A Geometric Approach to the Study of the Cartesian Stiiness Matrix

The stiiness 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 stiiness matrix. We derive an expression that is independent of the parameterization of the motion of the rigid body using methods of diierential 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 aane connection on the Lie group, SE(3). In addition, we show that the deenition of the Cartesian stiiness matrix used in the literature 2, 6] implicitly assumes an asymmetric connection and this results in an asymmetric stiiness matrix in a general loaded connguration. We prove that by choosing a symmetric connection we always obtain a symmetric Cartesian stiiness matrix. Finally, we derive stiiness matrices for diierent connections and illustrate the calculations using numerical examples.