High precision interpolation algorithm for 3D parametric curve generation Dimitris Kiritsis The paper presents an algorithm for general 3D and 2D parametric curve interpolation. This is an incremental step algorithm, and it uses two simple principles: firstly, each selected step has to followa given direction along the curve, and, secondly, each selected step has to be at the closest distance from the curve. The mathematical formulation of the algorithm permits the interpolation of any kind of parametric curve, which is an important aspect of modern CAD/CAM systems. Its software implementation on RISC workstations and the in-house academic tests to date have shown promising results, and it is hoped that it can be appliedin the next generation of computerized numerical control systems. Keywords: interpolation algorithms, numerical control, toollmths Toolpaths generated by part programming are realized by computerized numerical control (CNC) machine tools by means of very small steps along the machines axes. The sizes of these elementary steps are of the order of micrometres, and they specify the machine resolution, which is a factor of its performance in terms of precision. The CNC system of the machine tool calculates the necessary steps and produces pulses that activate the corresponding drivers of the axes motors. The part of the CNC system that is responsible for the step calculation is the so-called interpolator, and the resulting curve is a scale-like one that approximates the designed curve by interpolating the calculated steps around the curve. There are accurate interpolators for straight line segments and circular arcs, and in some cases for parabolic arcs. A circle interpolated by a 4-point interpolation algorithm is shown in Figure 1, and a circle interpolated by an 8-point interpolation algorithm is shown in Figure 2. The step is magnified to show how such algorithms work. Incremental step interpolators were realized in hardware from the very beginning of the NC era (DDA techniques). D6partement de M6canique, Institut de M6canique Appliqu6e et de Construction des Machines, Laboratoire de Conception Assist6e par Ordinateur, Ecole Polytechnique F6d6rale de Lausanne, ME-Ecublens, CH-1015 Lausanne, Switzerland Paper received." 24 May 1993. Revised" 22 September 1993 Recently, with the increasinglevels of performance of computers at an acceptable price, interpolators have been written in software. We can say that a modern CNC unit comprises a computer and software issuing appropriate commands (signals) to guide motor drivers and the other hardware. Straight line and circle interpolators cover a wide range of machining applications, although there is a need for precise machining of other, more complicated shapes. Actually, all other curves and shapes, for example a 2D freeform contour, are first approximated by means of straight line segments which are then used to create a part programme as the input to the CNC unit. Cutter location (CL) data are calculated and passed to the NC unit in predetermined 'sampling periods '1-4. It is clear that the internal interpolator of the NC unit will calculate the steps necessary to generate the straight line segments by means of basic length units (BLUs) and not the curve itself, while the NC unit also takes care of the velocity control. In other words, we accept a lower level of machining accuracy for the finished workpieces. In most of the cases, such finished parts require more treatment (frequently manual treatment) to achieve their designed quality. In this paper, a new high precision algorithm is proposed for the generation of 3D or 2D parametric curves of any kind. It is an incremental step algorithm, and it can thus exploit the machine resolution for finishing operations of complex part shapes. L I T E R A T U R E R E V I E W A N D D I S C U S S I O N As mentioned above, we have to distinguish incremental step interpolators from the 'sampled data', 'reference word '2'5or 'point-to-point' interpolators, which are based on a first approximation of the curve by means of straight line segments connecting points on the curve. The most advanced method for calculatingpoints of parametric curves under machining precision and velocity and acceleration constraints is presented in papers by Huang and Yang 6'7. Their method of command generation of parametric curves is based on the analytical model of a machine tool proposed by Chou and Yang s, and it follows the 'sampled data' interpolation principle. 850 0010-4485/94/11/0850-07 O 1994 Butterworth-Heinemann Ltd Computer-Aided Design Volume 26 Number 11 November 1994