Research Papers

Extensible-Link Kinematic Model for Characterizing and Optimizing Compliant Mechanism Motion

[+] Author and Article Information
Justin Beroz

Mechanosynthesis Group
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: jberoz@mit.edu

Shorya Awtar

Precision Systems Design Laboratory,
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: awtar@umich.edu

A. John Hart

Mechanosynthesis Group
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: ajhart@mit.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 9, 2013; final manuscript received December 12, 2013; published online January 10, 2014. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 136(3), 031008 (Jan 10, 2014) (11 pages) Paper No: MD-13-1115; doi: 10.1115/1.4026269 History: Received March 09, 2013; Revised December 12, 2013

We present an analytical model for characterizing the motion trajectory of an arbitrary planar compliant mechanism. Model development consists of identifying particular material points and their connecting vectorial lengths in a manner that represents the mechanism topology; whereby these lengths may extend over the course of actuation to account for the elastic deformation of the compliant mechanism. The motion trajectory is represented within the model as an analytical function in terms of these vectorial lengths, whereby its Taylor series expansion constitutes a parametric formulation composed of load-independent and load-dependent terms. This adds insight to the process for designing compliant mechanisms for high-accuracy motion applications because: (1) inspection of the load-independent terms enables determination of specific topology modifications that reduce or eliminate certain error components of the motion trajectory; and (2) the load-dependent terms reveal the polynomial orders of principally uncorrectable error components in the trajectory. The error components in the trajectory simply represent the deviation of the actual motion trajectory provided by the compliant mechanism compared to the ideally desired one. A generalized model framework is developed, and its utility demonstrated via the design of a compliant microgripper with straight-line parallel jaw motion. The model enables analytical determination of all geometric modifications for minimizing the error trajectory of the jaw, and prediction of the polynomial order of the uncorrectable trajectory components. The jaw trajectory is then optimized using iterative finite elements simulations until the polynomial order of the uncorrectable trajectory component becomes apparent; this reduces the error in the jaw trajectory by 2 orders of magnitude over the prescribed jaw stroke. This model serves to streamline the design process by identifying the load-dependent sources of trajectory error in a compliant mechanism, and thereby the limits with which this error may be redressed by topology modification.

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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]
Tian, Y., Shirinzadeh, B., and Zhang, D., 2010, “Design and Dynamics of a 3-DOF Flexure-Based Parallel Mechanism for Micro/Nanomanipulation,” Microelectron. Eng., 87(2), pp. 230–241. [CrossRef]
Chen, S. C., and Culpepper, M. L., 2006, “Design of a Six-Axis Micro-Scale Nanopositioner—mu HexFlex,” Precis. Eng., 30(3), pp. 314–324. [CrossRef]
Lantz, M. A., Rothuizen, H. E., Drechsler, U., Haberle, W., and Despont, M., 2007, “A Vibration Resistant Nanopositioner for Mobile Parallel-Probe Storage Applications,” J. Microelectromech. Syst., 16(1), pp. 130–139. [CrossRef]
Ho, C. M., and Tai, Y. C., 1998, “Micro-Electro-Mechanical-Systems (MEMS) and Fluid Flows,” Annu. Rev. Fluid Mech., 30, pp. 579–612. [CrossRef]
Beeby, S. P., Tudor, M. J., and White, N. M., 2006, “Energy Harvesting Vibration Sources for Microsystems Applications,” Meas. Sci. Technol., 17(12), pp. R175–R195. [CrossRef]
Hu, Y. H., Lin, K. H., Chang, S. C., and Chang, M., 2008, “Design of a Compliant Micromechanism for Optical-Fiber Alignment,” Measurement Technology and Intelligent Instruments VIII, W.Gao, Y.Takaya, Y.Gao, M.Krystek, eds., Trans Tech Publications Ltd, Stafa-Zurich, pp. 141–144.
Chen, S. C., Culpepper, M. L., and Jordan, S., “Six-Axis Compliant Mechanisms For Manipulation of Micro-Scale Fiber Optics Components—Art. No. 64660P,” Proc. Conference on MOEMS and Miniaturized Systems VI, SPIE-Int Soc Optical Engineering, pp. P4660–P4660.
Lee, C. W., and Kim, S. W., 1997, “An Ultraprecision Stage For Alignment of Wafers in Advanced Microlithography,” Precis. Eng., 21(2–3), pp. 113–122. [CrossRef]
Meli, F., and Thalmann, R., 1998, “Long-Range AFM Profiler Used for Accurate Pitch Measurements,” Meas. Sci. Technol., 9(7), pp. 1087–1092. [CrossRef]
Kim, D., Lee, D. Y., and Gweon, D. G., 2007, “A New Nano-Accuracy AFM System for Minimizing Abbe Errors and the Evaluation of its Measuring Uncertainty,” Ultramicroscopy, 107(4–5), pp. 322–328. [CrossRef] [PubMed]
Qin, D., Xia, Y. N., Rogers, J. A., Jackman, R. J., Zhao, X. M., and Whitesides, G. M., 1998, “Microfabrication, Microstructures and Microsystems,” Microsyst. Technol. Chem. Life Sci., 194, pp. 1–20. [CrossRef]
Howell, L. L., and Midha, A., 1995, “Parametric Deflection Approximations for End-Loaded Large-Deflection Beams in Compliant Mechanisms,” ASME J. Mech. Des., 117(1), pp. 156–165. [CrossRef]
Howell, L. L., and Midha, A., 1994, “A Method for The Design of Compliant Mechanisms With Small-Length Flexural Pivots,” ASME J. Mech. Des., 116(1), pp. 280–290. [CrossRef]
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]
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]
Awtar, S., and Sen, S., 2010, “A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Load-Displacement Formulation,” ASME J. Mech. Des., 132(8), p. 081008. [CrossRef]
Frecker, M.I., Ananthasuresh, G. K., Nishiwaki, S., Kikuchi, N., and Kota, S., 1997, “Topological Synthesis of Compliant Mechanisms Using Multi-Criteria Optimization,” ASME J. Mech. Des., 119(2), pp. 238–245. [CrossRef]
Saxena, A., and Ananthasuresh, G. K., 2000, “On an Optimal Property of Compliant Topologies,” Struct. Multidiscip. Optim., 19(1), pp. 36–49. [CrossRef]
Carbonari, R. C., Silva, E. C. N., and Nishiwaki, S., 2005, “Design of Piezoelectric Multi-Actuated Microtools Using Topology Optimization,” Smart Mater. Struct., 14(6), pp. 1431–1447. [CrossRef]
Matsui, K., and Terada, K., 2004, “Continuous Approximation of Material Distribution for Topology Optimization,” Int. J. Numer. Methods Eng., 59(14), pp. 1925–1944. [CrossRef]
Tai, K., and Chee, T. H., 2000, “Design of Structures and Compliant Mechanisms by Evolutionary Optimization of Morphological Representations of Topology,” J. Mech. Des., 122(4), pp. 560–566. [CrossRef]
Lu, K. J., and Kota, S., 2006, “Topology and Dimensional Synthesis of Compliant Mechanisms Using Discrete Optimization,” ASME J. Mech. Des., 128(5), pp. 1080–1091. [CrossRef]
Dimiduk, D. M., Uchic, M. D., and Parthasarathy, T. A., 2005, “Size-Affected Single-Slip Behavior of Pure Nickel Microcrystals,” Acta Mater., 53(15), pp. 4065–4077. [CrossRef]
Loh, O., Vaziri, A., and Espinosa, H., 2009, “The Potential of MEMS for Advancing Experiments and Modeling in Cell Mechanics,” Exp. Mech., 49(1), pp. 105–124. [CrossRef]
Ahn, S. K., Kasi, R. M., Kim, S. C., Sharma, N., and Zhou, Y. X., 2008, “Stimuli-Responsive Polymer Gels,” Soft Matter, 4(6), pp. 1151–1157. [CrossRef]
Cao, A. Y., Dickrell, P. L., Sawyer, W. G., Ghasemi-Nejhad, M. N., and Ajayan, P. M., 2005, “Super-Compressible Foamlike Carbon Nanotube Films,” Science, 310(5752), pp. 1307–1310. [CrossRef] [PubMed]
Tsai, Y. C., Lei, S. H., and Sudin, H., 2005, “Design and Analysis of Planar Compliant Microgripper Based on Kinematic Approach,” J. Micromech. Microeng., 15(1), pp. 143–156. [CrossRef]
Nah, S. K., and Zhong, Z. W., 2007, “A Microgripper Using Piezoelectric Actuation for Micro-Object Manipulation,” Sens. Actuators A, 133(1), pp. 218–224. [CrossRef]
Zubir, M. N. M., Shirinzadeh, B., and Tian, Y. L., 2009, “A New Design of Piezoelectric Driven Compliant-Based Microgripper for Micromanipulation,” Mech. Mach. Theory, 44(12), pp. 2248–2264. [CrossRef]
Zubir, M. N. M., and Shirinzadeh, B., 2009, “Development of a High Precision Flexure-Based Microgripper,” Precis. Eng., 33(4), pp. 362–370. [CrossRef]
Goldfarb, M., and Celanovic, N., 1999, “A Flexure-Based Gripper for Small-Scale Manipulation,” Robotica, 17, pp. 181–187. [CrossRef]
Tian, Y., Shirinzadeh, B., Zhang, D., and Zhong, Y., 2010, “Three Flexure Hinges For Compliant Mechanism Designs Based on Dimensionless Graph Analysis,” Precis. Eng., 34(1), pp. 92–100. [CrossRef]
Ando, T., Sato, K., Shikida, M., Yoshioka, T., Yoshikawa, Y., Kawabata, T., and Ieee, I., 1997, “Orientation-Dependent Fracture Strain in Single-Crystal Silicon Beams Under Uniaxial Tensile Conditions,” Mhs'97: Proceedings of 1997 International Symposium on Micromechatronics and Human Science.


Grahic Jump Location
Fig. 2

(a) The distance, l, between any two material points on a continuum body admits the decomposition of Eq. (1) upon deformation due to loading. Design insights are gained by considering the distances between (b) compliant hinges in lumped-compliance shapes, and (c) end-points of compliant beams. Given a compliant mechanism, these distances, l, therefore represent the comprising compliant segments.

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Fig. 3

Example (a) lumped compliance and (b) distributed compliance mechanisms sharing the same (c) topology, or equivalently, (d) analogous kinematic model at undeformed state, ξ0.

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Fig. 4

Compliant mechanism design procedure utilizing the ELKM

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Fig. 1

Representative compliant mechanism design scenario for high-accuracy motion applications

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Fig. 5

Displacement-constraint design problem for parallel-jaw gripping mechanism, where mirror-image topologies fill right and left mechanism areas

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Fig. 6

The typical topology solution within the mechanism area (Fig. 5) is (a) a parallelogram, where (b) the trajectory of P1 is constrained by a beam having extensible length l

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Fig. 8

Modifications of geometric parameters ΔΨ and Δg0 correct 1st- and 2nd-order P1 trajectory errors, respectively. The trajectory is plotted as: Δy = yx)−k0,0.

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Fig. 9

Lumped-compliance gripper with straight-line parallel-jaw trajectory based on the Hoekens four-bar linkage

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Fig. 10

Demonstration of reduced jaw trajectory error in (a) jaw link rotation over the jaw stroke, Δx, between initial (*) and optimized (o) FE simulations; and (b) P1 trajectory over jaw stroke between initial (*), 1st-order optimized only (Δ), and fully optimized (o) FE simulations. (c) The optimized jaw trajectory exhibits a residual 3rd-order error, as predicted. The trajectories in (b,c) are plotted as: Δy = yx) − k0,0.

Grahic Jump Location
Fig. 7

(a) Mechanism solution for straight-line horizontal jaw displacement, which utilizes (b) a four-bar mechanism based on the Hoeken's linkage to determine the motion of P1



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