{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T15:47:30Z","timestamp":1758124050082},"reference-count":34,"publisher":"Cambridge University Press (CUP)","issue":"8","license":[{"start":{"date-parts":[[2013,6,7]],"date-time":"2013-06-07T00:00:00Z","timestamp":1370563200000},"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":[[2013,12]]},"abstract":"<jats:title>SUMMARY<\/jats:title><jats:p>This paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary, and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group <jats:italic>SE<\/jats:italic>(2). An application of Pontryagin's maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals, and the expression for the position is reduced to quadratures. Reachable sets are defined in the arbitrary speed case, and a numerical plot of the time-limited reachable sets is presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment.<\/jats:p>","DOI":"10.1017\/s0263574713000519","type":"journal-article","created":{"date-parts":[[2013,6,7]],"date-time":"2013-06-07T12:39:27Z","timestamp":1370608767000},"page":"1285-1297","source":"Crossref","is-referenced-by-count":5,"title":["Path planning for simple wheeled robots: sub-Riemannian and elastic curves on SE(2)"],"prefix":"10.1017","volume":"31","author":[{"given":"Craig","family":"Maclean","sequence":"first","affiliation":[]},{"given":"James D.","family":"Biggs","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,6,7]]},"reference":[{"key":"S0263574713000519_ref19","volume-title":"A Mathematical Introduction to Robotic Manipulation","author":"Murray","year":"1994"},{"key":"S0263574713000519_ref12","first-page":"552","volume-title":"Proceedings of the 47th IEEE Conference on Decision and Control","author":"Mahmoudian","year":"2008"},{"key":"S0263574713000519_ref31","volume-title":"Proceedings of the International Conference on Intelligent Autonomous Systems","author":"Mazer","year":"1993"},{"key":"S0263574713000519_ref11","volume-title":"Cooperative Path Planning of Unmanned Aerial Vehicles","author":"Tsourdos","year":"2011"},{"key":"S0263574713000519_ref22","first-page":"1","volume-title":"Nonholonomic Motion Planning","author":"Brockett","year":"1993"},{"key":"S0263574713000519_ref32","volume-title":"Technical Report UU-CS-1995-22","author":"Svetska","year":"1995"},{"key":"S0263574713000519_ref10","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1990.145.367"},{"key":"S0263574713000519_ref13","first-page":"997","volume-title":"Proceedings of the IEEE International Conference on Intelligent Robots and Systems","author":"Scheuer","year":"1997"},{"key":"S0263574713000519_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511546877"},{"key":"S0263574713000519_ref27","volume-title":"NIST Handbook of Mathematical Functions","author":"Olver","year":"2010"},{"key":"S0263574713000519_ref15","first-page":"3722","volume-title":"Proceedings of International Conference on Robotics and Automation","author":"Fraichard","year":"2001"},{"key":"S0263574713000519_ref23","volume-title":"Principles of Robot Motion \u2013 Theory, Algorithms, and Implementation","author":"Choset","year":"2005"},{"key":"S0263574713000519_ref1","first-page":"52","volume-title":"Geometric Control Theory","author":"Jurdjevic","year":"1997"},{"key":"S0263574713000519_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/b97376"},{"key":"S0263574713000519_ref3","doi-asserted-by":"publisher","DOI":"10.1109\/9.412625"},{"key":"S0263574713000519_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2012.02.002"},{"key":"S0263574713000519_ref5","volume-title":"Proceedings of TAROS 2011","author":"Biggs","year":"2011"},{"key":"S0263574713000519_ref6","first-page":"7","article-title":"Towards a theory of optimal processes","volume":"110","author":"Boltyanskii","year":"1956","journal-title":"Reports Acad. 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