Passive realization of generalized springs

Elastic interactions between two rigid bodies are described by a potential function V. It is shown that there is no natural positive definite metric on the group of rigid body motions SE(3). This fact suggests that it is misleading to identify force vectors with infinitesimal displacement vector. The definition of spring stiffness must be based on group structure since Euclidean metric is not available. Spring stiffness is defined on the basis of exponential coordinates, and it is proved that all stiffness matrices, including unstable ones, may be realized from stable line springs. a related problem of realizing generalized dampers from passive elements is shown to be more difficult because dampers (unlike springs) do not allow energy storage.<>