Achieving an Arbitrary Spatial Stiffness with Springs Connected in Parallel

Abstract

In this paper, the synthesis of an arbitrary spatial stiffness matrix is addressed. We have previously shown that an arbitrary stiffness matrix cannot be achieved with conventional translational springs and rotational springs (simple springs) connected in parallel regardless of the number of springs used or the geometry of their connection. To achieve an arbitrary spatial stiffness matrix with springs connected in parallel, elastic devices that couple translational and rotational components are required. Devices having these characteristics are defined here as screw springs. The designs of two such devices are illustrated. We show that there exist some stiffness matrices that require 3 screw springs for their realization and that no more than 3 screw springs are required for the realization of full-rank spatial stiffness matrices. In addition, we present two procedures for the synthesis of an arbitrary spatial stiffness matrix. With one procedure, any rank-m positive semidefinite matrix is realized with m springs of which all may be screw springs. With the other procedure, any positive definite matrix is realized with 6 springs of which no more than 3 are screw springs.

Realization of point planar elastic behaviors using revolute joint serial mechanisms having specified link lengths

“…The synthesis of an arbitrary spatial stiffness matrix with a parallel system has also been addressed [2]. It was shown [2] that in order to realize an arbitrary stiffness, screw springs that couple the elastic behavior in translation and rotation in the same direction must be used.…”

“…It was shown [2] that in order to realize an arbitrary stiffness, screw springs that couple the elastic behavior in translation and rotation in the same direction must be used. The eigen-structure of a stiffness was analyzed using the eigenscrew decomposition of the stiffness matrix [17].…”

“…It is known that for any specified elastic behavior, an infinite number of parallel [1,2] or serial [3,4] mechanisms having springs at each joint can achieve the desired behavior. Proper compliant wrist design involves selecting the geometry and configuration of the mechanism and selecting the spring rates at each joint.…”

Compliance Analysis and Synthesis of a Serial Manipulator with Redundant DOFs Used in Tokamak

“…Similar to (6) and (7), the Cartesian compliance of the whole serial manipulator contains ten joints can be obtained by:…”

“…Notably, Huang and Schimmels presented the bounds and realization of an isotropic spatial stiffness [4] and compliance matrix [5], respectively. Then they put forward the concept of screw springs and achieved an arbitrary spatial stiffness [6]. However, all these algorithms are associated with the choice of coordinate systems.…”