{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:49:04Z","timestamp":1772297344401,"version":"3.50.1"},"reference-count":73,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2007,11,1]],"date-time":"2007-11-01T00:00:00Z","timestamp":1193875200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotica"],"published-print":{"date-parts":[[2007,11]]},"abstract":"<jats:title>SUMMARY<\/jats:title><jats:p>There have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out.<\/jats:p><jats:p>Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical constructions that imitate well-known constructions from classical approximation theory. Part III reviews a class of methods where classical<jats:italic>criteria<\/jats:italic>are extended to curves in Riemannian manifolds, including several recent mathematical results that have not yet found their way into the literature.<\/jats:p>","DOI":"10.1017\/s0263574707003669","type":"journal-article","created":{"date-parts":[[2007,8,1]],"date-time":"2007-08-01T13:11:12Z","timestamp":1185973872000},"page":"691-701","source":"Crossref","is-referenced-by-count":12,"title":["Geometry for robot path planning"],"prefix":"10.1017","volume":"25","author":[{"given":"Lyle","family":"Noakes","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomasz","family":"Popiel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2007,11,1]]},"reference":[{"key":"S0263574707003669_ref28","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015716"},{"key":"S0263574707003669_ref73","doi-asserted-by":"publisher","DOI":"10.1109\/70.704225"},{"key":"S0263574707003669_ref63","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2006.06.004"},{"key":"S0263574707003669_ref61","unstructured":"61. Popiel T. and Noakes L. , \u201cB\u00e9zier curves and C 2 interpolation in Riemannian manifolds,\u201d J. Approx. Theory (Oct. 2006) to be published."},{"key":"S0263574707003669_ref59","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009863"},{"key":"S0263574707003669_ref58","unstructured":"58. Silva-Leite F. , Camarainha M. and Crouch P. , \u201cTwo Higher Order Variational Problems on Riemannian manifolds and the Interpolation Problem,\u201d Proceedings of the 3rd European Control Conf. (1995) pp. 3269\u20133274."},{"key":"S0263574707003669_ref57","doi-asserted-by":"publisher","DOI":"10.1007\/BF01901476"},{"key":"S0263574707003669_ref54","doi-asserted-by":"crossref","unstructured":"54. Noakes L. , \u201cAsymptotics of null Lie quadratics in E 3,\u201d SIAM J on Appl. Dyn. Sys. in-press 2007.","DOI":"10.1137\/070686755"},{"key":"S0263574707003669_ref69","unstructured":"69. Zefran M. , Kumar V. and Croke C. , \u201cChoice of Riemannian metrics for rigid body dynamics,\u201d Proceedings of the ASME Design Engineering Technical Conference and Computers in Engineering Conf., Irvine, CA (Aug. 18\u201322, 1996) pp. 1\u201311."},{"key":"S0263574707003669_ref66","unstructured":"66. Sprott K. and Ravani B. , \u201cRuled surfaces, Lie groups and mesh generation,\u201d Proceedinds of the ASME Design Engineering Technology Conference (1997) No. DETC97\/DAC-3966."},{"key":"S0263574707003669_ref52","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-004-7621-4"},{"key":"S0263574707003669_ref50","doi-asserted-by":"publisher","DOI":"10.1063\/1.1803609"},{"key":"S0263574707003669_ref46","doi-asserted-by":"crossref","unstructured":"46. Noakes L. , \u201cAccelerations of Riemannian Quadratics,\u201d Proceedings of the American Mathematical Society 127 (1999) pp. 1827\u20131836.","DOI":"10.1090\/S0002-9939-99-04809-1"},{"key":"S0263574707003669_ref10","unstructured":"10. Camarinha M. , The Geometry of Cubic Polynomials on Riemannian Manifolds, Ph.D. Thesis (Coimbra, Portugal: University of Coimbra, 1996)."},{"key":"S0263574707003669_ref43","doi-asserted-by":"publisher","DOI":"10.1007\/1-4020-3858-8_5"},{"key":"S0263574707003669_ref42","first-page":"145","article-title":"Animated Rotations using Quaternions and Splines on a 4D Sphere","volume":"18","author":"Nielson","year":"1992","journal-title":"Programm. Comput. Softw"},{"key":"S0263574707003669_ref37","doi-asserted-by":"publisher","DOI":"10.1115\/1.2826869"},{"key":"S0263574707003669_ref36","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/dni014"},{"key":"S0263574707003669_ref34","unstructured":"34. Kim M.-J. , Kim M.-S. and Shin S. , \u201cA C2 Continuous B-Spline Quaternion Curve Interpolating a Given Sequence of Solid Orientations,\u201d Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (1995) pp. 369\u2013376."},{"key":"S0263574707003669_ref49","doi-asserted-by":"publisher","DOI":"10.1063\/1.1537461"},{"key":"S0263574707003669_ref29","doi-asserted-by":"crossref","unstructured":"29. H\u00fcper K. and Silva-Leite F. , \u201cOn the geometry of rolling and interpolation curves on Sn , SOn and Grassmann manifolds,\u201d J. Dynam. Control Syst. (2007) to be published.","DOI":"10.1007\/s10883-007-9027-3"},{"key":"S0263574707003669_ref30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03118-6"},{"key":"S0263574707003669_ref68","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2005.06.003"},{"key":"S0263574707003669_ref72","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4485(97)00060-2"},{"key":"S0263574707003669_ref32","volume-title":"Geometric Control Theory (Cambridge Studies in Advanced Mathematics)","volume":"51","author":"Jurdjevic","year":"1997"},{"key":"S0263574707003669_ref19","volume-title":"A Practical Guide to Splines","author":"Boor","year":"2001"},{"key":"S0263574707003669_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(93)90037-4"},{"key":"S0263574707003669_ref40","volume-title":"Annals of Mathematical Studies","volume":"51","author":"Milnor","year":"1963"},{"key":"S0263574707003669_ref70","doi-asserted-by":"crossref","unstructured":"70. Zefran M. and Kumar V. , \u201cPlanning of Smooth Motions on SE(3),\u201d Proceedings of the IEEE International Conf. on Robotics and Automation, Minneapolis, MN (1996).","DOI":"10.1109\/ROBOT.1996.503583"},{"key":"S0263574707003669_ref39","volume-title":"(Texts in Applied Mathematics)","volume":"17","author":"Marsden","year":"1994"},{"key":"S0263574707003669_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s00498-003-0139-3"},{"key":"S0263574707003669_ref22","unstructured":"22. Duff T. , \u201cQuaternion Splines for Animating Orientation,\u201d Proceedings of the USENIX Association 2nd Computer Graphics Workshop, Monterey, CA (1985) pp. 54\u201362."},{"key":"S0263574707003669_ref26","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/19.4.445"},{"key":"S0263574707003669_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0110205"},{"key":"S0263574707003669_ref4","doi-asserted-by":"publisher","DOI":"10.1243\/0954406021524909"},{"key":"S0263574707003669_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S1571-0661(04)00312-3"},{"key":"S0263574707003669_ref8","doi-asserted-by":"publisher","DOI":"10.1145\/502122.502124"},{"key":"S0263574707003669_ref44","first-page":"319","volume-title":"Curves and Surfaces with Applications in CAGD","author":"Noakes","year":"1997"},{"key":"S0263574707003669_ref13","first-page":"134","article-title":"The numerical evaluation of B-splines","volume":"10","author":"Cox","year":"1972","journal-title":"J. IMA"},{"key":"S0263574707003669_ref71","unstructured":"71. Zefran M. and Kumar V. , \u201cTwo Methods for Interpolating Rigid Body Motions,\u201d Proceedings of the IEEE International Conf. on Robotics and Automation, Leuven, Belgium (1996)."},{"key":"S0263574707003669_ref65","doi-asserted-by":"publisher","DOI":"10.1145\/325165.325242"},{"key":"S0263574707003669_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0926-2245(01)00054-7"},{"key":"S0263574707003669_ref60","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2005.11.003"},{"key":"S0263574707003669_ref23","volume-title":"State of the Art in Image Synthesis, SIGGRAPH '86 Course Notes","author":"Duff","year":"1986"},{"key":"S0263574707003669_ref45","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018940112654"},{"key":"S0263574707003669_ref3","article-title":"On the computation of rigid body motion","volume":"2","author":"Belta","year":"2002","journal-title":"Electron. J. Computat. Kinematics"},{"key":"S0263574707003669_ref62","doi-asserted-by":"crossref","unstructured":"62. Popiel T. and Noakes L. , \u201cElastica in SO(3),\u201d J. Aust. Math. Soc. (2006) to be published.","DOI":"10.1017\/S1446788700036417"},{"key":"S0263574707003669_ref2","doi-asserted-by":"publisher","DOI":"10.1145\/142920.134086"},{"key":"S0263574707003669_ref5","volume-title":"Interdisciplinary Applied Mathematics","volume":"24","author":"Bloch","year":"2003"},{"key":"S0263574707003669_ref38","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.1980.4766968"},{"key":"S0263574707003669_ref67","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-006-0638-3"},{"key":"S0263574707003669_ref47","doi-asserted-by":"publisher","DOI":"10.1023\/A:1016277023669"},{"key":"S0263574707003669_ref7","first-page":"49","volume-title":"Mathematical Methods for Curves and Surfaces","author":"Brunnett","year":"1995"},{"key":"S0263574707003669_ref9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6602"},{"key":"S0263574707003669_ref12","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/12.4.399"},{"key":"S0263574707003669_ref25","doi-asserted-by":"publisher","DOI":"10.1115\/1.2919446"},{"key":"S0263574707003669_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02254638"},{"key":"S0263574707003669_ref16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021770717822"},{"key":"S0263574707003669_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(72)90080-9"},{"key":"S0263574707003669_ref35","unstructured":"35. Krakowski K. , Geometrical Methods of Inference Ph.D. Thesis (Perth, Australia: University of Western Australia, 2002)."},{"key":"S0263574707003669_ref55","doi-asserted-by":"publisher","DOI":"10.1115\/1.2826114"},{"key":"S0263574707003669_ref24","first-page":"1","volume-title":"State of the Art in Image Synthesis, SIGGRAPH '85 Course Notes","author":"Gabriel","year":"1985"},{"key":"S0263574707003669_ref14","doi-asserted-by":"crossref","unstructured":"14. Crouch P. and Silva-Leite F. , \u201cGeometry and the Dynamic Interpolation Problem,\u201d Proceedings of the American Control Conference, Boston, U.K. (1991) pp.1131\u20131136.","DOI":"10.23919\/ACC.1991.4791552"},{"key":"S0263574707003669_ref41","doi-asserted-by":"crossref","unstructured":"41. Nielson G. , \u201cSmooth Interpolation of Orientation,\u201d Proceedings of Computer Animation '93: Models and Techniques in Computer Animation (1993) pp. 75\u201393.","DOI":"10.1007\/978-4-431-66911-1_8"},{"key":"S0263574707003669_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02519034"},{"key":"S0263574707003669_ref48","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/6.4.465"},{"key":"S0263574707003669_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009863"},{"key":"S0263574707003669_ref33","volume-title":"Space Kinematics and Lie Groups","author":"Karger","year":"1985"},{"key":"S0263574707003669_ref64","unstructured":"64. Popiel T. , Geometrically Defined Curves in Riemannian Manifolds Ph.D. Thesis (Perth, Australia: University of Western Australia, 2007)."},{"key":"S0263574707003669_ref51","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/dni040"},{"key":"S0263574707003669_ref56","doi-asserted-by":"publisher","DOI":"10.1145\/256157.256160"},{"key":"S0263574707003669_ref53","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/hal002"},{"key":"S0263574707003669_ref31","doi-asserted-by":"publisher","DOI":"10.2307\/2375037"}],"container-title":["Robotica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0263574707003669","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T16:59:03Z","timestamp":1556729943000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0263574707003669\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,11]]},"references-count":73,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2007,11]]}},"alternative-id":["S0263574707003669"],"URL":"https:\/\/doi.org\/10.1017\/s0263574707003669","relation":{},"ISSN":["0263-5747","1469-8668"],"issn-type":[{"value":"0263-5747","type":"print"},{"value":"1469-8668","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,11]]}}}