JPH09258957A - Arithmetic method and arithmetic device - Google Patents

Abstract

PROBLEM TO BE SOLVED: To easily detect how much degree of error is contained in numeric data being an arithmetic result by always executing a numeric arithmetic and an error arithmetic in a pair based on a numeric arithmetic rule and an error arithmetic rule. SOLUTION: An arithmetic means 20 executes the numeric arithmetic and the error arithmetic based on the numeric arithmetic rule and the error arithmetic rule, which are read from a rule storage means 10, on numeric data and error data, which are inputted from outside, or numeric data and error data, which are read from a data storage means 30, and generates new numeric data and new error data. Thus, new numeric data and new error data, which are thus generated, are outputted outside the arithmetic device 100 or stored in the data storage means 30 for using it for the next arithmetic. Thus, how much degree of error is contained in the final arithmetic result can immediately be learnt.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an arithmetic method and an arithmetic device for performing numerical arithmetic on numerical data, and more particularly to an arithmetic method and an arithmetic device characterized by management of error information.

[0002]

2. Description of the Related Art Numerical data obtained by measuring with a sensor or the like inevitably contains an error, and a study on the error has been conventionally made. However, the error evaluation is usually extremely complicated, and the error cannot be evaluated easily. Here, as an example, the error of the measurement value by the laser range finder (LRF) will be considered.

FIG. 7 shows a laser range finder (LR).
FIG. 8 is a schematic diagram showing a state of distance measurement by F), and FIG.
It is explanatory drawing of the distance measurement principle by F. As shown in FIG. 7, the optical pulse as shown in FIG. 8A is repeatedly emitted from the LRF 1 in the direction of the predetermined elevation angle φ. This light pulse travels toward the measurement object after being emitted from LRF1, the component irradiated on the measurement object and reflected by the measurement object returns to LRF1, and the returning light is as shown in FIG. 8C, for example. Detected by. Here, when the optical pulse PO of the plurality of optical pulses repeatedly emitted from the LRF1 returns to the LRF1 with a delay of the time T from the emission time,
By measuring this time T, the time T can be converted into the distance x shown in FIG. 7.

However, here, the optical pulse is as shown in FIG.
Since the signal is repeatedly emitted as shown in FIG. 8A, the received signal also has a repetitive waveform as shown in FIG.
The received light signal corresponding to the optical pulse P0 shown in FIG.
It is generally unknown whether the received light signal is the received light signal PO or the received light signal P1 shown in (c). Therefore, by some method, the distance x is between the distance x _{min at} which the light receiving signal shown in FIG. 8B is obtained and the distance x _{max at} which the light receiving signal shown in FIG. 8D is obtained. Assuming that it is known, instead of measuring the time T, the phase difference between the waveform of FIG.
The method of converting to is adopted. In this way, each distance x for a plurality of angles φ is obtained.

Each distance x for each angle φ thus measured is converted into a height map. That is, the distance r in the horizontal direction and the height z in the vertical direction are obtained from the angle φ and the distance x as r = xcosφ (1) z = xsinφ (2) as shown in FIG. Such measurement and calculation are performed by changing the angle of LRF1 also in the direction perpendicular to the paper surface of FIG.
By doing so, a height map (contour map) can be obtained.

Here, how the measurement error due to such LRF propagates when obtaining the height map is considered. For simplicity, we make the following assumptions. (Assumption 1) When the true distance between the LRF and the measured object is x, the distance x ₀ measured by the LRF is

[0007]

[Equation 1]

[0008] However, x _min , x _max
Is the minimum value and the maximum value of the measurable distance range of the LRF as described above, and n is the resolution (this LRF represents the distance measurement value by b
When expressed in bits, the resolution n is n = 2 ^b ),
[...] is a Gaussian symbol indicating that only the integer part of ... Is extracted. At this time, the relationship between the measured value x ₀ and the true value x is x = x ₀ + ε (4) However, the lower and upper limits of the error ε are fixed, and 0 ≦ ε <(x _max −x _min ) / n (5).

That is, the equation (3) is obtained by calculating the minimum distance x _min
It means that a quantization error ε occurs when the distance between the maximum distance x _max and the maximum distance x _max is represented by the resolution n. (Assumption 2) above (1), as can be seen from equation (2), when performing the distance measurement, together with the measured distance x _0, by measuring the angle φ of the time, measured angle and measured distance x ₀ Must be associated. Here, when the true angle when the distance measurement is performed is φ, the measured angle φ ₀
, Φ = φ ₀ + d (6) However, the lower limit and the upper limit of the error d are set, and it is assumed that d _min ≦ d ≦ d _max (7).

Let us consider how the error assumed in (Assumption 1) and (Assumption 2) propagates when converted to a height map. Substituting the above (Assumption 1) and (Assumption 2) into the equations (1) and (2), and letting d ≦ 1 ... (8), r = x ₀ cosφ ₀ + ρ (6) z = x ₀ sinφ ₀ + ξ (7) However, ρ, ξ is the error _{ρ = -dx 0 sinφ 0 + εcosφ} 0 +0 (d 2) ... (8) ξ = dx 0 cosφ 0 + εsinφ 0 +0 (d 2) ... (9). Here, 0 (...) Is a Landau symbol and indicates a value of the order of.

Observing the above situation, it can be seen that the error propagates in a complicated manner depending on the measured values φ ₀ and x ₀ .

[0012]

DISCLOSURE OF THE INVENTION Problems to be Solved by the Invention As shown in this example,
When various calculations (calculations based on formulas (1) and (2) in this example) are performed based on the measured values (numerical data), how the errors contained in the numerical data before the calculation are caused by the calculation. Propagation is extremely complicated, and it is not easy to find out how much error the numerical data resulting from the operation contains.

In view of the above-mentioned circumstances, the present invention provides an arithmetic method capable of easily knowing how an error included in numerical data before arithmetic operation propagates to numerical data after arithmetic operation, and implementation of the arithmetic method. It is an object of the present invention to provide a suitable arithmetic unit.

[0014]

According to the arithmetic method of the present invention for achieving the above object, the error contained in the numerical data of the result of the numerical operation based on the numerical operation rule is corrected corresponding to each numerical operation rule. An error calculation rule for calculation is set, and the numerical data of the numerical data associated with each other and the error data representing the error included in the numerical data are subjected to the numerical calculation based on the predetermined numerical calculation rule. At that time, new numerical data is obtained by performing numerical calculation on the numerical data based on a predetermined numerical calculation rule, and the predetermined numerical calculation rule is added to the error data associated with the numerical data before the numerical calculation. Error calculation based on the error calculation rule corresponding to to obtain new error data,
It is characterized in that these new numerical data and new error data are associated with each other.

According to the calculation method of the present invention, the calculation of numerical data and the calculation of error propagation are always performed as a pair. Therefore, when the final calculation result is obtained, the state of error propagation to the calculation result is determined. It can be easily grasped. Here, in the above-mentioned calculation method of the present invention, the error calculation rule is that the error data is obtained by performing error calculation based on the error calculation rule on the error data representing the minimum value and the maximum value of the true value of the numerical data. This is a calculation rule for obtaining new error data representing the minimum and maximum true values of new numerical data obtained by performing numerical calculation on the corresponding numerical data based on the numerical calculation rule corresponding to the error calculation rule. It is preferable.

This is because it is rational to keep the minimum and maximum true values. However, if the error is statistical, the maximum and minimum values may not be determined statistically theoretically, but the minimum and maximum values that are rationally defined may be adopted. For example, the positions of 5% and 95% may be set as the minimum value and the maximum value. Further, in the above-described calculation method of the present invention, the above-mentioned error calculation rule defines that the probability distribution of the error between the minimum value and the maximum value of the true values of the numerical data is uniform, depending on the nature of the error. It may be an operation rule for obtaining error data, or the error operation rule is such that the probability distribution of the error between the minimum value and the maximum value of the true values of the numerical data is between the minimum value and the maximum value. It may be a calculation rule that obtains new error data assuming that the data is uniform in each area when divided into a plurality of areas.

Further, in the case of the error calculation rule corresponding to the numerical integration calculation rule, the plurality of data having respective values between the minimum value and the maximum value of the true value of the numerical data before the numerical integration calculation is performed are Numerical integration operation based on the rules for numerical integration operation is performed, and the minimum value and the maximum value of the new data obtained as a result of the numerical integration operation are respectively the minimum value and the maximum value of the true value of the new numerical data. It is preferable that

In addition, the arithmetic unit of the present invention which achieves the above object stores a plurality of numerical operation rules, and numerical data obtained as a result of performing a numerical operation corresponding to each numerical operation rule based on the numerical operation rule. Calculation rule storing means for storing an error calculation rule for calculating an error included in, numerical data associated with each other, and numerical data of error data representing an error included in the numerical data, A new numerical data is generated by performing the numerical calculation based on the numerical calculation rule stored in the calculation rule storing means, and the calculation rule storing means stores the error data associated with the numerical data before the numerical calculation. A new error data is generated by performing an error calculation based on an error calculation rule that corresponds to the stored numerical calculation rule that is the basis of the numerical calculation. Means, a new numerical data and the new error data generated by the calculating means, characterized by comprising a data storage means for storing in association with each other.

[0019]

BEST MODE FOR CARRYING OUT THE INVENTION Embodiments of the present invention will be described below. FIG. 1 is a block diagram showing an embodiment of the arithmetic unit of the present invention. The arithmetic unit 100 shown in FIG. 1 includes a rule storage unit 10, an arithmetic unit 20, and a data storage unit 30.

The rule storage means 10 stores a plurality of numerical calculation rules and an error calculation rule corresponding to each of the plurality of numerical calculation rules. Numerical calculation rules refer to calculation rules when performing addition, subtraction, multiplication, etc. operations on numerical data, and error calculation rules refer to numerical data resulting from numerical calculations based on the numerical calculation rules. It is a rule for calculating an included error, and is defined for each numerical calculation rule.

Table 1 below is a table showing the numerical calculation rules and the error calculation rules stored in the rule storage means 10. Further, FIG. 2 is an explanatory diagram of symbols in Table 1. In Table 1, x _obs is an observed value (numerical data according to the invention), x
_min and x _max are the minimum and maximum values of the true value of the observed value, respectively. Subscripts 1 and 2 are the two observed values or maximum value and maximum value before calculation, and the subscript 0 is the calculation result. It means that. Also, _max (x ₁ , x ₂ , ..., x
_n ) is the maximum value of x ₁ , x ₂ , ..., X _n , _min
(X ₁ , x ₂ , ..., X _n ) represents the minimum value of x ₁ , x ₂ , ..., X _n . Here, as shown in FIG. 2, it is assumed that the observed value x _obs is observed with a uniform appearance probability in a region between the minimum value x _min and the maximum value _max .

[0022]

[Table 1]

The arithmetic means 20 shown in FIG. 1 reads out from numerical data and error data (here, observed values, x _obs and minimum value x _min , maximum value x _max ) input from the outside, or data storage means 30. Numerical data (observed values) and error data (minimum value x _min , maximum value x)
_max ) is subjected to a numerical operation and an error operation based on the numerical operation rule and the error operation rule read from the rule storage means 10, respectively, to generate new numerical data and new error data. The new numerical data and the new error data thus generated are output to the outside of the arithmetic unit 1 or stored in the data storage means 30 for use in the next arithmetic operation. In this way, the intermediate storage result is temporarily stored in the data storage means 30 or the final storage result is stored for saving.

As described above, in the arithmetic unit 100, the table
In numerical calculation rule and error calculation rule as illustrated in 1.
Numerical and error operations based on always performed in pairs
Therefore, when the final calculation result is obtained,
You can immediately see if there is a degree of error
You. This allows, for example, reliable operation with small
Only the calculation result is saved, and the calculation result with large error and low reliability is
The fruits are preserved by discarding them without storing them.
The memory capacity for memory can be reduced. Or one
Of multiple events using multiple measurement methods or multiple sensors,
Perform the calculation based on those measured values to obtain the desired calculation result.
When the fruit is obtained, or the original measurements are the same
Even if you want to perform the desired operation by performing operations using multiple calculation methods
When a result is obtained, one of the measured values or
Error range of the result of the calculation based on the other calculation method
Is x_min 01~ X_max 01, The other measured value or the other performance
The error range of the calculation result based on the calculation method is x_min 
02~ X_max 02And, for example, x_min 01<X_min 02<X
_max 01<X_max 02, The error range is x_min 02~ X
_max 01A book that can be limited to a narrow range such as
High accuracy and high reliability of data by applying the invention
It can also be achieved.

In the above-described embodiment, the occurrence probability of the observed value x _obs is the minimum value x _min and the maximum value x _min as shown in FIG.
_Although it is assumed to be uniform between _max and _max, it may be better to assume that the appearance probability of the observed value x _obs follows a normal distribution depending on the nature of the error. FIG. 3 is an explanatory diagram of an error calculation method when the appearance probability of an observed value has a normal distribution.

Since the normal distribution error is inconvenient for error calculation if the normal distribution function is used as it is, the area between the minimum value x _min and the maximum value x _max is divided into a plurality of areas as shown in FIG. Then, the appearance probability is treated as uniform. For example, when the observed value x _obs is within the range of x to x + 8x, x
_It is assumed that the appearance probability is h regardless of the position of _{obs in} the area of x to x + dx.

In this way, the normal distribution curve is approximated stepwise, and when the appearance probability of the region of x to x + dx is h, x is x
_The calculation is performed in the same manner as in the case of the uniform distribution in Table 1 with _min and x + dx as x _max, and the calculation result x _{min 0} ,
in x region sandwiched _{max 0 x mi n 0 ~x max} 0,
It is assumed that the appearance probability of the observed value x _obs after the calculation is h. By repeating this for each area shown in FIG.
The error probability distribution after calculation can be obtained.

Although the normal distribution curve theoretically extends from infinity to infinity, the minimum value x _min and the maximum value x _max of the entire distribution shown in FIG. Considering, for example, the minimum value x with the value of occurrence probability 1%
_min , maximum value x _max , etc. are appropriately determined. FIG.
FIG. 9 is an explanatory diagram of an error calculation method in the case of numerical integration calculation.

The function value (observed value) at a certain variable value t is
x_obst, The minimum and maximum true values of the function
x_min t , X_max t And multiply this by the variable value t + dt
Divided by observation x_obstIs the observed value x_obst,_{t + dt}Has changed to
I do. At this time, the minimum value x_{min t} , Maximum x_max t The figure
4 are integrated along broken lines a1 and a2, or
Due to the non-linear effect of integration, for example, the alternate long and short dash lines b1 and b2
The value may change along with it, so the minimum value x before calculation_min 
t , Maximum x_max t The value of the result of integrating
Observed value x after calculation_obst,_{t + dt}Minimum value of x_min ,_{t + dt}, Up
Large value x_max ,_{t + dt}May not be. So the integral
For calculation, the minimum value x before calculation_min tAnd the maximum value x_max t
 Put multiple values (marked x in Fig. 4) in the area sandwiched between
Then, an integration operation is performed for each value and
Of the calculated results (marked with Δ in Fig. 4)
Large value is the observed value x after calculation_obst,_{t + dt}Minimum x of the true value of
_{mi n} ,_{t + dt}, Maximum x_max ,_{t + dt}And To do this
As a result, non-linear operations like
The difference range can be obtained.

[0030]

EXAMPLES Hereinafter, the laser range finder (L
The simulation result of (RF) will be described. Table 2 shows the LRF used for the simulation here.
2 is a table showing the specifications of.

[0031]

[Table 2]

Further, here, the error factors of this LRF are assumed as shown in Table 3. The details of each error are described in the above (Assumption 1) and (Assumption 2).

[0033]

[Table 3]

Using the specifications shown in Table 2 and the LRF assuming the errors shown in Table 3, a distance measurement simulation as shown in FIG. 7 is performed for a certain terrain, and from the measured distances, (1), (2) The height map was calculated according to the formula. still,
Here, a uniform distribution error as shown in FIG. 2 is assumed.
5 and 6 are views of the height map obtained as described above, taken from a certain cross section. That is, it corresponds to the vertical section of the original terrain. 5 and 6 show the difference in angular accuracy. A certain measured value exists for the original data (true topography), and the measured value is sandwiched between the maximum and minimum values from above and below.

According to the invention, the error range is thus immediately known.

[0036]

As described above, according to the present invention,
It is possible to easily know how the error included in the numerical data before calculation propagates to the calculation result.

[Brief description of drawings]

FIG. 1 is a block diagram showing an embodiment of a computing device of the present invention.

FIG. 2 is an explanatory diagram of symbols in Table 1.

FIG. 3 is an explanatory diagram of an error calculation method when the appearance probability of an observed value has a normal distribution.

FIG. 4 is an explanatory diagram of an error calculation method in the case of numerical integration calculation.

FIG. 5 is a diagram drawn by taking a cross section having a height map obtained by simulation.

FIG. 6 is a diagram drawn by taking a cross section having a height map obtained by simulation.

FIG. 7 is a schematic diagram showing how distance measurement is performed by a laser range finder (LRF).

FIG. 8 is an explanatory diagram of a principle of distance measurement by LRF.

[Explanation of symbols]

 1 Laser Range Finder (LRF) 10 Rule Storage Means 20 Computing Means 30 Data Storage Means 100 Computing Devices

Claims (6)

[Claims] 1. An error calculation rule for calculating an error included in numerical data resulting from a numerical calculation based on the numerical calculation rule is defined corresponding to each numerical calculation rule, and they are associated with each other. Numerical calculation based on the predetermined numerical calculation rule when performing numerical calculation on the numerical data of the numerical data and the error data representing the error included in the numerical data based on the predetermined numerical calculation rule To obtain new numerical data, and the error data associated with the numerical data before the numerical calculation is subjected to error calculation based on the error calculation rule corresponding to the predetermined numerical calculation rule. An arithmetic method characterized by obtaining error data and associating these new numerical data and new error data with each other. 2. The error calculation rule applies error calculation based on the error calculation rule to error data representing the minimum value and the maximum value of the true value of the numerical data, thereby calculating the numerical data corresponding to the error data. A calculation rule for obtaining new error data representing the minimum and maximum true values of new numerical data obtained by performing a numerical calculation based on the numerical calculation rule corresponding to the error calculation rule. The calculation method according to item 1. 3. The error calculation rule is a calculation rule for obtaining the new error data assuming that the probability distribution of the error between the minimum value and the maximum value of the true values of the numerical data is uniform. The calculation method according to claim 1. 4. The error calculation rule is such that when the probability distribution of the error between the minimum value and the maximum value of the true values of the numerical data divides the minimum value and the maximum value into a plurality of regions. 2. The calculation method according to claim 1, wherein the calculation rule is a calculation rule for obtaining the new error data assuming that it is uniform in each area. 5. The error calculation rule corresponding to the numerical integration calculation rule is such that the numerical values are set for each of a plurality of data having respective values between the minimum and maximum true values of the numerical data before the numerical integration calculation is performed. Numerical integration operation based on the integration operation rule is performed, and the minimum value and the maximum value of a plurality of new data obtained as a result of the numerical integration operation are respectively the minimum value and the maximum value of the true value of the new numerical data. The calculation method according to claim 1, wherein: 6. An error calculation for storing a plurality of numerical calculation rules and calculating an error corresponding to each of the numerical calculation rules and included in numerical data as a result of performing a numerical calculation based on the numerical calculation rules. Numerical operation stored in the operation rule storage means in numerical value data of numerical data and error data representing an error included in the numerical data, which are associated with each other A new numerical data is generated by performing a numerical operation based on a rule, and the error data associated with the numerical data before the numerical operation is stored in the calculation rule storage means and is the basis of the numerical operation. Calculating means for performing error calculation based on the error calculation rule corresponding to the numerical calculation rule to generate new error data; Numerical data and the new error data, arithmetic apparatus characterized by comprising a data storage means for storing in association with each other such.