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Interpolation principle of digital increment

linear interpolation has no approximation error

when interpolating a curve, when approaching with an inscribed chord, the relationship between interpolation error, interpolation period T, feed speed F and the radius of curvature of the curve is:

it can be seen that interpolation period T is related to feed speed F, approximation error and radius of curvature

when f and are constant, the smaller t is, the smaller it is

when it is certain, the smaller t is, the larger f is

therefore, the smaller the T, the better. However, the selection of T is limited by the interpolation operation time and position control cycle

in the actual system, t is fixed, and the trajectory is required by the computer. At this time, in order to meet the error requirements, the value of F must be limited

2. The relationship between interpolation period and interpolation operation time

after the interpolation algorithm of each linear of the system is designed, the longest interpolation operation time of the system is determined. The interpolation period must be greater than the maximum time of the interpolation operation. For time-sharing CNC, the interpolation cycle should generally be more than twice the longest interpolation operation time

3. The relationship between the interpolation cycle and the position control cycle

the interpolation cycle is either equal to the position control cycle or an integral multiple of the position control cycle

second, line interpolation algorithm

in order to simplify the design of the program, the origin of the coordinate system of interpolation calculation is selected at the beginning of the interpolated line

set the straight line OP, O (0, 0) as the starting point. P (Xe, ye) is the end point, the feed speed is f, the feed is along OP, and the interpolation cycle is t, then the synthetic feed L in t is:

l=ft/60 (UM)

let P (Xi, Yi) be an interpolation point, and P (xi+ 1, yi+1) be the next interpolation point, then it can be seen from the geometric relationship:

which of the above two equations is better can be analyzed as follows:

at that time, algorithm (1) should be used, and at that time, algorithm (2) should be used. That is, in the interpolation calculation, always calculate the large coordinate increment first, and then calculate the small coordinate increment. Considering different quadrants, there will be 8 groups of interpolation calculation formulas. In order to facilitate program design, the concept of guide coordinate is introduced, that is, in the interpolation cycle, the coordinate with large feed increment value is defined as guide coordinate g, and the other is non guide coordinate n. After the introduction of guide coordinates, the eight groups of interpolation calculation formulas can be reduced to a group of

III. arc interpolation algorithm

the basic method of arc interpolation using time division interpolation is that the internal chord approximates the arc. As long as the feed speed f is reasonably selected according to the radius, the approximation accuracy can meet the requirements

select the origin of the interpolation calculation coordinate system on the center of the interpolated arc, and take the first quadrant clockwise as an example to discuss the principle of arc interpolation

p (Xi, Yi) is an interpolation point a on the circle, P (xi+1, y i+1) is the next interpolation C, the straight line segment AC (= L) is the composite feed rate of this time, and D is the midpoint of AC, which is the approximation error of this interpolation. From the geometric relationship:

ABC ∽ odym

then there is i= + i/2

then there is cos I =cos (+ i/2) =ym/(r-) = (Yi Yi/2)/(r-)

because Yi and unknown, the following approximate treatment is carried out:

because l is very small, I-1 can be used to replace Yi; Because of R, R can replace r-. Therefore:

cos i = (Yi is the embodiment of the strong alliance of enterprises in which scientific research institutions are stationed - yi-1/2)/r starting point Y0 is obtained by DDA method: y0= L y0/r

how to use algorithms (1) and (2) can be analyzed similar to straight-line interpolation. The conclusion is: first calculate the large coordinate increment, and then calculate the small coordinate increment

reduce the inlet force (load) and outlet load appropriately. Similarly, introduce the concept of guide coordinate, and the 16 groups of interpolation calculation formulas considering forward and reverse and different quadrants can be summarized into two groups:

the case where the formulas for forward and reverse circular interpolation are used in each quadrant

in the derivation of the interpolation formula, approximate calculation is used, and the cos I value will inevitably produce deviation, and the obtained interpolation value will have error. This error: for the trajectory accuracy, because the formula is used in the algorithm, the interpolation point () can always be guaranteed to be on the circle, so it has no impact on the trajectory accuracy

will lead to the fluctuation of synthetic feed rate and uneven speed; It affects the approximation error. When the actual I is less than the accurate I, the approximation error is greater than the given one. However, the nonuniformity coefficient of fluctuation is the largest: Max 0.35%, and the influence is very small

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