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# Interpolation

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### Interpolation

1. 1. Interpolations for CNC Machining April 29, 2009 Md. Shafayet Hossain Bhuiya MME, UWO
2. 2. Outlines <ul><li>Linear Interpolation </li></ul><ul><li>Circular Interpolation </li></ul><ul><li>Parametric Interpolation </li></ul><ul><li>Parametric versus Linear Interpolator </li></ul><ul><li>Spline </li></ul><ul><li>NURBS </li></ul><ul><li>NURBS versus other Interpolators </li></ul><ul><li>Remarks </li></ul>
3. 3. Linear Interpolation <ul><li>The idea is to approximate a given curved tool path by a set of line segments while satisfying desired tolerance and feed rate. </li></ul><ul><li>The gap between the original curve and chordal line segments is called the maximum position error. </li></ul>
4. 4. Advantages with Linear Interpolation <ul><li>A curved cutter path is approximated by straight line segments connecting a finite set of cutter location points on the curve. </li></ul><ul><li>Along with other information, such as the feed rate, spindle speed and lubrication requirements , these data are sent to the controller of a CNC machine. </li></ul><ul><li>Linear interpolation technique has been widely used in CNC machining for its simplicity. </li></ul><ul><li>Linear interpolation requires less computation . </li></ul>
5. 5. Problems with Linear Interpolation <ul><li>In the early days of NC, this method was just fine . As CNC technology has progressed, however, the linear interpolation paradigm has come to be a limiting factor on the contour or free form machining process . </li></ul><ul><li>It is not accurate- forcing the machinist to work from an approximation of the true work piece surface geometry rather than the real thing. So before the part program ever gets out of the CAD/CAM department, the tool path has already deviated from the ideal geometry. </li></ul><ul><li>It is not smooth —what should be a continuous contour is instead machined as a series of facets. It will require laborious hand finishing that degrades final accuracy even more. </li></ul>
6. 6. <ul><li>It is not fast enough- The NC programmer sets the tolerance small to keep the line segments short and make the path as accurate and smooth as possible . But since each line segment corresponds to a block in the part program, size increases. Some sort of external memory buffer at the machine tool is required or blocks can not be executed fast enough to keep up with desirable feed rate. It will hamper the cutting speed. If it runs intermittently, surface finish will degrade. </li></ul>Problems with Linear Interpolation
7. 7. Circular Interpolation <ul><li>Circular Interpolation is one way, CNC’s employ toolpath that are not straightline . </li></ul><ul><li>When CNC executes circular interpolation blocks, feed rates of individual axes are not constant . They vary continuously throughout the move. So, the resulting path of the tool tip traces some portion of a circle. So, a single block can convey all the information necessary for the move. </li></ul><ul><li>In contrast, using lines to define a fair representation of some circular arc- a quarter of a circle, would demand several program blocks in sequence. </li></ul>
8. 8. Parametric Interpolation <ul><li>For accurate reproduction, one interpolator has received considerable attention is parametric interpolator. </li></ul><ul><li>Unlike other interpolators, this generates real time position commands that lie on the original curves . </li></ul><ul><li>Since segmentation is not required, memory and communication bottlenecks can be avoided . </li></ul>
9. 9. Parametric Versus Linear Interpolator <ul><li>Memory Size Requirement: </li></ul><ul><li>Linear interpolation needs to allocate major portion of its memory for the storage of CL data. For parametric interpolator, memory size required is proportional to the number of individual curve segments. </li></ul><ul><li>When the tolerance requirement is tight , in terms of memory size, parametric interpolator is greatly favoured.The difference of memory size, in this magnitude is very significant. When tolerance is very large , the advantages of the parametric interpolator gradually diminishes. </li></ul>
10. 10. Parametric Versus Linear Interpolator <ul><li>Feedrate fluctuation: </li></ul><ul><li>From the experiment, it was found that feedrate fluctuation of linear interpolator can be as high as 12.5%, while that of our parametric interpolator is only about 0.2%. </li></ul><ul><li>CPU Time: </li></ul><ul><li>Linear interpolator requires a shorter calculation time (simple algorithm) than parametric interpolators. This simplicity made linear interpolator feasible when computer was slower in the past. Now, computing speed is no longer a barrier to the implementation of complex interpolator such as parametric interpolator . </li></ul>
11. 11. Parametric Versus Linear Interpolator <ul><li>Position Tracking Error: </li></ul><ul><li>In case of linear interpolator, servo system instructed to track along a series of line segments. At the junction of line segment, tracking velocity is subject to a sudden change of direction. So, commands generated with the linear interpolator always lead to a larger tracking error. </li></ul><ul><li>For parametric interpolator does not cause this type of problem as it evenly distributes the change of direction along the entire curve segment. So, it is obvious that the magnitude of jerk (tweak or pull) is more for linear interpolator. </li></ul>
12. 12. B-Spline Curves
13. 13. NURBS <ul><li>A NURBS curve can be defined by- </li></ul>
14. 14. NURBS
15. 15. Reasons for Using NURBS <ul><li>It permits manipulation of weight and control points . So, one common mathematical forms for both standard analytical shapes and free formed shapes. </li></ul><ul><li>NURBS uses fewer parameters (control points, knots, vector and weights) to represent the designed curves (or surfaces), so that CNC memory can be saved. </li></ul><ul><li>Less data transmission between CAD/CAM and CNC machine results in faster manufacturing speed such that the goal of high-speed machining can be achieved. </li></ul><ul><li>Local controls and geometric intuitiveness are also noticeable. It is a generalization of non-rational B-splines and non-rational and rational Bezier curves and surfaces. </li></ul>
16. 16. NURBS Versus Other Interpolators <ul><li>NURBS interpolation overcomes the problem of linear interpolation ( slow execution, dense data file ) by defining complex geometry with fewer blocks. </li></ul><ul><li>In many cases where comparable programs defined through line segments and NURBS are run through a high speed controller capable of executing either program effectively, NURBS program allows a higher average feed rate. </li></ul><ul><li>When CNC executes a circular interpolation block, the feed rate of individual axes are not constant. They vary continuously throughout the move. </li></ul>
17. 17. NURBS Versus Other Interpolators <ul><li>NURBS interpolation accomplishes something similar. Feed rate of each individual axis also varies continously throughout a NURBS block, so the precise co-ordination between the varying axes results in tool tip motion that follows the intended curve. </li></ul><ul><li>Circular interpolation is generally limited to two axis moves, NURBS interpolation can run for three axes or more. </li></ul><ul><li>NURBS interpolation is not only limited to circular moves. </li></ul>
18. 18. NURBS Versus Other Interpolators <ul><li>For chord version, acceleration or deceleration from one chord to the next introduces a cycle time delay. The constraints are less constraining for NURBS where feed rate changes more gradually resulting less cycle time for machining. </li></ul>
19. 19. Types of Parametric NURBS Interpolator <ul><li>Uniform Increment Interpolator: Uniformly increases u. But obtaining optimum increment size is difficult. To overcome this, small increment of u is preferred. But this will increase machining time. </li></ul><ul><li>Constant Feedrate Interpolator: Uses Taylor’s series. Constant feedrate should be selected in such a way that Chord error can be avoided . </li></ul><ul><li>Variable Feedrate Interpolator: Variable feedrate based on the instantaneous curvature is used. It is preferred for constant material removal rate . </li></ul>
20. 20. Remarks <ul><li>If we are machining geometrically simple parts, NURBS interpolation can not help us. As long as the line segments are short enough for linear interpolation, they can deliver same accuracy as NURBS. But when accuracy is required, NURBS can deliver it in a more efficient way. </li></ul><ul><li>Currently, aerospace, automobile, die and mold and jewelry industries are the prominent users of free form shapes designed by NURBS .This will expand other applications also. </li></ul><ul><li>Present research is focused on removing the drawbacks of parametric interpolators with various models. Another area requiring more attention is machining curves and surfaces with 5 axis CNC machining. </li></ul>