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Kinematic and Kinetostatic Analysis of Parallel Manipulators with Emphasis on Position, Motion, and Actuation Singularities
- M. Kemal Ozgoren
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This paper provides a contribution to the singularity analysis of the parallel manipulators by introducing the position singularities in addition to the motion and actuation singularities. The motion singularities are associated with the linear velocity mapping between the task and joint spaces. So, they are the singularities of the relevant Jacobian matrices. On the other hand, the position singularities are associated with the nonlinear position mapping between the task and joint spaces. So, they are encountered in the position-level solutions of the forward and inverse kinematics problems. In other words, they come out irrespective of the velocity mapping and the Jacobian matrices. Considering these distinctions, a kinematic singularity is denoted here by one of the four acronyms, which are PSFK (position singularity of forward kinematics), PSIK (position singularity of inverse kinematics), MSFK (motion singularity of forward kinematics), and MSIK (motion singularity of inverse kinematics). There may also occur an actuation singularity (ACTS) concerning the kinetostatic relationships that involve forces and moments. However, it is verified that an ACTS is the same as an MSFK. Each singularity induces different consequences in the joint and task spaces. A PSFK imposes a constraint on the active joint variables and makes the end-effector position indefinite and uncontrollable. Therefore, it must be avoided. An MSFK imposes a constraint on the rates of the active joint variables and makes the end-effector motion indefinite and easily perturbable. Besides, since it is also an ACTS, it causes the actuator torques or forces to grow without bound. Therefore, it must also be avoided. On the other hand, a PSIK imposes a constraint on the end-effector position but provides freedom for the active joint variables. Similarly, an MSIK imposes a constraint on the end-effector motion but provides freedom for the rates of the active joint variables. A PSIK or MSIK need not be avoided if the constraint it imposes on the position or motion of the end-effector is acceptable or if the task can be planned to be compatible with that constraint. Besides, with such a compatible task, a PSIK or MSIK may even be advantageous, because the freedom it provides for the active joint variables can sometimes be used for a secondary purpose. This paper is also concerned with the multiplicities of forward kinematics in the assembly modes of the manipulator and the multiplicities of inverse kinematics in the posture modes of the legs. It is shown that the assembly mode changing poses of the manipulator are the same as the MSFK poses, and the posture mode changing poses of the legs are the same as the MSIK poses.
Interaction of a deep-water wave with a vertical cylinder: effect of self-excited vibrations on quantitative flow patterns
- M. OZGOREN, D. ROCKWELL
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- Journal:
- Journal of Fluid Mechanics / Volume 572 / February 2007
- Published online by Cambridge University Press:
- 23 January 2007, pp. 189-217
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Interaction of a deep-water wave with a cylinder gives rise to ordered patterns of the flow structure, which are quantitatively characterized using a technique of high-image-density particle image velocimetry. When the cylinder is stationary, the patterns of instantaneous flow structure take on increasingly complex forms for increasing Keulegan--Carpenter number KC. These patterns involve stacking of small-scale vorticity concentrations, as well as large-scale vortex shedding. The time-averaged consequence of these patterns involves, at sufficiently high KC, an array of vorticity concentrations about the cylinder.
When the lightly damped cylinder is allowed to undergo bidirectional oscillations, the trajectories can be classified according to ranges of KC. At low values of KC, the trajectory is elliptical, and further increases of KC allow, first of all, both elliptical and in-line trajectories as possibilities, followed by predominantly in-line and figure-of-eight oscillations at the largest value of KC.
Representations of the quantitative flow structure, in relation to the instantaneous cylinder position on its oscillation trajectory, show basic classes of patterns. When the trajectory is elliptical, layers of vorticity rotate about the cylinder surface, in accordance with rotation of the relative velocity vector of the wave motion with respect to the oscillating cylinder. Simultaneously, the patterns of streamline topology take the form of large-scale bubbles, which also rotate about the cylinder. When the cylinder trajectory is predominantly in-line with the wave motion, generic classes of vortex formation and shedding can be identified; they include sweeping of previously shed vorticity concentrations past the cylinder to the opposite side. Certain of these patterns are directly analogous to those from the stationary cylinder.
Space–time development of the onset of a shallow-water vortex
- J.-C. LIN, M. OZGOREN, D. ROCKWELL
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- Journal:
- Journal of Fluid Mechanics / Volume 485 / 25 May 2003
- Published online by Cambridge University Press:
- 24 June 2003, pp. 33-66
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An impulsively started jet in shallow water gives rise to vortices having a characteristic diameter larger than the water depth. A technique of high-image-density particle image velocimetry allows characterization of the space–time development of the instantaneous flow patterns along planes representing the quasi-two-dimensional and three-dimensional vortex structure. The quasi-two-dimensional patterns exhibit different categories of vortex development and interaction, depending upon the depth of the shallow water layer. Despite these distinctions, the variations of normalized vortex position, diameter, and circulation, as well as peak vorticity within the vortex, are very similar for sufficiently small water depth.
These quasi-two-dimensional patterns are, in turn, related to specific forms of three-dimensional flow structure, which is highly ordered. A prevalent feature is a vortex orthogonal to, and just ahead of, the primary, quasi-two-dimensional vortex. Its streamline topology, on a plane parallel to the axis of the quasi-two-dimensional vortex, exhibits a separation bubble with a well-defined separatix at the bottom (bed) surface. Moreover, its vorticity can exceed that of the quasi-two-dimensional pattern by a factor of two. This feature is consistent for all values of water depth. When the depth becomes sufficiently large, however, the three-dimensional vortex pattern involves an array of vorticity concentrations, which extends across the entire depth of the water.
On a plane very close to the bottom surface (bed), global instantaneous distributions of velocity and vorticity exhibit large gradients; they are associated with small-scale vorticity concentrations characteristic of rapid transition. The corresponding streamline topology of the averaged flow close to the bed, however, exhibits a stable focus and is a direct indicator of the topology well above the bed.
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