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Research Papers

Exact Constraint Design of a Two-Degree of Freedom Flexure-Based Mechanism1

[+] Author and Article Information
Dannis M. Brouwer

e-mail: d.m.brouwer@utwente.nl

Ronald G. K. M. Aarts

Mechanical Automation and Mechatronics,
Faculty of Engineering Technology,
University of Twente,
Enschede, P.O. Box 217,
7500 AE, The Netherlands

1Some preliminary results were presented in DETC2012-70377 [17].

2Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received February 8, 2013; final manuscript received July 19, 2013; published online September 11, 2013. Assoc. Editor: Philippe Wenger.

J. Mechanisms Robotics 5(4), 041011 (Sep 11, 2013) (10 pages) Paper No: JMR-13-1037; doi: 10.1115/1.4025175 History: Received February 08, 2013; Revised July 19, 2013

We present the exact constraint design of a two degrees of freedom cross-flexure-based stage that combines a large workspace to footprint ratio with high vibration mode frequencies. To maximize unwanted vibration mode frequencies the mechanism is an assembly of optimized parts. To ensure a deterministic behavior the assembled mechanism is made exactly constrained. We analyze the kinematics of the mechanism using three methods; Grüblers criterion, opening the kinematic loops, and with a multibody singular value decomposition method. Nine release-flexures are implemented to obtain an exact constraint design. Measurements of the actuation force and natural frequency show no bifurcation, and load stiffening is minimized, even though there are various errors causing nonlinearity. Misalignment of the exact constraint designs does not lead to large stress, it does however decrease the support stiffness significantly. We conclude that designing an assembled mechanism in an exactly constrained manner leads to predictable stiffnesses and modal frequencies.

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References

Eastman, F. S., 1935, “Flexure Pivots to Replace Knife Edges and Ball Bearings,” Engineering Experiment Station Bulletin 86, University of Washington.
Hale, L. C., 1999, “Principles and Techniques for Designing Precision Machines,” Ph. D. thesis, University of California, Livermore, CA.
Soemers, H. M. J. R., 2010, Design Principles for Precision Mechanisms, T-Pointprint, Enschede.
Haringx, J. A., 1949, “The Cross-Spring Pivot as a Constructional Element,” Appl. Sci. Res., Sect. A, 1(1), pp. 313–332. [CrossRef]
Jones, R. V., 1956, “A Parallel-Spring Cross-Movement for an Optical Bench,” J. Sci. Instrum., 33(7), pp. 279–280. [CrossRef]
Paros, J. M., and Weisbord, L., 1965, “How to Design Flexure Hinges,” Mach. Des., 37(25), pp. 151–156.
Eijk, J. V., 1985, “On the Design of Plate Spring Mechanism,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Jones, R. V., 1988, Instruments and Experiences, Papers on Measurement and Instrument Design, Wiley, New York.
Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Taylor & Francis, London, England.
Howell, L. L., 2001, Compliant Mechanisms, Wiley, New York.
Zelenika, S., and de Bona, F., 2002, “Analytical and Experimental Characterisation of High-Precision Flexural Pivots Subjected to Lateral Loads,” Precis. Eng., 26(4), pp. 381–388. [CrossRef]
Henein, S., Spanoudakis, P., Droz, S., Myklebust, L. I., and Onillon, E., 2003, “Flexure Pivot for Aerospace Mechanisms,” Proceedings of the 10th ESMATS/ESA, Vol. 10, pp. 285–288.
Awtar, S., and Slocum, A. H., 2007, “Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms,” ASME J. Mech. Des., 129(8), pp. 816–830. [CrossRef]
Hopkins, J. B., and Culpepper, M. L., 2010, “Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts Via Freedom and Constraint Topology (FACT)—Part I: Principles,” Precis. Eng., 34(4), pp. 259–270. [CrossRef]
Brouwer, D. M., Meijaard, J. P., and Jonker, J. B., 2013, “Large Deflection Stiffness Analysis of Parallel Prismatic Leaf-Spring Flexures,” Precis. Eng., 37(3), pp. 505–521. [CrossRef]
Wiersma, D. H., Boer, S. E., Aarts, R. G. K. M., and Brouwer, D. M., 2012, “Large Stroke Performance Optimization of Spatial Flexure Hinges,” Proceedings of the 1st Biennial International Conference on Dynamics for Design, No. DETC2012-70502.
Folkersma, K. G. P., Boer, S. E., Brouwer, D. M., Herder, J. L., and Soemers, H. M. J. R., 2012, “A 2-DOF Large Stroke Flexure-Based Positioning Mechanism,” Proceedings of the 36th Mechanisms and Robotics Conference, No. DETC2012-70377.
Blanding, D., 1999, Exact Constraint Machine Design Using Kinematic Principles, ASME Press, New York.
Brouwer, D. M., Boer, S. E., Meijaard, J. P., and Aarts, R. G. K. M., 2013, “Optimization of Release Locations for Small Stress Large Stiffness Flexure Mechanisms,” Mech. Mach. Theory, 64, pp. 230–250. [CrossRef]
Awtar, S., Shimotsu, K., and Sen, S., 2010, “Elastic Averaging in Flexure Mechanisms: A Three-Beam Parallelogram Flexure Case Study,” ASME J. Mech. Rob., 2(4), p. 041006. [CrossRef]
Meijaard, J. P., Brouwer, D. M., and Jonker, J. B., 2010, “Analytical and Experimental Investigation of a Parallel Leaf Spring Guidance,” Multibody Syst. Dyn., 23, pp. 77–97. [CrossRef]
Maxwell, J. C., 1864, “On the Calculation of the Equilibrium and Stiffness of Frames,” Philos. Mag., 27(4), pp. 294–299 [CrossRef].
Chebychev, P. L., 1907, Sur les parallélogrammes, Oeuvres, Tome II, pp. 85–106.
Grübler, M., 1883, “Allgemeine Eigenschaften der Zwangläufigen ebenen kinematischen Ketten,” Civilingenieur, 29, pp. 167–200.
Kutzbach, K., 1929, “Mechanische Leitungsverzweigung, ihre Gesetze und Anwendungen,” Maschinenbau. Betrieb, 8, pp. 710–716.
Besseling, J. F., 1979, “Trends in Solid Mechanics 1979,” Proceedings of the Symposium Dedicated to the 65th Birthday,” W. T.Koiter, J.Besseling, and A.van der Heijden, eds., Delft University Press, pp. 53–78.
Pellegrino, S., and Calladine, C. R., 1986, “Matrix Analysis of Statically and Kinematically Indeterminate Frameworks,” Int. J. Solids Struct., 22(4), pp. 409–428. [CrossRef]
Angeles, J., and Gosselin, C., 1989, “Détermination du degré de liberté des chaînes cinématiques,” Transactions de la Société Canadienne de Génie Mécanique, 12, pp. 219–226.
Aarts, R. G. K. M., Meijaard, J. P., and Jonker, J. B., 2012, “Flexible Multibody Modelling for Exact Constraint Mechatronic Design of Compliant Mechanisms,” Multibody Syst. Dyn., 27(1), pp. 119–133. [CrossRef]
Nelder, J. A., and Mead, R., 1965, “A Simplex Method for Function Minimization,” Comput. J., 7(4), pp. 308–313. [CrossRef]
Ryu, J. W., Gweon, D.-G., and Moon, K. S., 1997, “Optimal Design of a Flexure Hinge Based XY Wafer Stage,” Precis. Eng., 21(1), pp. 18–28. [CrossRef]
Culpepper, M. L., and Anderson, G., 2004, “Design of a Low-Cost Nano-Manipulator Which Utilizes a Monolithic, Spatial Compliant Mechanism,” Precis. Eng., 28(4), pp. 469–482. [CrossRef]
Yao, Q., Dong, J., and Ferreira, P. M., 2007, “Design, Analysis, Fabrication, and Testing of a Parallel-Kinematic Micropositioning XY Stage,” Int. J. Mach. Tools Manuf., 47(6), pp. 946–961. [CrossRef]
de Jong, B. R., Brouwer, D. M., de Boer, M. J., Jansen, H. V., Soemers, H. M. J. R., and Krijnen, G. J. M., 2010, “Design and Fabrication of a Planar Three-DOFs MEMS-Based Manipulator,” J. Microelectromech. Syst., 19(5), pp. 1116–1130. [CrossRef]
Werner, C., Rosielle, P. C. J. N., and Steinbuch, M., 2010, “Design of a Long Stroke Translation Stage for AFM,” Int. J. Mach. Tools Manuf., 50(2), pp. 183–190. [CrossRef]
Boer, S. E., Aarts, R. G. K. M., Brouwer, D. M., and Jonker, J. B., 2010, “Multibody Modelling and Optimization of a Curved Hinge Flexure,” The 1st Joint International Conference on Multibody System Dynamics, Lappeenranta, pp. 1–10.
Wijma, W., Boer, S. E., Aarts, R. G. K. M., Brouwer, D. M., and Hakvoort, W. B. J., 2013, “Modal Measurements and Model Corrections of a Large Stroke Compliant Mechanism,” ECCOMAS Multibody Dynamics 2013, July 1–4, 2013, University of Zagreb, Croatia, pp. 831–843.

Figures

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Fig. 1

Two-DOF mechanism allowing for base mounted actuators; (a) shows the motion resulting from driving actuator 1 and constraining actuator 2; (b) shows the motion resulting from driving actuator 2 and constraining actuator 1; (c) shows a drawing of the mechanism with top plate removed

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Fig. 2

Constraint analysis by opening loop h1–b1–h10–b7–h11 –b2–h2–base. The DOFs (solid vectors) and constraints (dashed vectors) are shown for points A and B.

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Fig. 3

Analysis of the constraints on end-effector b8. The DOFs (solid vectors) and constraints (dashed vectors) are shown for points C, D, and E. Actuators 1 and 2 are considered to be constrained. For displaying purposes hinges h4 and h10, and hinges h5 and h11 are spaced apart, this does not influence the constraint analysis.

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Fig. 4

The three von Mises stress distributions resulting from the three overconstraints in loop h1–b1–h10–b7–h11–b2–h2–base

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Fig. 5

The stress distribution in loop h1–b1–h10–b7–h11–b2–h2–base with bending released at both ends of bar b7; (a) the bending stress component. (b) The shear stress component. Please note that the von Mises stress scale differs from the bending and shear stress scale.

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Fig. 6

The von Mises stress distributions due to the three overconstraints due to the addition of h4–b4–h7–b8–h8–b5–h5 on the exact constrained loop h1–b1–h10–b7–h11–b2–h2–base

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Fig. 7

The von Mises stress distributions due to the three overconstraints due to the addition of h3–b3–h6–b6–h9 on the end-effector b8

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Fig. 8

Photo of the two-DOF mechanism with skeleton frame showing the nine release locations and orientations, double arrows indicating rotational DOFs

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Fig. 10

Torsionally and bending compliant arm

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Fig. 11

(a) Force-position measurement with one of the actuators blocked, (b) unfiltered and filtered force-residual displacement measurement with one of the actuators blocked

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Fig. 12

Beams h7 and h10 showed misalignment after several experiments

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Fig. 13

Mode shape measurement setup

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Fig. 14

Measurements of the first unwanted natural frequencies

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