Toolpath Interpolation and Smoothing for Computer Numerical Control Machining of Freeform Surfaces: A Review

Abstract

Driven by the ever increasing demand in function integration, more and more next generation high value-added products, such as head-up displays, solar concentrators and intra-ocular-lens, etc., are designed to possess freeform (i.e., non-rotational symmetric) surfaces. The toolpath, composed of high density of short linear and circular segments, is generally used in computer numerical control (CNC) systems to machine those products. However, the discontinuity between toolpath segments leads to high-frequency fluctuation of feedrate and acceleration, which will decrease the machining efficiency and product surface finish. Driven by the ever-increasing need for high-speed high-precision machining of those products, many novel toolpath interpolation and smoothing approaches have been proposed in both academia and industry, aiming to alleviate the issues caused by the conventional toolpath representation and interpolation methods. This paper provides a comprehensive review of the state-of-the-art toolpath interpolation and smoothing approaches with systematic classifications. The advantages and disadvantages of these approaches are discussed. Possible future research directions are also offered.

Spline Curves Formation Given Extreme Derivatives

“…Vector derivatives form scalar invariants at each regular point of curve p = p(t), p (1) (t) = dp dt = 0. Scalar invariants formed by vector derivatives p (1) ,p (2) ,p (3) ,...,p (m) have to meet certain requirements. They have to be coordinate system independent, i.e., invariant with respect to coordinate system transformation; they have to be independent of transformations of parameter t = f (t * ), where f is an analytic function of form p(t) = p( f (t * )) = p * (t * ), dp * (t * ) = dp dt f (1) , f (1) = d f dt * = 0.…”

“…3. Determine the vector coefficients (2) i A in the segment equation 2 s based on matrix (21). In this case, the boundary conditions of the segment 2 s are the vector derivatives 2 P′ , 2 P ′′ , and 2 P ′′′ at the knot 2 P determined according to item 1 and the given vector derivatives 3 P′ , 3 P ′′ , and 3 P ′′′ at the knot 3 P .…”

“…In machine-building cutting tool trajectory calculation through Hermite cubic spline interpolation is applied in pocket machining of surfaces of arbitrary shape. In this regard, it is noted that that cutting tool acceleration is not continuous, which results in machining defects [2].…”

“…i in the segment equation s 1 based on the matrix (2). In this case, the boundary conditions of the segment s 1 are the given vector derivatives P 1 and P 1 at the knot P 1 and the vector derivatives P 2 and P 2 at the knot P 2 determined according to item 1.…”

An attempt to relate dynamic tracking error to occurring situation based on additional rectilinear motion for five-axis machine tools

“…Furthermore, high values of the velocity, acceleration, and jerk of each motion axis may negatively impact the machining precision. Hence, a lookahead 23 function is widely used for CNC machine tools to adjust the velocity, acceleration, and jerk of each axis, 24,25 as follows: where v i , a i and J i are the velocity, acceleration and jerk of each motion axis. Due to the look-ahead function, the actual tool movement velocity is not completely set by the NC command but is also affected by the adjustments of velocities, accelerations and jerks, so the time series is not a perfect media of establishing relationship between dynamic tracking error and occurring situation.…”