Research Papers: Design of Mechanisms and Robotic Systems

Eliminating Underconstraint in Double Parallelogram Flexure Mechanisms

[+] Author and Article Information
Robert M. Panas

Materials Engineering Division,
Lawrence Livermore National Laboratory,
7000 East Ave, L-229,
Livermore, CA 94551
e-mail: panas3@llnl.gov

Jonathan B. Hopkins

Mechanical and Aerospace Engineering,
University of California, Los Angeles,
Los Angeles, CA 90095
e-mail: hopkins@seas.ucla.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 8, 2014; final manuscript received May 18, 2015; published online July 17, 2015. Assoc. Editor: Oscar Altuzarra.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Mech. Des 137(9), 092301 (Jul 17, 2015) (9 pages) Paper No: MD-14-1409; doi: 10.1115/1.4030773 History: Received July 08, 2014

We present an improved flexure linkage design for removing underconstraint in a double parallelogram (DP) linear flexural mechanism. This new linkage alleviates many of the problems associated with current linkage design solutions such as static and dynamic performance losses and increased footprint. The improvements of the new linkage design will enable wider adoption of underconstraint eliminating (UE) linkages, especially in the design of linear flexural bearings. Comparisons are provided between the new linkage design and existing UE designs over a range of features including footprint, dynamics, and kinematics. A nested linkage design is shown through finite element analysis (FEA) and experimental measurement to work as predicted in selectively eliminating the underconstrained degrees-of-freedom (DOF) in DP linear flexure bearings. The improved bearing shows an 11 × gain in the resonance frequency and 134× gain in static stiffness of the underconstrained DOF, as designed. Analytical expressions are presented for designers to calculate the linear performance of the nested UE linkage (average error < 5%). The concept presented in this paper is extended to an analogous double-nested rotary flexure design.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Awtar, S. , Slocum, A. H. , and Sevincer, E. , 2007, “Characteristics of Beam-Based Flexure Modules,” ASME J. Mech. Des., 129(6), pp. 625–639. [CrossRef]
Krijnen, B. , and Brouwer, D. M. , 2014, “Flexures for Large Stroke Electrostatic Actuation in MEMS,” J. Micromech. Microeng., 24(1), p. 015006. [CrossRef]
Saggere, L. , Kota, S. , and Crary, S. B. , 1994, “A New Design for Suspension of Linear Microactuators,” International Mechanical Engineering Congress and Exposition: Dynamic Systems and Control, Chicago, pp. 671–676.
Suh, N. P. , 2001, Axiomatic Design: Advances and Applications, Oxford University, New York.
Slocum, A. , 1992, Precision Machine Design, Prentice-Hall, Inc., Eaglewood Cliffs, NJ.
Blanding, D. L. , 1999, Exact Constraint: Machine Design Using Kinematic Principles, ASME, New York.
Smith, S. T. , 2000, Flexures: Elements of Elastic Mechanisms, CRC, Boca Raton, FL.
Smith, S. T. , 1992, Foundations of Ultraprecision Mechanism Design, CRC, Boca Raton, FL.
Hopkins, J. B. , and Culpepper, M. L. , 2010, “Synthesis of Multi-Degrees-of-Freedom, Parallel Flexure System Concepts Via Freedom and Constraint Topology (FACT)—Part I: Principles,” Precis. Eng., 34(2), pp. 259–270. [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 II: Practice,” Precis. Eng., 34(2), pp. 271–278. [CrossRef]
Panas, R. M. , Cullinan, M. A. , and Culpepper, M. L. , 2012, “Design of Piezoresistive-Based MEMS Sensor Systems for Precision Microsystems,” Precis. Eng., 36(1), pp. 44–54. [CrossRef]
Zhao, H. , Bi, S. , and Yu, J. , 2012, “A Novel Compliant Linear-Motion Mechanism Based on Parasitic Motion Compensation,” Mech. Mach. Theory., 50, pp. 15–28. [CrossRef]
Hongzhe, Z. , Bi, S. , Yu, J. , and Guo, J. , 2012, “Design of a Family of Ultra-Precision Linear Motion Mechanisms,” J. Mech. Rob., 4(4), p. 041012. [CrossRef]
Howell, L. L. , 2001, Compliant Mechanisms, Wiley, New York.
Duarte, R. M. , Howells, M. R. , Hussain, Z. , Lauritzen, T. , McGill, R. , Moler, E. J. , and Spring, J. , 1997, “Linear Motion Machine for Soft X-Ray Interferometry,” Proc. SPIE, 3132, Optomechanical Design and Precision Instruments, pp. 224–232.
Brouwer, D. M. , Otten, A. , Engelen, J. B. C. , Krijnen, B. , and Soemers, H. M. J. R. , 2010, “Long-Range Elastic Guidance Mechanisms for Electrostatic Comb-Drive Actuators,” International Conference of the European Society for Precision Engineering and Nanotechnology (EUSPEN), May 31–Jun. 4, pp. 41–50.
Hubbard, N. B. , Wittwer, J. W. , Kennedy, J. A. , Wilcox, D. L. , and Howell, L. L. , 2004, “A Novel Fully Compliant Planar Linear-Motion Mechanism,” ASME International Design Engineering Technical Conference and Computers and Information in Engineering Conference (IDETC/CIE), Salt Lake City, UT, pp. 1–5.
Chang, S. H. , and Li, S. S. , 1999, “A High Resolution Long Travel Friction-Drive Micropositioner With Programmable Step Size,” Rev. Sci. Instrum., 70(6), pp. 2776–2782. [CrossRef]
Awtar, S. , 2004, Synthesis and Analysis of Parallel Kinematic XY Flexure Mechanisms, Sc.D. thesis, Massachusetts Institute of Technology, Cambridge MA.
Brouwer, D. M. , de Jong, B. R. , and Soemers, H. M. J. R. , 2010, “Design and Modeling of a Six DOFs MEMS-Based Precision Manipulator,” Precis. Eng., 34(2), pp. 307–319. [CrossRef]
Xu, Q. , and Li, Y. , 2010, “Novel Design of a Totally Decoupled Flexure-Based XYZ Parallel Micropositioning Stage,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), IEEE, Montreal, Canada, pp. 866–871.
Seggelen, J. K. V. , Rosielle, P. C. J. N. , Schellekens, P. H. J. , Spaan, H. A. M. , Bergmans, R. H. , and Kotte, G. J. W. L. , 2005, “An Elastically Guided Machine Axis With Nanometer Repeatability,” CIRP Ann.—Manuf. Technol., 54(1), pp. 487–490. [CrossRef]
Li, Y. , and Xu, Q. , 2009, “Design and Analysis of a Totally Decoupled Flexure-Based XY Parallel Micromanipulator,” IEEE Trans. Rob., 25(3), pp. 645–657. [CrossRef]
Parmar, G. , Barton, K. , and Awtar, S. , 2014, “Large Dynamic Range Nanopositioning Using Iterative Learning Control,” Precis. Eng., 38(1), pp. 48–56. [CrossRef]
Hao, G. , Kong, X. , and Meng, Q. , 2010, “Design and Modelling of Spatial Compliant Parallel Mechanisms For Large Range of Translation,” ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE), Vol. 44, QC, Canada.
Pusl, K. E. , “Folded Spring Flexure Suspension For Linearly Actuated Devices,” U.S. patent application 7550880B1, June 23, 2009.
Eijk, J. V. , 1985, On the Design of Plate Spring Mechanisms, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Zhou, G. , and Dowd, P. , 2003, “Tilted Folded-Beam Suspension for Extending the Stable Travel Range of Comb-Drive Actuators,” J. Micromech. Microeng., 13(2), pp. 178–183. [CrossRef]
Brouwer, D. M. , De Jong, B. R. , Soemers, H. M. J. R. , and Van Dijk, J. , 2006, “Sub-Nanometer Stable Precision MEMS Clamping Mechanism Maintaining Clamp Force Unpowered for TEM Application,” J. Micromech. Microeng., 16(6), pp. S7–S12. [CrossRef]
Chang, S. , Wang, C. S. , Xiong, C. Y. , and Fang, J. , 2005, “Nanoscale In-Plane Displacement Evaluation by AFM Scanning and Digital Image Correlation Processing,” Nanotechnology, 16(4), pp. 344–349. [CrossRef]
Krijnen, B. , and Brouwer, D. M. , 2011, “Position Control of a MEMS Stage With Integrated Sensor,” 11th International Conference on European Society for Precision Engineering and Nanotechnology (EUSPEN), Como, Italy, pp. 2–3.
Olfatnia, M. , Sood, S. , and Awtar, S. , 2012, “Note: An Asymmetric Flexure Mechanism for Comb-Drive Actuators,” Rev. Sci. Instrum., 83(11), p. 116105. [CrossRef] [PubMed]
Olfatnia, M. , Sood, S. , Gorman, J. J. , and Awtar, S. , 2013, “Large Stroke Electrostatic Comb-Drive Actuators Enabled by a Novel Flexure Mechanism,” J. Microelectromech. Syst., 22(2), pp. 483–494. [CrossRef]
Trutna, T. T. , and Awtar, S. , 2010, “An Enhanced Stability Model for Electrostatic Comb-Drive Actuator Design,” ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE), Montreal, Canada, pp. 1–9.
Ferreira, P. M. , Dong, J. , and Mukhopadhyay, D. , 2012, “High Precision Silicon-on-Insulator MEMS Parallel Kinematic Stages,” U.S. patent application 8310128B2.
Jaecklin, V. P. , Linder, C. , De Rooij, N. F. , Moret, J. M. , Bischof, R. , and Rudolf, F. , 1992, “Novel Polysilicon Comb Actuators for XY-Stages,” An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots, IEEE MEMS '92, Travemunde, Germany, pp. 147–149.
Jones, R. V. , Phil, D. , and Young, I. R. , 1956, “Some Parasitic Deflexions in Parallel Spring Movements,” J. Sci. Instrum., 33(1), pp. 11–15. [CrossRef]
Jones, R. V. , 1962, “Some Uses of Elasticity in Instrument Design,” J. Sci. Instrum., 39(5), pp. 193–203. [CrossRef]
Brouwer, D. , 2007, Design Principles for Six Degrees-of-Freedom MEMS-Based Precision Manipulators, Ph.D. thesis, University of Twente, Eindhoven, The Netherlands.
Plainevaux, J. E. , 1954, “Mouvement parasite vertical d'une suspension élastique symétrique à compensation et asservissement,”. Nuovo Cimento, 11(6), pp. 626–638. [CrossRef]
Plainevaux, J. E. , 1954, “Guidage rectiligne sur lames élastiques. Comparaison de divers types connus et nouveaux,” Nuovo Cimento, 12(1), pp. 37–59. [CrossRef]
Hopkins, J. B. , and Culpepper, M. L. , 1954, “Synthesis of Precision Serial Flexure Systems Using Freedom and Constraint Topologies (FACT),” Precis. Eng., 35(4), pp. 638–649. [CrossRef]
Franklin, G. F. , Powell, J. D. , and Naeini, A. E. , 2006, Feedback Control of Dynamic Systems, 5th ed., Prentice Hall, Upper Saddle River, NJ.
Smith, S. T. , 1987, Mechanical Systems in Nanometre Metrology, Ph.D. thesis, University of Warwick, Coventry, UK.
Grade, J. D. , Jerman, H. , and Kenny, T. W. , 2003, “Design of Large Deflection Electrostatic Actuators,” J. Microelectromech. Syst., 12(3), pp. 335–343. [CrossRef]
Chen, C. , and Lee, C. , 2004, “Design and Modeling for Comb Drive Actuator With Enlarged Static Displacement,” Sens. Actuators, A, 115(2–3), pp. 530–539. [CrossRef]
Jerman, J. H. , and Grade, J. D. , 2003, “Miniature Device With Translatable Member,” U.S. Patent 6664707 B2, Dec 16, 2003.
Valois, M. , “Linear Flexure Bearing,” U.S. Patent Application 20130015616 A1, Jan 17, 2013.
Genequand, P.-M. , “Device for the Guidance in Rectilinear Translation of an Object That is Mobile in Relation to a Fixed Object,” U.S. Patent 6059481 A, May 9, 2000.
Spanoudakis, P. , Schwab, P. , and Johnson, P. , 2003, “Design and Production of the METOP Satellite IASI Corner Cube Mechanisms,” 10th European Space Mechanisms and Tribology Symposium, Noordwijk, The Netherlands, pp. 97–103.
Henein, S. , Kjelberg, I. , and Zelenika, S. , 2002, “Flexible Bearings for High-Precision Mechanisms in Accelerator Facilities,” Proc. NANOBEAM 2002, Lausanne, Switzerland, pp. 103–110.
Hao, G. , Li, H. , He, X. , and Kong, X. , 2014, “Conceptual Design of Compliant Translational Joints for High-Precision Applications,” Front. Mech. Eng., 9(4), pp. 331–343. [CrossRef]
Hopkins, J. B. , and Panas, R. M. , 2013, “A Family of Flexures That Eliminate Underconstraint in Nested Large-Stroke Flexure Systems,” 13th International Conference on European Society for Precision Engineering and Nanotechnology, Berlin, Germany, pp. 1–4.


Grahic Jump Location
Fig. 1

Flexure DP with nested UE linkage. This linkage selectively removes the underconstraint inherent in the DP design by linking the motion of the intermediate and final stage.

Grahic Jump Location
Fig. 2

(a) DP linear bearing with components labeled. Possible motions for the structure are shown in an equivalent linkage model, drawn from pseudo-rigid-body models [1,12,14] in (b). The solid arrows show the nominal translational motion, with the kinematic errors canceled by geometry reversal. The dotted arrows show the y-axis DOF for the structure observed at large displacements, which is revealed as a drop in axial stiffness.

Grahic Jump Location
Fig. 3

(a) Double tilted-beam linear bearing shown on left with instant centers identified. The two possible motions for the structure are shown in equivalent linkage models in (b). The solid arrows show the nominal translational motion where the final stage does not rotate. Note that the kinematic errors are not canceled in this translation. The dotted arrows show the extra DOF of the structure where the final stage rotates without translating.

Grahic Jump Location
Fig. 4

(a) Exact constraint folded flexure utilizing an external linkage. The possible motion for the structure is shown in the equivalent linkage model in (b). The solid arrows show the nominal translational motion, which is equivalent to the DP flexure bearing alone.

Grahic Jump Location
Fig. 5

(a) Exact constraint folded flexure design with an improved, nested linkage for UE. The possible motion for the structure is shown in the equivalent linkage model in (b). The solid arrows show the nominal translational motion, which is equivalent to the DP flexure bearing alone.

Grahic Jump Location
Fig. 6

(a) Double-nested rotary flexure, (b) similar flexure but with a UE linkage inserted, and (c) the final stage is forced to rotate twice as many radians as the intermediate stage in the same direction

Grahic Jump Location
Fig. 7

FEA of the nested UE linkage design undergoing loading along the former DOF of the intermediate stage. A 1 N load is applied in the x-axis on the intermediate stage (red arrow), while the final stage and grounds are held in place. The UE linkage rigid body is warped by the loading, with the top horizontal beam in the linkage forming an “S” shape, and both of the side beams bowing in the same direction. The motion is amplified to clearly indicate the warping.

Grahic Jump Location
Fig. 8

Flexure DP with nested linkage, attached to optical table. This general setup was used to capture the static and dynamic properties of the main DOF as well as the underconstrained DOF.

Grahic Jump Location
Fig. 9

The Bode gain plot for the unaugmented DP flexure is compared to that of the DP flexure with the nested linkage. The intermediate stage resonance has been shifted from 60 Hz to 650 Hz, while the main stage resonance has been changed by only ≈5%.

Grahic Jump Location
Fig. 10

Logarithmic plot of axial stiffness versus nondimensional displacement, showing large displacement effects. Both simple DP and DP with UE stage displacements were simulated using large displacement FEA with displacement (hollow points) and force (solid points) control on the final stage. The DP results are compared to models presented in literature (LD DP Theory) [33].



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In