FANUC exponential interpolation: why rotary travel decreases over equal axial distances
The 16i/18i-MA exponential-interpolation relationship gives different rotary increments over equal axial intervals. Its formula constant is distinguished from the segment length at address K, and a linear approximation exposes angular interpolation error.
Contents
A constant-helix-angle groove on a tapered tool cannot simply reuse the constant-lead relationship for a cylinder. Equal axial advances may require progressively changing rotary travel. Exponential interpolation is one advanced function for this nonlinear relationship.
Separate the formula constant from the command address
The FANUC 16i/18i-MA machining-center operator's manual B-63014EN/02, §4.11, pages 71–74 (public mirror of the original manual) gives the exponential relationship for G02.3/G03.3. To avoid confusion with command address K, this article denotes the formula's angular scaling constant by c:
c=tanJ/tanI；θ(X)=c·ln[1+X·tanI/R].
I is the taper angle, J the helix angle and R the geometric constant defined for this function; θ is in radians. Positive rotation is assumed here. X must use the relationship's coordinate reference, rather than arbitrarily resetting zero at the start of every segment. Program address K specifies segmentation length on the linear axis and must not be replaced by c.
Three 5 mm axial intervals require different angles
An original example uses I=5°, J=30° and R=10 mm. The relationship gives θ(0)=0°, θ(5)≈16.188°, θ(10)≈31.712° and θ(15)≈46.623°. Table differences use unrounded results:
| X interval | Axial length | Rotary-angle increment |
|---|---|---|
| 0→5 mm | 5 mm | About 16.188° |
| 5→10 mm | 5 mm | About 15.524° |
| 10→15 mm | 5 mm | About 14.911° |
Incorrectly substituting X=5 anew for each segment gives 16.188° every time and a total of about 48.565°, exceeding the correct endpoint by about 1.942°. The error comes from treating a global function value as a local increment, not from decimal rounding.
The derivative explains the trend
Differentiation gives dθ/dX=tanJ/[R+XtanI]. With positive I and J in this example, the denominator grows with X, so the angle needed over an equal axial distance decreases. Converting the coordinate derivative to rotary-axis speed also requires multiplication by actual axial speed and conversion from radians to degrees. The program's combined feed F cannot simply be treated as X-axis speed.
The domain also matters: 1+XtanI/R must be positive. This is the logarithmic relationship's mathematical condition; the table's trend cannot be extended across a singularity.
Why segmentation affects the contour
For an offline approximation comparison only, connect the angle values at X=0 and 10 by a straight line. At X=5, the linear prediction is about 15.856°, approximately 0.332° below the correct 16.188°. This is angular interpolation deviation, not a directly established workpiece-surface error.
The referenced section says segment length affects form accuracy, while an excessively small length may cause stops during interpolation. “Smaller is always better” therefore does not follow. This example explains geometry and segmentation, without supplying a machine-entry K value, compensation mode or machining program. Applicability remains limited to that edition and the machine's configuration. Further analysis is in the OPTION section.