{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T09:52:40Z","timestamp":1777715560094,"version":"3.51.4"},"reference-count":18,"publisher":"SAGE Publications","issue":"10","license":[{"start":{"date-parts":[[1998,10,1]],"date-time":"1998-10-01T00:00:00Z","timestamp":907200000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The International Journal of Robotics Research"],"published-print":{"date-parts":[[1998,10]]},"abstract":"<jats:p>General planar motions with two degrees of freedom differ funda mentally from those with one degree of freedom, in that they may exhibit \"cross-caps\" singularities. At such singularities the trajec tories fail to exhibit four of the generic singularity types normally associated with such motions, namely, lips, beaks, and two of their degenerations. Provided cross-caps are avoided, the union of the bifurcation curves associated with the lips and beaks types are char acterized kinematically as envelopes of a two-parameter family of instantaneous singular lines. In the special class of composite planar motions, these lines are described in terms of the classical instanta neous centers of the constituent motions with 1 DOF, and it is shown that not all generic singularity types can appear on the trajectories. Computer-generated pictures illustrate the theory for examples of rational motions.<\/jats:p>","DOI":"10.1177\/027836499801701004","type":"journal-article","created":{"date-parts":[[2007,3,11]],"date-time":"2007-03-11T03:05:13Z","timestamp":1173582313000},"page":"1068-1080","source":"Crossref","is-referenced-by-count":4,"title":["Singular Aspects of General Planar Motions with Two Degrees of Freedom"],"prefix":"10.1177","volume":"17","author":[{"given":"C.G.","family":"Gibson","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences University of Liverpool Liverpool L69 3BX, Great Britain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D.","family":"Marsh","sequence":"additional","affiliation":[{"name":"Department of Mathematics Napier University Edinburgh EH14 1DJ, Scotland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Y.","family":"Xiang","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of York York, YO1 5DD, Great Britain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[1998,10,1]]},"reference":[{"key":"atypb1","volume-title":"Ebene Kinematik, Mathematische einzelschriften, Band 5","author":"Blaschke, W.","year":"1956"},{"key":"atypb2","volume-title":"Theoretical Kinematics","author":"Bottema, O.","year":"1979"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139172615"},{"key":"atypb4","volume-title":"Instantaneous singular sets for spatial motions","author":"Cocke, M.W."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1115\/1.3254977"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/BF00046883"},{"key":"atypb7","volume-title":"Trajectory singularities of general planar motions","author":"Donelan, P.S.","year":"1996"},{"key":"atypb8","volume-title":"Singular Points of Smooth Mappings. 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