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

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 is derived using the 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. 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.

[1]  R. Ball.  A treatise on the theory of screws, by Sir Robert Stawell Ball. , .

[2]  F. Dimentberg.  The screw calculus and its applications in mechanics , 1968 .

[3]  G. K. Matthew,et al.  Synthesis of Spring Parameters to Satisfy Specified Energy Levels in Planar Mechanisms , 1977 .

[4]  Daniel E. Whitney,et al.  Force Feedback Control of Manipulator Fine Motions , 1977 .

[5]  Daniel E. Whitney,et al.  Quasi-Static Assembly of Compliantly Supported Rigid Parts , 1982 .

[6]  Neville Hogan,et al.  Impedance Control: An Approach to Manipulation , 1984, 1984 American Control Conference.

[7]  Josip Loncaric,et al.  Geometrical analysis of compliant mechanisms in robotics , 1985 .

[8]  Mark R. Cutkosky,et al.  Active Control of a Compliant Wrist in Manufacturing Tasks , 1986 .

[9]  R. D. Hill,et al.  On the cone of positive semidefinite matrices , 1987 .

[10]  Josip Loncaric,et al.  Normal forms of stiffness and compliance matrices , 1987, IEEE Journal on Robotics and Automation.

[11]  H. Harry Asada,et al.  The Dynamic Analysis and Design of a High-Speed Insertion Hand Using the Generalized Centroid and Virtual Mass , 1990 .

[12]  Michael A. Peshkin,et al.  Programmed compliance for error corrective assembly , 1990, IEEE Trans. Robotics Autom..

[13]  Homayoon Kazerooni,et al.  Automated Robotic Deburring of Parts Using Compliance Control , 1991 .

[14]  Joseph Duffy,et al.  Kinestatic Control: A Novel Theory for Simultaneously Regulating Force and Displacement , 1991 .

[15]  Ashok Midha,et al.  Methodology for Compliant Mechanisms Design: Part II - Shooting Method and Application , 1992 .

[16]  Michael A. Peshkin,et al.  Admittance matrix design for force-guided assembly , 1992, IEEE Trans. Robotics Autom..

[17]  Joseph Duffy,et al.  Global stiffness modeling of a class of simple compliant couplings , 1993 .

[18]  Bernard Roth,et al.  Dimensional Synthesis of Closed-Loop Linkages to Match Force and Position Specifications , 1993 .

[19]  H. Harry Asada,et al.  Representation and learning of nonlinear compliance using neural nets , 1993, IEEE Trans. Robotics Autom..

[20]  H. Lipkin,et al.  Structure of Robot Compliance , 1993 .

[21]  Neville Hogan,et al.  The macro/micro manipulator : an improved architecture for robot control , 1993 .

[22]  Joseph M. Schimmels.  The use of compliance and constraint for improved robotic material removal processes , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[23]  Michael A. Peshkin,et al.  Force-assembly with friction , 1994, IEEE Trans. Robotics Autom..

[24]  A. Midha,et al.  Synthesis of Single-Input and Multiple-Output Port Mechanisms with Springs for Specified Energy Absorption , 1992 .

[25]  Larry L. Howell,et al.  A Method for the Design of Compliant Mechanisms With Small-Length Flexural Pivots , 1994 .

[26]  Shuguang Huang,et al.  A Passive Mechanism that Improves Robotic Positioning through Compliance and Constraint , 1996 .

[27]  Larry L. Howell,et al.  A Loop-Closure Theory for the Analysis and Synthesis of Compliant Mechanisms , 1996 .

[28]  Shuguang Huang.  The analysis and synthesis of spatial compliance , 1998 .