Multibody System Dynamics

, Volume 24, Issue 4, pp 441–472 | Cite as

On Cartesian stiffness matrices in rigid body dynamics: an energetic perspective

  • Melodie F. Metzger
  • Nur Adila Faruk Senan
  • Oliver M. O’Reilly
Open Access


Several Cartesian stiffness matrices for a single rigid body subject to a conservative force field are developed in this paper. The treatment is based on energetic arguments and an Euler angle parameterization of the rotation of the rigid body is employed. Several new representations for the stiffness matrix are obtained and the relation to other works on Cartesian stiffness matrices and Hessians is illuminated. Additional details are presented with respect to determining the Cartesian stiffness matrix for a pair of rigid bodies, as well as for a system of rigid bodies constrained to a plane.


Rigid body Rotation Stiffness matrix Cartesian stiffness matrix Dual Euler basis Euler angles Conservative force fields 


  1. 1.
    Andriacchi, T.P., Mikosz, R.P., Hampton, S.J., Galante, J.O.: Model studies of the stiffness characteristics of the human knee joint. J. Biomech. 16(1), 23–29 (1983). doi: 10.1016/0021-9290(83)90043-X CrossRefGoogle Scholar
  2. 2.
    Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. Dover, New York (1980) Google Scholar
  3. 3.
    Chen, S.F., Kao, I.: Conservative congruence transformation for joint and Cartesian stiffness matrices of robotic hands and fingers. Int. J. Robotics Res. 19(9), 835–847 (2000). doi: 10.1177/02783640022067201 CrossRefGoogle Scholar
  4. 4.
    Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard Bleick, M.: Analysis, Manifolds, and Physics, revised edn. North-Holland Physics Publishing, Amsterdam (1982) zbMATHGoogle Scholar
  5. 5.
    Ciblak, N., Lipkin, H.: Asymmetric Cartesian stiffness for the modelling of compliant robotic systems. In: Pennock, G.R., Angeles, J., Fichter, E.F., Freeman, R.A., Lipkin, H., Thompson, B.S., Wiederrich, J., Wiens, G.L. (eds.) Robotics: Kinematics, Dynamics and Controls, Presented at The 1994 ASME Design Technical Conferences—23rd Biennial Mechanisms Conference, Minneapolis, Minnesota, September 11–14, 1994, vol. DE-72, pp. 197–204. ASME, New York (1994) Google Scholar
  6. 6.
    Faruk Senan, N.A., O’Reilly, O.M.: On the use of quaternions and Euler–Rodrigues symmetric parameters with moments and moment potentials. Int. J. Eng. Sci. 47(4), 595–609 (2009). doi: 10.1016/j.ijengsci.2008.12.008 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gardner-Morse, M.G., Stokes, I.A.F.: Structural behavior of the human lumbar spinal motion segments. J. Biomech. 37(2), 205–212 (2004). doi: 10.1016/j.jbiomech.2003.10.003 CrossRefGoogle Scholar
  8. 8.
    Griffis, M., Duffy, J.: Global stiffness modeling of a class of simple compliant couplings. Mech. Mach. Theory 28(2), 207–224 (1993). doi: 10.1016/0094-114X(93)90088-D CrossRefGoogle Scholar
  9. 9.
    Howard, S., Žefran, M., Kumar, V.: On the 6×6 Cartesian stiffness matrix for three-dimensional motions. Mech. Mach. Theory 33(4), 389–408 (1998). doi: 10.1016/S0094-114X(97)00040-2 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kövecses, J., Angeles, J.: The stiffness matrix in elastically articulated rigid-body systems. Multibody Syst. Dyn. 18(2), 169–184 (2007). doi: 10.1007/s11044-007-9082-2 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Murray, R.N., Li, Z.X., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994) zbMATHGoogle Scholar
  12. 12.
    Nordenholz, T.R., O’Reilly, O.M.: A class of motions of elastic, symmetric Cosserat points: existence, bifurcation, and stability. Int. J. Non-Linear Mech. 36(2), 353–374 (2001). doi: 10.1016/S0020-7462(00)00021-4 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    O’Reilly, O.M.: The dual Euler basis: Constraints, potentials, and Lagrange’s equations in rigid body dynamics. ASME J. Appl. Mech. 74(2), 256–258 (2007). doi: 10.1115/1.2190231 zbMATHCrossRefGoogle Scholar
  14. 14.
    O’Reilly, O.M.: Intermediate Engineering Dynamics: A Unified Approach to Newton-Euler and Lagrangian Mechanics. Cambridge University Press, New York (2008). CrossRefGoogle Scholar
  15. 15.
    O’Reilly, O.M., Srinivasa, A.R.: On potential energies and constraints in the dynamics of rigid bodies and particles. Math. Probl. Eng. 8(3), 169–180 (2002). doi: 10.1080/10241230215286 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    O’Reilly, O.M., Metzger, M.F., Buckley, J.M., Moody, D.A., Lotz, J.C.: On the stiffness matrix of the intervertebral joint: application to total disk replacement. J. Biomech. Eng. 131(8), 63–87 (2009). doi: 10.1115/1.3148195 Google Scholar
  17. 17.
    Panjabi, M.M., Brand, R.A. Jr., White, A.A. III: Three-dimensional flexibility and stiffness properties of the human thoracic spine. J. Biomech. 9(4), 185–192 (1976). doi: 10.1016/0021-9290(76)90003-8 CrossRefGoogle Scholar
  18. 18.
    Pigoski, T., Griffis, M., Duffy, J.: Stiffness mappings employing different frames of reference. Mech. Mach. Theory 33(6), 825–838 (1998). doi: 10.1016/S0094-114X(97)00083-9 zbMATHCrossRefGoogle Scholar
  19. 19.
    Quennouelle, C., Gosselin, C.M.: Stiffness matrix of compliant parallel mechanisms. In: Lenarčič, J., Wenger, P. (eds.) Advances in Robot Kinematics: Analysis and Design, pp. 331–341. Springer, Dordrecht (2008). doi: 10.1007/978-1-4020-8600-7_35 CrossRefGoogle Scholar
  20. 20.
    Shuster, M.D.: A survey of attitude representations. J. Astronaut. Sci. 41(4), 439–517 (1993) MathSciNetGoogle Scholar
  21. 21.
    Žefran, M., Kumar, V.: A geometrical approach to the study of the Cartesian stiffness matrix. ASME J. Mech. Des. 124(1), 30–38 (2002). doi: 10.1115/1.1423638 CrossRefGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Melodie F. Metzger
    • 1
  • Nur Adila Faruk Senan
    • 2
  • Oliver M. O’Reilly
    • 2
  1. 1.Department of Orthopaedic SurgeryUniversity of California at San FranciscoSan FranciscoUSA
  2. 2.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA

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