Abstract
Discussed in this paper is the Cartesian stiffness matrix, which recently has received special attention within the robotics research community. Stiffness is a fundamental concept in mechanics; its representation in mechanical systems whose potential energy is describable by a finite set of generalized coordinates takes the form of a square matrix that is known to be, moreover, symmetric and positive-definite or, at least, semi-definite. We attempt to elucidate in this paper the notion of “asymmetric stiffness matrices”. In doing so, we show that to qualify for a stiffness matrix, the matrix should be symmetric and either positive semi-definite or positive-definite. We derive the conditions under which a matrix mapping small-amplitude displacement screws into elastic wrenches fails to be symmetric. From the discussion, it should be apparent that the asymmetric matrix thus derived cannot be, properly speaking, a stiffness matrix. The concept is illustrated with an example.
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References
Angeles, J.: Fundamentals of Robotic Mechanical Systems. Theory, Methods and Algorithms. Springer, New York (2007)
Angeles, J.: On twist and wrench generators and annihilators. In: Pereira, M.F.O.S., Ambrósio, J.A.C. (eds.) Computer-Aided Analysis of Rigid and Flexible Mechanical Systems, pp. 379–411. Kluwer Academic, Dordrecht (1994)
Bathe, K.-J.: Finite Element Procedures. Prentice Hall, New Jersey (1996)
Ciblak, N., Lipkin, H.: Asymmetric Cartesian stiffness for the modeling of compliant robotic systems. In: Proc. 23rd Biennial ASME Mechanisms Conference, Minneapolis, MN (1994)
Den Hartog, J.P.: Mechanical Vibrations, 4th edn. McGraw–Hill, New York (1956)
Geradin, M., Cardona, A.: Flexible Multibody Dynamics: A Finite Element Approach. Wiley, New York (2001)
Griffis, M., Duffy, J.: Global stiffness modeling of a class of simple compliant couplings. Mech. Mach. Theory 28(2), 207–224 (1993)
Howard, S., Zefran, M., Kumar, V.: On the 6×6 Cartesian stiffness matrix for three-dimensional motions. Mech. Mach. Theory 33(4), 389–408 (1998)
Loncaric, J.: Normal forms of stiffness and compliance matrices. IEEE J. Robot. Autom. RA-3(6), 567–572 (1987)
Majou, F., Gosselin, C., Wenger, P., Chablat, D.: Parametric stiffness analysis of the orthoglide. Mech. Mach. Theory 42(3), 296–311 (2007)
Meirovitch, L.: Fundamentals of Vibrations. McGraw–Hill, New York (2001)
Pigoski, T., Griffis, M., Duffy, J.: Stiffness mappings employing different frames of reference. Mech. Mach. Theory 33(6), 825–838 (1998)
Shabana, A.: Dynamics of multibody systems, 3rd edn. Cambridge University Press, New York (2005)
Synge, J.L., Griffith, B.A.: Principles of Mechanics. McGraw–Hill, New York (1959)
Timoshenko, S.: Vibration Problems in Engineering. Constable, London (1928)
Zefran, M., Kumar, V.: A geometrical approach to the study of the Cartesian stiffness matrix. ASME J. Mech. Des. 124, 30–38 (2002)
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Kövecses, J., Angeles, J. The stiffness matrix in elastically articulated rigid-body systems. Multibody Syst Dyn 18, 169–184 (2007). https://doi.org/10.1007/s11044-007-9082-2
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DOI: https://doi.org/10.1007/s11044-007-9082-2