[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127874-en":3,"doc-seo-127874-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},127874,2336474459895,"Aria","https://ap-avatar.wpscdn.com/avatar/22000baeef7a5ed0655?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786071322749376916",8,"Research & Report","Machine Learning Optimized Orthogonal Basis Piecewise Polynomial Approximation","Piecewise polynomials (PPs) approximate engineering position profiles from point clouds, while domain requirements such as Ck-continuity can be enforced via closed-form systems—yet such solutions limit flexibility in polynomial degree, basis choice, and additional constraints. When objectives become complex, gradient-based numerical optimization is needed. The approach combines PP models for 1D trajectory planning with modern machine-learning optimizers from TensorFlow, optimizing PP parameters directly without restricting usage to neural networks. Using Chebyshev polynomials of the first kind, an orthogonal-basis regularization improves convergence, and the method is validated in electronic cam design.","arXiv :2403 .08579v3 [ cs .LG] 8 May 2024  \nMachine Learning Optimized Orthogonal Basis Piecewise Polynomial Approximation⋆  \nHannes Waclawek 1 ,2 and Stefan Huber 1  \n1 Josef Ressel Centre for Intelligent and Secure Industrial Automation,  \nSalzburg University of Applied Sciences, Austria  \n2 Paris Lodron University Salzburg, Austria  \n{hannes.waclawek, [stefan.huber](stefan.huber}@fh-salzburg.ac.at)[}](stefan.huber}@fh-salzburg.ac.at)[@fh-salzburg.ac.at](stefan.huber}@fh-salzburg.ac.at)  \nAbstract. 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.  \nIn this paper, we show how an orthogonal polynomial basis contributes to improving approximation and continuity optimization performance.  \nUtilizing Chebyshev polynomials of the first kind, we develop a novel regularization approach enabling clearly improved convergence behavior.  \nWe 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.  \nKeywords: Piecewise Polynomials · Gradient Descent · Chebyshev Polynomials · Approximation · TensorFlow · Electronic Cams  \n⋆ This work was supported by the Christian Doppler Research Association (JRCISIA), the federal state of Salzburg WISS-FH project IAI and the European Interreg ¨Osterreich-Bayern project BA0100172 AI4GREEN. This preprint has not undergone peer review or any post-submission improvements or corrections.  \n2 H. Waclawek, S. Huber  \n1 Introduction  \n1.1 Motivation  \nPPs are of special interest in several science and engineering disciplines, where the latter are particularly interesting, as they come with additional physical constraints. Path and trajectory planning for machines in the field of mechatronics are just two examples. Path planning is the task of finding possible waypoints of a robot or automated machine to move through its environment without collision. Trajectory planning computes time-dependent positional, velocity or acceleration profiles that hold setpoints for controllers of robots or automated machines to move joints from one waypoint to the other. Electronic cams are a subfield of the latter, describing repetitive motion executed by servo drives within industrial machines. In this context, positional, velocity or acceleration profiles are defined as input point clouds and approximated by PPs, which are then processed by industrial servo drives, like B&R Industrial Automation’s ACOPOS series. Conventionally, the approximation target along with domain-specific requirements such as continuity, cyclicity or periodicity are formulated as a system of equations and a closed-form solution is computed. While","cbCaikURu5amLH42","https://ap.wps.com/l/cbCaikURu5amLH42","pdf",1278521,1,16,"English","en",105,"# Introduction\n## Motivation\n# Abstract\n## Keywords","[{\"question\":\"Why are piecewise polynomials used for trajectory planning in engineering?\",\"answer\":\"They approximate position profiles from point sets and can incorporate engineering constraints such as continuity requirements.\"},{\"question\":\"What limitation of closed-form PP solutions motivates the use of optimization methods?\",\"answer\":\"Closed-form results offer limited flexibility in choosing polynomial degrees, polynomial bases, and adding further domain-specific requirements.\"},{\"question\":\"How does the proposed method use TensorFlow in this work?\",\"answer\":\"It reuses TensorFlow’s gradient-based optimizers to directly optimize parameters of a PP model, without confining the approach to training artificial neural networks.\"}]","Machine Learning Optimized Orthogonal Basis Piecewise Polynomial Approximation | PDF",1785942471,40,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"machine-learning-optimized-orthogonal-basis-piecewise-polynomial-approximation","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-optimized-orthogonal-basis-piecewise-polynomial-approximation/127874/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-23","2026-08-05",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why are piecewise polynomials used for trajectory planning in engineering?","Question",{"text":76,"@type":77},"They approximate position profiles from point sets and can incorporate engineering constraints such as continuity requirements.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What limitation of closed-form PP solutions motivates the use of optimization methods?",{"text":81,"@type":77},"Closed-form results offer limited flexibility in choosing polynomial degrees, polynomial bases, and adding further domain-specific requirements.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the proposed method use TensorFlow in this work?",{"text":85,"@type":77},"It reuses TensorFlow’s gradient-based optimizers to directly optimize parameters of a PP model, without confining the approach to training artificial neural networks.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":29,"slug":119},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]