Point-to-Point Reference Trajectories Generation Using Frequency-Aware B-Splines/NURBS

Implementing Non-Uniform Rational B-Splines in motion controllers involves challenges like computational complexity, parameterization issues, high-fidelity interpolation needs, real-time processing requirements, hardware limitations, system integration, error management, and the demand for user-friendly design tools, all of which necessitate advanced algorithms, robust hardware, and effective interfaces for successful implementation. Expanding the standard sine profile allows for the creation of a generalized, symmetric, frequency-sensitive basis function for generating point-to-point motion trajectories, while other functions like polynomials, sigmoids, and harmonic models can also be utilized. This study employs the basis function at the jerk level to meet the motion system’s constraints, and time shifts create B-spline-like and NURBS-like profiles for CAD and motion control, eliminating the need for interpolation and curve fitting. Utilizing vector notation and algorithms reduces computational demands on both the CAD and the motion controller sides, enabling real-time implementation. Frequency-sensitive B-spline profiles enhance both repetitive and random walk motions, facilitate data collection, and help segment the workspace into regions with unique frequency characteristics. A model-free approach addresses position-dependent errors, and experimental results validate the effectiveness of these techniques. Robust statistical models based on Multivariate Analysis of Variance are developed to both characterize and assess the performance of a precision motion system under the proposed framework. For example, the mean positioning error of the motion system improved from $2.29\,\mu \mathrm {m}$ to $0.717\,\mu \mathrm {m}$ at the maximum operating frequency of $90\,\mathrm {rad/s}$ under the model-free approach. Note to Practitioners—This paper introduces novel definitions for B-spline and NURBS entities, which are widely used in computer-aided design, manufacturing, and robotics applications. Upon implementation, these new definitions enable practitioners to create computer-based parts, such as paths or artworks, while adhering to the intended machine capabilities for physical realization. By considering machine limitations, practitioners can anticipate how they will impact product quality. Modifications to parts or assigned machines can then be made, potentially reducing rework. However, it is important to note that this study focuses solely on the kinematic constraints of machines, without considering the computational overhead balance between computers and machines. Additionally, the goal is not to perfectly match the geometry obtained using standard B-splines and NURBS entities. Future investigations will explore issues related to contouring, tracking motions, and other aspects.