{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T09:41:38Z","timestamp":1777714898544,"version":"3.51.4"},"reference-count":17,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[1990,2,1]],"date-time":"1990-02-01T00:00:00Z","timestamp":633830400000},"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":[[1990,2]]},"abstract":"<jats:p>A form closure (or complete restraint) of a solid object is a finite set of wrenches (force-moment combinations) applied on the object, with the property that any other wrench acting on the object can be balanced by a positive combination of the original ones. Intuitively, form closure is a way of defining the notion of a \"firm grip\" of the object (when friction is not taken into account). It has been pointed out by Reuleaux (1875) and Somoff ( 1897), and more recently by Lakshmin arayana (1978), that the form closure of a two-dimensional object requires at least four wrenches, and that form closure of a three-dimensional object requires at least seven wrenches. It was also conjectured that these numbers can be achieved by wrenches realizable as forces normal to the surface of the object (such wrenches are called fingers). In this paper we prove this conjecture. In particular we show that form closure of any two-dimensional bounded object with piecewise smooth boundary (except a circle) can be achieved by four fingers. For three dimensions, we prove that form closure of any bounded object with piecewise smooth boundary can be achieved with 12 fingers if and only if the object does not have a rotational symmetry (in which case, of course, form closure is not achievable, since no moment along the axis of symmetry can be opposed). We also show that, under very general conditions, form closure of three-dimensional objects without rotational symmetries can be achieved with seven fingers. Finally, we show that when Coulomb friction is taken into account, under the most relaxed assumptions three fingers are necessary and sufficient in two dimensions, and four fingers in three dimensions.<\/jats:p>","DOI":"10.1177\/027836499000900102","type":"journal-article","created":{"date-parts":[[2007,3,4]],"date-time":"2007-03-04T20:24:06Z","timestamp":1173039846000},"page":"61-74","source":"Crossref","is-referenced-by-count":303,"title":["The Geometry of Grasping"],"prefix":"10.1177","volume":"9","author":[{"given":"Xanthippi","family":"Markenscoff","sequence":"first","affiliation":[{"name":"Department of Applied Mechanics and Engineering Science University of California La Jolla, California 92093"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"family":"Luqun Ni","sequence":"additional","affiliation":[{"name":"Institute of Mathematics Academia Sinica Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christos H.","family":"Papadimitriou","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering University of California La Jolla, California 92093"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[1990,2,1]]},"reference":[{"key":"atypb1","volume-title":"Proc. 1985 ACM Symp. on the Theory of Computing","author":"Baker, B.S."},{"key":"atypb2","volume-title":"Linear Programming and Extensions","author":"Dantzig, G.B.","year":"1959"},{"key":"atypb3","volume-title":"Proc. 7th ISIR","author":"Hanafusa, H."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1177\/027836498600400401"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.3138\/9781487589448"},{"key":"atypb6","volume-title":"The mechanics of form closure","author":"Lakshminarayana, K.","year":"1978"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1177\/027836498900800202"},{"key":"atypb8","volume-title":"Proc. FOCS Conference","author":"Megiddo, M."},{"issue":"4","key":"atypb9","first-page":"541","volume":"2","author":"Mishra, B.","year":"1988","journal-title":"Algorithmica"},{"key":"atypb10","first-page":"1368","volume":"3","author":"Nguyen, V.","year":"1986","journal-title":"Proc. IEEE Conf. Robot. Automat."},{"key":"atypb11","volume-title":"An extension of screw theory and its application to the automation of industrial assemblies. Ph.D. thesis","author":"Ohwovoriole, M.S.","year":"1980"},{"key":"atypb12","volume-title":"Combinatorial Optimization: Algorithms and Complexity","author":"Papadimitriou, C.H.","year":"1983"},{"key":"atypb13","volume-title":"Theoretische Kinematik","author":"Reuleaux, F.","year":"1875"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"Salisbury, J.K., and Roth, B. 1983. Kinematic and force analysis of articulated mechanical hands. J. Mech. Trans. Automat. 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