Third-order chord error estimation for freeform contour in computer-aided manufacturing and computer numerical control systems

The chord error employed in computer-aided manufacturing and computer numerical control systems is a crucial index to evaluate the machining accuracy of machined parts. It is usually estimated by the second-order method, that is, the osculating circle method. The second-order estimation only takes the curvature of the curve into account, which will bring about great estimation error when applying to freeform curves. In this article, a third-order method that estimates the chord error using conical helices is proposed. By investigating the geometric properties of the conical helix, it is found that there exists a conical helix that has third-order contact with the freeform curve. With the aid of this conical helix, a third-order model for estimating the chord error of freeform curves is developed. Numerical examples of three freeform curves are provided to verify the effectiveness of the proposed estimation model.

A METHOD OF REAL-TIME NURBS INTERPOLATION WITH CONFINED CHORD ERROR FOR CNC SYSTEMS

This paper presents a method of real-time CNC interpolation for free-form NURBS curves. The interpolation algorithm is based on second order Taylor’s expansion with the principle part being a formula for updating the parametric value u after each sampling period. With the updated value of u, a new interpolated point is calculated based on the DeBoor’s algorithm. In this paper, an efficient method of limiting chord error introduced by the interpolation algorithm is also presented with the basic idea of reducing machining feedrate at positions with a radius of curvature smaller than a critical value.

A chord error conforming tool path B-spline fitting method for NC machining based on energy minimization and LSPIA

Piecewise linear (G01-based) tool paths generated by CAM systems lack G1 and G2 continuity. The discontinuity causes vibration and unnecessary hesitation during machining. To ensure efficient high-speed machining, a method to improve the continuity of the tool paths is required, such as B-spline fitting that approximates G01 paths with B-spline curves. Conventional B-spline fitting approaches cannot be directly used for tool path B-spline fitting, because they have shortages such as numerical instability, lack of chord error constraint, and lack of assurance of a usable result. Progressive and Iterative Approximation for Least Squares (LSPIA) is an efficient method for data fitting that solves the numerical instability problem. However, it does not consider chord errors and needs more work to ensure ironclad results for commercial applications. In this paper, we use LSPIA method incorporating Energy term (ELSPIA) to avoid the numerical instability, and lower chord errors by using stretching energy term. We implement several algorithm improvements, including (1) an improved technique for initial control point determination over Dominant Point Method, (2) an algorithm that updates foot point parameters as needed, (3) analysis of the degrees of freedom of control points to insert new control points only when needed, (4) chord error refinement using a similar ELSPIA method with the above enhancements. The proposed approach can generate a shape-preserving B-spline curve. Experiments with data analysis and machining tests are presented for verification of quality and efficiency. Comparisons with other known solutions are included to evaluate the worthiness of the proposed solution. Highlights The presented B-spline tool path fitting method is chord-error conforming. It is numerically stable and hence industrial-strength. The proposed ELSPIA algorithm incorporates stretching energy into LSPIA algorithm. Includes actual machining experiments to validate the worthiness.