JPS63142215A - Multivariable polynomial type measured value calibrator - Google Patents

Abstract

PURPOSE:To simply calibrate the measured value based on a polynomial, by a method wherein data affected by a large number of parameters and having non-linear relation but showing smooth change is expressed by the polynomial of multivaliables and coefficient easily obtained is used. CONSTITUTION:The calibrator consists of an operation element X performing multiplication and an operation element + performing addition, and performs the operation based on a predetermined formula. In this case, coefficients C10-C19 for the calibration of data are ones calculated by the generalized method of a general inverse matrix calibration method and, therefore, the measured value subjected to calibration as the output of this circuit can be obtained.

Description

DETAILED DESCRIPTION OF THE INVENTION [Industrial Field] The present invention relates to a measurement value calibrator for calibrating multivariate polynomial type measurements.

[Prior Art] It is desirable that the output of various measuring instruments has a one-to-one correspondence with the quantity to be measured, and that they are connected in a linear relationship. However, when such a relationship is not realized, There are more.

In that case, the measured values are calibrated to match the initial purpose. For this purpose, y=ax+b...(0) is -σXc b=y-2x (n: number of data, X + V : xi', yl
(average value of ) (Masatsugu Ishida, Fundamentals of Data Analysis).

Morikita Publishing P35).

Similarly, y)=00+C,x,+C2x',+・-+C,x7
* * It is known that when expressed as an m-th degree polynomial such as (1) (i tsu 1, 2...n), it is sufficient to solve the following equation (P92 of the same book).

However, a calibration method when a measured variable is influenced by a large number of measured values is generally unknown.

Furthermore, it is assumed that the generalization of equation (1) is complex in expression and calculation is not easy.

On the other hand, in matrix theory, for the relationship expressed as equation (1), the solution method mainly when n < m is known as the general inverse matrix, and when the number of parameters is greater than the number of data, the maximum (Souvenir Shigeru general inverse matrix, Mathematical Science 5-5.68/72.1973
).

In the theory of general inverse matrices, the case where n > m has not been sufficiently investigated, and similarly the case where it is expressed in multiple variables has not been sufficiently investigated.

[Problems to be Solved by the Invention] -゛1 The purpose of the present invention is to make it possible to easily calibrate measured values based on the above polynomial using easily obtained coefficients that are influenced by a large number of parameters. It is in.

[Means for Solving the Problems] A measured value calibrator of the present invention for achieving the above object includes an arithmetic device that performs the arithmetic operation of the multivariate polynomial described above, and a coefficient for data calibration in the arithmetic device. The method is characterized in that coefficients calculated by a generalized method of calculating a general inverse matrix are set, and measured values are calibrated using these coefficients.

[Operation and Effect] Since coefficients calculated by a generalized method of calculating a general inverse matrix are used as coefficients for data calibration in the arithmetic device, coefficients of multivariate polynomials can be easily obtained. With a very simple circuit configuration using those coefficients,
Easily calibrate measured values “+-Cto+C++”t+C+z
ff++C+3"t"z"+Ft+C+sYt"CH6
X FCI 7! stomach! i◆018"i' 〇,,yj
+--bi=C20"21"i+C2241+C23
"i"24"1ffi+C25!i+C26"i+02
7”1ffi+Cz8”iYi+C29F”...(2
), but when expressed as a matrix, it becomes as follows.

C: l×2.

! L: Number of parameter terms)...(4) It can be expressed as

Multiplying both sides of equation (4) by X'' (T: indicates the transposed matrix) and taking the inverse matrix, C= (XT X)-1XT A,
−(5).

Here, when the number of data n is larger than the prose of the parameter, there generally exists an inverse tj column of XtX (sentence×column). If sufficient data cannot be collected for some reason, the inverse matrix does not exist, but in that case, the general inverse matrix (XT
T) 4″ (the above-mentioned reference) may be used.

In this way, using the coefficient C obtained from equation (5), it is already known that equation (2) can be used for any order and arbitrary variable (Tsunehiro Takeda, Three-dimensional position measurement Center of gravity measurement using device and original reaction force meter, Product Science Research Institute Research Report No.
94.37/48), and since the general inverse matrix becomes the maximum likelihood estimate that minimizes the sum of squared errors, equation (5) can be expected to generally represent a good data calibration coefficient. This will be illustrated by the experimental examples described below.

Note that the number and order of variables are determined by other a priori knowledge.

This data calibrator can be realized by the circuit configuration shown in FIG. The figure is a circuit diagram of a measured value calibrator using a two-variable quadratic polynomial, in which various calculation elements are arranged to perform the calculation of equation (2). It shows an arithmetic element, and + also shows an arithmetic element that performs addition. Data Next, an experimental example related to the present invention will be shown.

In the "eyeball refractive power measuring device" previously filed by the present inventors as Japanese Patent Application No. 146227/1980, the rotation angle of the eyeball is measured by the rotation angle of two mirrors.
Since the rotating mirror cannot be placed at the center of the spherical mirror, the rotation angle of each mirror does not directly correspond to the horizontal and vertical rotation angles of the eye.

In other words, when the model eye is placed on a stage that can be rotated left and right (X axis) and up and down (Y axis), and rotated in 5 degree increments, the output values of the two mirrors will be as shown in Figure 2, and will be smooth. Although it is a function, it does not have a relationship that can be expressed by the following equation. Therefore, aI: horizontal rotation angle of the model eye, bl: vertical rotation angle of the model eye, xl: rotation angle of the mirror Rotation angle, yI: Rotation angle of mirror Y, defined as, and expressed as, n = 35 (left and right ±15°, vertical ±10', 5
Take data in increments. ), 1,2,3°4.5
It can be seen that the sum of squared errors when expressed by the following polynomial equation monotonically decreases as the degree increases, as shown in Table 1, giving better approximation as the degree increases. - As an example, FIG. 3 shows the calibration result of the data in FIG. 2 when approximated by a cubic equation.

Table 1

[Brief explanation of the drawing]

Fig. 1 is a circuit diagram of the multivariate polynomial type measured value calibrator according to the present invention, and Fig. 2 shows the case where the model eye is moved by 15 degrees left and right and 10 degrees up and down in 5 degree increments (corresponding to the intersection points of the grid). )
Figure 3 is the result of calibrating the data in Figure 2 from 35 pieces of data using equation (5) (marked 0). FIG. Professor Takashi, Director, Product Science Research Institute, Agency of Industrial Science and Technology

Claims (1)

[Claims] 1. Equipped with an arithmetic device that expresses data that has a nonlinear relationship affected by many parameters but changes smoothly using a multivariate polynomial, and performs operations on this polynomial, and performs the above operations. A multivariable polynomial type measurement value calibration characterized in that coefficients calculated by a generalized method of calculating a general inverse matrix are set as coefficients for data calibration in the device, and measurement values are calibrated using this coefficient. vessel.