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These transformations fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. Moreover, we fully characterize the constraint varieties of dyads with revolute and prismatic joints in the dual quaternion model. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a \u201cnull quadrilateral.\u201d If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterize such dyads. We provide a complete characterization by introducing a new invariant, the \u201cStudy fibre projectivity,\u201d and we present examples that demonstrate its potential to explain hitherto not sufficiently well-understood phenomena.<\/jats:p>","DOI":"10.1017\/s0263574718000504","type":"journal-article","created":{"date-parts":[[2018,6,27]],"date-time":"2018-06-27T10:00:14Z","timestamp":1530093614000},"page":"1477-1492","source":"Crossref","is-referenced-by-count":3,"title":["The kinematic image of RR, PR, and RP dyads"],"prefix":"10.1017","volume":"36","author":[{"given":"Tudor-Dan","family":"Rad","sequence":"first","affiliation":[]},{"given":"Daniel F.","family":"Scharler","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2601-6695","authenticated-orcid":false,"given":"Hans-Peter","family":"Schr\u00f6cker","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2018,6,27]]},"reference":[{"key":"S0263574718000504_ref22","doi-asserted-by":"publisher","DOI":"10.4171\/138"},{"key":"S0263574718000504_ref9","doi-asserted-by":"publisher","DOI":"10.1115\/1.4029186"},{"key":"S0263574718000504_ref23","doi-asserted-by":"crossref","DOI":"10.5948\/9781614445166","volume-title":"Non-Euclidean Geometry","author":"Coxeter","year":"1998"},{"key":"S0263574718000504_ref1","volume-title":"Theoretical Kinematics","author":"Bottema","year":"1990"},{"key":"S0263574718000504_ref20","unstructured":"M. Pfurner , H.-P. Schr\u00f6cker and M. Husty , \u201cPath Planning in Kinematic Image Space Without the Study Condition,\u201d Proceedings of Advances in Robot Kinematics ( J. Lenar\u010di\u010d and J.-P. Merlet , eds.) (2016). https:\/\/hal.archives-ouvertes.fr\/hal-01339423."},{"key":"S0263574718000504_ref21","doi-asserted-by":"crossref","unstructured":"T.-D. Rad and H.-P. Schr\u00f6cker , \u201cThe Kinematic Image of 2R Dyads and Exact Synthesis of 5R Linkages,\u201d Proceedings of the IMA Conference on Mathematics of Robotics, ISBN 978-0-90-509133-4 (2015).","DOI":"10.19124\/ima.2015.001.16"},{"key":"S0263574718000504_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-007-4620-6_37"},{"key":"S0263574718000504_ref11","doi-asserted-by":"crossref","unstructured":"H.-P. Schr\u00f6cker , M. Husty and J. M. McCarthy , \u201cKinematic Mapping Based Evaluation of Assembly Modes for Planar Four-bar Synthesis,\u201d Proceedings of ASME 2005 29th Mechanism and Robotics Conference, Long Beach (2005).","DOI":"10.1115\/DETC2005-85037"},{"key":"S0263574718000504_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2006.02.001"},{"key":"S0263574718000504_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2013.05.010"},{"key":"S0263574718000504_ref2","volume-title":"Geometric Fundamentals of Robotics","author":"Selig","year":"2005"},{"key":"S0263574718000504_ref26","doi-asserted-by":"crossref","unstructured":"A. Purwar and J. Ge , \u201cKinematic Convexity of Rigid Body Displacements,\u201d Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC\/CIE, Montreal (2010).","DOI":"10.1115\/DETC2010-29064"},{"key":"S0263574718000504_ref3","doi-asserted-by":"crossref","unstructured":"M. Husty and H.-P. Schr\u00f6cker , \u201cKinematics and algebraic geometry,\u201d In Proceedings of the 21st Century Kinematics, The 2012 NSF Workshop ( J. M. McCarthy , ed.) (Springer, London, 2012), pp. 85\u2013123.","DOI":"10.1007\/978-1-4471-4510-3_4"},{"key":"S0263574718000504_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2013.08.004"},{"key":"S0263574718000504_ref19","unstructured":"T. Stigger , Algebraic Constraint Equations of Simple Kinematic Chains Master Thesis (University of Innsbruck, 2015)."},{"key":"S0263574718000504_ref24","doi-asserted-by":"crossref","unstructured":"Z. Li , J. Schicho and H. -P. Schr\u00f6cker , \u201c7R Darboux Linkages by Factorization of Motion Polynomials,\u201d Proceedings of the 14th IFToMM World Congress ( S.-H. Chang , ed.) (2015).","DOI":"10.1115\/1.4031806"},{"key":"S0263574718000504_ref7","doi-asserted-by":"publisher","DOI":"10.1243\/09544062JMES166"},{"key":"S0263574718000504_ref5","unstructured":"K. Brunnthaler , H.-P. Schr\u00f6cker and M. Husty \u201cA New Method for the Synthesis of Bennett Mechanisms,\u201d Proceedings of CK 2005, International Workshop on Computational Kinematics, Cassino (2005)."},{"key":"S0263574718000504_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-00398-6_11"},{"key":"S0263574718000504_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2014.09.035"},{"key":"S0263574718000504_ref25","doi-asserted-by":"publisher","DOI":"10.1115\/1.3267417"},{"key":"S0263574718000504_ref10","unstructured":"L. Zhu , P. Zhao , B. Zi , A. Purwar and Q. J. Ge , \u201cSimultaneous Type and Dimensional Synthesis Approach with Expandable Solution Space for Planar Linkages,\u201d Proceedings of the 14th IFToMM World Congress (2015). http:\/\/www.iftomm2015.tw\/IFToMM2015CD\/PDF\/OS2-029.pdf."},{"key":"S0263574718000504_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2010.10.001"},{"key":"S0263574718000504_ref4","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-658-07618-4","volume-title":"Geometric Modelling and Chain Geometries with Application in Kinematics","author":"Klawitter","year":"2015"},{"key":"S0263574718000504_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-4941-5_41"},{"key":"S0263574718000504_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s10440-011-9634-6"}],"container-title":["Robotica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0263574718000504","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,3]],"date-time":"2023-09-03T10:59:55Z","timestamp":1693738795000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0263574718000504\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,27]]},"references-count":26,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2018,10]]}},"alternative-id":["S0263574718000504"],"URL":"https:\/\/doi.org\/10.1017\/s0263574718000504","relation":{},"ISSN":["0263-5747","1469-8668"],"issn-type":[{"value":"0263-5747","type":"print"},{"value":"1469-8668","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,27]]}}}