The stiffness of a rigid body subject to conservative forces and moments is described by a tensor, whose components are best described by a 6×6 Cartesian stiffness matrix. We derive an expression that is independent of the parameterization of the motion of the rigid body using methods of differential geometry. The components of the tensor with respect to a basis of twists are given by evaluating the tensor on a pair of basis twists. We show that this tensor depends on the choice of an affine connection on the Lie group, In addition, we show that the definition of the Cartesian stiffness matrix used in the literature [1,2] implicitly assumes an asymmetric connection and this results in an asymmetric stiffness matrix in a general loaded configuration. We prove that by choosing a symmetric connection we always obtain a symmetric Cartesian stiffness matrix. Finally, we derive stiffness matrices for different connections and illustrate the calculations using numerical examples.
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March 2002
Technical Papers
A Geometrical Approach to the Study of the Cartesian Stiffness Matrix
Milosˇ Zˇefran,
Milosˇ Zˇefran
Dept. of Electrical and Computer Engineering, University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607
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Vijay Kumar
Vijay Kumar
GRASP Laboratory, Dept. of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, 3401 Walnut St., Philadelphia, PA 19104-6228
Search for other works by this author on:
Milosˇ Zˇefran
Dept. of Electrical and Computer Engineering, University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607
Vijay Kumar
GRASP Laboratory, Dept. of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, 3401 Walnut St., Philadelphia, PA 19104-6228
Contributed by the Design Automation Committee for publication in JOURNAL OF MECHANICAL DESIGN. Manuscript received Oct. 1998. Associate Editor: H. Lankarani.
J. Mech. Des. Mar 2002, 124(1): 30-38 (9 pages)
Published Online: October 1, 1998
Article history
Received:
October 1, 1998
Citation
Zˇefran, M., and Kumar, V. (October 1, 1998). "A Geometrical Approach to the Study of the Cartesian Stiffness Matrix ." ASME. J. Mech. Des. March 2002; 124(1): 30–38. https://doi.org/10.1115/1.1423638
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