Research Papers

Deterministic Design for a Compliant Parallel Universal Joint With Constant Rotational Stiffness

[+] Author and Article Information
Yan Xie

Robotics Institute,
Beihang University,
Beijing 100191, China
e-mail: xieyan@buaa.edu.cn

Jingjun Yu

Robotics Institute,
Beihang University,
Beijing 100191, China
e-mail: jjyu@buaa.edu.cn

Hongzhe Zhao

Robotics Institute,
Beihang University,
Beijing 100191, China
e-mail: hongzhezhao@gmail.com

1Corresponding author.

Manuscript received June 18, 2017; final manuscript received December 11, 2017; published online March 30, 2018. Assoc. Editor: Guimin Chen.

J. Mechanisms Robotics 10(3), 031006 (Mar 30, 2018) (12 pages) Paper No: JMR-17-1182; doi: 10.1115/1.4039065 History: Received June 18, 2017; Revised December 11, 2017

Compliant universal joints have been widely employed in high-precision fields due to plenty of good performance. However, the stiffness characteristics, as the most important consideration for compliant mechanisms, are rarely involved. In this paper, a deterministic design for a constraint-based compliant parallel universal joint with constant rotational stiffness is presented. First, a constant stiffness realization principle is proposed by combination of the freedom and constraint topology (FACT) method and beam constraint model (BCM) to establish a mapping relationship between stiffness characteristics and topology configurations. A parallel universal joint topology is generated by the constant stiffness realization principle. Then, the analytical stiffness model of the universal joint with some permissible approximations is formulated based on the BCM, and geometrical prerequisites are derived to achieve the desired constant rotational stiffness. After that, finite element analysis (FEA), experimental testing, and detailed stiffness analysis are carried out. It turns out that the rotational stiffness of the universal joint can keep constant with arbitrary azimuth angles even if the rotational angle reaches up to ±5 deg. Meanwhile, the acceptable relative errors of rotational stiffness are within 0.53% compared with the FEA results and 2.6% compared with the experimental results, which indicates the accuracy of the theoretical stiffness model and further implies the feasibility of constant stiffness realization principle on guiding the universal joint design.

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Mills, A. , 2007, “ Robert Hooke's ‘Universal Joint’ and Its Application to Sundials and the Sundial-Clock,” Notes Rec. R. Soc., 61(2), pp. 219–236. [CrossRef]
Hans-Christoph, S. T. , and Friedrich, S. , and, Erich, A. , 2006, Universal Joints and Driveshafts, Springer, Berlin.
Paros, J. , 1965, “ How to Design Flexure Hinges,” Mach. Des., 37, pp. 151–156.
Lobontiu, N. , 2002, Compliant Mechanisms, CRC Press, Boca Raton, FL. [CrossRef]
Howell, L. L. , 2013, Compliant Mechanisms, 21st Century Kinematics, Springer, London.
Loney, G. C. , 1990, “ Design of a Small-Aperture Steering Mirror for High Bandwidth Acquisition and Tracking,” Opt. Eng., 29(11), p. 1360. [CrossRef]
Boynton, R. , Wiener, P. K. , Kennedy, P. , Rathbun, B. , and Engineer, A. C. , 2003, “ Static Balancing a Device With Two or More Degrees of Freedom (the Key to Obtaining High Performance on Gimbaled Missile Seekers),” 62nd Annual Conference of Society of Allied Weight Engineers, (SAWE), New Haven, CT, May 19–21, pp. 24–28.
Hongzhe, Z. , and Shusheng, B. , 2010, “ Accuracy Characteristics of the Generalized Cross-Spring Pivot,” Mech. Mach. Theory, 45(10), pp. 1434–1448. [CrossRef]
Dong, X. , Raffles, M. , Cobos-Guzman, S. , Axinte, D. , and Kell, J. , 2015, “ A Novel Continuum Robot Using Twin-Pivot Compliant Joints: Design, Modeling, and Validation,” ASME J. Mech. Rob., 8(2), p. 021010. [CrossRef]
Walker, I. D. , 2013, “ Continuous Backbone ‘Continuum’ Robot Manipulators,” ISRN Rob., 2013, pp. 1–19. [CrossRef]
Smith, S. T. , 2000, Flexures: Elements of Elastic Mechanisms, CRC Press, Boca Raton, FL.
Lobontiu, N. , and Garcia, E. , 2003, “ Two-Axis Flexure Hinges With Axially-Collocated and Symmetric Notches,” Comput. Struct, 81(13), pp. 1329–1341. [CrossRef]
Trease, B. P. , Moon, Y.-M. , and Kota, S. , 2005, “ Design of Large-Displacement Compliant Joints,” ASME J. Mech. Des., 127(4), pp. 788–798. [CrossRef]
Stark, J. A. , 1958, “ Flexible Couplings,” U.S. Patent No. US2860495.
Tanık, E. , and Parlaktaş, V. , 2012, “ Compliant Cardan Universal Joint,” ASME J. Mech. Des., 134(2), p. 021011. [CrossRef]
Farhadi Machekposhti, D. , Tolou, N. , and Herder, J. L. , 2015, “ A Review on Compliant Joints and Rigid-Body Constant Velocity Universal Joints Toward the Design of Compliant Homokinetic Couplings,” ASME J. Mech. Des., 137(3), p. 032301. [CrossRef]
Chen, C.-T. , 1995, Linear System Theory and Design, Saunders College Publishing, Philadelphia, PA.
Schetzen, M. , 1980, The Volterra and Wiener Theories of Nonlinear Systems, Krieger Publishing Co., Melbourne, FL.
Rugh, W. J. , 1981, Nonlinear System Theory, Johns Hopkins University Press, Baltimore, MD.
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]
Slocum, A. H. , 1992, Precision Machine Design, Prentice Hall, Upper Saddle River, NJ.
Awtar, S. , 2004, “ Synthesis and Analysis of Parallel Kinematic XY Flexure Mechanisms,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Brouwer, D. M. , de Jong, B. R. , and Soemers, H. , 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), Montreal, QC, Canada, July 6–9, pp. 866–871.
Seggelen, J. K V. , Rosielle, P. , Schellekens, P. H. J. , Spaan, H. A. M. , Bergmans, R. H. , and Kotte, G. , 2005, “ An Elastically Guided Machine Axis With Nanometer Repeatability,” CIRP Ann. Manuf. Technol., 54(1), pp. 487–490. [CrossRef]
Yangmin, L. , and Qingsong, X. , 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]
Wittrick, W. H. , 1951, “ The Properties of Crossed Flexure Pivots, and the Influence of the Point at Which the Strips Cross,” Aeronaut. Q., 2(04), pp. 272–292. [CrossRef]
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]
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. , Shusheng, B. , Jingjun, Y. , and Jun, G. , 2012, “ Design of a Family of Ultra-Precision Linear Motion Mechanisms,” ASME J. Mech. Rob., 4(4), p. 041012. [CrossRef]
Sarajlic, E. , Yamahata, C. , Cordero, M. , and Fujita, H. , 2010, “ Three-Phase Electrostatic Rotary Stepper Micromotor With a Flexural Pivot Bearing,” J. Microelectromech. Syst., 19(2), pp. 338–349. [CrossRef]
Sarajlic, E. , Yamahata, C. , Cordero, M. , and Fujita, H. , 2009, “ Electrostatic Rotary Stepper Micromotor for Skew Angle Compensation in Hard Disk Drive,” IEEE 22nd International Conference on Micro Electro Mechanical Systems (MEMS), Sorrento, Italy, Jan. 25–29, pp. 1079–1082.
Sung, E. , 2011, “ Design and Analysis of Diagnostic Machines Utilizing Compliant Mechanisms,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Sung, E. , Slocum, A. H. , Ma, R. , Bean, J. F. , and Culpepper, M. L. , 2011, “ Design of an Ankle Rehabilitation Device Using Compliant Mechanisms,” ASME J. Med. Device, 5(1), p. 011001. [CrossRef]
Shi, H. , Su, H. J. , and Dagalakis, N. , 2014, “ A Stiffness Model for Control and Analysis of a MEMS Hexapod Nanopositioner,” Mech. Mach. Theory, 80(4), pp. 246–264. [CrossRef]
Teo, T. J. , Chen, I. M. , and Yang, G. , 2014, “ A Large Deflection and High Payload Flexure-Based Parallel Manipulator for UV Nanoimprint Lithography—Part II: Stiffness Modeling and Performance Evaluation,” Precis. Eng., 38(4), pp. 872–884. [CrossRef]
Kang, B. H. , Wen, T. Y. , Dagalakis, N. G. , and Gorman, J. J. , 2005, “ Analysis and Design of Parallel Mechanisms With Flexure Joints,” IEEE Trans. Rob. Autom., 21(6), pp. 1179–1185. [CrossRef]
Xie, Y. , Pan, B. , Pei, X. , and Yu, J. , 2016, “ Design of Compliant Universal Joint With Linear Stiffness,” ASME Paper No. DETC2016-59110.
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(2), pp. 259–270. [CrossRef]
Su, H.-J. , Dorozhkin, D. V. , and Vance, J. M. , 2009, “ A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms,” ASME J. Mech. Rob., 1(4), p. 041009. [CrossRef]
Yu, J. J. , Li, S. Z. , Pei, X. , Su, H. , Hopkins, J. B. , and Culpepper, M. L. , 2010, “ Type Synthesis Principle and Practice of Flexure Systems in the Framework of Screw Theory—Part I: General Methodology,” ASME Paper No. DETC2010-28783.
Qiu, C. , Yu, J. J. , Li, S. Z. , Su, H. , and Zeng, Y. Z. , 2011, “ Synthesis of Actuation Spaces of Multi-Axis Parallel Flexure Mechanisms Based on Screw Theory,” ASME Paper No. DETC2011-48252.
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]
Awtar, S. , and Sen, S. , 2010, “ A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation,” ASME J. Mech. Des., 132(8), p. 081009. [CrossRef]
Hao, G. , and Li, H. , 2015, “ Nonlinear Analytical Modeling and Characteristic Analysis of a Class of Compound Multibeam Parallelogram Mechanisms,” ASME J. Mech. Rob., 7(4), p. 041016. [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]
Awtar, S. , and Parmar, G. , 2013, “ Design of a Large Range XY Nanopositioning System,” ASME J. Mech. Rob., 5(2), p. 021008. [CrossRef]
Hao, G. , and Kong, X. , 2012, “ A Novel Large-Range XY Compliant Parallel Manipulator With Enhanced Out-of-Plane Stiffness,” ASME J. Mech. Des., 134(6), p. 061009. [CrossRef]
Panas, R. M. , and Hopkins, J. B. , 2015, “ Eliminating Underconstraint in Double Parallelogram Flexure Mechanisms,” ASME J. Mech. Des., 137(9), p. 092301. [CrossRef]


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Fig. 1

Visualization of constant stiffness realization principle: (a) a cantilever beam under arbitrary loads within actuation space, (b) a cantilever beam with the free-end restricted to the primary motion, and (c) a serial layout to eliminate the axial load

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Fig. 2

Visualization of constant stiffness in basic flexure modules: (a) parallelogram mechanism, (b) double parallelogram mechanism with mirror symmetry, (c) equivalent double parallelogram mechanism of constant translational stiffness, (d) generalized cross-spring pivot, (e) double cross-spring pivot with mirror symmetry, and (f) equivalent double cross-spring pivot of constant rotational stiffness

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Fig. 3

Conceptual design of constraint-based universal joint: (a) freedom and constraint space of universal joint, and (b) beam-based universal joint design (left) and blade-based universal joint design (right)

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Fig. 4

Constant stiffness realization of the beam-based universal joint: (a) physical embodiment of a constraint line with two noncoplanar blade flexures, (b) physical embodiment of a constraint plane with eight dual blade flexures in parallel, and (c) physical embodiment of a universal joint

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Fig. 5

A generalized blade flexure

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Fig. 6

Load distribution on the moving stage

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Fig. 7

Configuration of flexure module I

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Fig. 8

The rotational deformation of flexure module I: (a) rotational deformation of upper part in flexure module I and (b) partial enlarged view for end-displacements

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Fig. 9

The influence of geometric parameters α1 and λ on rotational stiffness kr

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Fig. 10

The rotational deformation of universal joint around x-axis except flexure modules II and IV

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Fig. 11

The translational deformation of upper part in flexure module I

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Fig. 12

The influence of geometric parameter α1 and external load f on displacement s: (a) isometric view and (b) top view

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Fig. 13

The load-rotation of ACE V: (a) the rotational deformation of ACE V around x-axis and (b) the influence of geometric parameter ξ and rotational angle θ on equivalent load msum

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Fig. 14

Detailed design of the universal joint: (a) scheme of the flexure module and (b) assembly scheme of the universal joint

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Fig. 15

Proof-of-concept prototypes: (a) flexure module prototype, (b) universal joint prototype, and (c) experimental setup for stiffness test

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Fig. 16

Stiffness test of the universal joint with an azimuth angle ψ = 0: (a) moment-rotational angle curves and (b) stiffness-rotational angle and relative error-rotational angle curves

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Fig. 17

Stiffness test of the universal joint with azimuth angles varying from 0 deg to 360 deg: (a) average stiffness under different azimuth angles (0 deg, 15 deg, 30 deg, and 45 deg) and (b) average stiffness and relative errors with azimuth angles varying from 0 to 360 deg




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