Abstract In CNC machining, G01 codes are widely used to represent the tool path. Directly interpolating these G01 codes is time-consuming and may cause discontinuities. In this paper, we propose a time-spline curve fitting method that combines tool path fitting and feedrate scheduling into a single step for five-axis CNC machining. The input for this method consists of a three-dimensional linear path of the tool tip in the workpiece coordinate system and two-dimensional tool orientations in the machine coordinate system (MCS). The output is a fitted tool path in the MCS represented by a five-dimensional smooth time-parametric B-spline curve, simply referred to as the time-spline curve. The time-spline curve provides not only position information but also kinematic information, including velocity, acceleration, and jerk for each axis, directly derived from the first, second, and third derivatives of the curve. To meet fitting error constraints and axial kinematic constraints, our objective is to find the time-spline curve that is time-optimal. We formulate the optimization problem as a nonlinear optimization model and design a recursive algorithm to solve it. The resulting time-spline curve demonstrates high accuracy and fully utilizes the machine’s kinematic capabilities. Along the tool path defined by the time-spline curve, exact interpolation points can be straightforwardly obtained according to the interpolation period. Simulations and experimental results indicate that the proposed method yields a time-optimal time-spline curve with the desired precision and kinematic constraints. 

Cubic time-spline fitting and interpolation for five-axis CNC machining

Interpolating non-uniform rational B-spline (NURBS) curves often leads to undesirable feedrate fluctuations due to the non-analytic relationship between the spline parameter and arc length. In this paper, we introduce a feedback method that effectively reduces feedrate fluctuations while maintaining simplicity in the interpolation process. Our approach begins by defining the chord length-parameter ratio (CPR), which we then predict for the current interpolation point using Newton’s divided difference interpolation polynomial and historical CPR data. Subsequently, we calculate the compensation length based on tangent and chord directions, and we correct the initial parameter using the second-order Runge–Kutta method. Importantly, our method involves only two calculations of the first derivative of the NURBS curve, making it computationally efficient compared to most approximation methods. To validate our proposed method, we conduct numerical simulations and compare its performance against several approximation methods and a recent feedback method. The results demonstrate that our approach excels in terms of both efficiency and accuracy, making it a promising solution for improving NURBS curve interpolation. 

A novel method for calculating interpolation points of NURBS curves based on chord length-parameter ratio

A novel feedrate scheduling method using modifying S-shaped feedrate profile with a round-off error elimination approach for CNC machining

/pdf/a-c3-continuous-double-circumscribed-corner-rounding-method-16i526uc.pdf

A C3 continuous double circumscribed corner rounding method for five-axis linear tool path with improved kinematics performance

Abstract Non-uniform rational B-spline (NURBS) curve has been widely used in computer numerical control machining due to its superior properties, such as global smoothness and local support. However, because of the nonanalytic relationship between the spline parameter and the arc length, it can be challenging to obtain accurate parameters when the arc length given. Approximation methods are commonly used because of their simplicity, but they can result in feedrate fluctuations that are difficult to avoid. This paper presents a feedback method to reduce feedrate fluctuations while maintaining simplicity. To obtain the initial parameter value of the next interpolation point, the chord length-parameter ratio (CPR) is defined first, and Newton's divided difference interpolation polynomial is applied to predict the CPR of the current interpolation point based on the CPRs of the historical interpolation points. Then, the initial parameter is corrected using the Second-order Runge-Kutta method with the compensation length, which is derived according to the tangent and chord directions at the initial parameter. The proposed method only requires two calculations of the first derivative of the NURBS curve, which is comparable to the computational complexity of most approximation methods. Numerical simulations are conducted to compare the performance of the proposed method with several approximation methods and a recent feedback method. The results demonstrate that the proposed method performs exceptionally well in terms of both efficiency and accuracy. 

A novel feed rate scheduling method with acc-jerk-continuity and round-off error elimination for non-uniform rational B-spline interpolation

An integrated trajectory smoothing method for lines and arcs mixed toolpath based on motion overlapping strategy

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A novel feed rate scheduling method with acc-jerk-continuity and round-off error elimination for non-uniform rational B-spline interpolation

An integrated trajectory smoothing method for lines and arcs mixed toolpath based on motion overlapping strategy