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<p>Smooth tool orientation planning is essential for improving the machining precision and efficiency in 5-axis CNC machining. A widely adopted strategy is to first designate a set of key tool orientations at selected toolpath positions and then interpolate the rest, thereby transforming the full-path planning problem into a more manageable task with reduced computational complexity. Classical quaternion-based interpolation techniques, such as Spherical Linear Interpolation (SLERP), are fundamental in this field but cannot ensure high-order continuity in multi-orientation interpolation, which is a major barrier to industrial adoption in 5-axis CNC machining. In this paper, we propose a novel Smooth B-spline-based Quaternion Interpolation (SBQI) method with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^3\)</EquationSource> </InlineEquation> continuity to enhance the tool orientation smoothness in 5-axis CNC machining, and compare its performance with conventional quaternion-based methods. The proposed method begins by mapping the key tool orientations to unit quaternions, representing rotational transformations from a reference configuration. The obtained quaternions are subsequently transformed into Rotation Vectors (RVs) via quaternion logarithmic and exponential mappings through Lie algebra operations, enabling more efficient manipulation in reduced-dimensional space. Based on this mapping, the interpolation is thus formulated as a smooth transition task between RVs using a quintic B-spline curve, achieved by efficiently solving a linear system that minimizes the curve’s bending energy. The resultant interpolated RVs are then converted back to quaternion space through inverse mapping, ultimately reconstructing smooth Cartesian-space tool orientations. Comparative simulation and experimental results confirm that the proposed SBQI method outperforms conventional quaternion interpolation in improving machining kinematics and accuracy through enhanced tool movement smoothness and reduced cycle times.</p>

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SBQI: a smooth b-spline-based quaternion interpolation method for \(C^3\)-continuous tool orientation planning in 5-axis CNC machining

  • 期刊论文

  • 发表日期：

    2025年12月9日

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  • Lei Wu,
  • Huilin Lyu,
  • Jinting Xu,
  • Yuwen Sun

摘要

Smooth tool orientation planning is essential for improving the machining precision and efficiency in 5-axis CNC machining. A widely adopted strategy is to first designate a set of key tool orientations at selected toolpath positions and then interpolate the rest, thereby transforming the full-path planning problem into a more manageable task with reduced computational complexity. Classical quaternion-based interpolation techniques, such as Spherical Linear Interpolation (SLERP), are fundamental in this field but cannot ensure high-order continuity in multi-orientation interpolation, which is a major barrier to industrial adoption in 5-axis CNC machining. In this paper, we propose a novel Smooth B-spline-based Quaternion Interpolation (SBQI) method with \(C^3\) continuity to enhance the tool orientation smoothness in 5-axis CNC machining, and compare its performance with conventional quaternion-based methods. The proposed method begins by mapping the key tool orientations to unit quaternions, representing rotational transformations from a reference configuration. The obtained quaternions are subsequently transformed into Rotation Vectors (RVs) via quaternion logarithmic and exponential mappings through Lie algebra operations, enabling more efficient manipulation in reduced-dimensional space. Based on this mapping, the interpolation is thus formulated as a smooth transition task between RVs using a quintic B-spline curve, achieved by efficiently solving a linear system that minimizes the curve’s bending energy. The resultant interpolated RVs are then converted back to quaternion space through inverse mapping, ultimately reconstructing smooth Cartesian-space tool orientations. Comparative simulation and experimental results confirm that the proposed SBQI method outperforms conventional quaternion interpolation in improving machining kinematics and accuracy through enhanced tool movement smoothness and reduced cycle times.