The comparison of the algorithm of interpolation of two kinds of parabola that is based on time split plot design considers to reach N to divide a pace to grow algorithm

The summary is algorithmic to be based on the interpolation of two kinds of parabola of time split plot design in interpolation precision and interpolation operation time two respects had trade off study, proved the validity that maths analyses through the experiment, put forward N to divide a pace to grow interpolation algorithm, this algorithm applies to a variety of curves, raise interpolation precision effectively. Keyword: N of split plot design of interpolation algorithm time divides a pace to grow the part with 1 complex to any foreword, its contour line can be comprised by a few simple geometry elements, wait like linear, circular arc, parabola. Although body of a few freedom curve, curved surface also can be undertaken planning to close by batten curve, still want finally to use linear, circular arc or be approached, realize its CNC Machining, complete the work of interpolation operation namely. The algorithm of interpolation is very much, but common characteristic is to must be below the premise that assures to machine precision to ask decide the biggest interpolation span, calculate an each interpolation paragraph remove dead centre to be in the coordinate value that is gotten on by contour line of treatment spare parts. The athletic contrail that time split plot design is the sampling cycle according to systematic set and set is tangential speed, decide every pace place takes a space, calculate respectively again every unit time two axes feed, time split plot design applies extensively at machine tool of contemporary numerical control the CNC system of intermediate above [1, 2] . Current, those who be based on time split plot design is linear had been studied extensively with the interpolation algorithm of circular arc, but to resembling parabola the interpolation problem of a kind more complex curve returns little somebody to consider, cause the algorithm with effective lack, cause process designing complex, interpolation precision is low problem, those who affect numerical control system is practical. Rate of 32 machine processing is rapid, operation capability is strong, facilitating people studies further the interpolation problem of complex curve. The article passes algorithmic to be based on the interpolation of two kinds of parabola of time split plot design trade off study, from interpolation precision and interpolation operation time two respects studied further parabola interpolation problem; Go up in this foundation, put forward N to divide a pace to grow interpolation algorithm, this algorithm develops the property of 32 computers adequately, can apply at a lot of curve interpolation at the same time, raised interpolation precision effectively. The algorithm of interpolation of two kinds of parabola that is based on time split plot design gives out above all below 2 interpolation algorithm. 2.

1 interpolation is algorithmic one to ＝ of parabola equation Y - Ax2 ＋ Bx, (A ＞ 0, b ＞ 0) . F ＝ FT, f is feed, f is feed speed, t is sampling time, a (Xi, yi) nod for this second interpolation, b (Xi ＋ 1, yi ＋ 1) for next time interpolation dot, Δ Xi, Δ Yi is distance of feed of axis of X, Y. Ki ＝ - 1 ＝ of ＋ of Xi of 　 of 　 of 　 of 　 of 2axi ＋ Bi 1 ＝ of ＋ of Yi of 　 of 　 of Xi of Xi ＋ Δ - 1 ＋ of Ax2i ＋ 1 　 of Bxi ＋ ＋ of Yi of ＝ of 　 Δ Yi 1 - Yi(1)2.

2 interpolation are algorithmic 2 [1] 　 parabola equation is Y ＝ - Ax2 ＋ Bx (A ＞ 0, b ＞ 0) , a (Xi - 1, yi - 1) , B (Xi, yi) nod for two foregone interpolation, c (Xi ＋ 1, yi ＋ 1) nod to issue one interpolation, f is feed, f is feed speed, t is interpolation time. In parabola acme left: Yi of Δ of Δ Yi-1 ＝ - 1 - 2a Δ X2i - ＋ of Xi of 　 of 　 of 1 　 　 1 ＝ of ＋ of Yi of 　 of 　 of Xi of Δ of ＋ of 1 ＝ Xi - 1 ＋ of Ax2i ＋ 1 　 of Bxi ＋ ＋ of Yi of ＝ of 　 Δ Yi 1 - Yi (2) 　 : Yi of Δ of Δ Yi-1 ＝ - 1 - 2a Δ X2i - ＋ of Xi of 　 of 　 of 1 　 　 1 ＝ of ＋ of Yi of 　 of 　 of Xi of Δ of ＋ of 1 ＝ Xi - 1 ＋ of Ax2i ＋ 1 　 of Bxi ＋ ＋ of Yi of ＝ of 　 Δ Yi 1 - Yi (3) 3, the key is chord replaces arc length, and arc length should be F of ＝ of F of synthetic interpolation feed.

T. Every time what interpolation takes chord and F difference actually the value is less, corresponding interpolation precision is taller. Right above two kinds of algorithm that put forward make following comparison. Set parabola equation to be Y ＝ - Ax2 ＋ Bx (A ＞ 0, b ＞ 0) , consideration interpolation dot is in apical and left circumstance, see a picture 1. Graph the A in plan of plan of trade off study of algorithm of two kinds of 1 interpolation (Xi - 1, yi - 1) , B (Xi, yi) nod for two foregone interpolation, c is dot of below one ideal interpolation, AE makes parabola tangent to cross A to choose place, α of Tg of ＝ of its slope K 1, α 1 for AE and X axial included angle, join AB is lengthened, hand in the circle that is radius for the centre of a circle, F with B to be nodded at C1, a, C1 all is the dot on the circle. Cross C1 to make Y axis parallel, hand in parabola to be nodded at C ′ , lengthen C1C ′ , had handed in B to nod place to make X axis parallel be nodded at D1. C2 nods parabola tangent to fall the point on Yu Yuan to cross B, β 1 for BC2 at X axial included angle. Lengthen BC ′ to hand in a circle to be nodded at C3, β 2 for BC3 and X axial included angle, cross C2, C3, C to make Y axis parallel respectively, hand in BD1 extension line to be nodded at D2, D3, D. Can see C2 selects the interpolation site that is the first kind of algorithm, its interpolation measures ｜ of BD2 of ｜ of Δ Xi ＝ , ｜ of C2D2 of ｜ of ＝ of Δ Yi ′ , c3 is nodded for the 2nd kind of algorithmic interpolation, its interpolation measures ｜ of C3D3 of ｜ of ＝ of Δ Yi ″ , and C is nodded for ideal interpolation, its interpolation measures ｜ of BD of ｜ of Δ Xi ＝ , ｜ of CD of ｜ of Δ Yi ＝ . The slope of BC2 and BC3 is tried below. See A drop first, choose relatively apical place from hypothesis equation and A knowable AE slope is more than AB slope surely, namely 1 ＞ of α of KAE ＝ Tg α of KAB ＝ Tg 2. Because of A, B, C1 at 3 o'clock in all line, so α of Tg of ＝ of BC1 slope KBC1 2, consider point of A, B respectively, to A the dot has: Y ′ I - 1 ＝ - 2axi - 1 ＋ B; To B the dot has: Yi ′ ＝ - 2axi ＋ B has 　 Yi ′ so - Y ′ I - 1 ＝ - 2a (Xi - Xi - 1) ＝ - I of ′ of 2a Δ X - 1: I of ′ of Y of Yi ′ ＝ - 1 - I of ′ of 2a Δ X - 1 ＝ KAB - I of ′ of 2a Δ X - 1 and nod to B, I of ″ of Y of ＝ of ｜ of D1 of ′ of C having ｜ - 1 - 2ax ″ I - 12 　 are right last time interpolation, can assume, I of Δ Y ′ - I of ″ of Y of 1 ＝ Δ - 1, I of Δ X ′ - I of ″ of X of 1 ＝ Δ - 1, have then: 1 ＝ of Tg β KBC2 ＝ KAE - I of ′ of 2a Δ X - 2 ＝ of β of Tg of 　 of 　 of 1 　 　 KBC3 ＝ KAB - I of ′ of 2a Δ X - 1 　 because KAE ＞ KAB, so 　 of KBC2 ＞ KBC3 considers B to nod local coordinate department now, have B(0, 0) , round equation is X ′ 2 ＝ of ′ of 2 ＋ Y F2. Nod to crossing B and the hands in Yu Yuan point-blank nodical coordinate that slope is K, can get have C3 of ′ of X of ＜ of X ′ C2. And in ＝ of the K inside this limits - ＞ of 2ax ＋ B 0, so C3 of ′ of Y of ＜ of Y ′ C2, namely 　 　 can be seen by afore-mentioned derivation, algorithmic the synthetic interpolation of 2 is actual feed compares algorithm of one more approach at synthesizing interpolation ideal feed, that is to say, algorithmic the interpolation precision of 2 is theoretic than algorithm the interpolation precision of one is tall. But through comparing two kinds of algorithmic and basic ideas, can see algorithm 2 than algorithm one complex, namely algorithmic 2 move actually should slower. The test has on 66MHzPC-486 type, two kinds of algorithm are apart from next is opposite in different ideal synthesis interpolation every time the actual distance of interpolation made count average. Watch 1 listed two kinds of algorithm synthesize interpolation actually below different condition to measure statistical average. Express algorithm of interpolation of two kinds of 1 parabola to synthesize interpolation actually to measure average 0.

020.

040.

060.

08a=0.

01, b=00.

0199560.

0396690.

0596070.

0792930.

0199550.

0396650.

0595890.

079272a=0.

1, b=00.

0199870.

0396820.

0597440.

0793450.

0199840.

0396800.

0597420.

079339a=1, b=00.

0199920.

0397100.

0598510.

0794520.

0199860.

0397090.

0597480.

079447 　 are measured in the experiment give two kinds of algorithm to insert complementary operation time every time, algorithmic one T1 ＝ 0. 000013s; Algorithmic 2 T2 ＝ 0. 000021s. By above experiment data can be seen, experimental result agrees with theoretical analysis, namely the operation speed of one compares algorithm algorithmic the operation rate of 2 is rapid. But algorithmic fast sex and interpolation precision compare the divide evenly of 2 algorithm is close friends. 4 　 N divides what the pace grows to improve algorithm to be used extensively as 32 personal computer, provided the operation capability with efficient high speed, raise complex rate CNC interpolation algorithm is OK and appropriately, and achieve the goal that raises interpolation precision. The improvement that put forward the N below to divide a pace to grow for this is algorithmic. Grow the basic idea that shows this algorithm with 2 minutes of paces, the parabola interpolation algorithm that above face introduces is exemple, see a picture 2. Graph 2 minutes of 2 conditions are chief improve algorithmic sketch map to set parabola equation to be Y ＝ - Ax2 ＋ Bx (A ＞ 0, b ＞ 0) , a (Xi, yi) nod for foregone interpolation, f synthesizes interpolation quantity for theory, AC ′ has been the parabola tangent that A nods, a, c ′ , b ′ 1 at 3 o'clock in all line, 1 ＝ of the ′ that make AB F, c ′ , c, d ′ at 3 o'clock in all line, this line is parallel at Y axis, ′ of A, D, D at 3 o'clock in all line, parallel at X axis. ／ of F of ＝ of ｜ of ｜ AC ′ 2. C nods the node that is ′ of C ′ D and parabola, CB ″ nods parabola tangent to cross C, b ″ D and parabola are handed in at B dot, ／ of F of ＝ of ｜ of ｜ CB ″ 2. 2 minutes of paces grow basic idea and algorithm one identical, but 2 minutes of paces grow every pace to calculate twice, every time with F ／ 2 for ideal synthesis interpolation is measured. Two hind cipher out is actual interpolation nods B. Pair of algorithm are narrated before the basis one, the trade off study of 2 is same method, can conclude, the precision that uses 2 minutes of paces to grow a law to grow a law to undertake interpolation than synchronization wants tall. Same, 2 minutes of conditions are chief can popularize the N mark that till N cent condition is chief,to 3 minutes of paces 4 minutes of long, paces grow, corresponding interpolation precision can rise. But also can see at the same time, from N cent pace grows N+1 to divide a pace to grow, interpolation time can increase, through be opposite on 66MHz486 type algorithm is mixed algorithmic 2 undertake N-1 divide a pace to grow algorithmic experiment to the N of N-10, reach the following every pace is actual and average interpolation time, see a table 2. Express 2 　 N to divide a pace to grow 　 of interpolation time 　 (Ms) N12345678910 is algorithmic one 0.

0130.

0260.

0370.

0460.

0580.

0690.

0790.

0910.

1000.

112 algorithm 2 0.

0210.

0370.

0510.

0680.

0860.

1010.

1160.

1340.

1490.

164 　 to raise interpolation precision, can consult express 2 medium data, according to the interpolation operation time of real need, choose appropriate N value. Among them interpolation is algorithmic applicably the time split plot design at other curve is basic interpolation is algorithmic. Graph 3 give out flow chart of algorithm of long interpolation of N cent pace. Graph long interpolation of pace of cent of 3 　 N is algorithmic flow chart  -- can use algorithm of interpolation of linear, circular arc, parabola 5 conclusion adopt formula the article is right the trade off study of algorithm of interpolation of two kinds of parabola, knowable algorithm 2 than algorithm the interpolation precision of one wants tall, but algorithmic the operation rate of one is rapidder; In the meantime, grow the research of interpolation algorithm according to dividing a pace to N, can see through choosing proper N, raise interpolation precision effectively. CNC Milling