{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:48:37Z","timestamp":1760240917381,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,10,19]],"date-time":"2019-10-19T00:00:00Z","timestamp":1571443200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MCA"],"abstract":"<jats:p>A numerical procedure based on the spectral Tau method to solve nonholonomic systems is provided. Nonholonomic systems are characterized as systems with constraints imposed on the motion. The dynamics is described by a system of differential equations involving control functions and several problems that arise from nonholonomic systems can be formulated as optimal control problems. Applying the Pontryagins maximum principle, the necessary optimality conditions along with the transversality condition, a boundary value problem is obtained. Finally, a numerical approach to tackle the boundary value problem is required. Here we propose the Lanczos spectral Tau method to obtain an approximate solution of these problems exploiting the Tau toolbox software library, which allows for ease of use as well as accurate results.<\/jats:p>","DOI":"10.3390\/mca24040091","type":"journal-article","created":{"date-parts":[[2019,10,21]],"date-time":"2019-10-21T03:40:29Z","timestamp":1571629229000},"page":"91","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Solving Nonholonomic Systems with the Tau Method"],"prefix":"10.3390","volume":"24","author":[{"given":"Alexandra","family":"Gavina","sequence":"first","affiliation":[{"name":"Laborat\u00f3rio Engenharia Matem\u00e1tica, Instituto Superior Engenharia Porto, R. Dr. Ant\u00f3nio Bernardino de Almeida 431, 4200-072 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3512-5930","authenticated-orcid":false,"given":"Jos\u00e9 M. A.","family":"Matos","sequence":"additional","affiliation":[{"name":"Laborat\u00f3rio Engenharia Matem\u00e1tica, Instituto Superior Engenharia Porto, R. Dr. Ant\u00f3nio Bernardino de Almeida 431, 4200-072 Porto, Portugal"},{"name":"Centro de Matem\u00e1tica, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal"}]},{"given":"Paulo B.","family":"Vasconcelos","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal"},{"name":"Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, s\/n, 4200-464 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1002\/rnc.813","article-title":"Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control","volume":"13","author":"Fontes","year":"2003","journal-title":"Int. J. Robust Nonlinear Control IFAC Affil. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1109\/37.476384","article-title":"Developments in nonholonomic control problems","volume":"15","author":"Kolmanovsky","year":"1995","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_3","unstructured":"Athans, M., and Falb, P.L. (2007). Optimal Control: An Introduction to the Theory and Its Applications, Courier Corporation."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF02243435","article-title":"An operational approach to the Tau method for the numerical solution of non-linear differential equations","volume":"27","author":"Ortiz","year":"1981","journal-title":"Computing"},{"key":"ref_5","unstructured":"Angeles, J., and Kecskemethy, A. (2014). Kinematics and Dynamics of Multi-Body Systems, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"De Luca, A., Oriolo, G., and Samson, C. (1998). Feedback control of a nonholonomic car-like robot. Robot Motion Planning and Control, Springer.","DOI":"10.1007\/BFb0036073"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1093\/imamat\/17.1.85","article-title":"The Lanczos tau-method","volume":"17","author":"Coleman","year":"1976","journal-title":"IMA J. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1002\/sapm1938171123","article-title":"Trigonometric interpolation of empirical and analytical functions","volume":"17","author":"Lanczos","year":"1938","journal-title":"J. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s11786-016-0269-x","article-title":"Towards a Lanczos\u2019Tau-method toolkit for differential problems","volume":"10","author":"Trindade","year":"2016","journal-title":"Math. Comput. Sci."},{"key":"ref_10","first-page":"305","article-title":"Spectral Lanczos\u2019 Tau Method for Systems of Nonlinear Integro-Differential Equations","volume":"Volume 1","author":"Constanda","year":"2017","journal-title":"Integral Methods in Science and Engineering"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s11786-016-0265-1","article-title":"Improving the Accuracy of Chebyshev Tau Method for Nonlinear Differential Problems","volume":"10","author":"Gavina","year":"2016","journal-title":"Math. Comput. Sci."},{"key":"ref_12","unstructured":"Yih, C.C., and Ro, P.I. (1996, January 22\u201328). Near-optimal motion planning for nonholonomic systems using multi-point shooting method. Proceedings of the IEEE International Conference on Robotics and Automation, Minneapolis, MN, USA."}],"container-title":["Mathematical and Computational Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2297-8747\/24\/4\/91\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:27:49Z","timestamp":1760189269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2297-8747\/24\/4\/91"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,19]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["mca24040091"],"URL":"https:\/\/doi.org\/10.3390\/mca24040091","relation":{},"ISSN":["2297-8747"],"issn-type":[{"type":"electronic","value":"2297-8747"}],"subject":[],"published":{"date-parts":[[2019,10,19]]}}}