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Research Papers

Wrench Uncertainty Quantification and Reconfiguration Analysis in Loosely Interconnected Cooperative Systems

[+] Author and Article Information
Javad Sovizi

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: javadsov@buffalo.edu

Rahul Rai

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: rahulrai@buffalo.edu

Venkat Krovi

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: vkrovi@buffalo.edu

1Corresponding author.

Manuscript received August 30, 2016; final manuscript received June 5, 2017; published online October 3, 2017. Assoc. Editor: Yan Wang.

ASME J. Risk Uncertainty Part B 4(2), 021002 (Oct 03, 2017) (12 pages) Paper No: RISK-16-1110; doi: 10.1115/1.4037122 History: Received August 30, 2016; Revised June 05, 2017

Loosely interconnected cooperative systems such as cable robots are particularly susceptible to uncertainty. Such uncertainty is exacerbated by addition of the base mobility to realize reconfigurability within the system. However, it also sets the ground for predictive base reconfiguration in order to reduce the uncertainty level in system response. To this end, in this paper, we systematically quantify the output wrench uncertainty based on which a base reconfiguration scheme is proposed to reduce the uncertainty level for a given task (uncertainty manipulation). Variations in the tension and orientation of the cables are considered as the primary sources of the uncertainty responsible for nondeterministic wrench output on the platform. For nonoptimal designs/configurations, this may require complex control structures or lead to system instability. The force vector corresponding to each agent (e.g., pulley and cable) is modeled as random vector whose magnitude and orientation are modeled as random variables with Gaussian and von Mises distributions, respectively. In a probabilistic framework, we develop the closed-form expressions of the means and variances of the output force and moment given the current state (tension and orientation of the cables) of the system. This is intended to enable the designer to efficiently characterize an optimal configuration (location) of the bases in order to reduce the overall wrench fluctuations for a specific task. Numerical simulations as well as real experiments with multiple iRobots are performed to demonstrate the effectiveness of the proposed approach.

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Figures

Grahic Jump Location
Fig. 1

Cooperative aerial towing example: gray regions show the uncertainty ellipses. (c) shows a safer configuration that minimizes the probability of collision for this specific task compared to nonoptimal configuration in(b).

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Fig. 2

Schematic of a planar cable-robot. Solid lines show the nominal (mean) state of the system (tension and orientation of the cables) and dashed lines show the perturbed state.

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Fig. 3

Three planar systems with m=4, 5, and 6 cables and their corresponding statistical information used for Monte Carlo (MC) validation of the closed-form solutions given by Eqs. (21)(24), (27), and (28).

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Fig. 4

Optimization of the tension and orientation of the cables to minimize (a) the tension factor, (b) Var(Fx) with two cables, (c) Var(Fx) with six cables, (d) Var(Fy) with two cables, and (e) Var(Fy) with six cables. The prescribed task is to provide the static wrench Wd=[45 N, 60 N, 0 N⋅m]T at the end effector. Each base can move 2π rad as shown by dashed lines (constraint (33)). The minimum and maximum traction of the cables are set to be 0 N and 100 N for all cables (constraint (34)). Additionally, λθi=100 and σTi=2 N ∀i=1,…,m.

Grahic Jump Location
Fig. 5

Optimization of the tension and orientation of the cables to ((a) and (b)) maximize the tension factor and ((c) and (d)) minimize Var(M). The desired static wrench is Wd=[45 N, 60 N, 120 N⋅m]T. Each base can move along the arcs shown by dashed lines whose lengths are π rad. The minimum and maximum tension of the cables are set to be and 0 N and 40 N, respectively. Also, λθi=100 and σTi=2 N ∀i=1,…,m.

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Fig. 6

Experimental setup. Three iRobots are tied to the force transducer located at the origin of the global coordinate system x-y. The task is to provide a predefined output force at the end effector.

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Fig. 7

The traction force provided by each iRobot corresponding to different motion commands. Three motion commands are sent consecutively with a pause after first and second command to separate the data points. The average traction values are shown above the corresponding data points for each motion command. The data acquisition frequency is 50 Hz and we collect the force data for 50 s, i.e., 2500 data points for each motion command.

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Fig. 8

Orientational uncertainty of the force vectors. The data points corresponding to three motion commands are shown by blue, black, and red dots and their mean direction is shown by a solid line drawn from the center of the circles. The points shown by green are the rotated data points in order to use MLE approach for estimating λθi's (refer to the text for more details; refer to the web version of the article for interpretation of the references to color in this figure).

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Fig. 9

Experimental application of the optimization scenarios. First row corresponds to TF maximization. Second and third rows show the configuration and cable tensions that minimize Var(Fx) and Var(Fy), respectively.

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