{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T09:56:01Z","timestamp":1777715761808,"version":"3.51.4"},"reference-count":17,"publisher":"SAGE Publications","issue":"6","license":[{"start":{"date-parts":[[2000,6,1]],"date-time":"2000-06-01T00:00:00Z","timestamp":959817600000},"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":[[2000,6]]},"abstract":"<jats:p>Kinematic metric functions, which allow us to make metric measurements such as \u201cdistance,\u201d \u201clength,\u201d and \u201cangle,\u201d are often needed in robotics. However, some commonly used metric functions are not well-defined because of their dependence on choice of reference frame. This paper presents formal and practical conditions for kinematic metric functions to be well-defined. To properly analyze this issue, we introduce an intrinsic formulation of the configuration space of a rigid body that does not depend on frames. Based on this intrinsic definition, the principle of objectivity, or observer indifference, is introduced to derive a formal condition for well-definedness of kinematic metric functions. The principle of objectivity is further used to gain physical insight into left, right, and bi-invariances on the Lie group SE(3). The abstract notion of objectivity is then related to the more practically useful notion of frame invariance, which is shown to be necessary and sufficient for objectivity. Thus, frame invariance can be used as a practical condition for determining objective functions, and we give an explicit formula to determine frame invariance of a given metric function. Finally, by use of examples, we present several frame-invariant norms of rigid body velocities and wrenches. These examples demonstrate that norms with interesting physical interpretations need not be derived from inner products of velocities or wrenches.<\/jats:p>","DOI":"10.1177\/027836490001900605","type":"journal-article","created":{"date-parts":[[2004,12,17]],"date-time":"2004-12-17T20:23:10Z","timestamp":1103314990000},"page":"612-625","source":"Crossref","is-referenced-by-count":30,"title":["Objective and Frame-Invariant Kinematic Metric Functions for Rigid Bodies"],"prefix":"10.1177","volume":"19","author":[{"given":"Qiao","family":"Lin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joel W.","family":"Burdick","sequence":"additional","affiliation":[{"name":"Division of Engineering and Applied Science, California Institute of                         Technology, Pasadena, CA 91125, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2000,6,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"Abraham, R., Marsden, J. E., and Ratiu, T. S. 1988. Manifolds, Tensor Analysis, and Applications. 2d ed. New York: Springer-Verlag .","DOI":"10.1007\/978-1-4612-1029-0"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/rob.4620070202"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"Etzel, K. R., and McCarthy, J. M. 1996. A metric for spatial displacements using biquaternions on SO(4) . Proc. Int. Conf. on Robotics and Automation, Minneapolis, MN, pp. 3185-3190 .","DOI":"10.1109\/ROBOT.1996.509197"},{"key":"atypb4","unstructured":"Ferrari, C., and Canny, J. 1992 (May). Planning optimal grasps . Proc. Int. Conf. on Robotics and Automation, Nice, France, pp. 2290-2295 ."},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"Kazerounian, K., and Rastegar, J. 1992 (September). Object norms: A class of coordinate and metric independent norms for displacements . Flexible Mechanisms, Dynamics and Analysis: ASME Design Technical Conferences, 22nd Biennial Mechanisms Conference, Scottsdale, AZ, vol. 47, pp. 271-275 .","DOI":"10.1115\/DETC1992-0390"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"Kirkpatrick, D. G., Mishra, B., and Yap, C. K. 1990 (May). Quantitative Steinitz's theorems with applications to multifingered grasping . Proc. 20th ACM Symp. on Theory of Computing, Baltimore, MD, pp. 341-351 .","DOI":"10.1145\/100216.100261"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1115\/1.2826735"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"Latombe, J.C. 1991. Robot Motion Planning. New York: Kluwer Academic .","DOI":"10.1007\/978-1-4615-4022-9"},{"key":"atypb9","unstructured":"Li, Z. 1990. Geometrical considerations of robot kinematics . IEEE J. of Robotics and Automation 5(3): 139-145 ."},{"key":"atypb10","unstructured":"Lin, Q. 1998 (May).\n                      Mechanics and Planning of Workpiece Fixturing and Robotic Grasping.\n                      Ph. D. thesis, Department of Mechanical Engineering, California Institute of Technology, Pasadena."},{"key":"atypb11","unstructured":"Loncaric, J. 1985.\n                      Geometric Analysis of Compliant Mechanisms in Robotics.\n                      Ph. D. thesis, Harvard University, Cambridge, MA."},{"key":"atypb12","unstructured":"Marsden, J. E., and Hughes, T.J.R. 1983. Mathematical Foundations of Elasticity. Englewood Cliffs, NJ: Prentice Hall ."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1115\/1.2826115"},{"key":"atypb14","unstructured":"Murray, R. M., Li, Z., and Sastry, S. S. 1994. A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC ."},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"Park, F. C. 1995. Distance metrics on the rigid body motions with applications to mechanism design . ASME J. of Mechanical Design 17(1): 48-54 .","DOI":"10.1115\/1.2826116"},{"key":"atypb16","unstructured":"Pogorelov, A. V. 1966. Lecutures on the Foundations of Geometry. Translated from the 2d Russian edition by L. F. Boron. Groningen, the Netherlands: P. 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