Research Papers

Wrench Accuracy for Parallel Manipulators and Interval Dependency

[+] Author and Article Information
Leila Notash

Department of Mechanical and
Materials Engineering,
Queen's University,
Kingston, ON K7L 3N6, Canada
e-mail: Leila.Notash@queensu.ca

Manuscript received July 11, 2016; final manuscript received November 1, 2016; published online December 2, 2016. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 9(1), 011008 (Dec 02, 2016) (9 pages) Paper No: JMR-16-1201; doi: 10.1115/1.4035221 History: Received July 11, 2016; Revised November 01, 2016

In this paper, the wrench accuracy for parallel manipulators is examined under variations in parameters and data. The solution sets of actuator forces/torques are investigated utilizing interval arithmetic (IA). Implementation issues of interval arithmetic to analyze the performance of manipulators are addressed, including the consideration of dependencies in parameters and the design of input vectors to generate the required wrench. Specifically, the effect of the dependency within and among the entries of the Jacobian matrix is studied, and methodologies for reducing and/or eliminating the overestimation of solution set are presented. In addition, the subset of solution set that produces platform wrenches within the required lower and upper bounds is modeled. Furthermore, the formulation of solutions that provide any platform wrench within the defined interval is examined. Intersection of these two sets, if any, results in the given interval platform wrench. Implementation of the methods to identify the solution for actuator forces/torques is presented on example parallel manipulators.

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Hansen, E. , 1992, Global Optimization Using Interval Analysis, 1st ed., Marcel Dekker, New York.
Moore, R. , Kearfott, R. B. , and Cloud, M. J. , 2009, Introduction to Interval Analysis, SIAM, Philadelphia, PA.
Kearfott, R. B. , 1996, Rigorous Global Search: Continuous Problems, Kluwer Academic, Dordrecht, The Netherlands.
Notash, L. , 2015, “ Analytical Methods for Solution Sets of Interval Wrench,” ASME Paper No. DETC2015-47575.
Shary, S. P. , 1992, “ A New Class of Algorithms for Optimal Solution of Interval Linear Systems,” Interval Comput., 2(4), pp. 18–29.
Hartfiel, D. , 1980, “ Concerning the Solution Set of Ax = b Where PAQ and p ≤ b ≤ q,” Numer. Math., 35(3), pp. 355–359. [CrossRef]
Popova, E. D. , and Krämer, W. , 2008, “ Visualizing Parametric Solution Sets,” BIT Numer. Math., 48(1), pp. 95–115. [CrossRef]
Nazari, V. , and Notash, L. , 2016, “ Motion Analysis of Manipulators With Uncertainty in Kinematic Parameters,” ASME J. Mech. Rob., 8(2), p. 021014. [CrossRef]
Rao, R. S. , Asaithambi, A. , and Agrawal, S. K. , 1998, “ Inverse Kinematic Solution of Robot Manipulators Using Interval Analysis,” J. Mech. Des., 120(1), pp. 147–150. [CrossRef]
Merlet, J. , 2004, “ Solving the Forward Kinematics of a Gough-Type Parallel Manipulator With Interval Analysis,” Int. J. Rob. Res., 23(3), pp. 221–235. [CrossRef]
Daney, D. , Andreff, N. , Chabert, G. , and Papegay, Y. , 2006, “ Interval Method for Calibration of Parallel Robots: Vision-Based Experiments,” Mech. Mach. Theory, 41(8), pp. 929–944. [CrossRef]
Carricato, M. , 2013, “ Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With Three Cables,” ASME J. Mech. Rob., 5(3), p. 031008. [CrossRef]
Notash, L. , 2016, “ On the Solution Set for Positive Wire Tension With Uncertainty in Wire-Actuated Parallel Manipulators,” ASME J. Mech. Rob., 8(4), p. 044506. [CrossRef]
Fieldler, M. , Nedoma, J. , Ramik, J. , and Rohn, J. , 2006, Linear Optimization Problems With Inexact Data, Springer, New York.
Elishakoff, I. , and Miglis, Y. , 2012, “ Overestimation-Free Computational Version of Interval Analysis,” Int. J. Comput. Methods Eng. Sci. Mech., 13(5), pp. 319–328. [CrossRef]
Popova, E. D. , 2013, “ On Overestimation-Free Computational Version of Interval Analysis,” Int. J. Comput. Methods Eng. Sci. Mech., 14(6), pp. 491–494. [CrossRef]
Shary, S. P. , 1992, “ On Controlled Solution Set of Interval Algebraic Systems,” Interval Comput., 4(6), pp. 66–75.
Rump, S. M. , 1999, “ INTLAB—INTerval LABoratory,” Developments in Reliable Computing, T. Csendes , ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 77–104.
Notash, L. , 2016, “ Investigation of Wrench Accuracy for Parallel Manipulators,” ASME Paper No. DETC2016-59425.


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Fig. 1

Planar parallel manipulators with (a) two actuators and (b) three actuators

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Fig. 2

Parameters of 2DOF parallel manipulators

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Fig. 3

Solution sets of Example 1: (a) solution using six parameters and (b) and (c) lines that characterize tolerance solution

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Fig. 4

Solution sets of Example 1 using optimum bounds of Jacobian matrix: (a) solution using six parameters and (b) and (c) tolerance solution using closed half-planes

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Fig. 5

Solution sets of Example 1: (a) with ai and p as independent parameters, (b) three solution sets, and (c) tolerance solution using 256 lines and its vertices

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Fig. 6

Solution sets of Example 2: (a) tolerance and control solutions using 16 lines and discrete method, (b) zoomed-in tolerance solution using closed half-planes, and (c) control solution using closed half-planes in each quadrant

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Fig. 7

Solution sets of Example 3: (a) solution and (b) control solution using discrete method and intersections of closed half-planes




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