Machine Learning Optimized Orthogonal Basis Piecewise Polynomial Approximation
Hannes Waclawek, Stefan Huber
TL;DR
The paper addresses 1D trajectory approximation with piecewise polynomials under $\mathcal{C}^k$ continuity constraints in electronic cams. It proposes a gradient-based optimization framework on an orthogonal Chebyshev basis within TensorFlow, employing a combined loss $\ell = \alpha \ell_{CK} + (1-\alpha) \ell_2$ and a local continuity enforcement CKMIN. Key contributions include a derivative-aware regularization for $\ell_{CK}$, evidence that Chebyshev bases outperform power bases across optimizers, and a local, strictly enforcing continuity strategy suitable for cam profiles, validated on datasets requiring $\mathcal{C}^3$-continuity with degree $d=7$. The approach yields a flexible, explainable method for high-precision cam-like trajectory generation and offers guidance for initialization and regularization in gradient-based polynomial optimization.
Abstract
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.