JPS6115446B2 - - Google Patents

Description

[Detailed description of the invention]

The present invention is applicable to automatic drafting machines in which movement of a marking head is controlled along an X-Y coordinate system, and numerically controlled machine tools in which movement of a tool holder is controlled along an X-Y coordinate system and a Y-Z coordinate system. The present invention relates to a cubic interpolation method in contour control such as the following. In general, contour control in automatic drafting machines, numerically controlled machine tools, etc. is performed by interpolating the moment-to-moment trajectories drawn by marking members, tools, etc. using straight lines, circular arcs, or arbitrary function curves. The more complex the trajectory becomes, the more the control speed becomes slow. Therefore, in conventional automatic drafting machines and numerically controlled machine tools, etc., when the trajectory drawn by a marking member or tool has a complex function shape, it is possible to draw it without using a complex function curve as long as the accuracy allows. , converting this trajectory into straight line or circular arc data by approximately dividing the trajectory by straight lines or circular arcs,
Interpolation is performed by digital control based on these data. However, even with this method, the number of straight line or circular arc data obtained from the trajectory becomes enormous, especially when interpolating between a large number of point sequences with minute intervals on the trajectory. In addition, the control speed is slow. The present invention was made in order to eliminate such drawbacks found in the interpolation method of the trajectory of marking members, tools, etc. in conventional automatic drafting machines and numerically controlled machine tools. For cubic interpolation, which is one of the methods for connecting with The present invention provides a novel method that enables high-speed control by adapting the execution time of one pulse distribution by the microcomputer to the maximum speed of movement of the marking head and the tool holder. The basic idea of the cubic interpolation method in the present invention is that when interpolating between the previously given point P _i-1 and the currently given point P _i , the next given point P _i+1 and the point P The shape of the cubic curve to be interpolated is determined by taking _into account the point P _{i -2 that} was , the direction of the tangent at the end point P _i matches the direction of the line segment from point P _i-1 to point P _i+1 . The details of the present invention will be explained below based on the drawings. Figure 1 shows three points positioned with respect to the reference XX coordinate system, namely the previously given point P _i-
1 , Point _{P i} given this time and interpolated so far this time
and is an exemplary diagram of a cubic curve passing through the next given point P _i+1 . Now, in this figure, point P _i-1
If the origin is taken as the ξ-axis, and the straight line connecting this origin P _i-1 and point P _i is taken as the ξ-axis, and the straight line passing through the origin P _i-1 and perpendicular to this ξ-axis is taken as the η-axis, then the cubic curve is this ξ-η Since it passes through the origin of the coordinate system, it is expressed by the following cubic equation: η=aξ+bξ ² +cξ ³ (1). The coefficients a, b, and c in equation (1) are obtained as follows. That is, as shown in FIG. 2,
Translate the X coordinate system in parallel, and set its origin to the first point P of the ξ-η coordinate system, that is, the first point P through which the cubic curve passes.
_i-1 (the coordinates of this first point P _i-1 are
Regarding both the X-Y and ξ-η coordinate systems, both become [o, o]. ), and on the line in the third quadrant of the X-Y coordinate system of this cubic curve, that is, in front of the first point P _i-1 , with respect to the X-Y coordinate system, (Bx,
By) Take another point P _i-2 with coordinate values, and with respect to the X-Y coordinate system, take the second point P i-2 through which the cubic curve passes.
The coordinates of the third points P _i and P _i+1 are (x ₁ , y ₁ ) and (x ₂ , y ₂ ), respectively, and the angle between the X axis and the ξ axis (Y axis and η axis) is θ. If so, ξ in equation (1)
Regarding η and η, the following equations are obtained. ξ ₁ =Bxcosθ+Bysinθ ξ ₂ =0 ξ ₃ =x ₁ cosθ+y ₁ sinθ ξ ₄ =x ₂ cosθ+y ₂ sinθ η ₁ =−Bxsinθ+Bycosθ η ₂ =0 η ₃ =0 η ₄ =−x ₂ sinθ+y ₂ cosθ In these equations ξ ₁ and η ₁ are the coordinate values of point P _i-2 with respect to the ξ-η coordinate system, ξ ₂ and η ₂
are also the coordinate values of point P _i-1 with respect to the ξ-η coordinate system, ξ ₃ and η ₃ are also the coordinate values of point P _i with respect to the ξ-η coordinate system, and ξ ₄ and η ₄ are also the coordinate values of point P i-1 with respect to the ξ-η coordinate system.
These are the coordinate values of point P _i+1 regarding the η coordinate system, and these coordinate values are obtained in order to calculate each coefficient a, b, and c in equation (1). In equation (1), a is the direction of the right component indicated by vector V _i-1 at point P _i-1 , and point P _i-2
It is expressed by the following equation, assuming that the direction is the same as that from to point P _i . a=η ₃ −η ₁ /ξ ₃ −ξ ₁ Also, if the direction of the right component indicated by the vector V _i at the point P _i is a ₁ , this direction is from the point P _i-1 to the point P _i+1 It is expressed by the following formula, assuming that the direction is the same as the direction leading to . a ₁ = η ₄ −η ₂ /ξ ₄ −ξ _2Furthermore , since η ₃ =0 at point P _i , substitute ξ ₃ into equation (1) and get aξ ₃ + bξ ² ₃ + cξ ³ ₃ = η ₃ = 0, ξ ₃ ≠ 0 Therefore, a + bξ ₃ + cξ ² ₃ = 0 ...(2) Also, the direction of the right component at point P _i is obtained by differentiating equation (1) and substituting ξ ₃ into it. Therefore, the formula a+2bξ ₃ +3cξ ² ₃ =a ₁ ...(3) holds, so the other coefficient b in formula (1)
and c are obtained from these equations (2) and (3) as follows. b=-a ₁ -2a/ξ ₃ In general, the X-Y coordinate system as shown in Figure 1 and ξ-η
Between the coordinate systems, as shown in Figure 3, there are
If the angle between the axes (Y axis and η axis) is θ, then The following relational expression holds true. Furthermore, as shown in Figure 4, the relationship between the X-Y coordinate system and the ξ-η coordinate system is such that a unit vector of scalar amount 1 is taken on the ξ-axis, and the corresponding change in the η-axis direction is When △η is ₁ , these and the amount of change in the X-Y coordinate system corresponding to these △X ₁
Also holds between and △Y ₁ , The following relational expression is obtained. Similarly, (△X ₁ , △
The next unit vector in the ξ-axis direction with Y ₁ ) as the origin, the corresponding amount of change in the η-axis direction △η ₂ , and the corresponding amount of change in the X-Y coordinate system △
Also between X ₂ and △Y ₂ , The following relational expression holds true. Therefore, the j-th unit vector in the ξ-axis direction, the corresponding amount of change in the η-axis direction △ηj, and the corresponding amount of change in the X-Y coordinate system △
The relationship between Xj and △Yj is similar to equation (4). It is expressed by the following formula. △ηj in equation (5) is expressed as (j+
1) and by substituting j, Δηj=ηj+1−ηj=(a+b+c)+(2b+3c)j+3cj ² . Now, in this formula, replace (a+b+c) with b, (2b+3c) with m, and replace 3c with m, and j and
If we rewrite j ₂ as a recurrence formula, we get becomes. f(j-1) in this equation is expressed by the following equation. In order to clarify the meaning of this equation (6), △η ₁ , △
Concretely expressing η ₂ , Δη ₃ and Δη ₄ is as follows. △η ₁ =k++o+f(o)=k++m (f(o)=m) △η ₂ =k+2+2m+f(1)=k+2+
4m (f(1)=f(o)+2(o)+m=2m) △η ₃ =k+3+2(2m)+f(2)=k+3
+9m (f(2)=f(1)+2(m)+m=5m) △η ₄ =k+4+2(3m)+f(3)=k+4
+16m (f(3)=f(2)+2(2m)+m=10m) By substituting equation (6) like this into equation (5), The following formula is obtained. Each term in these equations (7) and (8) is cosθ, ksinθ, sinθ, msinθ, sinθ+
Since kcosθ, cosθ and mcosθ are in a constant or cumulative relationship, each
It is clear that there is no need to multiply by cos θ or sin θ, and the calculation processing time is shortened. Furthermore, if the amount of change in the X-Y coordinate system from the first point P _i -1 to the second point P _i in FIG. 1 is △X and △Y, cos θ and sin
Each of θ can be obtained from the following equations. In this way, from the unit vector on the ξ axis,
After obtaining △Xj and △Yj, add the fractions axj−1 and ayj−1 that remained from the previous time to these to obtain the movement components △X′j and △Y′j in the X-axis direction, and convert these into integers. It is separated into pulse portions N and M and fractional portions αxj and αyj. △Xj＋αXj−1=△X′j=N+αXj……(11) △Yj＋αyi−1=△Y′j=M+αyj……(12) In this way, pulse distribution can be easily performed algebraically. . Next, the manner in which such pulse distribution is performed will be explained based on a simple numerical example and with reference to FIG. In the figure, the amount of change in the j-1st X-Y coordinate system is △Xj-1 = 2.3 (the number of significant digits in these numbers is determined as appropriate), △Yj-1 = 3.6, and the same X-Y coordinate system as the j-th The amount of change in the Y coordinate system is △Xj = 0.8 △Yj = 2.1 Furthermore, up to j-2nd, the fraction (value after the decimal point) of the numerical value that has already been distributed in the X-Y coordinate system If the values of the registers are αxj−2=0.2 αyj−2=0.1, respectively,
The moving components at the j-1st position are △X′j-1=△Xj-1+αxj-2=2.5 and △Y′j-1=△Yj-1+αyj-2=3.7. Therefore, at this point, the pulses related to the Y-axis are output before those related to the X-axis, and the number of pulses in each is 3 and 2, respectively. As a result, the values of the j-1th fraction register are αxj-1=0.5 αyj-1=0.7, respectively. Similarly, the j-th moving component, ie, the pulse distribution amount, is ΔX′j=ΔXj+αxj−1=1.3 and ΔY′j=ΔYj+αyj−1=2.8, respectively. Therefore, at this point as well, the pulses related to the Y-axis are outputted earlier than those related to the X-axis, and the number of pulses is 2 and 1, respectively. As a result, the values of the j-th fraction register are αxj=0.3 αyj=0.8, respectively. The timing of such pulse distribution in the present invention is performed, for example, as follows. In other words, marking heads and tool holders in automatic drafting machines, numerically controlled machine tools, etc.
Along the axis and Y axis, V = 24 (m/min) =
It is driven at a speed of 24000/60 (mm/sec), and the driving distance per pulse is set to 0.01 mm. As a result, the pulse interval becomes ΔT=1/400×100=25 (μsec), and it is sufficient to output one pulse every 25 μsec. In such a case, if a microcomputer with a one-instruction execution time of 200 nsec is used, calculations can be repeated in 25000/200, or 125, program execution steps. Regarding the above-mentioned pulse distribution, regarding equations (7) and (8), 15 steps are required for calculating △Xj, 15 steps are required for calculating △Yj, and 15 steps are required for calculating △Yj. Fractions αx and α
Regarding the calculation of y, there are 6 steps each.
There are 12 steps in total, and there are about 30 steps for determining the speed, and the total number of steps is within 100, so it can be easily executed by the microcomputer. Therefore, conventionally, interpolation using such a straight line division method has been performed at speeds of 8 m/min to 12 m/min.
While interpolation was performed at a speed of m/min, according to the present invention, interpolation can be performed at a high speed of 24 m/min. Figure 6 shows the principle of the configuration of a device that actually performs pulse distribution using the above-mentioned specifications, and Figure 9 shows a flowchart of a computer program. The enclosed area indicates the execution range of the microcomputer. In the figure, 1 decodes the input data and calculates each coefficient a, b, and c in equation (1), k and m in equation (6), and cosθ and sinθ shown in equations (9) and (10). From, the eight operational elements cosθ, ksinθ, sinθ, msinθ, sin
A computing unit that calculates θ, kcosθ, cosθ, and mcosθ, respectively, and inputs the results to a microcomputer.The calculation of these computing elements is usually done by
It is performed on a minicomputer, and in the flowchart of FIG. 9, calculations are performed using blocks 11 to 16. 2 is based on the 8 calculation elements input from this calculation unit 1,
This is a calculator that calculates △Xj and △Yj shown by equations (7) and (8), and the calculation contents are shown in block 21 on the flowchart. For example, △Xj
about,

[expression] and

Accumulation (accumulation rate) register of [formula], f(j-1)
From the G (msinθ) register of sinθ (meaning the rank accumulation register of msinθ) and the constant registers of cosθ and ksinθ,
Similarly, regarding △Yj,

[expression] and

Accumulation (accumulation rate) register of [formula], f(j-1)
G (mcosθ) register of cosθ and sinθ
and kcosθ constant registers. With these registers of the calculator 2, the values of △Xj and △Yj calculated in block 21 on the flowchart are inputted to the calculator shown as 3. 32, as shown in equations (11) and (12), △X′j and △Y′j
The pulse values N and M corresponding to , and the fractions αxj and αyj at that point in time are calculated. The pulse values N and M(N
and M may also be O. ) is input to the output device 4, which distributes pulses according to the DDA method while synchronizing with the program execution step of the microcomputer, and the output signal is sent to the drive device 5. By being fed, the marking head, tool holder, etc. are properly driven. The status of such pulse distribution is monitored by a first command unit 6, and when the N and M pulse distributions are completed, the signal is sent to the second command unit 7. These command units 6,
7 is shown as blocks 61 and 71 on the flowchart. The command unit 7 calculates the sum of unit vectors in the ξ direction.

j is incremented by 1 until [Formula] reaches ξ ₃ , and the calculator 2 repeatedly calculates each unit vector in the ξ direction.

When [formula] reaches ξ ₃ , the output of the calculator 1 is temporarily stopped, and in order to perform calculations up to the next point to be passed, the data of the point where i is advanced one by one is read into the calculator 1. Then, the above operation is repeated. Although not shown in the apparatus and flowcharts of FIGS. 6 and 9, we would like to add a supplementary explanation regarding the number of execution steps in pulse distribution for cubic interpolation based on FIGS. 7 and 8. , as follows. The distance between the first point P _i-1 and the second _{point P i} along the ξ axis in _FIG . _Regarding the coefficients, if we set ξ ₃ α= _u ξ ₃ α ₁ =w, equation ( ¹ ) becomes
u+w) (ξ/ξ ₃ ) It can be rewritten as ³ . In this formula, if we set ξ/ξ ₃ = τ, then η = u τ - (2u + w) τ ² + (u + w) τ ³
...(13) becomes. The limit value of the function expressed by equation (13) is the value of τ when dη/dτ=0, that is, It is indicated by. The value of τ shown in equation (14) is all O
<τ<1, the cubic curve in Figure 1 will look like Figure 7a, and if one of the values of τ is 1 <
If τ is within the range, the result will be as shown in b in the same figure, and if one of the values of τ is in the range of τ<o, then the result will be as shown in c in the same figure. Now, when we obtain τ such that 0<τ<1, by substituting that τ into equation (13), we can obtain the maximum value η _nax of η between point P _i-1 and point P _i .
At this time, the cubic curve between the first point P _{i-1 and} the second point P _i is subdivided by straight lines as shown in FIG. Moving with _i is equivalent to moving along the straight line P _i-1 → (b) → (d) P _i while performing the same straight line division, and the moving distance is Locally, P _i-1 →(i)→
It is equal to the length of the straight line (c)→(e)→P _i . Therefore, if the maximum value of the η coordinate of the cubic curve between the first point P _i-1 and the second point P _i is η _nax , the number of execution steps in pulse distribution is L=(ξ ₃ +2ηmax) (cosθ+sinθ)
...(15) It is obtained by the formula. By this L, the maximum speed and the number of steps used for acceleration and deceleration when driving the marking head, tool holder, etc. can be appropriately determined. The above description has been made based on the X-Y coordinate system, but if control is also performed on the Y-Z coordinate system, such as in a numerically controlled machine tool, this Y-Z coordinate system may be used as the reference. In this way, the present invention solves the so-called cubic interpolation, which is a method of connecting a given point sequence with a smooth curve, by performing straight line division by checking the chord height, etc. Instead of using the conventional method, a microcomputer is used instead, and the execution time of one pulse distribution by this microcomputer can be reduced by the marking head of automatic drafting machines, numerically controlled machine tools, etc., and the movement of tool holders, etc. By adapting it to the maximum speed of

[Brief explanation of the drawing]

Figure 1 is an example of a cubic curve passing through three points P _i-1 , P _i and P _i+1 positioned with respect to the reference X-Y coordinate system, and Figure 2 is an illustration of the cubic curve shown in Figure 1. Figure 3 is a diagram showing the relationship between the X- _Y coordinate system and the ξ-η coordinate system in Figure ₁ . The figure shows the unit vector taken in the ξ-axis direction, the corresponding amount of change in the η-axis direction, and the corresponding X
- Diagram showing the relationship between the amount of change in the Y coordinate system, fifth
The figure shows the pattern of pulse distribution in cubic interpolation, Figure 6 shows the principle of the configuration of a device that performs pulse distribution using a microcomputer, and Figure 7
8 and 8 are diagrams for determining the number of execution steps in pulse distribution, and FIG. 9 is a flowchart of a computer program. DESCRIPTION OF SYMBOLS 1... Arithmetic unit, 2... Calculator, 3... Calculator, 4... Output device, 5... Drive device, 6, 7... Command device.

Claims (1)

[Claims] 1. In the contour control of the locus of a moving member that is controlled along two reference coordinate axes that are orthogonal to each other, a point that coincides with the coordinate values regarding the two coordinate axes that are given as input commands regarding the movement control of this member. As an interpolation curve to smoothly connect the rows of , η= aξ+bξ ²
+cξ Using a cubic curve expressed by the formula ³ , ξ
Continuously take a unit vector with a scalar amount of 1 on the axis, calculate the amount of change in the η-axis component that corresponds to the amount of change in this unit vector, and calculate the amount of change in these unit vectors and the η-axis component between the two coordinate axes. A cubic interpolation method characterized in that the position control of the movable member is performed in real time by converting the amount of change in each component into the amount of change in each component of these two coordinate axes and distributing the amount of change in each component in pulses.