Models for compliant robotic systems often use a symmetric 6 × 6 stiffness matrix. However, when subjected to external loads, the stiffness actually becomes asymmetric. For a compliant system modelled using line springs, a new and important theorem is presented that represents the skew-symmetric part in its simpliest form: the skew-symmetric part of the stiffness matrix is negative one-half the externally applied load expressed as a spatial cross product operator. Several corollaries follow including the obvious result that the stiffness matrix is symmetric if and only if it is at an unloaded equilibrium.

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