (21). <br />0≦m₂&lt;1   (21)

0130The spatial concept of m₂will now be described with reference to <figref idref="DRAWINGS">FIG. 1</figref>. When the coefficients of the components of the temporary position vector g in the principle axis directions of the distribution are denoted by g_αand g_βin decreasing order of the corresponding principle values and the coefficients of the components of the position vector f in the principle axis directions of the distribution are denoted by f_αand f_βin decreasing order of the corresponding principle values, the relationships between the temporary position vector g, the correction vector f, and m₂in this embodiment are expressed by the following equations (22) and (23).

0131<maths id="MATH-US-00008" num="00008"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><msub><mi>f</mi><mi>α</mi></msub><msub><mi>g</mi><mi>α</mi></msub></mfrac><mo>=</mo><mn>1</mn></mrow></mtd><mtd><mrow><mo>(</mo><mn>22</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><msub><mi>f</mi><mi>β</mi></msub><msub><mi>g</mi><mi>β</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>⁢</mo><msub><mi>m</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow></mrow><mrow><mrow><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>⁢</mo><msub><mi>m</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>-</mo><msubsup><mi>m</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>23</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0008.tif" />

0132The weighting factor f_α/g_αassociated with the component of the principle axis direction corresponding to the maximum principle value may be set to be less than “1”. In addition, if the following conditions are satisfied, any definition may be made for “m₂” determined such that the weighting factors continuously correspond to the ratios of principle values, without being limited to the definition of Equation (20).

0133The range of values of m₂is [0,1] or its subset. <ul id="ul0001" list-style="none"><li id="ul0001-0001" num="0000"><ul id="ul0002" list-style="none"><li id="ul0002-0001" num="0134">m₂≠1 when λ₂/λ₁=1</li><li id="ul0002-0002" num="0135">m₂=1 when λ₂/λ₁=0 if λ₂=0 is permitted</li><li id="ul0002-0003" num="0136">m₂decreases monotonically in a broad sense as λ₂/λ₁increases.</li></ul></li></ul>

0137Specifically, m₂may be defined, for example, by the following equation (24).

0138<maths id="MATH-US-00009" num="00009"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>sgn</mi><mo>⁢</mo><mrow><mo>{</mo><mrow><mrow><mn>2</mn><mo>⁢</mo><msup><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mi>s</mi></msup></mrow><mo>-</mo><mn>1</mn></mrow><mo>}</mo></mrow><mo>⁢</mo><msup><mrow><mo></mo><mrow><mrow><mn>2</mn><mo>⁢</mo><msup><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mi>s</mi></msup></mrow><mo>-</mo><mn>1</mn></mrow><mo></mo></mrow><mi>k</mi></msup></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>24</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0009.tif" />

0139where “s” and “k” are nonnegative real numbers and “sgn” is a sign function expressed by the following equation.

0140<maths id="MATH-US-00010" num="00010"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>sgn</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo><</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>25</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0010.tif" />

0141Since a very large amount of calculation is required to derive m₂defined by Equation (24), calculation results of Equation (24) with inputs λ₁and λ₂may be stored in a 2D lookup table and an approximate value of m₂may then be derived with reference to the 2D lookup table.

0142The indices “s” and “k” are set depending on embodiments since weighting effects vary depending on their values. <figref idref="DRAWINGS">FIGS. 9 and 10</figref> are graphs representing weighting effects that vary depending on the indices “s” and “k”. <figref idref="DRAWINGS">FIG. 9</figref> shows the relationship between the ratio of eigenvalues (λ₂/λ₁) and the weighting factor m₂when “s” is fixed to 1 and “k” is set to ¼, 1, and 4 in Equation (24). <figref idref="DRAWINGS">FIG. 10</figref> shows the relationship between the ratio of eigenvalues (λ₂/λ₁) and the weighting factor m₂when “k” is fixed to ¼ and “s” is set to ½, 1, and 2 in Equation (24).

0143When a magnetic sensor is mounted on a moving body or object such as a mobile phone or a Personal Digital Assistant (PDA), the attitude or posture of which changes at a high angular speed or velocity, it can be expected that the distribution of a magnetic data set stored in a certain period of time will be relatively wide on average. In case that the distribution of the magnetic data is not so wide, the accuracy of the offset would be rather degraded if the offset correction is carried out by significantly evaluating the magnetic data group of the principal axis direction having the small principal value since such a magnetic data group has low reliability. Therefore, in case that the invention is applied to a moving object where the distribution of the magnetic data group tends to become relatively wide, the values of the parameters s and k should be set so that the magnetic data group of the principal axis direction having the small principal value is evaluated significantly only if the distribution of the magnetic data is considerably wide. Here, as the value of m₂in the Equation (24) becomes smaller, it means that the weight in the principal axis direction of the distribution having the small principal value becomes greater. Consequently, in case that the Equation (24) is applied to a moving object which changes its posture at a fast angular velocity, it is desirable to set the parameter s with a relatively great value and set the parameter k with relatively small value.

0144On the other hand, when a magnetic sensor is mounted on a moving body or object such as an automobile, the attitude or posture of which changes at a low angular speed, it is assumed that the distribution of a magnetic data set stored in a certain period of time will be relatively narrow on average. In such a case, the accuracy of the offset would not be improved if the offset correction is carried out while the magnetic data group of the principal axis direction having the small principal value is not evaluated with a heavy weight although such a magnetic data group is not so reliable, when the distribution of the magnetic data is not so wide. Consequently, in case that the Equation (24) is applied to a moving object which changes its posture at a slow angular velocity, it is desirable to set the parameter s with a relatively small value and set the parameter k with relatively great value.

0145Derivation of New Offset

0146When it is difficult to derive a solution to the optimization problem under a specific constraint condition, a relaxation problem for solving the optimization problem by relaxing the constraint condition may be introduced. By applying this relaxation problem, this embodiment realizes a process for deriving a new offset c as the sum of the old offset c₀and a correction vector f that is obtained by weighting the coefficients g_αand g_βof the temporary position vector g (see <figref idref="DRAWINGS">FIG. 1</figref>) described above by weighting factors continuously corresponding to the ratios of principle values of the distribution of the data set of statistical population. The following are details of this process.

0147An unknown constant multiplier ρ is defined as a variable required for calculations during the process and c and ρ are grouped together into one vector “x” that is defined by the following equation (26).

0148<maths id="MATH-US-00011" num="00011"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>x</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>ρ</mi></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>26</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0011.tif" />

0149In addition, a matrix “B” is defined by Equation (27) and a vector “b” is defined by Equation (28).

0150<maths id="MATH-US-00012" num="00012"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>⁢</mo><msub><mi>u</mi><mn>2</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup></mrow></mtd><mtd><mrow><mfrac><mn>1</mn><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>27</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>b</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><msup><mi>X</mi><mi>T</mi></msup><mo>⁢</mo><mi>j</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>28</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0012.tif" />

0151The process for deriving a new offset at step S<b>208</b> is to find a solution to the following simultaneous equations (29). The vector x is uniquely specified since the matrix B must be nonsingular. <br />Bx=b   (29)

0152Finding a solution to the simultaneous equations (29) is equivalent to solving the optimization problem for minimizing the objective function of Equation (12) under a constraint condition that a new offset be obtained as the sum of the old offset c₀and a correction vector f whose components are values obtained by weighting the coefficients of the position vector g in the principle axis directions of the distribution corresponding to the principle values by factors f_α/g_αand f_β/g_βcontinuously corresponding to the ratios of principle values of the distribution of the data set of statistical population.

0153In the second embodiment, it is easy to develop or improve the offset derivation module <b>94</b> and the data size of the offset derivation module <b>94</b> is also decreased since there is no need to branch the new offset derivation process according to the distribution of the data set of statistical population as described above. In addition, the second embodiment increases the use efficiency with which the data set of statistical population is used by the offset derivation module <b>94</b> and also allows the orientation derivation module to correct magnetic data using the most probable offset since the old offset can be corrected in the principle axis directions of the distribution by distances continuously corresponding to the ratios of principle values of the data set of statistical population unless any of the principle values is zero.

C. OTHER EMBODIMENTS

0154The first aspect of the present invention is not limited to the above embodiments and various embodiments are possible without departing from the spirit of the present invention. For example, the first aspect of the present invention can also be applied to a magnetic sensor that is mounted on a PDA, a mobile phone, a two-wheeled motor vehicle, a ship, or the like.

0155Next, a principle and algorithm used in the second aspect of the invention is described below in detail with reference to <figref idref="DRAWINGS">FIG. 11</figref>. Key points of this algorithm are that a magnetic data set distributed in a principle axis direction with larger dispersion is estimated to be more significant elements of statistical population for use in updating the offset and a magnetic data set distributed in a principle axis direction with smaller dispersion is estimated to be less significant elements of statistical population for use in updating the offset. Details are as follows. Each of the old offset c₀, the new offset c, and the temporary offset, which corresponds to the position of the end point of “g” relative to the point of origin “0”, is 3D position vector data that is a linear combination of a fundamental vector set of magnetic data. That is, each of the offsets is vector data represented in the xyz coordinate system. The new offset c is derived based on the old offset c₀and a magnetic data set that is stored to update the old offset c₀with the new offset c.

0156The data set of statistical population, which is a magnetic data set stored to update the old offset with the new offset c, may include a magnetic data set that has been stored in a predetermined period of time and may include a magnetic data set including a predetermined number of magnetic data and may also include a magnetic data set including any number of magnetic data that has been stored at a certain time (for example, at the time when an offset update request is made).

0157The old offset c₀may be derived using the same method as the new offset c and may also be predetermined.

0158Although the temporary offset is defined to be derived from the data set of statistical population without using the old offset c₀, this definition is introduced to define the constraint condition under which the new offset c is derived and the temporary offset is actually not data which it is necessary to derive. If the temporary offset is actually derived from the data set of statistical population without using the old offset c₀, the temporary offset is a position vector of the center of a spherical surface near which the data set of statistical population is distributed. However, if the data set of statistical population is unevenly distributed at a portion of the spherical surface derived from the data set of statistical population, an error included in each element of the data set of statistical population greatly affects the derivation result of the spherical surface and thus there is a possibility that a temporary offset distant from the true offset is derived. Let us consider, for example, that a data set of statistical population is distributed in doughnut-shaped distribution with mutually orthogonal eigenvectors u₁, u₂, and u₃and the variance of the data set of statistical population is minimized in the direction of the eigenvector u₃corresponding to the minimum principle value as shown in <figref idref="DRAWINGS">FIG. 11</figref>. In this case, since the variance of the data set of statistical population is small in the direction of the eigenvector u₃of the distribution, there is a high probability that the position of a temporary offset derived from the data set of statistical population is distant from the position of the true offset in the direction of the eigenvector u₃. On the other hand, in this case, since the variance of the data set of statistical population is large in the direction of the eigenvector u₁of the distribution, there is a high probability that the position of the temporary offset derived from the data set of statistical population is near the position of the true offset in the direction of the eigenvector u₁.

0159Since the variances in the principle axis directions of the distribution can be expressed using the principle values λ₁, λ₂, and λ₃of the distribution as indicators of the distribution, this device estimates elements of statistical population distributed in the directions corresponding respectively to the principle values according to the ratios of the principle values λ₁, λ₂, and λ₃. Specifically, first, a correction vector f, which is a position vector of the new offset c relative to the old offset c₀, and a position vector g of the temporary offset relative to the old offset c₀can be defined in a coordinate system having coordinate axes α, β, and γ coincident with the principle axis directions of the distribution. That is, each of the correction vector f and the position vector g can be defined to be a linear combination of fundamental vectors of the principle axis directions of the distribution. This corresponds to conversion into principle axis values. If the components f_α, f_β, and f_γof the correction vector f are derived by weighting the components g_α, g_β, and g_γof the position vector g of the temporary offset relative to the old offset c₀according to the measures of the corresponding principle values u₁, u₂, and u₃of the distribution, it is possible to derive the correction vector f by increasing the reliability of the elements of statistical population in directions with large dispersion and decreasing the reliability of the elements of statistical population in directions with small dispersion. However, such definitions of the correction vector f and the position vector g are also introduced to define the constraint condition under which the new offset c is derived and each of the correction vector f and the position vector g is actually not data which it is necessary to derive.

0160By deriving the new offset c under a constraint condition that the new offset c be obtained as the sum of the old offset c₀and the correction vector f defined as described above, it is possible to derive the new offset while estimating that a magnetic data set distributed in a principle axis direction with larger dispersion is more significant elements of statistical population for use in updating the offset and a magnetic data set distributed in a principle axis direction with smaller dispersion is less significant elements of statistical population for use in updating the offset. One example technique for deriving the new offset in this manner is to formulate the distribution as an optimization problem. If a new offset is derived as an optimization problem of the distribution with a constraint condition, the new offset can be derived by solving simple simultaneous linear equations as described later in the embodiments. That is, no matter what the distribution of the data set of statistical population is, it is possible for this device to derive, through a simple process, the most probable new offset which can be derived from the data set of statistical population.

0161In the inventive magnetic data processing device, the constraint condition may be that a weighting factor of the position vector for the coefficient of one of the second fundamental vectors in one of the principle axis directions corresponding to a minimum one of the principle values be zero if a ratio of an intermediate one of the principle values to a maximum one of the principle values is higher than a first threshold and a ratio of the minimum principle value to the maximum principle value is equal to or less than a second threshold and that respective weighting factors of the position vector for both the coefficient of the second fundamental vector in the principle axis direction corresponding to the minimum principle value and the coefficient of another of the second fundamental vectors in one of the principle axis directions corresponding to the intermediate principle value be zero if the ratio of the intermediate principle value to the maximum principle value is equal to or less than the first threshold and the ratio of the minimum principle value to the maximum principle value is equal to or less than the second threshold.

0162This device discretely estimates the distribution of the data set of statistical population and discretely weights the coefficients of the position vector of the temporary offset relative to the old offset c₀according to the discrete estimation result. Specifically, the weight of the coefficient of the position vector g in the direction in which the value of the distribution is the minimum value is “0”. Also the weight of the coefficient of the position vector g in the direction in which the value of the distribution is the intermediate value is “0” if the value of the distribution is small in the direction. That is, no magnetic data is estimated in directions in which the value of the distribution is smaller than a threshold.

0163In the inventive magnetic data processing device, the constraint condition may be that the coefficients of the correction vector be values obtained by weighting the coefficients of the position vector by weighting factors continuously corresponding to the ratios of the principle values of the distribution of the data set of statistical population.

0164This device can increase the substantial use efficiency of the data set of statistical population since the weighting factors have a continuous association with the distribution. In addition, the device can simplify the offset update process since it is possible to derive the new offset without changing the process according to the distribution of the data set of statistical population.

0165In the inventive magnetic sensor offset derivation device, the respective weighting factors for the coefficients of the position vector may be normalized by setting, to one, the weighting factor for the coefficient of the second fundamental vector in the principle axis direction corresponding to the maximum principle value.

0166In the case where estimation indicators of the distribution of the magnetic data set of statistical population other than the principle values of the distribution are introduced, it is not necessary to normalize the weighting factors by setting the maximum weighting factor to one. For example, the maximum weighting factor may be set to less than one according to the ratio of the maximum distance between magnetic data in a principle axis direction (i.e., the principle direction) corresponding to the maximum principle value to the radius of a sphere that has been derived as a spherical surface, having a portion near which a magnetic data set of statistical population is distributed, based on the magnetic data set of statistical population.

0167In the inventive magnetic data processing device, the offset derivation part may derive “c” that minimizes the following objective function f(c) under the constraint condition: <br />f(c)=(Xc−j)^T(Xc−j)

0168where “X” and “j” are as follows when the magnetic data is represented by q_i=(q_ix,q_iy,q_iz) (i=0,1,2, . . .):

0169<maths id="MATH-US-00013" num="00013"><math overflow="scroll"><mrow><mi>X</mi><mo>=</mo><mrow><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr></mtable><mo>]</mo></mrow><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>j</mi></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁡</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>1</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></math></maths><img file="USRE44582E_D0013.tif" />

0170Since this device derives a new offset as an optimization problem of the distribution with a constraint condition, it is possible to derive the new offset by solving simple simultaneous linear equations as described later in the embodiments. That is, no matter what the distribution of the data set of statistical population is, it is possible for this device to derive, through a simple process, the most probable new offset which can be derived from the data set of statistical population. An inventive magnetic measurement apparatus comprises the magnetic data processing device described above; and the 3D magnetic sensor.

0171No matter what the distribution of the data set of statistical population is, it is possible for this device to derive, through a simple process, the most probable new offset which can be derived from the data set of statistical population.

0172The function of each of the plurality of parts included in the device of the present invention is realized by a hardware resource, the function of which is specified by its construction, a hardware resource, the function of which is specified by a program, or a combination of these resources. The function of each of the plurality of parts is not necessarily realized by a physically independent hardware resource. The present invention can be specified not only by a device but also by a program, a recording medium on which the program is recorded, and a method. The operations of the method described in the claims are not necessarily performed in the order as described in the claims and may be performed in any other order or at the same time, provided there is no technical impediment.

0173Embodiments of the second aspect of the present invention will be described in the following order.

A. FIRST EMBODIMENT

0174[1. General Description]

01751-1. Hardware Structure

01761-2. Software Structure

0177[2. Procedure]

01782-1. Overall Flow

01792-2. Buffer Update

01802-3. Estimation of Distribution

01812-4. Derivation of New Offset through Optimization Problem

01822-5. Constraint Condition when the Distribution is Two-Dimensional

01832-6. Constraint Condition when the Distribution is Substantially One-Dimensional

01842-7. Derivation of New Offset when the Distribution is Three-Dimensional

01852-8. Summary

0186B. Second Embodiment

0187Overview

0188Estimation of Distribution

0189Derivation of New Offset

0190C. Other Embodiments

0191[General Description]

01921-1. Hardware Structure

0193<figref idref="DRAWINGS">FIG. 12</figref> is a schematic diagram of an external appearance of a mobile phone <b>3</b> that is an example of a moving body to which the present invention is applied. The mobile phone <b>3</b> includes a 3-dimensional (3D) magnetic sensor <b>4</b>. The 3D magnetic sensor <b>4</b> detects the direction and intensity of a magnetic field by detecting respective intensities of the magnetic field in three orthogonal directions (x, y, z). A display <b>2</b> of the mobile phone <b>3</b> displays a variety of character or image information. For example, the display <b>2</b> displays a map and an arrow or characters representing the orientation (or azimuth).

0194<figref idref="DRAWINGS">FIG. 13</figref> is a block diagram of a magnetic measurement device which includes a 3D magnetic sensor <b>4</b> and a magnetic data processing device <b>1</b>. The 3D magnetic sensor <b>4</b> includes x, y, and z-axis sensors <b>30</b>, <b>32</b>, and <b>34</b> that detect x, y, and z direction components of a magnetic field vector due to terrestrial magnetism. Each of the x, y, and z-axis sensors <b>30</b>, <b>32</b>, and <b>34</b> includes a magnetic resistance element, a hall sensor, or the like, which may be any type of 1-dimensional magnetic sensor provided that it has directivity. The x, y, and z-axis sensors <b>30</b>, <b>32</b>, and <b>34</b> are fixed so that their sensitive directions are perpendicular to each other. Outputs of the x, y, and z-axis sensors <b>30</b>, <b>32</b>, and <b>34</b> are time-divided and input to a magnetic sensor interface (I/F) <b>22</b>. The magnetic sensor interface <b>22</b> analog-to-digital converts inputs from the x, y, and z-axis sensors <b>30</b>, <b>32</b>, and <b>34</b> after amplifying the inputs. Digital magnetic data output from the magnetic sensor interface <b>22</b> is input to the magnetic data processing device <b>1</b> through a bus <b>5</b>.

0195The magnetic data processing device <b>1</b> is a computer including a CPU <b>40</b>, a ROM <b>42</b>, and a RAM <b>44</b>. The CPU <b>40</b> controls overall operations of, for example, the mobile phone <b>3</b>. The ROM <b>42</b> is a nonvolatile storage medium which stores a magnetic data processing program or a variety of programs (for example, a navigation program) used to implement functions of the moving body, which are executed by the CPU <b>40</b>. The RAM <b>44</b> is a volatile storage medium which temporarily stores data to be processed by the CPU <b>40</b>. The magnetic data processing device <b>1</b> and the 3D magnetic sensor <b>4</b> may be constructed as a one-chip magnetic measurement device.

01961-2. Software Structure

0197<figref idref="DRAWINGS">FIG. 14</figref> is a block diagram of a magnetic data processing program <b>90</b>. The magnetic data processing program <b>90</b> is stored in the ROM <b>42</b> to provide orientation data to a navigation program <b>98</b>. The orientation data is 2D vector data representing the orientation of the Earth's magnetic field. As 3D vector data for attitude detection of, for example, a moving body, the orientation data may be provided to other applications. The magnetic data processing program <b>90</b> is constructed as a group of modules such as a buffer management module <b>92</b>, an offset derivation module <b>94</b>, and an orientation derivation module <b>96</b>.

0198The buffer management module <b>92</b> is a program part that receives a plurality of magnetic data sequentially output from the magnetic sensor <b>4</b> and stores the received magnetic data in a buffer in order to use the magnetic data in offset update. The buffer management module <b>92</b> allows the CPU <b>40</b>, the RAM <b>44</b>, and the ROM <b>42</b> to function as an input part and storage part. This buffer may be embodied not only in hardware but also in software. A magnetic data set stored in this buffer will now be referred to as a data set of statistical population.

0199The offset derivation module <b>94</b> is a program part that derives a new offset based on a data set of statistical population held by the buffer management module <b>92</b> and an old offset held by the offset derivation module <b>94</b> and updates the old offset with the new offset. The offset derivation module <b>94</b> allows the CPU <b>40</b>, the RAM <b>44</b>, and the ROM <b>42</b> to function as an offset derivation part. Since updating the old offset with the new offset causes the new offset to become an old offset, the “old offset” will be referred to simply as an “offset” in context in which it causes no misunderstanding. Actually, an offset used for orientation data correction is set in one variable and the new offset is derived as a different variable from that variable. When the new offset is derived, it is set in the variable used for orientation data correction. Therefore, the variable used for orientation data correction is that in which the old offset is stored.

0200The orientation derivation module <b>96</b> is a program part that corrects the magnetic data sequentially output from the magnetic sensor using the offset held by the offset derivation module <b>94</b> to create orientation data. The orientation derivation module <b>96</b> allows the CPU <b>40</b>, the RAM <b>44</b>, and the ROM <b>42</b> to function as an orientation derivation part. Specifically, the orientation derivation module <b>96</b> outputs, as orientation data, all or two of the 3 components obtained by subtracting the components of the offset from the components of the magnetic data which is 3D vector data.

0201The navigation program <b>98</b> is a known program that searches for a route to the destination and displays the route on a map. Since it is easy to recognize the map, the map is displayed such that the orientation of the map matches the real-world orientation. Accordingly, for example, when the mobile phone <b>3</b> is rotated, the map displayed on the display <b>2</b> is rotated relative to the display <b>2</b> such that the map is not rotated relative to the earth. The orientation data is used in this map display processing. Of course, the orientation data may be used only to display north, south, east and west by characters or arrows.

0202[2. Procedure]

02032-1. Overall Flow

0204<figref idref="DRAWINGS">FIG. 15</figref> is a flow chart illustrating a new offset derivation procedure. The CPU <b>40</b> performs the procedure of <figref idref="DRAWINGS">FIG. 15</figref> by executing the offset derivation module <b>94</b> when an offset update request has been made.

02052-2. Buffer Update

0206At step S<b>100</b>, all magnetic data stored in the buffer, in which a magnetic data set (data set of statistical population) used for deriving a new offset is stored, is deleted. That is, in this process, a data set of statistical population used for deriving the old offset is deleted.

0207At step S<b>102</b>, magnetic data used for deriving a new offset is input and stored in the buffer. When a plurality of magnetic data is sequentially input from the magnetic sensor <b>4</b> with almost no change in the attitude of the mobile phone <b>3</b>, the distance between two sequentially input magnetic data (or values) is small. Storing a plurality of near magnetic data in a buffer with a limited capacity wastes memory resources and causes unnecessary buffer update processes. In addition, if a new offset is derived based on a set of near magnetic data, there is a possibility that an inaccurate new offset is derived based on an unevenly distributed data set of statistical population. Whether or not it is necessary to update the buffer may be determined in the following manner. For example, if the distance between the last input magnetic data and magnetic data stored in the buffer immediately before the last input magnetic data is less than a given threshold, it is determined that it is unnecessary to update the buffer and the last input magnetic data is discarded without being stored in the buffer.

0208At step S<b>104</b>, it is determined whether or not a specified number of magnetic data required to derive an accurate new offset has been stored in the buffer. That is, the number of elements of the data set of statistical population is predetermined. Setting a small number of elements of the data set of statistical population improves response to the offset update request. The processes of steps S<b>102</b> and S<b>104</b> are repeated until the specified number of magnetic data is stored in the buffer.

02092-3. Estimation of Distribution

0210Once the specified number of magnetic data is stored in the buffer, the distribution of the data set of statistical population is estimated (S<b>106</b> and S<b>108</b>). The distribution is estimated based on principle values of the distribution. When the magnetic data set is expressed by the following Equation (31), the principle values of the distribution are eigenvalues of a symmetric matrix A defined by Equations (32), (33), and (34) using the sum of vectors starting from a center (average) of the data set of statistical population and ending with the respective magnetic data.

0211<maths id="MATH-US-00014" num="00014"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>=</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>ix</mi></msub><mo>,</mo><msub><mi>q</mi><mi>iy</mi></msub><mo>,</mo><msub><mi>q</mi><mi>iz</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi></mrow><mo>)</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>31</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>A</mi><mo>=</mo><mrow><msup><mi>X</mi><mi>T</mi></msup><mo>⁢</mo><mi>X</mi></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>32</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><mi>X</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>33</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mover><mi>q</mi><mi>_</mi></mover><mo>=</mo><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><msub><mi>q</mi><mi>i</mi></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>34</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0014.tif" />

0212The matrix A may also be rewritten as Equation (35).

0213<maths id="MATH-US-00015" num="00015"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>A</mi><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>35</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0015.tif" />

0214Let λ₁, λ₂, and λ₃be the eigenvalues of the matrix A in increasing order. Let u₁, u₂, and u₃be mutually orthogonal eigenvectors that correspond to λ₁, λ₂, and λ₃and have been normalized to size 1. The ranges of λ₁, λ₂, and λ₃handled in this specification are λ₁>0, λ₂>0, and λ₃≧0. When two or more eigenvalues of the matrix A are zero, i.e., when the rank of the matrix A is one or less, there is no need to consider it since the number of elements of the data set of statistical population is one or the distribution is a perfectly straight line. Each of the eigenvalues must be zero or a positive real number since the matrix A is a positive semi-definite matrix from its definition.

0215The distribution of the data set of statistical population is estimated based on the ratio λ₃/λ₁of the minimum eigenvalue to the maximum eigenvalue and the ratio λ₂/λ₁of an intermediate eigenvalue to the maximum eigenvalue.

0216At step S<b>106</b>, it is determined whether or not the distribution of the data set of statistical population is sufficiently three-dimensional. Specifically, the determination is affirmative when the following condition (36) is satisfied and negative when it is not satisfied. <br />λ₃/λ₁>t₁and λ₂/λ₁>t₂  (36)

0217Here, “t₁,” and “t₂” are predetermined constant values.

0218How to set the values of the values t₁and t₂is a design option and they can be set optionally based on how to determine derivation characteristics of the offset. When the condition (36) is satisfied, the data set of statistical population is distributed isotropically from the center of the data set of statistical population. The isotropic distribution of the data set of statistical population about the center indicates that the data set of statistical population is distributed evenly near a specific spherical surface.

0219At step S<b>108</b>, it is determined whether or not the distribution of the data set of statistical population is sufficiently two-dimensional. Specifically, the determination is affirmative when the following condition (37) is satisfied and negative when it is not satisfied. <br />λ₃/λ₁≦t₁and λ₂/λ₁>t₂  (37)

0220When the condition (37) is satisfied, the data set of statistical population is distributed isotropically from the center of the data set of statistical population in a range restricted near a specific plane. The isotropic distribution of the data set of statistical population about the center in a range restricted near a specific plane indicates that the data set of statistical population is unevenly distributed near the circumference of a sectional circle of a specific spherical surface.

0221When the determination of step S<b>108</b> is negative, the distribution of the data set of statistical population is substantially one-dimensional (i.e., linear). The substantially linear distribution of the data set of statistical population indicates that the data set of statistical population is unevenly distributed on a short arc of a sectional circle of a specific spherical surface or on both ends of a diameter of the sectional circle.

02222-4. Derivation of New Offset through Optimization Problem

0223An optimization problem for deriving a new offset will now be described.

0224When the data set of statistical population includes 4 magnetic data not present on the same plane, a spherical surface on which the data set of statistical population is distributed is uniquely specified without using a statistical technique. A position vector c=(c_x, c_y, c_z) of the center of this spherical surface is obtained by solving simultaneous equations (38). Although four equality constraints exist for three variables, the equations (38) must have a solution since one of the four equality constraints is redundant.

0225<maths id="MATH-US-00016" num="00016"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>3</mn></msub><mo>-</mo><mover><mi>q</mi><mi>_</mi></mover></mrow><mo>)</mo></mrow><mi>T</mi></msup></mtd></mtr></mtable><mo>]</mo></mrow><mo>⁢</mo><mi>c</mi></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁡</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>1</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>3</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>3</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>38</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><mi>R</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><msubsup><mi>q</mi><mi>i</mi><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mi>i</mi></msub></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>39</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0016.tif" />

0226When the number of elements of the data set of statistical population is 5 or more, “j” is defined by the following equation (40).

0227<maths id="MATH-US-00017" num="00017"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>j</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁡</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>0</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>1</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><msubsup><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>q</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mi>R</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>40</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0017.tif" />

0228Here, if simultaneous linear equations (41) for “c” have a solution, the solution is the center of a spherical surface on which the data set of statistical population is distributed. <br />Xc=j   (41)

0229However, if an inherent measurement error of the 3D magnetic sensor <b>4</b> is considered, it is practically impossible for Equations (41) to have a solution. A vector “e” defined by the following equation (42) is introduced to obtain a plausible solution through a statistical technique. <br />e=Xc−j   (42)

0230“c” which minimizes ∥e∥₂²(that is, e^Te) can be considered to be plausible as the center of a spherical surface closest to the distribution of the data set of statistical population. A problem for finding the value “c” minimizing ∥e∥₂²is an optimization problem for minimizing an objective function of the following equation (43) when the matrix A is nonsingular. <br />Objective Function: f(c)=(Xc−j)^T(Xc−j)→min   (43)

02312-5. Constraint Condition when the Distribution is Two-Dimensional

0232As shown in <figref idref="DRAWINGS">FIG. 16</figref>, when the distribution of the data set of statistical population is two-dimensional (i.e., planar), a new offset is derived by restricting directions in which the old offset is corrected to two orthogonal directions (S<b>112</b>). When the data set of statistical population is distributed near a specific plane and the distribution is discrete in a direction normal to the plane, the distribution of the data set of statistical population in a direction parallel to the plane is sufficiently reliable while the distribution of the data set of statistical population in the direction normal to the plane is unreliable. In this case, the old offset is not corrected in the direction normal to the plane, thereby preventing the offset from being updated based on unreliable information.

0233When the data set of statistical population is distributed near a specific plane and the distribution is discrete in a direction normal to the plane, the direction normal to the plane is coincident with the direction of an eigenvector u₃corresponding to the minimum eigenvalue λ₃and orthogonal directions parallel to the plane are coincident with the directions of eigenvectors u₁and u₂corresponding respectively to the maximum eigenvalueλ₁and the intermediate eigenvalue λ₂. Accordingly, in order to derive a new offset without correcting the old offset in the direction normal to the plane, a new offset c which minimizes the objective function of Equation (43) is found under a constraint condition expressed by the following equation (44). <br />c=c₀+β₁u₁+β₂u₂(β₁, β₂: real numbers)   (44)

0234Equation (44) is equivalent with the following equation (45). <br />u₃^T(c−c₀)=0   (45)

0235The equation for solving the optimization problem of Equation (43) under the constraint condition of Equation (45) can be modified to its equivalent simultaneous equations using the method of Lagrange multipliers. When an unknown constant multiplier p is introduced and “x” is defined by the following equation (46), simultaneous linear equations (47) of “x” are the above-mentioned simultaneous equations.

0236<maths id="MATH-US-00018" num="00018"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>x</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>ρ</mi></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>46</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>B</mi><mn>4</mn></msub><mo>⁢</mo><mi>x</mi></mrow><mo>=</mo><msub><mi>b</mi><mn>4</mn></msub></mrow></mtd><mtd><mrow><mo>(</mo><mn>47</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><msub><mi>B</mi><mn>4</mn></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mn>2</mn><mo>⁢</mo><mi>A</mi></mrow></mtd><mtd><msub><mi>u</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><msubsup><mi>u</mi><mn>3</mn><mi>T</mi></msubsup></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>48</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>b</mi><mn>4</mn></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mn>2</mn><mo>⁢</mo><msup><mi>X</mi><mi>T</mi></msup></mrow></mtd><mtd><mi>j</mi></mtd></mtr><mtr><mtd><msubsup><mi>u</mi><mn>3</mn><mi>T</mi></msubsup></mtd><mtd><msub><mi>c</mi><mn>0</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>49</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0018.tif" />

0237As can be understood from the above description, if the distribution of the data set of statistical population is two-dimensional, the process for deriving the new offset at step S<b>112</b> is to solve the simultaneous linear equations (47).

0238The solution “x” must be uniquely specified since the rank of the matrix B₄must be 4.

02392-6. Constraint Condition when the Distribution is Substantially One-Dimensional

0240As shown in <figref idref="DRAWINGS">FIG. 17</figref>, when the distribution of the data set of statistical population is substantially one-dimensional (i.e., linear), a new offset is derived by restricting directions in which the old offset is corrected to one principle direction of the distribution (S<b>110</b>). When the data set of statistical population is distributed near a specific line and the distribution is discrete in the direction of the line, the distribution of the data set of statistical population in the direction of the straight line is sufficiently reliable while the distribution of the data set of statistical population in other directions is unreliable. In this case, the old offset is not corrected in directions other than the direction of the line, thereby preventing the offset from being updated based on unreliable information.

0241When the data set of statistical population is distributed near a specific line and the distribution is discrete in the direction of the line, the direction of the line is coincident with the direction of an eigenvector u₁corresponding to the maximum eigenvalue λ₁and the other directions are coincident with the directions of eigenvectors u₂and u₃corresponding respectively to the intermediate eigenvalue λ₂and the minimum eigenvalue λ₃.

0242Accordingly, in order to derive a new offset only in the direction of the line, a new offset c which minimizes the objective function of Equation (43) is found under a constraint condition expressed by the following equation (50). <br />c=c₀+β₁u₁  (50)

0243Equation (50) is equivalent with the following equations (51). <br />u₂^T(c−c₀)=0 and u₃^T(c−c₀)=0   (51)

0244The equation for solving the optimization problem of Equation (43) under the constraint condition of Equation (51) can be modified to its equivalent simultaneous equations using the method of Lagrange multipliers. When unknown constant multipliers ρ₁and ρ₂are introduced and “x” is defined by the following equation (52), simultaneous linear equations (53) of “x” are the above-mentioned simultaneous equations.

0245<maths id="MATH-US-00019" num="00019"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>x</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>c</mi></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>52</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>B</mi><mn>5</mn></msub><mo>⁢</mo><mi>x</mi></mrow><mo>=</mo><msub><mi>b</mi><mn>5</mn></msub></mrow></mtd><mtd><mrow><mo>(</mo><mn>53</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>where</mi></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr><mtr><mtd><mrow><msub><mi>B</mi><mn>5</mn></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mn>2</mn><mo>⁢</mo><mi>A</mi></mrow></mtd><mtd><msub><mi>u</mi><mn>2</mn></msub></mtd><mtd><msub><mi>u</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msubsup><mi>u</mi><mn>3</mn><mi>T</mi></msubsup></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>54</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>b</mi><mn>5</mn></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mn>2</mn><mo>⁢</mo><msup><mi>X</mi><mi>T</mi></msup><mo>⁢</mo><mi>j</mi></mrow></mtd></mtr><mtr><mtd><mrow><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msubsup><mi>u</mi><mn>3</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>55</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0019.tif" />

0246As can be understood from the above description, if the distribution of the data set of statistical population is substantially one-dimensional, the process for deriving the new offset at step S<b>110</b> is to solve the simultaneous linear equations (53). The solution “x” must be uniquely specified since the rank of the matrix β₅must be 5.

02472-7. Derivation of New Offset when the Distribution is Three-Dimensional

0248When the distribution is three-dimensional, a new offset is derived without restricting directions in which the old offset is corrected (S<b>114</b>). When the distribution is three-dimensional, i.e., if the data set of statistical population is distributed in all directions to a certain extent when seen from the center of the data set of statistical population, the data set of statistical population is sufficiently reliable in all directions. Accordingly, in this case, to derive the new offset, it is unnecessary to use the old offset and thus the new offset can be derived based on the data set of statistical population without using the old offset. An algorithm for deriving a new offset based on the data set of statistical population without using the old offset may be an algorithm using one of a variety of statistical techniques that have been suggested and may also be an algorithm using no statistical technique as described in Japanese Patent Application Nos. 2005-337412 and 2006-44289 that have already been filed by the present applicant.

0249In this embodiment, a new offset is derived using a statistical technique. That is, at step S<b>114</b>, the new offset “c” is derived as a solution to the optimization problem for minimizing the objective function of Equation (43) without any constraint condition.

02502-8. Summary

0251The processes of steps S<b>110</b>, S<b>112</b>, and S<b>114</b> will now be described using spatial concepts with reference to <figref idref="DRAWINGS">FIGS. 1</figref>, <b>6</b>, and <b>7</b>. If it is assumed that the data set of statistical population is completely reliable, the new offset c is defined by the following equation (56) by considering the new offset c as the sum of the old offset c₀and a position vector g of the center of a spherical surface, derived from only the data set of statistical population, relative to the old offset c₀. <br />c=c₀+g   (56)

0252The position vector g derived as a solution to the optimization problem for minimizing the objective function of Equation (43) without any constraint condition is a linear combination of fundamental vectors in the same directions as the eigenvectors u₁, u₂, and u₃of the distribution. Therefore, a correction vector “f”, which corresponds to a vector corrected from the position vector “g” according to the respective degrees of reliability of the components of the position vector “g”, can be obtained by weighting coefficients g_α, g_β, and g_γof the position vector “g” according to the respective degrees of reliability of the data set of statistical population in the corresponding principle axis directions (see <figref idref="DRAWINGS">FIG. 11</figref>).

0253In the process of step S<b>112</b> which is performed when the distribution of the data set of statistical population is two-dimensional as shown in <figref idref="DRAWINGS">FIG. 16</figref>, the following constraint condition is imposed when deriving a new offset based on the old offset c₀and the data set of statistical population. The constraint condition is that the new offset c be obtained as the sum of the old offset c₀and a correction vector “f” that is obtained by weighting both a coefficient g_αof the position vector “g” in a principle axis direction of the distribution corresponding to the maximum principle value of the distribution (i.e., corresponding to the maximum eigenvalue λ₁) and a coefficient g_βin a principle axis direction of the distribution corresponding to the intermediate principle value of the distribution (i.e., corresponding to the intermediate eigenvalue λ₂) by a weighting factor of “1” of the position vector “g” and weighting a coefficient g_γin a principle axis direction of the distribution corresponding to the minimum principle value of the distribution (i.e., corresponding to the minimum eigenvalue X₃) by a weighting factor of “0” of the position vector “g”.

0254In the process of step S<b>110</b> which is performed when the distribution of the data set of statistical population is substantially one-dimensional as shown in <figref idref="DRAWINGS">FIG. 17</figref>, the following constraint condition is imposed when deriving a new offset based on the old offset c₀and the data set of statistical population. The constraint condition is that the new offset c be obtained as the sum of the old offset c₀and a correction vector “f” that is obtained by weighting a coefficient g_αof the position vector “g” in a principle axis direction (or a principle direction) of the distribution corresponding to the maximum principle value of the distribution (i.e., corresponding to the maximum eigenvalue λ₁) by a weighting factor of “1” of the position vector “g” and weighting both a coefficient g_βin a principle axis direction of the distribution corresponding to the intermediate principle value of the distribution (i.e., corresponding to the intermediate eigenvalue λ₂) and a coefficient g_γin a principle axis direction of the distribution corresponding to the minimum principle value of the distribution (i.e., corresponding to the minimum eigenvalue λ₃) by a weighting factor of “0” of the position vector “g”.

0255In the process of step S<b>110</b> which is performed when the distribution of the data set of statistical population is three-dimensional, any specific constraint condition is not imposed. That is, at step S<b>110</b>, the new offset c is obtained as the sum of the old offset c₀and the position vector “g” that is obtained as a solution to the optimization problem for minimizing the objective function of Equation (43) without any constraint condition.

B. SECOND EMBODIMENT

0256Overview

0257In the first embodiment, the distribution of the data set of statistical population is estimated discretely and, when the distribution is two-dimensional, the new offset “c” is derived by setting, to zero, the component of the correction vector “f” in the principle axis direction in which the principle value is the minimum value and, when the distribution is one-dimensional, the new offset “c” is derived by setting, to zero, the components of the correction vector “f” in the two principle axis directions in which the principle values are the intermediate and minimum values. In the second embodiment, a description will be given of a simple, highly accurate algorithm that can eliminate the need to perform different processes according to estimations of the distribution as in the first embodiment and can also derive a more probable new offset by efficiently using the data set of statistical population.

0258<figref idref="DRAWINGS">FIG. 18</figref> is a flow chart illustrating a new offset derivation process. In the same manner as in the first embodiment, the CPU <b>40</b> performs the procedure of <figref idref="DRAWINGS">FIG. 18</figref> by executing the offset derivation module <b>94</b> when an offset update request has been made. The process of step S<b>200</b> is the same as that of the process of step S<b>100</b> described above in the first embodiment. The process of step S<b>202</b> is the same as that of the process of step S<b>102</b> described above in the first embodiment. The process of step S<b>204</b> is the same as that of the process of step S<b>104</b> described above in the first embodiment.

0259Estimation of Distribution

0260At step S<b>206</b>, distribution indicators of the data set of statistical population are derived. That is, the distribution of the data set of statistical population is estimated as continuous values by deriving, as distribution indicators, m₂and m₃defined by the following equations (57) and (58).

0261<maths id="MATH-US-00020" num="00020"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mrow><mn>1</mn><mo>-</mo><msup><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><msub><mi>k</mi><mn>2</mn></msub></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>57</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>3</mn></msub><mo>=</mo><mrow><mn>1</mn><mo>-</mo><msup><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>3</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><msub><mi>k</mi><mn>3</mn></msub></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>58</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0020.tif" />

0262Here, “k₂” and “k₃” are predetermined positive constant numbers. The values of k₂and k₃determine the association between principle values and the degrees of reliability of corresponding principle axis directions of the data set of statistical population. Here, “m₂” and “m<b>3</b>” must satisfy the following conditions (59). <br />0≦m₂&lt;1 and 0≦m₃≦1   (59)

0263The values of coefficients k₂and k₃are adequately determined in accordance with embodiments of the invention, since the effect of the weighting depends on the values of the coefficients k₂and k₃. In case that the magnetic sensor is mounted on a portable object such as a portable telephone and PDA which change their posture at relatively fast angular velocity, it is expected that a distribution of a magnetic data group accumulated in a predetermined time interval becomes relatively wide in average. In case that the distribution of the magnetic data is not so wide, an accuracy of the offset would be rather degraded if the offset correction is carried out by significantly weighting the magnetic data group of the principal axis direction having the small principal value since such a data group has low reliability. Therefore, in case that the invention is applied to a portable object where the distribution of the magnetic data group tends to become wide, the values of the coefficients k₂and k₃should be set so that the magnetic data group of the principal axis direction having the small principal value is evaluated with heavy weight only when the distribution of the magnetic data is considerably wide.

0264On the other hand, in case that the magnetic sensor is mounted on a moving object such as a vehicle which changes its posture in a relatively slow angular velocity, it is expected that a distribution of a magnetic data group accumulated in a predetermined time interval becomes relatively narrow in average. An accuracy of the offset would not be improved if the offset correction is carried out while the magnetic data group of the principal axis direction having the small principal value is not evaluated with a heavy weight although such a magnetic data group is not so reliable, when the distribution of the magnetic data is not so wide. Therefore, in case that the invention is applied to a moving object where the distribution of the magnetic data group tends to become narrow, the values of the coefficients k₂and k₃should be set so that the magnetic data group of the principal axis direction having the small principal value is evaluated with heavy weight even when the distribution of the magnetic data is narrow.

0265The spatial concept of m₂and m₃will now be described with reference to <figref idref="DRAWINGS">FIG. 11</figref>. When the coefficients of the components of the position vector g in the principle axis directions of the distribution are denoted by g_α, g_β, and g_γin decreasing order of the corresponding principle values and the coefficients of the components of the position vector f in the principle axis directions of the distribution are denoted by f_α, f_β, and f_γin decreasing order of the corresponding principle values, the relationships between the position vector g, the correction vector f, and m₂and m₃are expressed by the following equations (60), (61), and (62).

0266<maths id="MATH-US-00021" num="00021"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><msub><mi>f</mi><mi>α</mi></msub><msub><mi>g</mi><mi>α</mi></msub></mfrac><mo>=</mo><mn>1</mn></mrow></mtd><mtd><mrow><mo>(</mo><mn>60</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><msub><mi>f</mi><mi>β</mi></msub><msub><mi>g</mi><mi>β</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>⁢</mo><msub><mi>m</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow></mrow><mrow><mrow><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>⁢</mo><msub><mi>m</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>-</mo><msubsup><mi>m</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>61</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><msub><mi>f</mi><mi>y</mi></msub><msub><mi>g</mi><mi>y</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>3</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>⁢</mo><msub><mi>m</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>3</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow></mrow><mrow><mrow><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>3</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>⁢</mo><msub><mi>m</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><mo>(</mo><mfrac><msub><mi>λ</mi><mn>3</mn></msub><msub><mi>λ</mi><mn>1</mn></msub></mfrac><mo>)</mo></mrow><mo>-</mo><msubsup><mi>m</mi><mn>3</mn><mn>2</mn></msubsup></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>62</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0021.tif" />

0267The relationship equations determined such that the weighting factors continuously correspond to the ratios of principle values are not limited to Equations (60), (61), and (62). In addition, the weighting factor f_α/g_αassociated with the component of the principle axis direction corresponding to the maximum principle value may be set to be lower than “1”.

0268Derivation of New Offset

0269When it is difficult to derive a solution to the optimization problem under a specific constraint condition, a relaxation problem for solving the optimization problem by relaxing the constraint condition may be introduced. By applying this relaxation problem, this embodiment realizes a process for deriving a new offset c as the sum of the old offset c₀and a correction vector f that is obtained by weighting the coefficients g_α, g_β, and g_γof the position vector g (see <figref idref="DRAWINGS">FIG. 11</figref>) described above by weighting factors continuously corresponding to the ratios of principle values of the distribution of the data set of statistical population. The following are details of this process.

0270Unknown constant multipliers ρ₁and ρ₂are defined as variables required for calculations during the process and c, ρ₁, and ρ₂are grouped together into one vector “x” that is defined by the following equation (63).

0271<maths id="MATH-US-00022" num="00022"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>x</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>c</mi></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>63</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0022.tif" />

0272In addition, a matrix “B” is defined by Equation (64) and a vector “b” is defined by Equation (65).

0273<maths id="MATH-US-00023" num="00023"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mn>2</mn><mo>⁢</mo><mi>A</mi></mrow></mtd><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>⁢</mo><msub><mi>u</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>m</mi><mn>3</mn></msub><mo>⁢</mo><msub><mi>u</mi><mn>3</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup></mrow></mtd><mtd><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>⁢</mo><msub><mi>λ</mi><mn>1</mn></msub></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>3</mn></msub><mo>⁢</mo><msubsup><mi>u</mi><mn>3</mn><mi>T</mi></msubsup></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>⁢</mo><msub><mi>λ</mi><mn>1</mn></msub></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>m</mi><mn>3</mn></msub><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>64</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>b</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mn>2</mn><mo>⁢</mo><msup><mi>X</mi><mi>T</mi></msup><mo>⁢</mo><mi>j</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>u</mi><mn>2</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mn>3</mn></msub><mo>⁢</mo><msubsup><mi>u</mi><mn>3</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>65</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><img file="USRE44582E_D0023.tif" />

0274The process for deriving a new offset at step S<b>208</b> is to find a solution to the following simultaneous equations (36). The vector x is uniquely specified since the matrix B must be nonsingular. <br />Bx=b   (66)

0275Finding a solution to the simultaneous equations (66) is equivalent to solving the optimization problem for minimizing the objective function of Equation (43) under a constraint condition that a new offset be obtained as the sum of the old offset c₀and a correction vector f whose components are values obtained by weighting the coefficients of the position vector g in the principle axis directions of the distribution corresponding to the principle values by factors f_α/g_α, f_β/g_β, and f_γ/g_γcontinuously corresponding to the ratios of principle values of the distribution of the data set of statistical population.

0276In the second embodiment, it is easy to develop or improve the offset derivation module <b>94</b> and the data size of the offset derivation module <b>94</b> is also decreased since there is no need to branch the new offset derivation process according to the distribution of the data set of statistical population as described above. In addition, the second embodiment increases the use efficiency with which the data set of statistical population is used by the offset derivation module <b>94</b> and also allows the orientation derivation module to correct magnetic data using the most probable offset since the old offset can be corrected in the principle axis directions of the distribution by distances continuously corresponding to the ratios of principle values of the data set of statistical population unless any of the principle values is zero.

C. OTHER EMBODIMENTS

0277The second aspect of the present invention is not limited to the above embodiments and various embodiments are possible without departing from the spirit of the present invention. For example, the present invention can also be applied to a magnetic sensor that is mounted on a PDA or a magnetic sensor that is mounted on a vehicle.