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 Pythagorean-hodograph Curve Interpolation for High Speed CNC Machines


­Abstract: The function of the real-time interpolator in a computer numerical control (CNC) machine is to compute a reference point in each sampling interval of the servo system, based on prescribed tool path and feedrate data. By comparing the actual instantaneous machine position, measured by encoders on the machine axes, with the reference point, accurate closed-loop control of the machine position and speed is possible. Modern CNC machines not only provide linear/circular interpolations, but also offer parametric interpolations for Bezier, B-spline, and NURBS curves. These parametric interpolations can reduce feedrate fluctuations, chord errors and machining time in comparison with linear/circular interpolations. These curves are not arc-length parametrized, which makes their interpolators inherently approximate in nature. By contrast, Pythagorean-hodograph (PH) curves admit closed-form analytic reductions of the interpolation integral, yielding real-time interpolators that are essentially exact, and remarkably versatile in terms of feedrate variations. In this project, an integrated look-ahead dynamics-based algorithm for PH curves is proposed. This algorithm considers geometric and servo errors simultaneously. The algorithm consists of three different modules: a sharp corner detection module, a jerk-limited module, and a dynamics module. The sharp corner detection module identifies sharp corners of the curve. The curve is then divided into segments according to sharp corners. The jerk-limited module plans the feedrate profile of each segment according to the constraints on feedrate, acceleration/deceleration, jerk, and chord errors. The module’s first task is to o­btain the feedrate at sharp corners. Both chord errors and curvature information of the curve should be taken into account to achieve this task. After the feedrate at sharp corners is obtained, the second task of the jerk-limited module is to plan the feedrate profile of each segment such that the constraints on feedrate, acceleration/deceleration, and jerk limits are satisfied. To ensure that the contour errors are bounded within the specified value, the dynamics module can further modify the feedrate profile using the contour error equation. The contour error equation is derived based on the closed-loop transfer function of the servo control system. Simulations and experiments will be performed to validate the proposed algorithm. ­

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