The bounds and realization of spatial stiffnesses achieved with simple springs connected in parallel

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 passive translational springs and rotational springs. The restriction on the stiffness matrices (the same restriction previously identified using Lie Algebra for a parallel network of active or passive translational springs) is derived using screw theory by investigating the compliant behavior of individual simple springs, We show that the restriction results from the fact that simple springs can only provide either a pure force or a pure torque to the suspended body (not a combination). We show that the 20-dimensional subspace of "realizable" spatial stiffness matrices achieved with parallel simple springs is defined by a linear necessary and sufficient condition on the positive semidefinite stiffness matrix. A procedure to synthesize an arbitrary full-rank stiffness matrix within this realizable subspace is provided. This procedure requires no more than seven simple springs.