Closed-Form Quintic B-Spline Reconstruction via Higher-Order Derivative Degeneration for Trajectory Smoothing

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Abstract
In industrial trajectory planning and real-time motion control, quintic B-splines are widely used for corner smoothing owing to their local support and high-order continuity. However, existing evaluation methods mainly rely on basis-function recursion or the de Boor algorithm, with limited attention paid to the analytical properties of fixed-topology continuity-constrained structures. This study reveals that, under geometric symmetry and C3 continuity constraints at the junction points, higher-order derivative control-point structures undergo progressive geometric degeneration, whereby second- and third-order derivatives reduce to one-dimensional forms governed by a single direction. Based on this degeneration property, a closed-form reconstruction method for fixed-topology quintic B-spline corner smoothing is developed, yielding unified closed-form expressions for curve position and first- to third-order derivatives. Mathematical analysis proves equivalence between the proposed reconstruction and the original quintic B-spline representation. Numerical validation and efficiency evaluation demonstrate machine-precision consistency with conventional B-spline evaluation while achieving an approximately 3–7-fold speedup in curve and derivative evaluation. System-level trajectory-planning simulations further confirm reduced geometric computation load. The proposed method provides an efficient analytical evaluation framework for real-time trajectory planning and demonstrates how continuity constraints can be exploited to derive efficient analytical spline representations.
Funding Information
  • National Key Research and Development Program of China (2024YFB4707700)
  • National Key Research and Development Program of China (2025YFC3408800)
  • National Natural Science Foundation of China (U25A2024)
  • National Natural Science Foundation of China (T2596042)
  • National Natural Science Foundation of China (62333021)
  • CAS Project for Young Scientists in Basic Research (YSBR-041)
  • United Foundation for Dalian Institute of Chemical Physics, Chinese Academy of Sciences and Shenyang Institute of Automation, Chinese Academy of Sciences (DICP&SIA UN202504)
  • Foundation of the State Key Laboratory of Robotics and Intelligent Systems (2025-Z02-08)
  • Foundation of the State Key Laboratory of Robotics and Intelligent Systems (2025-Z16)
  • Fundamental Research Project of SIA (2024JC1K02)

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