{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T10:07:08Z","timestamp":1777716428466,"version":"3.51.4"},"reference-count":25,"publisher":"SAGE Publications","issue":"5","license":[{"start":{"date-parts":[[1989,10,1]],"date-time":"1989-10-01T00:00:00Z","timestamp":623203200000},"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":[[1989,10]]},"abstract":"<jats:p>\n                    Parallel manipulators have a specific mechanical architecture where all the links are connected both at the base and at the gripper of the robot. By changing the lengths of these links we are able to control the position\n                    <jats:sup>1<\/jats:sup>\n                    of the gripper. In general, for a given set of links lengths there is only one position for the gripper. However, it may be suspected that in some cases more than one solution may be found for the position of the gripper: the robot is in a singular configuration. To determine these singular configurations the classical method is to find the roots of the determinant of the jacobian matrix. In our case the jacobian matrix is complex, and it seems not to be possible to find these roots. We propose here a new method based on Grassmann line geometry. The set of lines, P\n                    <jats:sup>3<\/jats:sup>\n                    , constitutes a linear variety of rank 6. We show that a singular configuration is obtained when the variety spanned by the lines associated to the robot links has a rank less than 6. An important feature of the varieties of this geometry is that they can be described by simple geometric rules. Thus to find the singular configurations of parallel manipulators we have to find the configuration in which the robot matches these rules. Such an analysis is performed on a special parallel manip ulator. We show that we find all the well-known singular configurations, and also some new ones.\n                  <\/jats:p>","DOI":"10.1177\/027836498900800504","type":"journal-article","created":{"date-parts":[[2007,3,4]],"date-time":"2007-03-04T20:24:06Z","timestamp":1173039846000},"page":"45-56","source":"Crossref","is-referenced-by-count":514,"title":["Singular Configurations of Parallel Manipulators and Grassmann Geometry"],"prefix":"10.1177","volume":"8","author":[{"given":"Jean-Pierre","family":"Merlet","sequence":"first","affiliation":[{"name":"INRIA, Centre de Sophia Antipolis 06560 Valbonne, France"}]}],"member":"179","published-online":{"date-parts":[[1989,10,1]]},"reference":[{"key":"atypb1","volume-title":"Proc. AGARD Conf. 249","author":"Baret, M."},{"key":"atypb2","volume-title":"Fundamentals of Mathematics: Geometry","author":"Behnke, H.","year":"1986","edition":"3"},{"key":"atypb3","volume":"3","author":"Bricard, R.","year":"1987","journal-title":"Journal de Math\u00e9matiques Pures et Appliqu\u00e9es"},{"key":"atypb4","author":"Cauchy, A.","year":"1813","journal-title":"Journal de l'Ecole Polytechnique"},{"key":"atypb5","author":"Crapo, H.","year":"1973","journal-title":"Colloquio Int. 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Application \u00e0 la commande de robots parall\u00e8les","author":"Merlet, J-P.","year":"1986"},{"key":"atypb14","volume-title":"Parallel manipulator, part 1: Theory, design, kinematics and control","author":"Merlet, J-P.","year":"1987"},{"key":"atypb15","volume-title":"Parallel manipulator, part 2: Singular configurations and Grassman geometry","author":"Merlet, J-P.","year":"1988"},{"key":"atypb16","volume-title":"Manipulator design vignettes","author":"Minsky, M.","year":"1972"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1115\/1.3258713"},{"key":"atypb18","volume-title":"Proc 3rd ISRR","author":"Reboulet, C."},{"key":"atypb19","volume-title":"Proc. of the Institution of Mechanical Engineers 1965-66","author":"Stewart, D."},{"key":"atypb20","volume-title":"Proc. Computational Mechanics '88","author":"Tanaka, M."},{"key":"atypb21","volume-title":"Generalized Clifford Parallelism","author":"Tyrrell, J.A.","year":"1971"},{"key":"atypb22","volume-title":"Projective Geometry","author":"Veblen, O.","year":"1910"},{"issue":"4","key":"atypb23","first-page":"12","volume":"46","author":"Watson, P.","year":"1984","journal-title":"Electron. Aust."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1115\/1.3258578"},{"key":"atypb25","volume-title":"Proc. Int. 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