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（PP）被应用于多个工程领域（如轨迹规划），用于逼近以点集形式给出的位置轮廓。尽管逼近目标与领域特定要求（如Ck连续性）可被表述为方程组并直接计算解，但这种封闭解在多项式次数、多项式基函数或添加更多领域特定要求方面灵活性有限。当优化目标足够复杂时，必然需要采用数值方法（如梯度下降）。由于梯度下降是人工神经网络（ANN）训练的核心，现代机器学习（ML）框架（如TensorFlow）提供了一套基于梯度的优化器，这些优化器可能适用于超越ANN训练任务的广泛优化问题。我们的方法在于利用PP模型的通用性，并将其与现代ML优化器的潜力相结合，用于电子凸轮设计背景下的一维轨迹规划中的函数逼近。我们直接使用ML框架TensorFlow中现有优化器（超出ANN范围）来优化PP模型的参数。本文展示了正交多项式基如何有助于提升逼近和连续性优化性能。通过采用第一类切比雪夫多项式，我们开发了一种新型正则化方法，显著改善了收敛行为。研究表明，使用该正则化方法时，切比雪夫基在组合逼近与连续性优化设置中，对全部相关优化器均优于幂基，并验证了所提方法在电子凸轮领域的实用性。