JP2018028511A - Method and device for measuring dynamic sensitivity matrix of inertia sensor - Google Patents

Abstract

PROBLEM TO BE SOLVED: To measure a dynamic sensitivity matrix by obtaining a non-linear item up to a high-order item according to target accuracy necessary for an inertia sensor.SOLUTION: A method for measuring a dynamic sensitivity matrix is provided in which a vibration table is controlled by a vibration table control device, translational vibration or rotational vibration is applied to a multi-axis inertia sensor that is fixed to the vibration table, and a signal processing calculator measures a dynamic sensitivity matrix from an output value of the inertia sensor that is obtained by the application of the vibration. The signal processing calculator selects an excitation amplitude vector in which a McLaughlin power-transformed vector becomes linearly independent, applies vibration to the inertia sensor by the selected excitation amplitude vector, obtains a non-linear item up to a high-order item according to target accuracy necessary for the inertia sensor, and measures the dynamic sensitivity matrix.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a method and apparatus for measuring a dynamic sensitivity matrix in a multi-axis inertial sensor.

  The multi-axis inertial sensor is an acceleration sensor, a gyro, or an IMU (Inertial Measurement System) that combines both, and can be said to be a multi-axis vector quantity sensor. The acceleration measured by the acceleration sensor is a vector quantity, and the angular velocity measured by the gyro is also a vector quantity. In the case of an IMU, a vector quantity of up to six axes is measured. The IMU, also called a combo sensor, also measures the earth's magnetic field, so it may measure nine axes. In recent years, MEMS (Micro-Electro-Mechanical System) sensors manufactured by silicon fine semiconductor processing technology have been widely used as such multi-axis inertial sensors.

  MEMS inertial sensors are expected to be applied to automated driving technology for automobiles, used for vibration detection for social infrastructure maintenance, and used for monitoring exposure to human body vibration. In such applications, it is important to ensure the measurement accuracy of the inertial sensor. However, at present, only the uniaxial acceleration standard is supplied, and the angular velocity standard is not supplied.

  In calibration of inertial sensor sensitivity, a uniaxial shaking table is used as a measurement device, and the direction in which vibration is applied to the inertial sensor coincides with the sensitivity axis of the inertial sensor, so acceleration is not considered a vector, and sensitivity is expressed as a complex number. It is only done. Therefore, there is a problem that even if the input acceleration (angular velocity) is measured by a laser interferometer, the nonlinearity cannot be measured because it approximates a sine wave. In addition, for an IMU that also measures the earth's magnetic field, the magnetic field leaking from the shaking table that generates acceleration and angular velocity is several orders of magnitude larger than the earth's magnetic field.

  The output of the inertial sensor is a signal in units of voltage, and the acceleration is expressed as a set of vectors obtained by converting this signal. The mathematical function of the inertial sensor is to project the real vector space to the signal vector space, and assuming linearity, only a matrix can convert a vector to a vector. Conventionally, only static linearity is taken into consideration, dynamic linearity is not taken into account, and only stationary measurement is performed. Therefore, Patent Documents 1 and 2 disclose a technique for representing dynamic sensitivity assuming linearity in a matrix.

Japanese Patent No. 485937 Japanese Patent No. 4924933

  However, the projection from the real vector space to the signal vector space is not always linear, and an actual inertial sensor has nonlinearity. For example, in some high-precision MEMS inertial sensors, the coefficient when static sensitivity is approximated by a fifth-order polynomial is expressed. The method of was not established. Therefore, there is a problem that not only dynamic linearity is considered but also a method for calibrating inertial sensors including nonlinear terms has not been established.

  The object of the present invention is to analyze the non-linear term up to the higher order term. As a result, the n-th order (n ≧ 2) sensitivities are collected in a matrix, and the higher-order order according to the target measurement accuracy required by the inertial sensor. It is to provide a method and apparatus for indicating and measuring that it is necessary to obtain a term.

  In an embodiment of the present invention, in order to achieve such an object, in one embodiment, a vibration table is controlled by a vibration table controller, and translational vibration or rotational vibration is applied to a multi-axis inertial sensor fixed to the vibration table. In the method in which the signal processing computer measures the dynamic sensitivity matrix from the output value of the inertial sensor obtained by applying the vibration, the signal processing computer expands the vector of the output value in a macrolin manner, The step of expanding to the Fourier series and the term having the same power of exp (jωt) expanded to the Fourier series are expressed as the product of the coefficient matrix and the excitation amplitude vector, respectively, and the excitation amplitude vector of the first order term is expressed as follows: A step of performing macro-Linning conversion into an excitation amplitude vector of a second-order or higher term, and an excitation amplitude vector in which the vector subjected to macro-Linning conversion is linearly independent are selected. By applying vibration to the inertial sensor according to the step and the selected excitation amplitude vector, the dynamic sensitivity matrix is obtained by obtaining nonlinear terms up to higher-order terms according to the target accuracy required by the inertial sensor. And a step of measuring.

  According to the present invention, not only dynamic linearity is taken into account, but also the non-linear term is obtained up to a higher-order term according to the target accuracy required by the inertial sensor, thereby dramatically improving the measurement accuracy of the inertial sensor. It becomes possible to improve.

It is a figure which shows the structure of the measuring device concerning one Embodiment of this invention. It is a block diagram which shows the triaxial 6 degrees of freedom required for the vibration stand of a measuring device. It is a flowchart which shows the measuring method concerning one Embodiment of this invention.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. First, an analysis method will be described, and then a specific measurement apparatus and measurement method will be described.

(Linear matrix sensitivity)
In the case of the IMU, the degree of freedom is 6 axes at the maximum, but the size of the matrix is large, which is inconvenient as compared with the size of the paper surface. The distribution in which acceleration is measured for all three axes or only one axis is measured and angular velocity is measured for the remaining two axes is not a problem in the following description.

The input to the inertial sensor is defined by (1-1), and an exponential function is introduced to display a complex number, and the input vector is defined by (1-2). The output of the inertial sensor is not written according to the input definition, but is written in a general way. Since it is reasonable to think that the functions fx, fy, and fz representing the output of the sensor are continuous and can be differentiated infinitely, when the equation (2) is expanded to Macrolin, the following equation is obtained.

If you write down,

It becomes. In the calculation of equation (4), it is assumed that partial differentiation can be replaced. The fourth term of the equation (4) is a linear region, and the third and subsequent terms are nonlinear regions. The reason is that, in the linear region, only the constant term and the first-order term of the input appear, whereas the third and subsequent terms show the second-order and higher-order terms of the input.

For y _out (t) and z _out (t), y _out (t) or z _out (t) can be obtained by replacing fx in the equation (4) with fy and fz. In the third and subsequent terms of equation (4), x = y = z = 0 in the sense of a value at the origin is omitted. When the second expression of the expression (1), which is an input vector, is substituted into the expression (4), the first [] can be expressed by exp (jωt), and the second [] can be expressed by exp (j2ωt). And the third [] can be wrapped with exp (j3ωt). Similarly, higher order terms are possible. Since frequency characteristics are basically considered, it is considered that x _out (t), y _out (t), and z _out (t) can be expanded into a Fourier series.

Therefore, when the terms with the same power of exp (jωt) in the equations (4) and (5) are compared, the following three equations are derived.

Expression (6) is an expression obtained as a result of calculation for the function fx related to the X axis. When the same calculation is performed for the function fy for the Y axis, the equation (9) is obtained, and when the calculation is performed for the function fz for the Z axis, the equation (10) is obtained.

In the expressions (6), (9), and (10), all the expressions of the first-order terms are obtained. Therefore, the following replacement is performed.

In Table 1, the left side of the subscript S is the output axis, and the right side is the input axis. Therefore, when the expressions (6), (9), and (10) are rewritten using the symbols in Table 1, the expression (11) is obtained. The value at the origin is not shown in the subscript.

The unknowns to be obtained here are Sxx, Sxy, Sxz, Syx, Syy, Syz, Szz, Szy, Szz. The coefficient matrix of the equation (11) is represented as S ₁ ³ .

Is called the excitation amplitude vector. The vector on the right side of equation (11) is a complex vector. Equation (11) represents one excitation. Since there are three equations and nine unknowns derived by one excitation, the number of equations and the number of unknowns match if the excitation is performed three times, so that the possibility of deriving a solution arises. Therefore, the symbols are defined as follows.

When the input / output relationship of the first excitation in Table 2 is substituted into the equation (11), the equation (11-1) is obtained, and when the input / output relationship of the second excitation is substituted into the equation (11) (11− Equation (2) is obtained, and when the third input / output relationship is substituted into Equation (11), Equation (11-3) is obtained.

When the equations (11-1), (11-2), and (11-3) are arranged with respect to the unknowns Sxx, Sxy, Sxz, Six, Syy, Syz, Szz, Szy, Szz, the following simultaneous linear equations are obtained. It is done.

In order for Equation (12) to be solved, the determinant of the coefficient matrix must not be zero. The coefficient determinant is calculated in consideration of the regular arrangement.

In the first line exchange of the equation (13), the second line and the fourth line are exchanged, and the third line and the seventh line are exchanged. In the next line exchange, the sixth and eighth lines are exchanged. The fact that the cube of the final determinant must not be zero means that the excitation amplitude vector of the input acceleration may be linearly independent. If vibration is performed with a linearly independent excitation amplitude vector, the linear matrix sensitivity corresponding to the first order term is uniquely determined.

(Macroline conversion)
Expression (7) is an expression obtained as a result of calculation regarding the function fx related to the X axis. If the same calculation is performed for the function fy for the Y axis, the equation (14) is obtained, and if the same calculation is performed for the function fz for the Z axis, the equation (15) is obtained.

  Here, symbols are newly defined as shown in Table 3.

Therefore, when Expression (7), Expression (14), and Expression (15) are put together, Expression (16) is obtained.

In equation (16), there are 18 unknowns S _{*, *} , and there are only 3 equations derived by one excitation. Therefore, in order to match the number of equations with the number of unknowns, 6 excitations are required. Have to do. The coefficient matrix of the equation (16) is represented by S ₂ ³ .

  To simplify the notation, the excitation amplitude vector

The conversion to be converted will be referred to as “Macrolyn 冪 conversion”. The reason for using macrolin is that the macrolin development is the starting point for discussion. The meaning of “冪” is that the 冪 of the vector component to be converted becomes an element of the converted vector. There is a condition that the order in which the components are arranged in the converted vector must correspond to the order in which the secondary partial differential coefficients are arranged in the Macrolin series expansion. At this time, the transformation is expressed as M ₂ ^{3 in the} sense that it is a second-order Macrolin- 冪 transformation for a three-component vector. The definition corresponding to Table 2 is Table 4.

Substituting this input / output relationship into equation (16) and organizing unknowns yields equation (17).

In the equation (17), since the Macrolin- 冪 transformation vector of the excitation amplitude vector has 6 components, the coefficient matrix is an 18 × 18 square matrix. 0 ₆ is a zero vector in which six zeros are arranged. Swap 2 and 4 rows, swap 3 and 7 rows, swap 12 and 16 rows, swap 15 and 17 rows. The coefficient matrix is

Next, the 4th line and the 10th line are exchanged, the 5th line and the 13th line are exchanged, and the 6th line and the 12th line are exchanged.

Further, the 10th and 7th lines are exchanged, the 13th and 8th lines are exchanged, the 11th and 14th lines are exchanged, and the 12th and 15th lines are exchanged.

Further, the 9th line and the 13th line are exchanged, and the 10th line and the 14th line are exchanged.

Finally, the 13th and 15th lines are exchanged, and the 13th and 14th lines obtained are exchanged.

Therefore, if the Macrolin- で あ れ ば transformation vectors of the six excitation amplitude vectors are linearly independent, the solution can be obtained. That is, it suffices to find six such three-component vectors. These six vectors need not be linearly independent.

  Further, as represented by the equations (13-3) and (17), the coefficient determinant is a square matrix. Next, it will be described that the equations (13) and (17) generally become simultaneous linear equations having a square matrix.

It is in the form of In order to be able to calculate the determinant and derive the solution condition, it is necessary to prove that the matrix A is always a square matrix. The matrix X has a form in which the components of the sensitivity matrix to be obtained are arranged in a vertical vector as unknowns. In the case of a first-order (linear) term, it is obvious that the matrix A is a square matrix. Here, it is shown that the matrix A is a square matrix even when a higher-order term is obtained.

Assume that the number of axes is n and an m-th order sensitivity matrix is obtained. The number of components in M _m ⁿ converted vector placing and k. Since the number of nonlinear matrix sensitivities per axis is also k, the number of all elements of the unknown vector is kn. As can be seen from the equation (17), the coefficient matrix has kn in the number of rows of the coefficient matrix because the matrix-transformation vector of the excitation amplitude vector is arranged stepwise.

  On the other hand, the number of row vectors of the coefficient matrix is equal to the number of components of vector B. Since the number of components of the vector B means the total number of equations, it is equal to the total number of unknowns and at the same time equal to the number of column vectors of the coefficient matrix. From the above, since the number of row vectors and the number of column vectors of the coefficient matrix are equal, the matrix is a square matrix.

  Table 5 shows the macrolin conversion of the quadratic term of the triaxial inertial sensor.

The primary independence is investigated.

Accordingly, since the Macrolin- 冪 conversion vector of the excitation amplitude vector in Table 5 is linearly independent, it can be solved. In the case of the first-order term, since it was necessary and sufficient that the excitation amplitude vector itself is linearly independent, the Macrolin- 冪 transform in the linear region is a unit matrix. From this fact, when formula (4) is generalized by mapping, it can be expressed by the following formula.

(21) in the formula

Is a linear combination representation of a linear map corresponding to the linear region and a nonlinear map corresponding to the nonlinear region. (21)

Alternatively, the DC component vector in equation (5) can be obtained by calculating the time average of the output signal.

  Table 6 describes the number of axes of the inertial sensor and the number of sensitivities from the primary term (linear term) to the nonlinear term on each axis.

In the case of a 6-axis inertial sensor, the description of the linear term has 6 sensitivities per axis, which is only 36 times the total number of axes, but 192 sensitivities appear in the fifth-order nonlinear term. The overall sensitivity is 6 times this, and 1152 sensitivities appear. The number of terms that need to be determined depends on the strength of nonlinearity and the required accuracy of measurement based on the sensor operating principle. The numerical value indicating the number of nonlinear sensitivities per axis is the number of components of the vector that has been subjected to the macrolin conversion, and the vector that has been subjected to the macrolin conversion must be linearly independent by that numerical value.

  Considering that any number of zero powers is 1, it is reasonable to define the 0th order Macrolin- 冪 transformation vector as shown in the equations (22) to (26) regardless of the values of the components. Is.

If the zero-order Macrolin- 冪 transform is defined as described above, the mathematical function of mapping of the inertial sensor can be expressed succinctly when the bias term in equation (21) is defined not by a vector but by a matrix. Become. The bias sensitivity is defined as follows.

If (bx, by, bz) is written, it is not written in the vector, but if it is defined by a matrix like the expression (28), the expression (32) is derived.

Expression (32) represents a mapping for converting a dynamic input into a dynamic output in both the linear region and the nonlinear region. In the case of 2 axes, the case of 4 axes, the case of 5 axes, and the case of 6 axes are also described below.

(Configuration of measuring device)
FIG. 1 shows a configuration of a measuring apparatus according to an embodiment of the present invention. The measurement device includes a measurement device main body incorporated in a housing 18 mounted on the vibration isolation device 8 and a control device 19. The measurement apparatus main body is provided with a vibration table 6 on which an inertial sensor 4 to be measured is installed, and the vibration table 6 is supported by six conductive actuators 1 to 3. The actuators 1 to 3 and the vibration table 6 are coupled by a link mechanism 7 (partially omitted) via a spherical bearing. The link mechanism 7 is designed so as not to mechanically restrain the vibration table 6.

  Each of the X-axis actuators 2a and 2b and each of the Y-axis actuators 3a and 3b are installed to face each other, and the center axis of the exciting rod is offset. For example, when the X-axis actuators 2a and 2b are operated by push-pull, translational vibration is applied to the vibration table 6, and when the X-axis actuators 2a and 2b are operated by push-push or pull-pull, rotational vibration is applied to the vibration table 6. Become. The Z-axis actuators 1 a and 1 b are also arranged with the center axis of the excitation rod offset, so that translational vibration and rotational vibration can be applied to the vibration table 6.

  With such a configuration, the vibration table 6 can vibrate in the direction of three axes and six degrees of freedom in the coordinate system shown in FIG. 2, and acceleration and angular velocity can be applied to the inertial sensor 4. The shaking table 6 firstly has no restriction on the degree of freedom of movement, that is, it can simultaneously generate movements up to six degrees of freedom, and secondly, it moves the origin of the coordinate system shown in FIG. It is required that the origin can be moved in the X-axis direction, the Y-axis direction, and the Z-axis direction in rotational vibration around the center or translational vibration in any direction. The reason why this function is necessary is that the origin of coordinates and the center of inertia need to coincide with the IMU to be measured. Further, for example, when a vibrating table is vibrated with a sine wave in the Z-axis direction, irregular vibration may occur in the X-axis direction or Y-axis direction corresponding to the vertical direction, which is called crosstalk. The crosstalk of the vibration table 6 is desirably 1/1000 or less.

  The control system including the control device 19 includes a digital computer in a feedback loop, has a function of measuring the motion of the shaking table 6 during excitation, and comparing the actual motion with the target value of the motion. Feedback control is performed based on the error between the motion of the robot and the target value of the motion. The movement of the shaking table 6 is measured using an optical element 5 such as a mirror and an optical measuring device 10 installed on the shaking table 6, or a control inertia sensor 9 and a signal conditioner 12 installed on the shaking table 6. And use. The outputs of the optical measuring device 10 and the signal conditioner 12 are input to the signal processing computer 15 via the transient signal recording device 13.

  A control inertial sensor 9 in which matrix sensitivity is defined is placed on the vibration table 6, and the output of the inertial sensor 4 is input to the signal processing computer 15 via the signal conditioner 11 and the transient signal recording device 13. To do. The shaking table control device 16 compares the control reference signal 14 (target value of motion) with the output of the signal processing computer 15 (actual motion), calculates a control value for each actuator, and a current amplifying device 17. The actuators are controlled via the.

  In FIG. 1, illustrations of auxiliary devices such as a cooling device and a temperature control device are omitted. Since the vibration of the vibration table 6 depends on the state of the measuring device body such as the temperature of the lubricating oil, it is desirable to control it within a predetermined measurement temperature range. In addition, in the feedback control described above, it is conceivable to implement an algorithm for calculating optimum control parameters taking into account environmental conditions such as temperature.

  In addition, it is desirable that the area of the vibration table 6 is large from the viewpoint of increasing the efficiency of measurement work of the inertial sensor. By synchronous operation with a robot apparatus that loads and unloads a tray in which a plurality of inertial sensors are set on the vibration table 6, the cost for measurement can be reduced, and the time required for measurement can also be reduced. .

(Measurement method and control algorithm)
As described in the principle of the analysis method above, the necessary and sufficient condition for obtaining the nonlinear term is to obtain the actual excitation amplitude vector corresponding to the necessary number of first-order independent Macrolin-transformed vectors. .

  The following method can be considered as a method for finding the required number of excitation amplitude vectors that are linearly independent when the Macrolin- 冪 conversion is performed. First, it is only necessary to calculate a determinant to determine whether or not the Macrolin- 冪 transformation vector of the excitation amplitude vector is linearly independent. This fact is clear from the fact that the coefficient matrix of the equations (13-3) and (17) is a square matrix.

  Next, a method for finding the necessary number of excitation amplitude vectors will be described. At present, it is not possible to a priorily derive the excitation amplitude vector that satisfies the conditions, so an algorithm that finds it by trial and error on a computer is described. The target is a six-axis inertial sensor. The three coordinates of translational acceleration are X, Y, and Z, and the three coordinate variables of angular velocity are P, Q, and R. Assume that the dynamic range of acceleration and angular velocity generated by the measuring device is 1000 times. When the excitation amplitude vector (X, Y, Z, P, Q, R) is written, when calculating the fifth-order nonlinear term, 192 vectors of 6 components are found as shown in Table 7. Is required.

  FIG. 3 shows a measurement method according to an embodiment of the present invention. The first method is a calculation method using a loop. In this method, 192 loops appear in 6 layers. This is because, as shown in Table 5, the fifth-order Macrolin transform of a six-component vector has 192 components. In the case of 6 axes, the number of axes is 126, but the number of axes is the same, so the calculation loop does not change with 6 layers. In the case of 4th order and 5th order, as can be seen from Table 5, 56 loops become a quadruple calculation loop. This calculation may be performed in the signal processing computer 15 of the measuring device, an external computer may be used, or cluster processing may be performed by a plurality of computers.

  The second method is a calculation method using random numbers. If the dynamic range is 500, a random number of [01] is generated for each component of the 192 six-component excitation amplitude vectors, multiplied by 500, and the excitation amplitude is 3 digits and 4 digits. Arrange numerical values according to the vector generator. When the arrangement is completed, a Macrolin-Long transformation vector corresponding to each excitation amplitude vector is calculated, a 192 × 192 square matrix is obtained, and a determinant is calculated. If the determinant is not zero, 192 excitation amplitude vectors are obtained. When the determinant becomes zero, it is not linearly independent, so the random number calculation is continued and the determinant calculation is continued.

  According to the present embodiment, it is possible to detect nonlinear resonance resulting from the mechanical structure of a multi-axis inertial sensor. The dynamic sensitivity matrix can be applied not only to consider dynamic linearity but also to quantify non-linearity, so the dynamic model of the inertial sensor becomes more elaborate, so the nonlinearity from the sensor output Since the input signal can be estimated in consideration of the above, the measurement accuracy of the inertial sensor can be dramatically improved.

DESCRIPTION OF SYMBOLS 1 Z-axis actuator 2 X-axis actuator 3 Y-axis actuator 4 Inertial sensor 5 Optical element 6 Shaking table 7 Link mechanism 8 Vibration isolator 9 Control inertia sensor 10 Optical measuring device 11, 12 Signal conditioner 13 Transient signal recording device 14 Control Reference signal 15 Signal processing computer 16 Shaking table control device 17 Current amplification device 18 Case 19 Control device

Claims (3)

The shaking table is controlled by a shaking table control device, and translational vibration or rotational vibration is applied to a multi-axis inertial sensor fixed to the shaking table, and the signal processing computer obtains the inertial sensor obtained by applying the vibration. In the method of measuring the dynamic sensitivity matrix from the output value of the signal processing computer, the signal processing computer,
Expanding the vector of the output values into a Maclaurin expansion and expanding it into a Fourier series;
The terms with the same power of exp (jωt) expanded to the Fourier series are expressed as the product of the coefficient matrix and the excitation amplitude vector, respectively. A macrolin conversion to an amplitude vector;
Selecting an excitation amplitude vector in which the Macrolin-transformed vector is linearly independent;
A step of measuring a dynamic sensitivity matrix by applying vibration to the inertial sensor according to a selected excitation amplitude vector and obtaining a nonlinear term up to a high-order term according to a target accuracy required by the inertial sensor. A method of measuring a dynamic sensitivity matrix characterized by performing and. In a device for measuring the dynamic sensitivity matrix of a multi-axis inertial sensor,
A vibration table controller for applying translational vibration or rotational vibration to an inertial sensor fixed to the vibration table;
A signal processing computer for measuring a dynamic sensitivity matrix from an output value of the inertial sensor obtained by applying vibration,
Expand the vector of the output value of the inertial sensor obtained by applying the vibration to Macrolin's expansion, expand it to Fourier series,
The terms with the same power of exp (jωt) expanded to the Fourier series are expressed as the product of the coefficient matrix and the excitation amplitude vector, respectively. Maclaurin conversion to amplitude vector
Select the excitation amplitude vector that makes the Macrolin-transformed vector linearly independent,
A signal for measuring the dynamic sensitivity matrix by applying vibration to the inertial sensor according to the selected excitation amplitude vector and obtaining a nonlinear term up to a higher order term according to the target accuracy required by the inertial sensor. An apparatus for measuring a dynamic sensitivity matrix characterized by comprising a processing computer.   The apparatus for measuring a dynamic sensitivity matrix according to claim 2, wherein the shaking table applies a vibration of six degrees of freedom to the inertial sensor.

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