FANUC involute interpolation: why base-circle radius cannot replace curvature radius in a speed-limit calculation
Using 30i-B involute interpolation, an original derivation separates base-circle radius, distance from the center and local curvature radius. Acceleration calculations at three positions explain why speed limits become more restrictive near the base circle.
Contents
An involute has a constant base-circle radius, but its local curvature changes continuously. Substituting the programmed R directly as a circular radius in an acceleration calculation can substantially underestimate acceleration near the base circle.
Which radius does the original function use?
The 30i/31i/32i-B machining-center manual B-64484EN-2/03, §3.1, pages 15–20 (public mirror of the original manual) describes G02.2/G03.2 involute interpolation: R specifies the base-circle radius, whereas feed control near that circle depends on the path's radius of curvature. This discussion concerns systems equipped with the relevant function; the same G-code designation does not establish support across all series.
Separate three lengths with an original curve
Place the base-circle center at the origin, use radius Rb=40 mm and start unwinding along positive X. Describe the involute with a radian variable u, taking u>0:
x=Rb(cos u+u sin u)，y=Rb(sin u−u cos u).
Differentiating with respect to u gives x′=Rb·u·cos u and y′=Rb·u·sin u, so arc length changes by Rb·u per unit u; the tangent direction angle is u. Hence the local curvature radius is ρ=ds/du=Rb·u. Meanwhile, distance from the point to the base-circle center is d=√(x²+y²)=Rb√(1+u²). Rb, d and ρ are different quantities.
| u, radians | Distance from center d | Curvature radius ρ | Normal acceleration at 20 mm/s |
|---|---|---|---|
| 0.1 | About 40.200 mm | 4 mm | 100 mm/s² |
| 0.4 | About 43.081 mm | 16 mm | 25 mm/s² |
| 0.8 | About 51.225 mm | 32 mm | 12.5 mm/s² |
Why the same F is harder to maintain near the base circle
The table converts 1200 mm/min to 20 mm/s and applies the kinematic relationship a=v²/ρ. Incorrectly using Rb=40 mm yields 10 mm/s² at all three positions, underestimating the first by a factor of ten. As an ideal constraint only, an allowable normal acceleration of 50 mm/s² requires the first position's speed to satisfy v≤√(50×4)=14.142 mm/s, or about 848.528 mm/min. This is a mathematical example, not a machine parameter recommendation.
Moving the curve's starting point away from the small-curvature-radius region near u=0 has a different effect from reducing F throughout the program. Whether the start may be changed depends on the part contour and toolpath joining requirements. Being “still 40 mm from the center” does not make this a gentle, large-radius bend.
Keep geometric calculations separate from actual control
This example excludes cutter-radius compensation and acceleration/deceleration transitions. The referenced section also distinguishes compensated feed from near-base-circle control. With compensation, identify the actual controlled path before applying any uncompensated contour's ρ. The point u=0 is a singular endpoint of this derivation; do not linearly extrapolate the table to it. Further advanced-function analysis is in the OPTION section.