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Research Papers

Dynamically Feasible Periodic Trajectories for Generic Spatial Three-Degree-of-Freedom Cable-Suspended Parallel Robots1

[+] Author and Article Information
Giovanni Mottola

Department of Industrial Engineering,
University of Bologna,
Bologna 40126, Italy,
e-mail: giovanni.mottola3@unibo.it

Clément Gosselin

Professor
Fellow ASME
Département de génie mécanique,
Université Laval,
Québec, QC G1V 0A6, Canada
e-mail: gosselin@gmc.ulaval.ca

Marco Carricato

Professor
Department of Industrial Engineering,
University of Bologna,
Bologna 40126, Italy
e-mail: marco.carricato@unibo.it

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 25, 2017; final manuscript received February 28, 2018; published online March 23, 2018. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 10(3), 031004 (Mar 23, 2018) (10 pages) Paper No: JMR-17-1326; doi: 10.1115/1.4039499 History: Received September 25, 2017; Revised February 28, 2018

Cable-suspended robots may move beyond their static workspace by keeping all cables under tension, thanks to end-effector inertia forces. This may be used to extend the robot capabilities, by choosing suitable dynamical trajectories. In this paper, we consider three-dimensional (3D) elliptical trajectories of a point-mass end effector suspended by three cables from a base of generic geometry. Elliptical trajectories are the most general type of spatial sinusoidal motions. We find a range of admissible frequencies for which said trajectories are feasible; we also show that there is a special frequency, which allows the robot to have arbitrarily large oscillations. The feasibility of these trajectories is verified via algebraic conditions that can be quickly verified, thus being compatible with real-time applications. By generalizing previous studies, we also study the possibility to change the frequency of oscillation: this allows the velocity at which a given ellipse is tracked to be varied, thus providing more latitude in the trajectory definition. We finally study transition trajectories to move the robot from an initial state of rest (within the static workspace) to the elliptical trajectory (and vice versa) or to connect two identical ellipses having different centers.

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References

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Figures

Grahic Jump Location
Fig. 1

(Left) Schematic of a 3DOF spatial CSPR. (Right) auxiliary vectors.

Grahic Jump Location
Fig. 2

An elliptical trajectory Γ with pC=(−1,1,2)T,a1=(2,1,0)T,a2=(−3,−2,0)T,a3=(−1,3,0)T and lying on a plane normal to ne=(1,2,3)T. In this special case, the cable exit points Ai are all at the same height and the trajectory is a circle with radius R = 1.2. The length units are arbitrary.

Grahic Jump Location
Fig. 3

The cable tensions divided by the mass of the end effector, given by a simplified model with stiff massless cables. For each cable, the solid line corresponds to ω=ωmax′, while the dashed line corresponds to ω=ωmax>ωmax′.

Grahic Jump Location
Fig. 4

Plane ψ¨−ψ˙2 with the three ellipses Ωi, rectangle Rψ (in gray) and curve Γψ (black line). Notice how Rψ∈Ω1∩Ω2∩Ω3 and Γψ∈Rψ.

Grahic Jump Location
Fig. 7

The vectors Φx,i, Φy,i and Φz,i, with their sum Φi, rotate in the qi,V/(2ω)−qi,W plane at angular velocity ω

Grahic Jump Location
Fig. 5

Cable tensions along a spatial trajectory

Grahic Jump Location
Fig. 6

Trajectory of the robot during the experiments (first part in attached video which is available under “Supplemental Data” tab for this paper on the ASME Digital Collection, simple periodic elliptical trajectory)

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