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.

1.
Ball, R. S., A Treaties on the Theory of Screws, Cambridge University Press, 1900.
2.
Barker, G. P., and Carlson, D., Cones of Diagonally Dominant Matrices. Pacific Journal of Mathematics, Vol. 57, No. 1, 1975.
3.
Bedford, A., and Fowler, W., Engineering Mechanics—Statics, Addison Wesley Publishing Company, Inc., 1995.
4.
Dimentberg, F. M., “The Screw Calculus and its Applications in Mechanics. Foreign Technology Division,” Wright-Patterson Air Force Base, Ohio. Document No. FTD-HT-23-1632-67, 1965.
5.
Golub, G. H., and Loan, C. F. V., Matrix Computations, The John Hopkins University Press, (3rd ed.) 1996.
6.
Griffis, M., and Duffy, J., “Kinestatic control: Novel theory for simultaneously regulating force and displacement,” ASME Journal of Mechanical Design, Vol. 113, No. 2, 1991.
7.
Hill, R. D., and Waters, S. R., “On the Cone of Positive Semidefinite Matrices,” Linear Algebra and its Applications, 90, 1987.
8.
Howell, L. L., and Midha, A., “A Method for the Design of Compliant Mechanisms With Small-Length Flexural Pivots,” ASME Journal of Mechanical Design, Vol. 116, No. 1, 1994.
9.
Howell, L. L., and Midha, A., “A Loop-Closure Theory for the Analysis and Synthesis of Compliant Mechanisms,” ASME Journal of Mechanical Design, Vol. 118, No. 1, 1996.
10.
Huang, C., and Roth, B., “Dimensional Synthesis of Closed-Loop Linkages to Match Force and Position Specifications,” ASME Journal of Mechanical Design, Vol. 115, No. 2, 1993.
11.
Huang, S., “The Analysis and Synthesis of Spatial Compliance,” PhD thesis, Marquette University, Milwaukee, WI, 1998.
12.
Huang, S., and Schimmels, J. M., “The Bounds and Realization of Spatial Stiffnesses Achieved With Simple Springs Connected in Parallel,” IEEE Transactions on Robotics and Automation, Vol. 14, No. 3, 1998.
13.
Loncaric, J., “Geometrical Analysis of Compliant Mechanisms in Robotics,” PhD thesis. Harvard University, Cambridge, MA, 1985.
14.
Loncaric, J., “Normal Forms of Stiffness and Compliance Matrices,” IEEE Journal of Robotics and Automation, Vol. 3, No. 6, 1987.
15.
Matthew, G. K., and Tesar, D., Synthesis of Spring Parameters to Satisfy Specified Energy Levels in Planar Mechanisms, ASME Journal of Engineering for Industry, May, 1977.
16.
Patterson, T., and Lipkin, H., Structure of Robot Compliance. ASME Journal of Mechanical Design, Vol. 115, No. 3, 1993.
17.
Schimmels
J. M.
, “
A Linear Space of Admittance Control Laws that Guarantees Force-Assembly With Friction
,”
IEEE Transactions on Robotics and Automation
, Vol.
13
, No.
5
, pp.
656
667
,
1997
.
18.
Schimmels, J. M., and Huang, S., “A Passive Mechanism that Improves Robotic Positioning Through Compliance and Constraint,” Robotics and Computer-Integrated Manufacturing, Vol. 12, No. 1, 1996.
19.
Schimmels
J. M.
, and
Peshkin
M. A.
, “
Admittance Matrix Design for Force Guided Assembly
,”
IEEE Transactions on Robotics and Automation
, Vol.
8
, No.
2
, pp.
213
227
,
1992
.
20.
Schimmels
J. M.
, and
Peshkin
M. A.
, “
Force-Assembly With Friction
,”
IEEE Transactions on Robotics and Automation
, Vol.
10
, No.
4
, pp.
465
479
,
1994
.
21.
Sturges, R. H., and Laowattana, S., “Design of an Orthogonal Compliance for Polygonal Peg Insertion,” ASME Journal of Mechanical Design, Vol. 118, No. 2, 1996.
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