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Research Papers

Dynamic Modeling and Adaptive Control of a Single Degree-of-Freedom Flexible Cable-Driven Parallel Robot

[+] Author and Article Information
Harsh Atul Godbole

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada
e-mail: harsh.godbole@mail.mcgill.ca

Ryan James Caverly

Department of Aerospace
Engineering and Mechanics,
University of Minnesota,
Minneapolis, MN 55455
e-mail: rcaverly@umn.edu

James Richard Forbes

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada
e-mail: james.richard.forbes@mcgill.ca

1Present address: CAE, Inc., Montreal, QC H4T 1G6, Canada.

2Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received August 6, 2018; final manuscript received April 2, 2019; published online May 17, 2019. Assoc. Editor: Heikki Handroos.

J. Dyn. Sys., Meas., Control 141(10), 101002 (May 17, 2019) (13 pages) Paper No: DS-18-1363; doi: 10.1115/1.4043427 History: Received August 06, 2018; Revised April 02, 2019

This paper investigates the dynamic modeling and adaptive control of a single degree-of-freedom flexible cable-driven parallel robot (CDPR). A Rayleigh–Ritz cable model is developed that takes into account the changes in cable mass and stiffness due to its winding and unwinding around the actuating winch, with the changes distributed throughout the cables. The model uses a set of state-dependent basis functions for discretizing cables of varying length. A novel energy-based model simplification is proposed to further facilitate reduction in the computational load when performing numerical simulations involving the Rayleigh–Ritz model. For control purposes, the massive payload assumption is used to decouple the rigid and elastic dynamics of the system, and a modified input torque and modified output payload rate are used to develop a passive input–output map for the naturally noncollocated system. A passivity-based adaptive control law is derived to dynamically adapt to changes in cable properties and payload inertia, and different forms of the adaptive control law regressor are proposed. It is shown through numerical simulations that the adaptive controller is robust to changes in payload mass and cable properties, and the selection of the regressor form has a significant impact on the performance of the controller.

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Figures

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Fig. 1

A single degree-of-freedom unconstrained CDPR (referred to as a half system)

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Fig. 2

The fully constrained single degree-of-freedom CDPR

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Fig. 3

Block diagrams showing (a) the closed-loop adaptive controller and (b) the proportional control prewrap and derivative control for μ-plot analysis

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Fig. 4

Comparison of open-loop system responses with cases 1 and 2, including (a) payload position, ρe (case 1) and ρa (case 2), versus time and (b) and a comparison of Ψ˙qe and Ψq˙e versus time

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Fig. 5

Bode plot of τ̂c↦ρ˙μ using (a) Mce with Ψoe and (b) Mce and Ψo

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Fig. 6

Bode plot of τ̂c↦ρ˙μ for the full system modeled usingthe lumped-mass (L.M.) method and Rayleigh-Ritz (R.R.) method with Mce

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Fig. 7

((a) and (b)) μ-plot for the range of 0.92≤μ≤1 and 2≤Kd≤150  N·s/m with ζα=5×10−10, ((c) and (d)) μ-plot for the range of 0.92≤μ≤1 and 2≤Kd≤150  N·s/m with ζα=5×10−6

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Fig. 8

Reference trajectory 1 in (a) and trajectory 2 in (b) with the closed-loop response of ρ using the proposed adaptive controller and the PD controller designed using exact or perturbed system parameters

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Fig. 9

Closed-loop simulation with trajectory 1 and exactly known parameters. Responses of (a) error in payload position versus time with the proposed adaptive controller (blue) and the PD controller (orange), (b) exact Mρρ (dashed) and estimated M̂ρρ (solid) versus time, and (c) exact Gnl (dashed) and estimated Ĝnl (solid) versus time.

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Fig. 10

Closed-loop simulation with trajectory 2 and exactly known parameters. Responses of (a) error in payload position versus time with the proposed adaptive controller (blue) and the PD controller (orange), (b) exact Mρρ (dashed) and estimated M̂ρρ (solid) versus time, and (c) exact Gnl (dashed) and estimated Ĝnl (solid) versus time.

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Fig. 11

Closed-loop simulation with Trajectory 1 and perturbed parameters. Responses of (a) error in payload position versus time with the proposed adaptive controller (yellow) and the PD controller (purple), (b) exact Mρρ (dashed) and estimated M̂ρρ (solid) versus time, and (c) exact Gnl (dashed) and estimated Ĝnl (solid) versus time.

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Fig. 12

Closed-loop simulation with trajectory 2 and perturbed parameters. Responses of (a) error in payload position versus time with the proposed adaptive controller (yellow) and the PD controller (purple), (b) exact Mρρ (dashed) and estimated M̂ρρ (solid) versus time, and (c) exact Gnl (dashed) and estimated Ĝnl (solid) versus time.

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Fig. 13

Plot of payload position tracking error ρ̃ versus time with trajectory 2 for different regressor forms within the time interval (a) 0–30 s and (b) 0–7 s

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