Huixuan Zhu

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.

Piezoelectric inkjet printing technology, known for its high precision and cost-effectiveness, has found extensive applications in various fields. However, the issue of residual vibration significantly limits its printing quality and efficiency. This paper presents a method for suppressing residual vibration based on the particle swarm optimization (PSO) algorithm. Initially, an improved PI model considering the nonlinear hysteresis characteristics of piezoelectric ceramics is established, and the model is identified through a strain gauge circuit to ensure its accuracy in describing the nonlinear hysteresis characteristics. Subsequently, a dynamic model of the piezoelectric inkjet printing system is constructed, with precise parameter identification achieved using the self-induction principle. This enables precise simulation of residual vibration. Finally, the driving waveform is optimized based on the PSO algorithm, with iterative calculations employed to find the optimal combination of driving waveform parameters, effectively suppressing residual vibration while ensuring sufficient injection energy. The results indicate that this method significantly reduces the amplitude of residual vibration, thereby effectively enhancing printing quality and stability. This research offers a novel solution for residual vibration suppression in piezoelectric inkjet printing technology, potentially advancing its applications in printing and biofabrication.