Consistent modeling resolves asymmetry in stiffness matrices

The Cartesian stiffness analysis of mechanisms and robotic manipulators is often performed using Jacobian-based stiffness analysis methods. In general it is necessary to consider the effect of loading, which introduces a Jacobian-derivative term. However, early examples of stiffness analyses of parallel manipulators in which the effect of loading was considered resulted in asymmetric stiffness matrices, which go against the definition of stiffness. By replicating those first analyses using an alternative formulation of the stiffness analysis, this paper finds that the earlier observed asymmetry can be explained as a modeling inconsistency. If that inconsistency is corrected, symmetric matrices are obtained, which supports the notion that the Jacobian-derivative term is an integral part of the stiffness analysis of parallel manipulators and mechanisms.