Principle of interpolation of digital integration method
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Digital integration method: Also call DDA law Digital Differential Analyzer) , it is to build the algorithm of a kind of interpolation on digital integrator foundation, its are the biggest the characteristic is to achieve linkage of much coordinate interpolation easily, it can realize the interpolation of planar and linear, circular arc not only, and the interpolation that still can realize dimensional curve, get applied extensively in system of outline control numerical control. Introduce the working principle of digital integrator above all below, what introduce digital integration method next is linear with method of circular arc interpolation. One, the working principle of digital integrator is like graph 3, 8 are shown, beg function to be in the calm integral of interval, the area that begs function to be surrounded with T axis place inside this interval namely, if will accumulate the divide into equal parts between partition,become a lot of villages namely between (among them) , criterion area S can regard as approximately for the sum of area of a lot of small rectangles, if will be taken,be namely most subsection time (namely time of cycle of a pulse) , namely, turned case into numeric cumulative operation because of the integral operation of this function, when enough hour, operation of cumulative sue for peace replaces the error that integral operation place introduces to need not exceed allows error. The principle graph of integral operation is like graph 3, 9 are shown, it by register of a function that be accumulated, an accumulator (call remainder register again) with a full adder, form. Appear every time a signal, will be accumulated the cost in the value in function register and accumulator is cumulative. If the capacity of accumulator regards an unit area as the value, it is in cumulative process of accumulator cumulative with the capacity that exceeds accumulator when, accumulator spills over a pulse, this pulse is value of an unit area namely, after cumulative end, accumulator spills over always the approximation that pulse number is place quadrature integral namely. Its graph 3, figure of principle of 9 integral operation is cumulative the frequency depends on the digit of register. 2, the linear interpolation of digital integration method is like graph 3, 10 are shown, those who set linear OA to be the first quadrant is linear, start is coordinate origin O(0, 0) , terminal coordinate is A. This linear equation the parameter equation that to will go up type melt into is parameter with time T: In type, k is coefficient of proportionality. Take differential to get to on two type: Beg on two type to get to the calm integral of A interval in O: (3, 7) type is counteracted respectively corresponding start and terminal time. Type is namely on beg X and Y to be in with digital integration method the calm integral of interval, integral value namely for O arrives the coordinate increment of D. Because integral start is coordinate origin, so this coordinate increment is terminal coordinate namely. type (3-7) is used cumulative and replace integral type to get (in 3-8) type, k, , all be constant. If take,be time-interval of a pulse, namely =1, criterion (3-9) by on type has Kn=1, when N (3 -10a) (3-10b) of K=1 ／ chooses K, should make namely every time increment and all be less than 1, a pulse does not exceed in order to make in each reference axis allocates feed every time when pulse (namely every time increment moves only equivalent of a pulse) , namely, (What 3-11) reachs is the biggest allow a value, get the limitation of register capacity. The word that sets register is long for N, what reach is the biggest allow a value to be. To satisfy type (3, the condition of 11) : Ask to be taken normally namely decided coefficient already so, assured to mix again all be less than the condition of 1. By type (3, 10a) , , cumulative number is reason: Take it is time-interval of a pulse (namely) , criterion by type (3, 8) has on will acting person two type, criterion (3, 12) type (3-12) makes clear, can finish planar and linear interpolation computation with two integrator, its are accumulated the case numerical value of function register is respectively and. To binary number, what be equivalent to namely is decimally left move N, because this is in N deposit in register with the number that deposit it is identical, should think only merely latter decimally in front of the most exalted. Accordingly, when the linear interpolation that has digital integration method is calculated, should part to undertake to terminus and terminus cumulative, accumulator every spill over a pulse, control a machine tool to be entered on corresponding reference axis to equivalent of a pulse. When cumulative second hind, z axis and the pace number just in time that Y axis takes are equal to the terminal coordinate of each axis. The terminal differentiate of linear interpolation, by the register capacity in capacity and integrator identical terminal subtration tally is finished, its are initiative the value is 0, cumulative one, terminal subtration tally decreases 1, when cumulative N second hind tally is 0, linear interpolation ends. To assure every time cumulative spill over only, pulse, the digit of accumulator and, the digit of register should identical, its long depend on dimension of the biggest treatment and precision. The program block diagram of the first quadrantal and linear interpolation is like digital integration method graph 3-11 place is shown. Below illustrate number integration method is linear the computational method of interpolation. Exemple 3-3 is set should machine linear 0A (to see graph 3-12) , start is coordinate origin 0(0, 0) , terminal coordinate is A(5, 2) , if be accumulated,the capacity of register of function register remainder and terminal tally all is register of 3 binary systems, cumulative number, before interpolation, , , all be 0, , deposit respectively, , if interpolation computation process expresses 3, 5 are shown. When linear interpolation, the numerical value in register of the function that be accumulated keeps changeless from beginning to end in interpolation process. Its interpolation contrail is like graph 3, 12 medium broken line are shown. This shows, through 8 cumulative hind coordinate of X, Y passes 5 to arrive at linear and terminal coordinate with 2 pulse respectively. Linear interpolation contrail and academic and linear the biggest error do not exceed equivalent of a pulse. Should be machined linear shorter, and when the digit of register and accumulator is longer, appear cumulative just produce to spill over for many times the phenomenon of pulse, right now feed speed is met very slow, affect productivity thereby. Reason will be mixed when process designing commonly magnify at the same time times, change namely spill over the position of pulse will raise feed rate. But right now terminal differentiate should make corresponding change. Because mix,magnify at the same time times, make spill over the place of pulse is right moved M, because this is cumulative,the frequency should decrease. Quadrantal to differring linear interpolation, if take the absolute value of terminal coordinate, calculate the process is same. If the feed direction of each reference axis is expressed 3 - 6 are shown. 3, the principle that the face of circular arc interpolation of digital integration method goes against a circle to be an interpolation of circular arc of discussion number integration method with the first quadrant. If graph 3-13 place is shown, set should machine circular arc to be AB, start is, terminus is, the centre of a circle is in coordinate origin, radius is R. To move a point, criterion the equation of circular arc AB to will go up type is begged to time T guide (3-13) in type, be in to move bit of P the cent speed of X direction; Be in to move bit of P the cent speed of Y direction. type (3, 13) is written into parameter equation, have (in 3-14) type, k is coefficient of proportionality. To type (3, 14) begs his to arrive in A the calm integral of B interval, and part corresponding start and terminal time, its integral value arrives for A the coordinate increment of B, namely, (3-15) type (3-15) is used cumulative and replace integral type to get, if take,be time-interval of a pulse, namely =1, criterion, this shows, as similar as linear interpolation, circular arc interpolation also can come true by integrator of word of two a cycle of songs in a traditional opera, the 3 that be like a graph, show 14 times. Both between what what differ is: Linear interpolation is accumulated function is constant (with) , and circular arc interpolation is accumulated function is variable (with) , and as spill over pulse and constant change. When start, deposit start coordinate to be worth respectively, . In interpolation process, the accumulator of Y integrator, every spill over a pulse, the register of X integrator should be added " 1 " (namely) ; Conversely, the accumulator of X integrator every spill over a pulse, the register of Y integrator should be decreased " 1 " (namely) . Graph 3-14 is useful " + " reach " , " express to amend move a dot to sit in addition, when circular arc interpolation, x coordinate is worth () spill over cumulatively the feed pulse that pulse regards Y as the axis, and Y coordinate is worth () spill over cumulatively the feed pulse that pulse regards X as the axis. When terminal differentiate, do not arrive at terminus at the same time certainly because of two coordinate of circular arc interpolation, reason should undertake terminal differentiate on two direction, its differentiate condition is respectively, achieve terminus when two coordinate only, just stop interpolation computation. To the suitable circular arc of different quadrant and the interpolation that go against circular arc, if take the absolute value of terminal coordinate, calculate the process is same, if the feed direction of each reference axis expresses 3, 7 are shown. CNC Milling