Real-time CNC interpolators for Pythagorean-hodograph curves

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 are length s and parameter xi for free-form parametric curves. With the Pythagorean-hodograph (PH) curves, however, the fact that s(xi) 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 are length s, We also consider the case in which V is stipulated to be inversely proportional to the local curvature kappa, and we discuss methods for direct interpolation of the offsets to PH curves.