In this paper, we identify the space of spatial compliant behavior that can be achieved through the use of simple springs connected in parallel to a single rigid body. Here, the expression “simple spring” refers to the set of compliant relations associated with purely translational springs and purely rotational springs.

We show that, regardless of the number of springs used, there exists a subspace within the 21 dimensional symmetric spatial stiffness matrix space that cannot be reached by a parallel simple spring system. The subspace of “realizable” spatial stiffness matrices achieved with parallel simple springs is 20 dimensional and is defined by a linear necessary and sufficient condition on the positive semidefinite stiffness matrix. We show that any full-rank spatial stiffness matrix satisfying the condition can be realized using only line springs.

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