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...
Haptic technology (sense of touch) along with 3D-virtual reality (VR) graphics, creating lifelike training simulations, was used to develop a dental training simulator system (PerioSim©). This preliminary study was designed to evaluate whether faculty considered PerioSim realistic and useful for training and evaluating basic procedural skills of students. The haptic device employed was a PHANToM™ and the simulator a Dell Xeon 530 workstation with 3D, VR oral models and instruments viewed on a stereoscopic monitor. An onscreen VR periodontal probe or explorer was manipulated by operating the PHANToM for sensing lifelike contact and interactions with the teeth and gingiva. Thirty experienced clinical dental and dental hygiene faculty judged the realism of the system. A PowerPoint presentation on one screen provided instructions for the simulator use with the 3D, VR simulator on a second stereoscopic monitor viewed with 3D goggles. Faculty/practitioners found the images very realistic for teeth and instruments, but less so for gingiva. Tactile sensation was realistic for teeth but not so for gingiva. The onscreen instructions were very useful with high potential for teaching. Faculty members anticipated incorporating this device into teaching and were enthusiastic about its potential for evaluating students' basic procedural skills. This study suggests that the preliminary "evidence-of-concept" was successful and PerioSim may aid students in developing necessary dental tactile skills.
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