Piecewise Polynomials (PPs) are utilized in several engineering disciplines, like trajectory planning, to approximate position profiles given in the form of a set of points. While the approximation target along with domain-specific requirements, like Ck -continuity, can be formulated as a system of equations and a result can be computed directly, such closed-form solutions posses limited flexibility with respect to polynomial degrees, polynomial bases or adding further domain-specific requirements. Sufficiently complex optimization goals soon call for the use of numerical methods, like gradient descent. Since gradient descent lies at the heart of training Artificial Neural Networks (ANNs), modern Machine Learning (ML) frameworks like TensorFlow come with a set of gradient-based optimizers potentially suitable for a wide range of optimization problems beyond the training task for ANNs. Our approach is to utilize the versatility of PP models and combine it with the potential of modern ML optimizers for the use in function approximation in 1D trajectory planning in the context of electronic cam design. We utilize available optimizers of the ML framework TensorFlow directly, outside of the scope of ANNs, to optimize model parameters of our PP model. In this paper, we show how an orthogonal polynomial basis contributes to improving approximation and continuity optimization performance. Utilizing Chebyshev polynomials of the first kind, we develop a novel regularization approach enabling clearly improved convergence behavior. We show that, using this regularization approach, Chebyshev basis performs better than power basis for all relevant optimizers in the combined approximation and continuity optimization setting and demonstrate usability of the presented approach within the electronic cam domain.

翻译：分段多项式（Piecewise Polynomials, PPs）在多个工程领域（如轨迹规划）中被用于逼近以点集形式给出的位置曲线。尽管通过将逼近目标与特定领域需求（如Ck连续）联立方程组可直接求解，但此类闭式解法对多项式阶次、多项式基或附加领域需求缺乏灵活性。当优化目标足够复杂时，需要借助数值方法（如梯度下降）实现。由于梯度下降是人工神经网络（ANNs）训练的核心，现代机器学习（ML）框架（如TensorFlow）提供了多种基于梯度的优化器，其适用范围可扩展至ANNs训练任务之外的广泛优化问题。本文方法旨在利用PP模型的通用性，结合现代ML优化器的潜力，用于电子凸轮设计中一维轨迹规划的函数逼近问题。我们直接使用ML框架TensorFlow中提供的优化器（而非通过ANNs框架）对PP模型参数进行优化。本文展示了正交多项式基如何提升逼近与连续性优化性能：通过采用第一类切比雪夫多项式，我们提出一种新型正则化方法，显著改善了收敛特性。实验证明，在组合逼近与连续性优化场景中，采用该正则化方法的切比雪夫基在所有相关优化器下均优于幂基，并验证了该方法在电子凸轮领域的实用性。