Real-time CNC interpolators for Pythagorean-hodograph curves Original Research Article

CNC machining requires “interpolation” algorithms that accurately and efficiently generate sequences of reference positions, distributed according to a prescribed feedrate function, along the tool paths. CNC interpolators have traditionally accommodated only linear and circular trajectories, on account of the absence of easily-calculable relations between the arc length s and parameter ξ for free-form parametric curves. With the Pythagorean-hodograph (PH) curves, however, the fact that s(ξ) is merely a polynomial function facilitates an analytic reduction of the interpolation problem to the solution of a sequence of monotone polynomial equations. A few Newton-Raphson iterations usually suffice to compute the reference points corresponding to the unique roots of these equations to high precision, allowing “real-time” execution. We present detailed formulations and analyses for instances where the feedrate V is specified as a constant, linear, or quadratic function of the arc length s. We also consider the case in which V is stipulated to be inversely proportional to the local curvature κ, and we discuss methods for direct interpolation of the offsets to PH curves.