Compliant Serial 3R Chain with Spherical Flexures

  • Farid Parvari RadEmail author
  • Rocco Vertechy
  • Giovanni Berselli
  • Vincenzo Parenti-Castelli
Part of the Springer Proceedings in Advanced Robotics book series (SPAR, volume 4)


A spherical flexure is a special kind of compliant hinge specifically conceived for spherical motion. It features an arc of a circle as centroidal axis and an annulus sector as cross-section, circle and annulus having a common center coinciding to that of the desired spherical motion. This paper investigates a compliant spherical 3R open chain that is obtained by the in-series connection of three identical spherical flexures having coincident centers and mutually orthogonal axes of maximum rotational compliance. The considered spherical chain is intended to be used as a complex flexure for the development of spatial parallel manipulators. The compliance matrix of the proposed chain is first determined via an analytical procedure. Then, the obtained equations are used in a parametric study to assess the influence of spherical flexure geometry on the overall stiffness performances of the considered 3R open chain.


Spherical flexures Compliance matrix Spherical mechanisms 


  1. 1.
    Belfiore, N.P., Balucani, M., Crescenzi, R., Verotti, M.: Performance analysis of compliant MEMS parallel robots through pseudo-rigid-body model synthesis. In: ASME ESDA 11th Biennial Conference on Engineering Systems Design and Analysis, pp. 329–334 (2012)Google Scholar
  2. 2.
    Carter Hale, L.: Principles and techniques for designing precision machines. Ph.D. thesis, Department of Mechanical Engineering, MIT, Cambridge, MA (1999)Google Scholar
  3. 3.
    Chen, S., Culpepper, M.L.: Design of a six-axis micro-scale nanopositioner \(\mu \)hexflex. Precis. Eng. 30(3), 314–324 (2006)CrossRefGoogle Scholar
  4. 4.
    Dong, W., Sun, L., Du, Z.: Stiffness research on a high-precision, large-workspace parallel mechanism with compliant joints. Precis. Eng. 32(3), 222–231 (2008)CrossRefGoogle Scholar
  5. 5.
    Dunning, A., Tolou, N., Herder, J.: A compact low-stiffness six degrees of freedom compliant precision stage. Precis. Eng. 37(2), 380–388 (2013)CrossRefGoogle Scholar
  6. 6.
    Greenberg, H., Gong, M., Magleby, S., Howell, L.: Identifying links between origami and compliant mechanisms. Mech. Sci 2(2), 217–225 (2011)CrossRefGoogle Scholar
  7. 7.
    Hanna, B.H., Lund, J.M., Lang, R.J., Magleby, S.P.: Waterbomb base: a symmetric single-vertex bistable origami mechanism. Smart Mater. Struct. 23(9), 094009 (2014)CrossRefGoogle Scholar
  8. 8.
    Hesselbach, J., Wrege, J., Raatz, A., Becker, O.: Aspects on design of high precision parallel robots. Assem. Autom. 24(1), 49–57 (2004)CrossRefGoogle Scholar
  9. 9.
    Hong, M.B., Jo, Y.H.: Design and evaluation of 2-DOF compliant forceps with force-sensing capability for minimally invasive robot surgery. IEEE Trans. Robot. 28(4), 932–941 (2012)CrossRefGoogle Scholar
  10. 10.
    Howell, L.L.: Compliant Mechanisms. Wiley, New York (2001)Google Scholar
  11. 11.
    Jacobsen, J.O., Chen, G., Howell, L.L., Magleby, S.P.: Lamina emergent torsional (LET) joint. Mech. Mach. Theory 44(11), 2098–2109 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Li, G., Chen, G.: Achieving compliant spherical linkage designs from compliant planar linkages based on prbm: a spherical young mechanism case study. In: 2012 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 193–197, IEEE (2012)Google Scholar
  13. 13.
    Lobontiu, N.: Compliant Mechanisms: Design of Flexure Hinges. CRC Press (2002)Google Scholar
  14. 14.
    Lobontiu, N., Paine, J., Garcia, E., Goldfarb, M.: Corner-filleted flexure hinges. J. Mech. Des. 123(3), 346–352 (2001)CrossRefGoogle Scholar
  15. 15.
    Love, A.: A Treatise on the Mathematical Theory. Dover Public (1944)Google Scholar
  16. 16.
    Lyse, I., Johnston, B.: Structural beams in torsion, 1934. Fritz Laboratory Reports (1934)Google Scholar
  17. 17.
    Machekposhti, D.F., Tolou, N., Herder, J.: A review on compliant joints and rigid-body constant velocity universal joints toward the design of compliant homokinetic couplings. J. Mech. Des. 137(3), 032301 (2015)CrossRefGoogle Scholar
  18. 18.
    Moon, Y., Choi, J.: A compliant parallel mechanism for needle intervention. In: 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 4875–4878 (2013)Google Scholar
  19. 19.
    Palmieri, G., Palpacelli, M.C., Callegari, M.: Study of a fully compliant u-joint designed for minirobotics applications. ASME J. Mech. Des. 134(11), 111003(9) (2012)CrossRefGoogle Scholar
  20. 20.
    Parlaktaş, V., Tanık, E.: Single piece compliant spatial slider-crank mechanism. Mech. Mach. Theory 81, 1–10 (2014)CrossRefGoogle Scholar
  21. 21.
    Paros, J.: How to design flexure hinges. Mach. Des. 37, 151–156 (1965)Google Scholar
  22. 22.
    Parvari Rad, F., Berselli, G., Vertechy, R., Parenti-Castelli, V.: Evaluating the spatial compliance of circularly curved-beam flexures. In: Computational Kinematics, pp. 329–336. Springer (2013)Google Scholar
  23. 23.
    Parvari Rad, F., Berselli, G., Vertechy, R., Parenti-Castelli, V.: Stiffness analysis of a fully compliant spherical chain with two degrees of freedom. In: Advances in Robot Kinematics, pp. 273–284. Springer (2014)Google Scholar
  24. 24.
    Parvari Rad, F., Berselli, G., Vertechy, R., Parenti-Castelli, V.: Design and stiffness analysis of a compliant spherical chain with three degrees of freedom. Precis. Eng. 47, 1–9 (2017)Google Scholar
  25. 25.
    Parvari Rad, F., Vertechy, R., Berselli, G., Parenti-Castelli, V.: Analytical compliance analysis and finite element verification of spherical flexure hinges for spatial compliant mechanisms. Mech. Mach. Theory 101, 168–180 (2016)CrossRefGoogle Scholar
  26. 26.
    Pham, H.H., Chen, I.M.: Stiffness modeling of flexure parallel mechanism. Precis. Eng. 29(4), 467–478 (2005)CrossRefGoogle Scholar
  27. 27.
    Polit, S., Dong, J.: Development of a high-bandwidth XY nanopositioning stage for high-rate micro-/nanomanufacturing. IEEE/ASME Trans. Mechatron. 16(4), 724–733 (2011)CrossRefGoogle Scholar
  28. 28.
    Ratchev, S.: Precision assembly technologies for mini and micro products. In: Proceedings of the IFIP TC5 WG5, 5 Third International Precision Assembly Seminar (IPAS’2006), 19–21 February 2006, Bad Hofgastein, Austria, vol. 198. Springer Science & Business Media (2006)Google Scholar
  29. 29.
    Rubbert, L., Renaud, P., Gangloff, J.: Design and optimization for a cardiac active stabilizer based on planar parallel compliant mechanisms. In: ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis, pp. 235–244. American Society of Mechanical Engineers (2012)Google Scholar
  30. 30.
    Sauceda-Carvajal, A., Kennedy-Cabrera, H.D., Hernndez-Torres, J., Herrera-May, A.L., Mireles, J.: Compliant MEMS mechanism to extend resolution in fourier transform spectroscopy. In: Proceedings of SPIE, Micromachining and Microfabrication Process Technology XIX 8973, 89,730S–89,730S–9 (2014)Google Scholar
  31. 31.
    Schotborgh, W., Kokkeler, F., Tragter, H., van Houten, F.: Dimensionless design graphs for flexure elements and a comparison between three flexure elements. Precis. Eng. 29(1), 41–47 (2005)CrossRefGoogle Scholar
  32. 32.
    Su, H.J.: Mobility analysis of flexure mechanisms via screw algebra. J. Mech. Robot. 3(4), 041010 (2011)CrossRefGoogle Scholar
  33. 33.
    Tanık, Ç.M., Parlaktaş, V., Tanık, E., Kadıoğlu, S.: Steel compliant cardan universal joint. Mech. Mach. Theory 92, 171–183 (2015)CrossRefGoogle Scholar
  34. 34.
    Teo, T.J., Chen, I.M., Yang, G.: 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), 872–884 (2014)CrossRefGoogle Scholar
  35. 35.
    Tian, Y., Shirinzadeh, B., Zhang, D., Zhong, Y.: Three flexure hinges for compliant mechanism designs based on dimensionless graph analysis. Precis. Eng. 34(1), 92–100 (2010)CrossRefGoogle Scholar
  36. 36.
    Tian, Y., Zhang, D., Shirinzadeh, B.: Dynamic modelling of a flexure-based mechanism for ultra-precision grinding operation. Precis. Eng. 35(4), 554–565 (2011)CrossRefGoogle Scholar
  37. 37.
    Trease, B., Moon, Y., Kota, S.: Design of large-displacement compliant joints. J. Mech. Des. 127(4), 788–798 (2005)CrossRefGoogle Scholar
  38. 38.
    Wilding, S.E., Howell, L.L., Magleby, S.P.: Spherical lamina emergent mechanisms. Mech. Mach. Theory 49, 187–197 (2012)CrossRefGoogle Scholar
  39. 39.
    Wu, T.L., Chen, J.H., Chang, S.H.: A six-dof prismatic-spherical-spherical parallel compliant nanopositioner. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55(12), 2544–2551 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Farid Parvari Rad
    • 1
    Email author
  • Rocco Vertechy
    • 1
  • Giovanni Berselli
    • 2
  • Vincenzo Parenti-Castelli
    • 1
  1. 1.Department of Industrial EngineeringUniversity of BolognaBolognaItaly
  2. 2.Department of Mechanical, Energy, Management and Transportation EngineeringUniversity of GenovaGenovaItaly

Personalised recommendations