Some isoperimetric inequalities and eigenvalue estimates Annales scientifiques de l'É.N.S. 4 e série, tome 13, n o 4 (1980), p. 419-435 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1980, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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Part of the Mechanical Engineering CommonsSuggested Citation: Zefran, M., V. Kumar and C. Croke. "On the generation of smooth three-dimensional rigid body motions." IEEE Transactions on Robotics and Automation, Vol. 14(4). p. 576-589.© 1998 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Recommended CitationŽefran, Miloš; Kumar, Vijay; and Croke, Christopher, "On the Generation of Smooth Three-Dimensional Rigid Body Motions" (1998) This paper addresses the problem of generating smooth trajectories between an initial and final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and find trajectories that minimize this functional. In order to ensure that the computed trajectories are independent of the parameterization of positions and orientations, we use the notions of Riemannian metric and covariant derivatives from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements, SE (3). We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest distance, minimum acceleration and minimum jerk trajectories. Analytical expressions for the smooth trajectories are derived for some special cases. We also provide several examples of the general case where the trajectories are computed numerically. Disciplines Engineering | Mechanical Engineering CommentsSuggested Citation: Zefran, M., V. Kumar and C. Croke. "On the generation of smooth three-dimensional rigid body motions." IEEE Transactions on Robotics and Automation, Vol. 14(4). p. 576-589.© 1998 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. AbstractThis paper addresses the problem of generating smooth trajectories between an initial and a nal position and orientation in space. The main idea is to de ne a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and nd trajectories that minimize this functional. In order to ensure that the computed trajectories are independent of the parameterization of positions and orientations, we use the notions of Riemannian metric and covariant derivative from di erentia...