The invention relates to a method and to a device for estimating an angular deviation between the magnetic axis of the magnetic moment of a magnetic object and a reference axis of said magnetic object.
The use of magnetic objects is known, particularly within the context of a system for recording the trace of a magnetic pencil on a writing medium. The magnetic object in this case is understood to be an object with which a non-zero magnetic moment is associated, for example, a permanent magnet attached to a non-magnetic pencil.
By way of an example, document WO 2014/053526 discloses a system for recording the trace of a pencil to which an annular permanent magnet is attached. The magnetic object, in this case the permanent magnet, comprises a magnetic material, for example a ferromagnetic or ferrimagnetic material, evenly distributed around a mechanical axis, called reference axis, which corresponds to its axis of rotation. The magnet is designed so that its magnetic moment is substantially co-linear to the reference axis.
The system for recording the trace of the pencil provided with the permanent magnet comprises an array of magnetometers capable of measuring the magnetic field generated by the permanent magnet. The magnetometers are attached to a writing medium.
However, the trace recording method assumes that the magnetic axis of the permanent magnet, which is defined as the axis passing through the magnetic moment, is effectively co-linear to the reference axis, or exhibits an acceptable angular deviation between the magnetic axis and the reference axis. Indeed, an angular deviation of several tens of degrees can result in an error in recording the trace, which can then exhibit a detrimental lack of precision. It then may be necessary for a prior estimate to be made of the angular deviation between the magnetic axis and the reference axis of the magnetic object.
The aim of the invention is to propose a method for estimating an angular deviation between a reference axis of a magnetic object and a magnetic axis co-linear to a magnetic moment of said magnetic object. To this end, the estimation method comprises the following steps:
Preferably, the estimation step comprises:
Preferably, during the identification sub-step, the minimum and maximum magnetic fields are respectively identified from the minimum and maximum values of the norm of the measurements of the magnetic field.
Preferably, said geometric parameters are the coordinates and the distance of the magnetometer relative to the magnetic object, in a plane passing through the reference axis and containing the magnetometer.
The angular deviation can be computed from a coefficient equal to the ratio of the norm of the vector formed from the subtraction of the minimum and maximum magnetic fields to the norm of the vector formed from the addition of the minimum and maximum magnetic fields, and from said geometric parameters.
The angular deviation can be computed from the following equation:
where d is the distance between the magnetometer and the magnetic object, z and r are the coordinates of the magnetometer relative to the magnetic object along an axis, respectively parallel and orthogonal to the reference axis, and where a is a predetermined coefficient.
During the rotation step, the magnetic object can complete at least one rotation about the reference axis.
Said at least one magnetometer comprises at least three axes for detecting the magnetic field, said detection axes being non-parallel to each other.
Preferably, said at least one magnetometer is a single tri-axis magnetometer.
Preferably, said at least one magnetometer is positioned outside the reference axis or outside the perpendicular to the reference axis passing through the magnetic object.
Said at least one magnetometer can be positioned relative to said magnetic object at a coordinate z along an axis parallel to the reference axis and at a coordinate r along an axis orthogonal to the reference axis, such that the coordinate z is greater than or equal to the coordinate r.
The invention also relates to a method for characterizing a magnetic object having an angular deviation between a reference axis of said magnetic object and a magnetic axis co-linear to a magnetic moment of said magnetic object, comprising the following steps:
Preferably, the amplitude of the magnetic moment is computed from the ratio between the norm of the maximum magnetic field and the norm of a magnetic field, for a unitary amplitude of said magnetic moment, expressed analytically using the following equation:
where
According to another embodiment, the estimation method comprises the following steps:
The step of estimating the mean angular deviation can comprise a sub-step of estimating an invariant vector rotating about the reference axis from the instantaneous magnetic moments, the estimation of the mean angular deviation also being performed from said invariant vector.
The estimation of the mean angular deviation can comprise computing an angular deviation amplitude between the instantaneous magnetic moments relative to the invariant vector.
During the step of measuring the magnetic field, the magnetic object can complete a whole number, greater than or equal to 1, of rotations about the reference axis.
The estimation method can comprise a step of homogenizing an angular distribution of the instantaneous magnetic moments about the reference axis throughout the measurement duration, with this homogenization step comprising a computation by interpolating a time series, called homogenized series, of instantaneous magnetic moments, from a time series, called initial series, of previously obtained instantaneous magnetic moments, so that the successive instantaneous magnetic moments of the homogenized time series exhibit a substantially constant angular deviation.
The homogenization step can comprise an iterative phase of computing interpolated magnetic moments, during which a magnetic moment interpolated at a considered iteration is obtained from a magnetic moment interpolated at a previous iteration, a predetermined threshold value and a unit vector defined from two successive instantaneous magnetic moments of the initial time series.
The positioning step can comprise:
The estimation method can comprise, prior to the positioning step, a step of measuring, using said magnetometers, an ambient magnetic field in the absence of the magnetic object, and can comprise a sub-step of subtracting the ambient magnetic field from the previously measured magnetic fields, so as to obtain the magnetic field generated by the magnetic object.
The estimation method can comprise a step of applying a low-pass filter to the values of components of the previously estimated instantaneous magnetic moments.
The low-pass filter can be a running mean on a given number of samples.
The number of samples can be predetermined so that a bias, which is defined as a difference between an angular deviation, called actual angular deviation, and the estimated mean angular deviation, is minimal, by absolute value, for a value called mean angular deviation threshold value.
The estimation method can comprise a step of sorting the magnetic object according to the difference in the estimated value of the mean angular deviation relative to a value called mean angular deviation threshold value.
The estimation method can comprise a step of estimating a parameter, called quality indicator, from the computation of a second dispersion parameter representing a dispersion of the values of a parameter, called instantaneous radius, computed as being the norm of an instantaneous vector defined between each instantaneous magnetic moment and the previously estimated invariant vector.
The estimation method can further comprise a step of comparing the second dispersion parameter to a predetermined threshold value.
Predetermining the threshold value can comprise the following sub-steps:
The invention also relates to a device for estimating a mean angular deviation between a magnetic axis and a reference axis of a magnetic object, comprising:
The invention also relates to an information recording medium, comprising instructions for implementing the estimation method according to any of the preceding features, said instructions being able to be executed by a processor.
Further aspects, aims, advantages and features of the invention will become more clearly apparent upon reading the following detailed description of preferred embodiments of the invention, which are provided by way of non-limiting examples, and with reference to the accompanying drawings, in which:
In the figure s and throughout the remainder of the description, the same reference signs represent identical or similar elements. Furthermore, the various elements are not shown to scale in order to enhance the clarity of the figure s. Moreover, the various embodiments and variations are not mutually exclusive and can be combined together. Unless otherwise stated, the terms “substantially”, “approximately”, “of the order of” mean to the nearest 10%, and preferably to the nearest 5%, even to the nearest 1%.
The invention relates to a device and to a method for estimating the mean angular deviation between the magnetic axis associated with the magnetic moment of a magnetic object and a mechanical reference axis of this object.
The magnetic object comprises a material having a magnetic moment that is spontaneous, for example. It preferably involves the material of a permanent magnet. The magnetic object can be a cylindrical, for example, annular, permanent magnet, as shown in the previously cited document WO 2014/053526, in which case the reference axis corresponds to an axis of symmetry of the magnet, for example the axis of rotation of the magnet. It can also involve a utensil or a pencil equipped with such a magnet or comprising a different permanent magnet, for example integrated into the body of the pencil, in which case the reference axis can correspond to the longitudinal axis along which the pencil extends, passing through the writing tip or lead of the pencil. The term pencil is to be understood in the broadest sense and can encompass pens, felt-tipped pens, paintbrushes or any other writing or drawing instrument.
The magnetic material is preferably ferrimagnetic or ferromagnetic. It has a non-zero spontaneous magnetic moment, even in the absence of an external magnetic field. It can have a coercive magnetic field that is greater than 100 A·m−1 or 500 A·m−1 and its intensity is preferably greater than 0.01 A·m2, even 0.1 A·m2. The permanent magnet is hereafter considered to be approximated by a magnetic dipole. The magnetic axis of the object is defined as being the axis co-linear to the magnetic moment of the object. The reference axis in this case corresponds to an axis of symmetry of the magnetic object. The angular difference between the reference axis and the magnetic axis is denoted angular deviation.
The estimation device 1 is capable of measuring the magnetic field at different measurement instants, throughout a measurement duration T, in a coordinate system (er, eθ, ez), and of estimating the value of the angular deviation
A direct three-dimensional orthogonal coordinate system (er, eθ, ez) is defined herein, where the axis ez is co-linear to the reference axis Aref and where the axes er and eθ are orthogonal to the axis ez.
The magnet 2 is intended to be positioned at the center of the coordinate system (er, eθ, ez), so that the position Po, of the magnet 2 has the coordinates (o, o, o) in this coordinate system. The position Po of the magnet 2 corresponds to the coordinates of the geometric center of the magnet 2, i.e. to the unweighted barycenter of all the points of the magnet 2. Thus, the magnetic moment m of the magnet 2 has the components (mr, mθ, mz) in the coordinate system (er, eθ, ez). Its norm, also called intensity or amplitude, is denoted ∥m| or m. The magnet is intended to be oriented so that the axis ez, corresponding to the axis of rotation of the magnet 2, coincides with the reference axis Aref thereof. Thus, when the magnet 2 will be set into rotation about its reference axis Aref, the magnetic moment m will rotate about the direction ez.
The estimation device 1 comprises a magnetic field measurement sensor with at least three distinct measurement axes d1, d2, d3 in pairs, i.e. the measurement axes are not parallel to each other, and can thus comprise at least one tri-axis magnetometer.
The tri-axis magnetometer M is placed at a defined position PM (r,θ,z) facing the position Po of the magnet and therefore the center of the coordinate system (er, eθ, ez). This position is known and constant throughout the entire measurement duration T. Knowing the position can allow for a certain tolerance margin, for example of the order of 10%, if a relative uncertainty of the order of 20% is accepted on the estimated value of the angular deviation, or less, for example of the order of 1% for a relative uncertainty of the order of 2% on the value of the angular deviation.
The magnetometer M therefore measures the amplitude and the direction of the magnetic field B disrupted by the magnet 2. More specifically, it measures the norm of the orthogonal projection of the magnetic field B along the measurement axes d1, d2, d3. The disrupted magnetic field B is understood to be the ambient magnetic field Ba, i.e. not disrupted by the magnet 2, to which the magnetic field Bd generated by the magnet 2 is added.
The estimation device 1 further comprises a computation unit 4 able to store the measured values of the magnetic field throughout the measurement duration and to determine the angular deviation
To this end, the magnetometer M is connected to the computation unit 4, in an electrical or other manner, using an information transmission bus (not shown). The computation unit 4 comprises a programmable processor 5 able to execute instructions stored on an information recording medium. It further comprises a memory 6 containing the instructions required to implement certain steps of a method for estimating the angular deviation
The computation unit 4 implements a mathematical model associating the measurements of the magnetometer M with the magnetic field and the position of the magnetometer M facing the magnet 2 in the coordinate system (er, eθ, ez). This mathematical model is constructed from equations of the electromagnetism, particularly the magnetostatic, and is particularly configure d by geometric parameters representing the position of the magnetometer M facing the magnet 2 in the coordinate system (er, eθ, ez).
In order to be able to approximate the permanent magnet 2 on a magnetic dipole, the distance between the permanent magnet 2 and the magnetometer M is greater than 2 times, 3 times, even 5 times, the largest dimension of the permanent magnet 2. This dimension can be less than 20 cm, even less than 10 cm, even less than 5 cm.
The estimation device 1 also comprises a retention and rotation component 7 capable of retaining the permanent magnet 2 relative to the magnetometer M in a known and constant position throughout the measurement duration.
It is also able to rotate the magnet 2 along its reference axis Aref during the measurement duration. Furthermore, during the measurement duration, the position of the permanent magnet 2 is fixed and only its orientation about the reference axis Aref varies over time. Thus, the reference axis Aref remains fixed in the coordinate system (er, eθ, ez). The retention and rotation component 7 comprises a motor 8 connected to an arm 9. The arm 9 thus can receive and retain the permanent magnet 2 and rotates said permanent magnet. The arm 9 is made of a non-magnetic material and the motor 8 is far enough away from the permanent magnet 2 and from the magnetometer M so as not to cause any disruption to the measured magnetic field. By way of a variation, the device 1 may not comprise a motor 8 and can be adapted so that the rotation of the arm 9 is performed manually.
The method comprises a step 100 of previously measuring the ambient magnetic field Ba, i.e. in this case the magnetic field not disrupted by the presence of the magnet 2. To this end, the magnetometer M is positioned at its measurement position PM and measures the magnetic field Bia in the absence of the magnet 2, i.e. the acquisition of the projection of the magnetic field Ba on each of the acquisition axes d1, d2, d3.
The method then comprises a step no of measuring the magnetic field B disrupted by the magnet 2 rotating about its reference axis Aref throughout a measurement duration T.
To this end, during a sub-step in, the permanent magnet 2 is positioned by the retention and rotation component 7 at the position Po facing the magnetometer M, such that the axis of rotation coincides with the reference axis Aref of the magnet 2. The magnetic moment m of the permanent magnet 2 is not co-linear to the reference axis Aref and forms, with respect to this axis, an angular deviation
During a sub-step 112, the permanent magnet 2 is rotated by the component 7 about the reference axis Aref. The reference axis Aref is static throughout the measurement duration T, in other words its position and its orientation in the coordinate system (er, eθ, ez) do not vary during the duration T. The rotation speed of the permanent magnet 2, denoted {dot over (θ)}, is preferably constant throughout the duration T.
Throughout the rotation, during the duration T=[t1; tN], a single tri-axis magnetometer M measures the magnetic field B disrupted by the presence of the permanent magnet 2, at a sampling frequency fe=N/T. N measurement instants tj are thus obtained. The magnetometer M, at the instant tj, measures the projection of the magnetic vector field B along the acquisition axes d1, d2, d3. A time series {B(tj)}N of N measurements of the disrupted magnetic field B is thus obtained.
The method then comprises a step 130 of estimating the magnetic angular deviation
Previously, during a sub-step 121, the computation unit 4 deduces the magnetic field Bd (tj) generated by the permanent magnet 2, at each instant tj, from the measurements of the ambient magnetic field Ba and of the disrupted magnetic field B(tj). To this end, Bd (tj)=B(tj)−Ba is computed. A time series {Bd (tj)}N is thus obtained of N measurements of the generated magnetic field Bd.
During a sub-step 131, the computation unit 4 identifies a magnetic field, called minimum magnetic field Bmind, and a magnetic field, called maximum magnetic field Bmaxd, from the measurements Bd (tj) of the magnetic field Bd. To this end, the norm ∥Bd (tj)∥ of each measurement Bd (tj) of the generated magnetic field Bd is computed, then the minimum field Bmind is identified as being that for which the norm is smallest, i.e. Bmind such that ∥Bmind∥=min({∥Bd (tj)∥}), and the maximum field Bmaxd is identified as being that for which the norm is largest, i.e. Bmaxd such that |Bmaxd∥=max ({∥Bd (tj)∥}). The minimum Bmind and maximum Bmaxd magnetic fields are thus obtained.
These sub-steps 121 of subtracting the ambient magnetic field Ba and 131 of identifying the minimum Bmind and maximum Bmaxd magnetic fields can be performed continuously, during the sub-step 112 of acquiring the measurements of the magnetic field B.
During the sub-steps 132 and 133, the computation unit then computes the angular deviation
As shown in
More specifically, the minimum field Bmind in is associated with the magnetic moment m when said moment belongs to the half-plane P′ of the plane P defined by the reference axis Aref and not containing the magnetometer PM (
The minimum Bmind in and maximum Bmaxd ax fields can be expressed analytically, within the context of the dipolar hypothesis. Indeed, the magnetic field Bd generated by the magnet 2, when it belongs to the plane P of the coordinate system (er, ep74 , ez), can be expressed using the following equation (1):
where the magnetic moment m=mu is expressed as the product of an amplitude m and of a directional unit vector u, and where the position vector PM-Po=dp of the magnetometer M in the plane (er, ez) is expressed as the product of a distance d and of a directional unit vector p, with the magnetic field in this case being expressed in microtesla.
From the equation (1), the minimum field Bmind, associated with the moment m contained in the half-plane P′ can be expressed analytically using the following equation (2):
where (r, z) are the coordinates PM of the magnetometer M in the plane P relative to the magnet 2, with r being the coordinate along an axis orthogonal to the reference axis Aref and z being the coordinate along an axis parallel to the reference axis Aref, and where d=√{square root over (r2+z2)} is the distance between the magnetometer and the magnetic object.
Similarly, from the equation (1), the maximum field Bmaxd associated with the moment m contained in the half-plane P″ can be expressed analytically using the following equation (3):
During a sub-step 132, based on equations (2) and (3), it is possible to compute a coefficient CV no longer dependent on the amplitude m of the magnetic moment, but only dependent on the angular deviation
This coefficient CV advantageously is a coefficient, called variation coefficient, which in this case is expressed as the ratio of the norm of the vector formed by the subtraction of the minimum field Bmind (α
It thus appears that the variation coefficient CV no longer depends on the amplitude m of the magnetic moment and that it is formed by the product of a first term that is only dependent on the angular deviation
Furthermore, the variation coefficient CV also can be computed from the previously identified minimum Bmind in and maximum Bmaxd magnetic fields, as expressed using the following relation:
Moreover, during a sub-step 133, the angular deviation
Thus, the estimation method according to this first embodiment allows the angular deviation
Preferably, the magnetometer M is positioned relative to the magnet 2 so that the coordinate z of the magnetometer in the plane P is greater than or substantially equal to the coordinate r. The coordinates r and z advantageously are substantially equal, thus allowing the precision of the estimation of the angular deviation
Furthermore, the magnetometer M is preferably located outside the reference axis Aref corresponding to the axis of rotation of the magnet 2, and outside the axis orthogonal to the reference axis Aref and passing through the position Po of the magnet 2. This arrangement allows the minimum Bmind and maximum Bmaxd magnetic fields to be clearly distinguished from each other. Furthermore, in order to obtain a good signal-to-noise ratio, the magnetometer is arranged facing the magnet 2 outside the deviation cone formed by the rotating magnetic moment m.
By way of a variation, a plurality of magnetometers Mi can be used to estimate the angular deviation
By way of a variation of the previously described analytical approach, it is possible to estimate the angular deviation from the fields Bmind in and Bmaxd identified during the sub-step 131, using a plurality of magnetometers that are distinct from each other in terms of positioning. Thus, if the magnetic sensor comprises a plurality of tri-axis magnetometers, the estimate of the angular deviation
where {circumflex over (α)} is the estimated value of the angular deviation, {circumflex over (m)} is the estimated amplitude of the magnetic moment, and where ri and zi are the coordinates of each magnetometer Mi. The term Wi in this case is a weighting term, for example dependent on the inverse of the noise associated with each magnetometer Mi. Similarly, this expression of minimization can be adapted to the case in which the magnetic sensor comprises a plurality of scalar magnetometers that measure the norm of the magnetic field. In this case, the square error is minimized between the norm of the magnetic fields. These examples are provided solely by way of an illustration and other approaches are possible.
Furthermore, preferably, the sampling frequency, the direction and/or the rotation speed are selected so as to improve the quality of the estimate of the angular deviation
Furthermore, within the context of a method for characterizing the magnetic object, it can be advantageous for the amplitude m of the magnetic moment m to also be determined, herein in the advantageous case where a single tri-axis magnetometer is used.
The amplitude m thus can be determined from the estimated angular deviation
More specifically, the amplitude m of the magnetic moment m can be computed from the ratio between the norm ∥Bmaxd∥ of the identified maximum magnetic field Bmaxd and the norm ∥{circumflex over (B)}maxd (
where
The amplitude m is thus computed from the following equation (8):
which then allows the magnet 2 to be characterized by the value of the amplitude m, on the one hand, and by the value of the angular deviation
By way of a variation, the amplitude m can be computed from the identified minimum magnetic field Bmind in and by its analytical expression of the equation (2). The use of the maximum magnetic field Bmaxd nevertheless allows better precision to be obtained.
The estimation device 1 is capable of estimating the instantaneous magnetic moment at different measurement instants throughout a measurement duration T, in an XYZ coordinate system. More specifically, the device 1 allows the position of the permanent magnet 2, and its magnetic moment, to be estimated at different instants, in the XYZ coordinate system. In other words, the device 1 allows the position and the orientation of the permanent magnet 2 to be located at different instants in the XYZ coordinate system.
In this case, and throughout the remainder of the description, a direct three-dimensional coordinate system (X, Y, Z) is defined where the axes X and Y form a plane parallel to the measurement plane of the array of magnetometers, and where the axis Z is oriented substantially orthogonal to the measurement plane. Throughout the remainder of the description, the terms “vertical” and “vertically” are understood to be relating to an orientation substantially parallel to the Z axis, and the terms “horizontal” and “horizontally” are understood to be relating to an orientation substantially parallel to the plane (X, Y). Furthermore, the terms “lower” and “upper” are understood to be relating to a growing position when moving away from the measurement plane in the direction +Z.
The position Pd of the permanent magnet 2 corresponds to the coordinates of the geometric center of the magnet 2. The geometric center is the unweighted barycenter of all the points of the permanent magnet 2. The magnetic moment m of the magnet 2 has the components (mx, my, mz) in the XYZ coordinate system. Its norm, or intensity, is denoted ∥m|.
The device 1 comprises an array of magnetometers Mi distributed facing each other so as to form a measurement plane Pmes. The number of magnetometers Mi can be, for example, greater than or equal to 2, preferably greater than or equal to 16, for example, equal to 32, particularly when tri-axis magnetometers are involved. The array of magnetometers nevertheless comprises at least 3 measurement axes that are remote from each other and are not parallel in pairs.
The magnetometers Mi are attached to a protection plate 3 and can be located on the rear face of the plate 3, which is produced from a non-magnetic material. The term attached is understood to mean that they are assembled on the plate 3 without any degree of freedom. In this case, they are aligned in rows and columns, but can be mutually positioned in a substantially random manner. The distances between each magnetometer and its neighbors are known and constant over time. For example, they can be between 1 cm and 4 cm.
The magnetometers Mi each have at least one measurement axis, for example three axes, denoted xi, yi, zi. Each magnetometer therefore measures the amplitude and the direction of the magnetic field B disrupted by the permanent magnet. More specifically, each magnetometer Mi measures the norm of the orthogonal projection of the magnetic field B along the axes xi, yi, zi of the magnetometer. The sensitivity of the magnetometers Mi can be 4.10−7 T. The term disrupted magnetic field B is understood to be the ambient magnetic field Ba, i.e. not disrupted by the magnet, to which the magnetic field Bd generated by the magnet is added.
The estimation device 1 further comprises a computation unit 4 able to compute the position and the orientation of the magnetic moment of the magnet 2 in the XYZ coordinate system from the measurements of the magnetometers Mi. It also allows the mean angular deviation of the permanent magnet 2 to be determined from the measurements of the magnetic moment.
To this end, each magnetometer Mi is electrically connected to the computation unit 4 using an information transmission bus (not shown). The computation unit 4 comprises a programmable processor 5 able to execute instructions stored on an information recording medium. It further comprises a memory 6 containing the instructions required to implement certain steps of a method for estimating the mean angular deviation using the processor 5. The memory 6 is also adapted to store the information computed at each measurement instant.
The computation unit 4 implements a mathematical model associating the position of the permanent magnet 2 in the XYZ coordinate system, as well as the orientation and the intensity of the magnetic moment, with the measurements of the magnetometers M. This mathematical model is constructed from the equations of the electromagnetism, particularly the magnetostatic, and is particularly configure d using the positions and orientations of the magnetometers in the XYZ coordinate system.
Preferably, in order to be able to approximate the permanent magnet 2 with a magnetic dipole, the distance between the permanent magnet 2 and each magnetometer Mi is greater than 2, and even 3 times the largest dimension of the permanent magnet 2. This dimension can be less than 20 cm, even less than 10 cm, even less than 5 cm.
The estimation device 1 also comprises a retention and rotation component 7 capable of retaining the permanent magnet 2 facing the measurement plane, in any position facing the measurement plane. The permanent magnet can be located above, i.e. facing, the array of magnetometers Mi, in a vertical position along the Z axis that is constant during the measurement duration T. It is also capable of rotating the magnet 2 along its reference axis Aref during the measurement duration. Furthermore, during the measurement duration, the vertical position of the permanent magnet 2 is fixed and only its angular position varies due to the rotation along the reference axis Aref. The retention and rotation component 7 comprises a motor 8 connected to an arm 9. The arm 9 thus can receive and retain the permanent magnet 2 and rotates said permanent magnet. The arm 9 is made of a non-magnetic material and the motor 8 is far enough away from the measurement plane Pmes and the permanent magnet 2 so as not to disrupt the measured magnetic field.
The method comprises a step 100 of previously measuring the ambient magnetic field Ba, i.e. in this case the magnetic field not disrupted by the presence of the magnet 2. To this end, each magnetometer Mi measures the magnetic field Bia in the absence of the magnet 2, i.e. the projection of the magnetic field Ba on each acquisition axis xi, yi, zi of the various magnetometers Mi.
The method then comprises a step 110 of measuring the magnetic field disrupted by the magnet 2 rotating about its reference axis Aref throughout a measurement duration T.
During a sub-step in, the permanent magnet 2 is positioned facing the measurement plane Pines using the retention and rotation component 7, which defines the orientation of the reference axis Aref of the permanent magnet 2 facing the measurement plane Pmes in the XYZ coordinate system. In this example, the permanent magnet 2 is positioned above the measurement plane Pines, but any other position is possible. The axis Aref can be oriented in any manner, but advantageously can be oriented substantially orthogonal to the measurement plane Pmes. The magnetic moment m of the permanent magnet 2 is not co-linear to the reference axis Aref and forms, with respect to this axis, an angular deviation α to be determined.
During a sub-step 112, the permanent magnet 2 is rotated by the component 7 about the reference axis Aref. The reference axis Aref is static throughout the measurement duration T, in other words its position and its orientation in the XYZ coordinate system do not vary during the duration T. The rotation speed of the permanent magnet 2, denoted {dot over (θ)}, is preferably constant throughout the duration T.
Throughout the rotation, during the duration T=[ti; tN], each magnetometer Mi measures the magnetic field Bi(tj) disrupted by the presence of the permanent magnet 2, at a sampling frequency fe=N/T. N measurement instants tj are thus obtained. Each magnetometer Mi at the instant measures the projection of the magnetic field B along the one or more acquisition axis/axes xi, yi, zi. A time series {Bi(tj)}N of N measurements of the disrupted magnetic field is thus obtained.
The method subsequently comprises a step 120 of estimating the instantaneous magnetic moment m(tj) of the permanent magnet 2, for each measurement instant tj, from the measurements of the disrupted magnetic field Bi(tj).
During a sub-step 121, the computation unit 4 deduces the magnetic field Bid(tj) generated by the permanent magnet 2, for each magnetometer Mi and at each instant tj, from measurements of the ambient magnetic field Bia and of the disrupted magnetic field Bi(tj). To this end, Bid(tj)=Bi(tj)−Bia is computed.
During a sub-step 122, the computation unit 4 estimates the position Pd of the permanent magnet 2, as well as its instantaneous magnetic moment m(tj), at each instant tj, from the previously computed magnetic field Bid(tj). To this end, the computation unit 4 addresses a mathematical model of equations of the electromagnetism associating the position and the magnetic moment of the permanent magnet 2 with the magnetic field that it generates Bid(tj)=f(Pd; m(tj)). Thus, at each measurement instant, the computation unit 4 determines the coordinates of the position of the permanent magnet 2, as well as the components mx, my, mz of the instantaneous magnetic moment, in the XYZ coordinate system. A time series {m(tj)}N of N instantaneous vectors of the magnetic moment m(tj) is thus obtained.
As shown in
The method then comprises a step 230 of estimating the mean angular deviation
During a sub-step 231, the computation unit estimates an invariant vector mo during the rotation of the permanent magnet 2. To this end, the time mean is determined, on the N measurement instants tj, for each coordinate mx(tj), my(tj), mz(tj) of the instantaneous magnetic moment m(tj) in the XYZ coordinate system. Thus, mo=(<mx(tj)>N; <my(tj)>N; <mz(tj)>N)=<m(tj)>N. The operator < >N in this case corresponds to the arithmetical time mean, which is optionally weighted, on the N measurement instants. A vector mo is thus obtained that is substantially co-linear to the reference axis Aref, which extends between the position Pa of the magnet 2 and the position Po of the center of the circle C.
During a sub-step 232, the computation unit estimates a parameter representing a mean amplitude of the angular difference of the instantaneous magnetic moments relative to the reference axis. In this case, this parameter is the mean radius
During a sub-step 233, the unit determines the mean angular deviation
Of course, other equivalent computations can be performed. Thus, by way of a variation, during the sub-step 233, the estimate of the mean angular deviation can be obtained from the arcsine of the ratio between the previously estimated mean radius
It is also possible, by way of a variation, to estimate, during the sub-step 232, the instantaneous radius R(tj) of the circle at each measurement instant tj and to compute, during the sub-step 233, the corresponding instantaneous angular deviation α(tj), then to determine the mean angular deviation
It is also possible, by way of a variation, to estimate, during the sub-step 232, the mean radius
By way of an example, the magnetic object is a permanent magnet with rotational symmetry about its reference axis Aref. It is positioned, for example, 4 cm on the vertical of an array of 32 tri-axis magnetometers with sensitivity of 4.10−7 T separated by 4 cm, for example. The rotation speed is π/5 rad/s (1 revolution in 10 seconds) and the sampling frequency is 140 Hz. During a 10 second duration T, the magnetic object completes a single rotation and the determination device acquires 1400 measurement instants. The computation unit estimates, at 0.165 A·m2, the mean intensity of the magnetic moment for an RMS (Root Mean Square) noise of 5.10−4 A·m2. The determination device estimates, at 0.01°, the mean angular deviation between the reference axis of the magnet and its magnetic axis.
Preferably, the measurement duration T and the rotation speed {dot over (θ)} are selected so that, during the duration T, the magnetic object has performed a whole number of complete revolutions, for example a single revolution. Furthermore, in order to obtain a substantially homogeneous angular distribution of the measurement data about the reference axis Aref throughout the duration T, the basic rotation Δθj between two measurement instants tj and tj+1 is substantially constant.
As now described, in the event that the measurement data does not exhibit substantially homogeneous angular distribution about the reference axis Aref during the duration T, an angular homogenization step can be implemented.
This step is advantageous when, as shown in
In this case, the vector mo, computed during the sub-step 231 described above, may not be rotationally invariant, i.e. may not be co-linear to the reference axis Aref. This error in the computation of the vector mo can introduce a bias on the estimated value of the mean angular deviation
With reference to
Then, during a sub-step 142 that is also optional, the estimate of the dispersion of the angular differences Δβj can be obtained by computing the standard deviation σΔβ, or an equivalent parameter, on the estimated angular differences Δβj. If the value of the standard deviation σΔβ is below a predetermined threshold, the measurement data exhibit a substantially homogeneous angular distribution. Otherwise, the angular distribution of the instantaneous magnetic moments about the reference axis Aref throughout the measurement duration T is homogenized.
During a sub-step 143, the initial time series {m(tj)}N is re-sampled by interpolation in order to obtain a time series, called homogenized series {mh(tk)}N′, of N′ interpolated magnetic moments, the angular differences Δβo of which are substantially constant. This involves over-sampling when N′>N, but the number N′ can be less than or equal to N.
To this end, an approach shown in
Thus, two magnetic moments m(tj+1) and m(tj) are considered that are measured at the successive instants tj and tj+1, the angular difference Δβj of which in this case has a value that is greater than the threshold Δβo. A unit vector ej,j+1=(m(tj+1)−m(tj))/|m(tj+1)−m(tj)| is determined. Then, one or more interpolated magnetic moments is/are computed such that: {tilde over (m)}j,k+1={tilde over (m)}j,k+Δβ0·ej,j+1, with {tilde over (m)}j,k=0=m(tj). The iteration on k continues as long as the angular deviation between m(tj+1) and {tilde over (m)}j,k+1 is greater than Δβo. By way of an illustration, the magnetic moments m(t1) and m(t2) are considered that are measured at the successive instants t1 and t2, for which the angular difference Δβ1 in this case exhibits a value that is greater than the threshold Δβo. The unit vector e1,2=(m(t2)−m(t1))/∥m(t2)−m(t1)∥ is determined. Then, one or more interpolated magnetic moments is/are computed such that: {tilde over (m)}1,k+1={tilde over (m)}1,k+Δβ0·e1,2, with {tilde over (m)}1,k=0=m(tj=1) and as long as the angular deviation between m(t2) and {tilde over (m)}i,k+i is greater than Δβ0. This sub-step 143 can be applied to all the pairs of magnetic moments m(tj) and m(tj+1) measured at successive instants, for which Δβj>Δβ0. By way of an example, in
Thus, a homogenized time series {mh}N′ of N′ instantaneous magnetic moments {mh}N′ is obtained corresponding to the N magnetic moments of the initial time series {m(tj)}N estimated during the sub-step 122 to which the interpolated magnetic moments {tilde over (m)} are added.
By way of a variation (not shown), a homogenized time series {mh}N′ can be constructed without having to incorporate, as previously, the initial time series {m(tj)}N therein. Thus, interpolated magnetic moments N′ can be computed such that: {tilde over (m)}k+1={tilde over (m)}k+Δβ0·ej,j+1, with {tilde over (m)}k=o=m(t1=1) and with the unit vector ej,j+i=(m(tj+1)−m(tj))/∥m(tj+1)−m(tj)∥. The iteration on k continues as long as the angular difference between {tilde over (m)}k+1 and m(tj+1) is greater than Δβ0, and as long as the product of the number k of iterations and of Δβ0 is less than one or several times 2π. Furthermore, the unit vector ej,j+1 is defined as j and j+1 so that the angular differences between m(tj) and {tilde over (m)}k+1, on the one hand, and between {tilde over (m)}k+1 and m(tj+1), on the other hand, are less than the angular deviation Δβj between m(tj) and m(tj+1). The homogenized time series {mh}N′ is thus obtained of N′ interpolated magnetic moments.
Prior to the sub-step 143 of computing the homogenized time series {mh}N′, it is possible for over-sampling of the initial time series to be performed, for example, {m(tj)}N, preferably by polynomial interpolation, optionally using splines, to thus obtain a new time series to be homogenized by means of the previously described sub-step 143.
Thus, the standard deviation computed on the angular deviations Δβ′ associated with the homogenized time series {mh}N′ is then minimal or substantially zero insofar as they are substantially equal to the value Δβo. The instantaneous magnetic moments mh(tk) then exhibit a substantially homogeneous angular distribution about the reference axis Aref throughout the measurement duration T.
The homogenized time series {mh}N′ corresponds to re-sampling of the measurement data at a new frequency fe′=N′/T, so that each instantaneous magnetic moment mh(tk) of the homogenized time series {mh}N′ can be viewed as an estimate of the magnetic moment at different measurement instants tk, with k={1, 2, . . . , N′}.
The method for estimating the mean angular deviation
As will now be described, it can be advantageous for a step 150 to be performed of low-pass filtering of the components of the instantaneous magnetic moments in the XYZ coordinate system, in order to reduce the time dispersion, in other words, the measurement noise, that they can exhibit, for example, using an arithmetical or exponential moving mean computed on a number K of samples. This is particularly advantageous when the mean angular deviation is less than 0.2° and/or when the noise associated with the components of the magnetic moments is the same order of magnitude as the value of the mean radius R of the circle C.
This dispersion of the values of the components m(tj) over time can cause a bias in the estimate of the mean angular deviation
In order to reduce the time dispersion of the values of the components m(tj), the step 150 of low-pass filtering using a running mean, or a moving mean, can be performed (
During an optional sub-step 151, the unit computes the standard deviation σ(∥m(tj)∥) on the N values of the norm ∥m(tj)∥ of the instantaneous magnetic moments, or an equivalent parameter representing the dispersion of the values of the norm ∥m(tj)∥ of the instantaneous magnetic moments. If the value of the standard deviation σ(∥m(tj)∥) is greater than a predetermined threshold, then the 3N values of components m(tj) are filtered.
During a sub-step 152, the unit applies a filter HK(tj) of the low-pass type, in this case by a running mean on K samples, for example an arithmetical or exponential mean, or an equivalent type of filter, on the time series of 3N values {m(tj)}N of the components of the instantaneous magnetic moments in the XYZ coordinate system. There is therefore a new time series such that {{circumflex over (m)}(tj)}=HK(tj). {m(tj)}, where {circumflex over (m)}(tj) represents the filtered values of the components of the instantaneous magnetic moment in the XYZ coordinate system, at the instant tj. Thus, a time series is obtained, called filtered series {{circumflex over (m)}(tj)}N, of instantaneous magnetic moments, which is then taken into account for executing the step 230 of estimating the mean angular deviation
Of course, this step 150 of low-pass filtering of the components of the instantaneous magnetic moments can be applied to the time series {m(tj)}N obtained during the sub-step 122, as in the homogenized time series {mh(tk)}N′.
Preferably, the number K of samples is selected so that the product of K with the mean angular difference <Δβj> of the considered time series, for example, the series {m(tj)}N originating from the sub-step 122, is less than a given value, for example, 45°, and preferably 10°, even 5°, and preferably 1°. Alternatively, the mean angular difference <Δβj> can be obtained by the rotation speed of the rotation component multiplied by the sampling frequency. This thus avoids applying excessive filtering that would restrict the considered time series, which would risk concealing the dynamics of the rotation signal.
It is advantageous for the homogenization step 140 to be performed before the step 231 of estimating the invariant vector mo. Furthermore, it is advantageous for the filtering step 150 to be performed before the step 232 of estimating the mean radius
As schematically shown in
When the method for estimating the mean angular deviation
The classification sub-step involves comparing the estimated value of the mean angular deviation
By way of an illustration, for a reference value of 0.5°, it can be said that the permanent magnet 2 is usable when its estimated angular deviation is less than or equal to the reference value αth and that it is to be rejected when it is greater than the reference value. Indeed, for an estimated value
Furthermore, the method for estimating the angular deviation can comprise a prior step 90 of vertically positioning the permanent magnet 2 facing the measurement plane Pmes, described with reference to
This positioning step comprises a sub-step 91 of estimating a parameter representing the dispersion of the values of the intensity of the instantaneous magnetic moment, for different vertical positions along the Z axis above the measurement plane Pmes.
To this end, the steps 110 and 120 are performed for different vertical positioning values of the object along the Z axis. For each vertical position, a time series of instantaneous magnetic moments is thus obtained, for example the time series {m(tj)}N obtained during the sub-step 122, and the standard deviation σ(|m(tj)∥) is deduced, for example, on the N values of the intensity ∥m(tj)| of the instantaneous magnetic moments. With reference to
During a sub-step 92, a vertical position value ZP is selected, for which value the value of the standard deviation σ(|m(tj)∥) on the intensity of the instantaneous magnetic moment is less than a threshold value σref. Then, the magnetic object is positioned so that it occupies the vertical position Zp.
The method for estimating the mean angular deviation
Furthermore, the method can comprise an additional step of estimating a quality indicator of the measurements performed during the measurement duration T. This step preferably can occur following the execution of step 120 or of step 150.
To this end, the computation unit 4 determines the standard deviation σR on the N values of the instantaneous radius that is estimated from the instantaneous magnetic moments m(tj). The instantaneous radius can be computed using the relation R(tj)=|m(tj)−mo∥. In this case, it is computed from the time series {m(tj)}N obtained during the sub-step 122. The quality indicator in this case is the magnitude σR/√N that allows the precision of the series of measurements to be characterized. Other quality indicators can be used from the standard deviation σR, for example a sin((σR/√N)/(<|m(tj)∥>N)) or a tan((σR/√N)/(∥m0∥)).
In order to detect a potential anomaly that has occurred during the series of measurements, the standard deviation σR on the N values of the instantaneous radius can be compared to a previously determined reference value σ(
Number | Date | Country | Kind |
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1656785 | Jul 2016 | FR | national |
1661693 | Nov 2016 | FR | national |
Filing Document | Filing Date | Country | Kind |
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PCT/FR2017/051870 | 7/10/2017 | WO |
Publishing Document | Publishing Date | Country | Kind |
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WO2018/011492 | 1/18/2018 | WO | A |
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Number | Date | Country | |
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20190301848 A1 | Oct 2019 | US |