An analytical method for corner smoothing of five-axis linear paths using the conformal geometric algebra

The chain of linear path segments is widely used in five-axis machining, but the tangential discontinuity of the path reduces machining efficiency and accuracy. This paper proposes an analytical continuous corner smoothing method based on the conformal geometric algebra. Based on the locations of the tooltip point and a tool axis point, the toolpath is represented in the 6-dimensional Euclidean space. With the help of the representation of conformal transformations and circles in conformal geometric algebra, the path in the 6-dimensional space is smoothed by continuous circle-based splines under the constraints of the maximum deviation error tolerance. The proposed approach can generate a smooth toolpath that passes through the discrete cutter locations given in the original linear segments analytically. The cycle time of machining is improved thanks to the small curvature and continuity of the toolpath. The effectiveness and efficiency of the proposed method are validated by simulations and experiments.