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Research Papers

# The Synthesis of Spherical Motion Generators in the Presence of an Incomplete Set of Attitudes

[+] Author and Article Information
Khalid Al-Widyan

Mechatronics Engineering Department,
Hashemite University,
Zarqa 13115, Jordan
e-mail: alwidyan@hu.edu.jo

Jorge Angeles

Department of Mechanical Engineering
and Centre for Intelligent Machines,
McGill University,
e-mail: angeles@cim.mcgill.ca

A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Talyor series in a neighborhood of every point—Weisstein, Eric W., “Real Analytic Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/RealAnalyticFunction.html

The CPM Ej of vector $ej∈IR3$ is defined, for any vector $v∈IR3$, as $Ej=CPM(ej)≡(∂(ej×v)/∂v)⇔ej×v=Ejv$.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received October 25, 2012; final manuscript received July 22, 2013; published online April 15, 2014. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 6(3), 031008 (Apr 15, 2014) (8 pages) Paper No: JMR-12-1174; doi: 10.1115/1.4025297 History: Received October 25, 2012; Revised July 22, 2013

## Abstract

Proposed in this paper is a general methodology applicable to the synthesis of spherical motion generators in the presence of an incomplete set of finitely separated attitudes. The spherical rigid-body guidance problem in the realm of four-bar linkage synthesis can be solved exactly for up to five prescribed attitudes of the coupler link, and hence, any number of attitudes smaller than five is considered incomplete in this paper. The attitudes completing the set are determined to produce a linkage whose performance is robust against variations in the unprescribed attitudes. Robustness is needed in this context to overcome the presence of uncertainty due to the selection of the unspecified attitudes, that many a time are specified implicitly by the designer upon choosing, for example, the location of the fixed joints of the dyads. A theoretical framework for model-based robust engineering design is thus, recalled, and a methodology for the robust synthesis of spherical four-bar linkages is laid down. An example is included here to concretize the concepts and illustrate the application of the proposed methodology.

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Topics: Linkages , Design , Generators

## References

Tsai, L., and Roth, B., 1972, “Design of Dyads With Helical, Cylindrical, Spherical, Revolute and Prismatic Joints,” Mech. Mach. Theory, 7(1), pp. 85–102.
Chiang, C., 1988, Kinematics of Spherical Mechanisms, Cambridge University Press, New York.
McCarthy, J., and Soh, G., 2010, Geometric Design of Linkages, 2nd ed., Springer-Verlag, New York.
Bodduluri, R., and McCarthy, J., 1992, “Finite Position Synthesis Using the Image Curve of a Spherical Four-Bar Motion,” ASME J. Mech. Des., 114(1), pp. 55–60.
Larochelle, P., and Tse, D., 2000, “Approximating Spatial Locations With Spherical Orientations for Spherical Mechanism Design,” ASME J. Mech. Des., 122(4), pp. 457–463.
Brunnthaler, K., Schröcher, H., and Husty, M., 2006, “Synthesis of Spherical Four-Bar Mechanisms Using Spherical Kinematic Mapping,” Advances in Robot Kinematics, J.Lenarčič and B.Roth, eds., Springer, The Netherlands, pp. 377–384.
Ketchel, J., and Larochelle, P., 2007, “Computer-Aided Manufacturing of Spherical Mechanisms,” Mech. Mach. Theory, 42(4), pp. 131–146.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, Dover Publications, New York.
Suh, C., and Radcliffe, C., 1978, Kinematics and Mechanisms Design, John Wiley & Sons, New York.
Phadke, M., 1989, Quality Engineering Using Robust Design, Prentice-Hall, Englewood Cliffs, NJ.
Wu, Y., and Wu, A., 2000, Taguchi Methods for Robust Design, ASME Press, New York.
Tsui, K., 1992, “An Overview of Taguchi Method and Newly Developed Statistical Methods for Robust Design,” IIE Trans., 24(5), pp. 44–57.
Nair, N., 1992, “Taguchi's Parameter Design: A Panel Discusion,” Technometrics, 34(1), pp. 127–161.
Al-Widyan, K., and Angeles, J., 2005, “A Model-Based Formulation of Robust Design,” ASME J. Mech. Des., 127(3), pp. 388–396.
Lu, X., Li, H., and Chen, C., 2012, “Model-Based Probabilistic Robust Design With Data-Based Uncertainty Compensation for Partially Unknown System,” ASME J. Mech. Des., 134(2), pp. 195–205.
Apley, D., Liu, J., and Chen, W., 2006, “Understanding the Effects of Model Uncertainty in Robust Design With Computer Experiments Model-Based Probabilistic Robust Design With Data-Based Uncertainty Compensation for Partially Unknown System,” ASME J. Mech. Des., 128(4), pp. 945–959.
Du, X., Venigella, P. K., and Liu, D., 2009, “Robust Mechanism Synthesis With Random and Interval Variables,” Mech. Mach. Theory, 44(7), pp. 1321–1337.
Kota, S., and Chiou, S., 1993, “Use of Orthogonal Arrays in Mechanism Synthesis,” Mech. Mach. Theory, 28(6), pp. 777–794.
Kunjur, A., and Krishnamury, S., 1997, “A Robust Multi-Criteria Optimization Approach,” Mech. Mach. Theory, 32(7), pp. 797–811.
Da Lio, M., 1997, “Robust Design of Linkages-Synthesis by Solving Non-Linear Optimization Problems,” Mech. Mach. Theory, 32(8), pp. 921–932.
Al-Widyan, K., and Angeles, J., 2010, “The Robust Linkage Synthesis for Planar Rigid-Body Guidance,” ASME Paper No. DETC2010-28308.
Mahableshwarkar, S., and Kramer, S., 1990, “Sensitivity Analysis of the Burmester Equations of Planar Motion,” ASME J. Mech. Des., 112, pp. 299–306.
Kalnas, R., and Kota, S., 2001, “Incorporating Uncertainty Into Mechanism Synthesis,” Mech. Mach.ine Theory, 46, pp. 843–851.
Cramer, C., 1946, Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ.
Householder, A., 1964, The Theory of Matrices in Numerical Analysis, Dover Publication, Inc., New York.
Golub, G. H., and Van Loan, C., 1996, Matrix Computations, The Johns Hopkins University Press, Baltimore.
Rao, S., 1996, Engineering Optimization, John Wiley and Sons, Inc., New York.
Angeles, J., 2007, Fundamentals of Robotic Mechanical Systems, 3rd ed., Springer-Verlag, New York.
Palm, W., 2000, MATLAB for Engineering Applications, Prentice-Hall, Inc., Upper Saddle River, NJ.
Balli, S., and Chand, S., 2002, “Defects in Link Mechnaism and Solution Rectification,” Mech. Mach. Theory, 37, pp. 851–876.
Angeles, J., and Rojas, A., 1983, “An Optimisation Approach to the Branchig Problem of Plane Linkage Synthesis,” Proceedings of 6th World Congress on the Theory of Machines and Mechanisms, pp. 120–123.

## Figures

Fig. 1

Fig. 2

A spherical RR dyad at a reference and at the jth attitude

Fig. 3

The synthesized linkage at the pick attitude

Fig. 4

The synthesized linkage at the place attitude

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