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Research Papers

Reliable and Failure-Free Workspaces for Motion Planning Algorithms for Parallel Manipulators Under Geometrical Uncertainties

[+] Author and Article Information
Hiparco L. Vieira

Department of Mechanical Engineering,
São Carlos School of Engineering,
University of São Paulo,
São Carlos 13566-590, São Paulo, Brazil
e-mail: hiparcolins@usp.br

João V. C. Fontes

Department of Mechanical Engineering,
São Carlos School of Engineering,
University of São Paulo,
São Carlos 13566-590, São Paulo, Brazil
e-mail: joao.fontes@usp.br

André T. Beck

Department of Mechanical Engineering,
São Carlos School of Engineering,
University of São Paulo,
São Carlos 13566-590, São Paulo, Brazil
e-mail: atbeck@sc.usp.br

Maíra M. da Silva

Department of Mechanical Engineering,
São Carlos School of Engineering,
University of São Paulo,
São Carlos 13566-590, São Paulo, Brazil
e-mail: mairams@sc.usp.br

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 30, 2018; final manuscript received November 9, 2018; published online January 7, 2019. Assoc. Editor: Yan Wang.

J. Comput. Nonlinear Dynam 14(2), 021005 (Jan 07, 2019) (9 pages) Paper No: CND-18-1239; doi: 10.1115/1.4042015 History: Received May 30, 2018; Revised November 09, 2018

Manufacturing tolerances and other uncertainties may play an important role in the performance of parallel manipulators since they can affect the distance to a singular configuration. Motion planning strategies for parallel manipulators under uncertainty require decision making approaches for classifying reliable regions within the workspace. In this paper, we address fail free and reliable motion planning for parallel manipulators. Failure is related to parallel kinematic singularities in the motion equations or to ill-conditioning of the Jacobian matrices. Monte Carlo algorithm is employed to compute failure probabilities for a dense grid of manipulator workspace configurations. The inverse condition number of the Jacobian matrix is used to compute the distance between each configuration and a singularity. For supporting motion planning strategies, not only failure maps are constructed but also reliable and failure-free workspaces are obtained. On the one hand, the reliable workspace is obtained by minimizing the failure probabilities subject to a minimal workspace area. Differently, a failure-free workspace is found by maximizing the workspace area subject to a probability of failure equal to zero. A 3RRR manipulator is used as a case study. For this case study, the usage of the reliable strategy can be useful for robustifying motion planning algorithm without a significant reduction of the reliable regions within the workspace.

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Figures

Grahic Jump Location
Fig. 1

Geometrical scheme of the 3RRR manipulator

Grahic Jump Location
Fig. 2

ICN maps of each orientation: (a) α1=−(π/6) rad, (b) α2=−(π/12) rad, (c) α3 = 0 rad, (d) α4=(π/12) rad, and (e) α5=(π/6) rad

Grahic Jump Location
Fig. 3

Failure probability maps for each orientation: (a) α1=−(π/6) rad, (b) α2=−(π/12) rad, (c) α3 = 0 rad, (d) α4=(π/12) rad, and (e) α5=(π/6) rad

Grahic Jump Location
Fig. 4

Probability of failure versus nominal ICN for each configuration in the workspace for different α values: (a) α1=−(π/6) rad, (b) α2=−(π/12) rad, (c) α3 = 0 rad, (d) α4=(π/12) rad, and (e) α5=(π/6) rad

Grahic Jump Location
Fig. 5

Probability of failure versus nominal ICN for each failure mode, considering α = 0 rad: (a) ICN failures and (b) workspace failures

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Fig. 6

Failure map for each failure mode, considering α = 0 rad: (a) ICN failures and (b) workspace failures

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Fig. 7

Compromise between the workspace area and the average probability of failure

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Fig. 8

Workspace area (%) for different reliable CICN and α values

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Fig. 9

Average probability of failure for different reliable CICN and α values

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