# Interpolation, the art of connecting the dots: understanding CNC begins with interpolation. Engineers, machinists, and operators need to keep returning to interpolation to make sense of contemporary developments.

CNC workers are served by three ideas about interpolation.

First there is path interpolation. The CNC directs the cutting tool to interpolate a path in the work envelope of the machine tool. This is akin to the idea that interpolation seeks to find a path that satisfies clearly described geometric constraints, such as the path has to pass through points.

Second, there is the idea that interpolation seeks a finer resolution by finding points between known points. If the eight oz. drink is 50 cents and the 16 oz. drink is a dollar, you have just done a linear interpolation if you think the 12 oz. drink is 75 cents.

Third, the CNC samples an interpolation function on a time grid--called the IPO cycle--and outputs incremental setpoints to the position servos. This third meaning is probably what most CNC workers have in mind when they point to the CNC and say, "That thing interpolates."

I prefer to point to the work envelope of the machine tool and say, "That thing that you call CNC directs the cutting tool to interpolate a path."

Not a Good Idea

An idea that does not serve the CNC worker is one that regards interpolation as two or more axes in motion at the same time. Certainly this is a prerequisite to interpolate a path in 2- or 3-space but the mere fact that two or more axes are in motion at the same time is not necessarily interpolation.

In turning, a cut-off tool can be in motion while the X and Z servos interpolate a path. Helical interpolation is interpolation in the XY plane while the Z-axis is positioning simultaneously.

Three Plus Two May Not Equal Five

Today, 5-axes--aerospace--interpolation is really 3-axes interpolation and 2-axes synchronous positioning. The X, Y, and Z axes trace out a path at the programmed feedrate while simultaneously, the orientation axes position to their block end points. This is called 3-2 interpolation. It is the reason that a block that contains path axes and orientation axes can be programmed in G94 inches/minute feed instead of G93 inverse time feed.

The path axes are specified with the command FGROUP. FGROUP(X, Y, Z) means that any axes programmed in a block that are not the X, Y, or Z axis will be synchronously positioned. In the aforementioned 5-axes aerospace contouring, if A and B are the orientation axes, and if you want to interpolate in 5-space, the FGROUP command would be FGROUP(X, Y, Z, A, B).

However, if you program this feed group, welcome back to memory lane and the way things were done decades ago when the way out of the erratic behavior of the orientation axis was to program the inverse of the time--commanded with G93--to complete a block's distance to go.

There is nothing wrong with 5-D feed groups and G93, but the current and future development in CNC functionality for aerospace contouring is 3-2 because it allows a more complete synthesis of electrical servo control with the mechanical linkages for faster, smoother, and more accurate machining.

Interpolation in Common

Aerospace machining is usually associated with interpolating paths of constantly changing curvature. This is true of mold and die as well, except that the latter is just as often not done with tool orientation.

Historically, in either case, curves of constantly changing curvature have been represented as polylines and posted to G01 blocks. G01 is an explicit representation of a straight line interpolation function.

In modern CNC, you can specify a spline algorithm to find the interpolation functions. Instead of specifying the interpolation function you specify the algorithm. The algorithm acts on the geometric data of the part program to interpolate a curve that is rendered as a sequence of parametric polynomial interpolation functions that blend smoothly with one another.

A controller such as the Sinumerik 840D CNC by Siemens samples the parametric polynomial on the IPO time grid to output setpoints to the position servos. This is what Sinumerik CNC workers mean when they say the 840D does polynomial interpolation.

Rendering Polynomials

With regards to the spline algorithms, the most commonly used algorithm is the so-called compressor function CompCad. CompCad is programmed ahead of a sequence of the linear blocks of a polyline. It interpolates a curve so all specified points are within a tolerance band and CompCad renders this curve as a sequence of parametric polynomials. Thus, CompCad makes one interpolation function out of many linear blocks. This happens on the fly as the program is executed. CompCad allows the shops to have one leg in the old ways of linear block programming and one in the new ways of spline.

It is important to note that NC/CNC was on the scene long before computer-aided geometric design was developed. In this regard, NC/CNC got the cart before the horse, since, after all, design does come before manufacture.

The Heart of the '80s

By the late 1980s, spline algorithms, and especially the nurb-spline, had become the heart of computer-aided design. The time had come for CNC to adapt to the geometry of design and the 840D by Siemens rose to this challenge with a reinvention of CNC from the inside-out on a foundation of polynomial methods.

Spline algorithms are software engines that interpolate curves. The contemporary development of numerically-directed interpolation is the inclusion of these algorithms to enable a seamless integration of the geometry of CNC to the geometry of design.

As "Easy" as Connecting the Dots

As you can see, a sophisticated understanding of contemporary CNC requires a grasp of interpolation as the act of finding a path that satisfies clearly specified geometric constraints. Anyone who has ever connected the dots of a page of a child's workbook has done interpolation. It can be as simple as that, or it can be more sophisticated like finding a smooth curve that passes within a tolerance band of any one dot.

Norman Bleier manages Siemens machine tool applications engineering in support of U.S. OEMs. His special interest is technology on the shop floor.

Norman Bleier

Siemans Energy and Automation, Machine Tool Business

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Controlling Interest is a new feature of MAN. It is an information column by Norman Bleier, one of the acknowledged pioneers and leaders in the CNC field.