JPH0241764B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to a function generator that easily and quickly generates an approximate value of three times the square root of an integer.

INDUSTRIAL APPLICATION This invention is suitably utilized for control which makes the position of a controlled object follow a target position, for example in the servo control of robots, etc.

[Conventional technology]

When a controlled object is moved with a constant acceleration, the speed of the controlled object is proportional to the square root of its position. The position of the controlled object can be read with an encoder or the like, and the speed can be obtained by calculating the square root based on an integer value indicating the read position. Although the accuracy of this calculation does not need to be high, high speed is required in order to generate the numerical value of the target position in real time.

Conventionally, special high-speed hardware has been used to calculate the square root, or previously calculated square roots have been stored in a function table.

[Problem that the invention seeks to solve]

However, there is a problem in that the cost of the entire device increases due to the provision of special hardware that operates at high speed. Furthermore, in order to store the function table, a large capacity memory is required, which again raises the cost of the device and causes problems in that the scale of the device is large.

In view of the above-mentioned problems in the prior art, an object of the present invention is to calculate the power value of an integer at high speed using a general-purpose microprocessor, and thereby calculate an approximate value of three times the square root of the integer using a small-scale hardware processor. The object of the present invention is to provide a function generating device that can be obtained quickly using hardware.

[Means for solving problems]

In order to solve the above problems, the present invention provides a method for storing a given integer in an initial state and converting the integer to lower bits according to a first shift command signal given in a first cycle. A first shift register that shifts one bit at a time in the direction, stores an integer with the same value as an integer in the initial state, and shifts the integer to lower bits in response to a second shift command signal given at twice the period of the first period. a second shift register that stores the integer 2 in the initial state and shifts the integer 2 one bit at a time in the direction of the upper bit in response to the second shift command signal; A zero judgment circuit detects that the contents of the first shift register become zero, and according to the detection output of the zero judgment circuit, the contents of the second shift register and the contents of the third shift register are added. This function generating device is provided with an adder circuit that obtains an approximate value three times the square root of a given integer in an initial state as the output of the adder circuit.

[Effect]

The zero judgment circuit judges that the integer x is between 2 ^2j and 2 ^2j+2 , and then the approximate value x/ of y=3√
The first term of 2 ^2j +2 ^2j+1 exists in the second shift register, the second term exists in the third shift register, and the approximate value of the sum is calculated by the adder circuit. .

〔Example〕

Embodiments of the present invention will be described below with reference to the drawings.

FIG. 1 is a block diagram showing a function generator according to an embodiment of the present invention. In the figure, in the initial state, integer x is stored in right shift registers 1 and 2, and integer 2 is stored in left shift register 3.
The contents of the right shift register 1 are shifted one bit at a time toward the lower bits in response to each clock signal of the shift command signal _C1 of period _T1 . Shifting one bit in the direction of the lower bit is equivalent to multiplying the contents of the shift register by 1/2, so after the shift by the 2jth clock signal of shift command signal _C1 , the contents of shift register 1 is x/2 ^2j
It is becoming.

The contents of the right shift register 2 are shifted one _bit at a time toward the lower bits in response to each clock signal of the shift command signal _C2 having a period _T2 that is twice the period T1. Therefore, after shifting by the j-th clock signal of the shift command signal C ₂ , which occurs at the same timing as the 2j-th clock signal of the shift command signal C ₁ , the contents of the shift register 2 become x/2 ^j . There is.

The contents of left shift register 3 are the shift command signal
The bits are shifted one bit at a time in the direction of the upper bit in response to each clock signal of _C2 . A shift of 1 bit towards the upper bit changes the contents of the shift register by 2.
Since it is equivalent to multiplying the shift command signal
After the shift by the jth clock signal of _C2 , the contents of shift register 3 are 2j ⁺¹ .

The content of the right shift register 1 is judged by a zero judgment circuit 4 whether it has become zero or not, and when it is judged that the content has become zero, an enable signal EN is given to the adder 5. When the adder 5 receives the enable signal EN, it adds the contents of the right shift register 2 and the left shift register 3 and outputs the result. In other words, if the contents of the right shift register 1 become zero due to the 2jth clock signal of the shift command signal C ₁ , then the contents of the right shift register 2 at that time are x/ ^2j and the contents of the left shift register 3. 2 ^j+1 is added, and the output is y=x/2 ^j +2 ^j+1 (1).

Next, it will be explained that the value y shown in the above equation (1) is an approximate value of three times the square root of the integer x.

FIG. 2 is a graph showing an example of the relationship between the position and speed of a controlled object in servo control of robots, which is one of the fields of application of the present invention. As shown in Fig. 2, when the current position of the controlled object is r ₀ and it _is moved to _the target position _r Control is usually performed such that the target position is moved at a constant speed from r ₁ to r ₂ and the speed is gradually reduced at a constant deceleration from r ₂ to r _x to reach the target position r _x .

In this control, it is necessary to calculate the speed as described above. When acceleration is a, position x and velocity y
The relationship is y=√2...(2).

Figure 3 shows that in equation (2), when √2=3,
In other words, it is a graph of y=3√. In Figure 3, the horizontal axis is 2 to the power of 2, that is, 2 ²ⁿ (n=
0, 1, 2, ..., j). Therefore, the horizontal axis is 2 ⁰ = 1, ^{2 2} = 4, ^{2 4} = 16, 2 ⁶ = 64,...
It is divided in terms of... When the value of x is 2 ^2j , the value of y is 3·2 ^j+1 , and when the value of x is 2 _2j+2 , the value of y is 3·2 _j+1 . The value of y corresponding to the section between points 2 ^2j and 2 ^2j+2 on the horizontal axis is the straight line connecting the points (2 ^2j , 3・2 ^j ) and (2 ^2j+2 , 3・2 ^j+1 ) It is approximated by the value of y above. This straight line is y=3(2 ^j+1 −2 ^j )/2 ^2j+2 −2 ^2j (x−2 ^2j
)+3・2 ^j = 1/2j+2 ^j+1 (3).

It is noted that equation (1) and equation (3) are the same.
In other words, equation (1) means that x is
This represents an approximate value of y=3√ when it is between 2j and 2 ^2j+2 .

In FIG. 1, the role of the right shift register 1 and the zero determination circuit 4 is to determine in which interval on the horizontal axis of FIG. 3 the integer x is located. That is,
If x is shifted to the right by 2 bits and becomes zero, x becomes 2 ²
If it is a smaller number, i.e. 1, 2, or 3, and is shifted to the right by 4 bits and becomes zero, then x is 2 ⁴
It is a smaller number, i.e. 1, 2,..., 15. Generally, if x is shifted to the right by 2j bits and becomes zero, then x is a number smaller than ²² . Therefore, by sequentially shifting x to the right by 2 bits and checking whether it becomes zero, it can be determined whether x is included in any of the intervals divided by 2 ^2j .

FIG. 4 is a flowchart showing the operation of the apparatus shown in FIG. In Figure 4, step S ₁
Integer x is set in registers 1 and 2, and integer 2 is set in register 3. In step _S2 , the contents of register 1 are shifted to the right by 2 bits. step
In _S3 , it is determined whether the contents of register 1 are zero. If it is zero, the contents of register 2 and register 3 are added in step _S4 . If it is not zero in step _S3 , the contents of register 2 are shifted to the right by 1 bit, and the contents of register 3 are shifted to the left by 1 bit in step _S5 . Step until the contents of register 1 become zero.
The loop of S2, step _S3 , and step _S5 is repeated.

Next, the error e from the above approximate calculation is determined.

The error e of y in the interval where the value of x is between 2 ^2j and 2 ^2j+2 is expressed as e=3√−(x/2 ^j +2 ^j+1 ). Therefore, to maximize the error in this interval, It's time. When formula (4) is solved, the error becomes maximum when x=9/42 ^2j (5).

The ratio of the error e to 3√, that is, the relative error E is e/3√. The relative error E for the value of x expressed by equation (5) is Therefore, it is 1/185.56% in both sections, which is unrelated to j.

In the actual calculation by the apparatus shown in FIG. 1, the right shift register 2 shifts j bits toward the lower bits and cuts off the decimal part, so the relative error is slightly larger than the above, but in any case, In servo control of robots and the like, this degree of error is within the permissible range.

From the above explanation, it can be seen that the apparatus shown in FIG. 1 provides an approximate value of three times the square root of the integer x at its output.

Generally, to obtain an approximate value of K√ (K is a constant), you can obtain an approximate value of 3√ using the device shown in Figure 1, and then multiply this result by K/3 using a multiplier, etc. .

〔Effect of the invention〕

As explained above, according to the present invention, by dividing integers into multiples of 2 to the power of 2, approximation of the square root of an integer can be performed using small-scale hardware using only shift operations and additions that are quick to execute on a computer. , a function generator can be obtained that can be realized at high speed and can keep errors to approximately the same level in any interval of integers.

[Brief explanation of the drawing]

FIG. 1 is a block diagram showing a function generator according to an embodiment of the present invention, and FIG. 2 shows an example of the relationship between the position and speed of a controlled object in third-party control of a robot, which is one of the fields of application of the present invention. Graph, 3rd
The figure is a graph for y=3√, and FIG. 4 is a flowchart showing the operation of the apparatus shown in FIG. 1...Right shift register, 2...Right shift register, 3...Left shift register, 4...Zero determination circuit, 5...Addition circuit, _C1 , _C2 ...Shift command signal.

Claims (1)

[Claims] 1. A first shift in which a given integer is stored in an initial state and the integer is shifted one bit at a time in the direction of lower bits in response to a first shift command signal given in a first cycle. A register, which stores an integer having the same value as the integer in an initial state, and shifts the integer one bit at a time toward the lower bits in response to a second shift command signal given at a cycle twice as long as the first cycle. a third shift register that stores an integer 2 in an initial state and shifts the integer 2 bit by bit in the direction of higher bits in response to the second shift command signal; a zero judgment circuit that detects that the content has become zero; and an addition circuit that adds the contents of the second shift register and the third shift register according to the detection output of the zero judgment circuit. A function generating device, comprising: obtaining an approximate value three times the square root of a given integer in the initial state as the output of the adding circuit.