Research Papers

Analytical Evaluation of the Double Stewart Platform Tensile Truss Stiffness Matrix

[+] Author and Article Information
Michael P. Hennessey

e-mail: mphennessey@stthomas.edu
School of Engineering,
University of St. Thomas,
100 O'Shaughnessy Science Hall,
2115 Summit Avenue,
St. Paul, MN 55105-1079

1Present address: Senior Engineer, at the 3M Company of Maplewood, MN.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July 31, 2012; final manuscript received August 22, 2013; published online November 6, 2013. Assoc. Editor: Yuefa Fang.

J. Mechanisms Robotics 6(1), 011003 (Nov 06, 2013) (18 pages) Paper No: JMR-12-1113; doi: 10.1115/1.4025470 History: Received July 31, 2012; Revised August 22, 2013

The 6 × 6 stiffness matrix for a single Stewart platform tensile truss is well known. This work extends the methodology used to determine the stiffness matrix of a double Stewart platform system, in which one Stewart platform is stacked on top of another, in serial fashion. A double Stewart platform may offer advantages for some applications in terms of increased stiffness in certain directions. Using principles of statics and considering small displacement perturbations in three-dimensional space of both mobile platforms (middle and bottom) from their weighted equilibrium locations, displacements can be related in a linear manner to application loading, implying a stiffness matrix. Scripts are then developed and executed in matlabtm to determine the stiffness matrix of a specific system. The matlabtm result is validated using single and double Stewart platform physical models and measuring system compliance responses to external forces and moments.

Copyright © 2014 by ASME
Topics: Cables , Stiffness
Your Session has timed out. Please sign back in to continue.


Stewart, D., 1965–1966, “A Platform With Six Degrees of Freedom,” Proc. Inst. Mech. Eng., 180–1(15), pp. 371–386.
Merlot, J.-P., 2006, Parallel Robots, 2nd ed., Springer, Dordrecht, The Netherlands.
Dagalakis, N. G., Albus, J. S., Wang, B. L., Unger, J., and Lee, J. D., 1989, “Stiffness Study of a Parallel Link Robot Crane for Shipbuilding Applications,” Trans. ASME J. Offshore Mech. Arct. Eng., 111, pp. 183–193. [CrossRef]
Unger, J., and Dagalakis, N., 1988, “Optimum Stiffness Study for a Parallel Link Robot Crane Under Horizontal Force,” ASME Proceedings of the International Symposia on Robotics and Manufacturing, pp. 1037–1046.
Lee, M. K., 1995, “Design of a High Stiffness Machining Robot Arm Using Double Parallel Mechanisms,” IEEE International Conference on Robotics and Automation, pp. 234–240.
Kozak, K., Zhou, Q., and Wang, J., 2004, “Static Analysis of Cable-Driven Manipulators With Non-Negligible Cable Mass,” Proceedings of the IEEE Conference on Robotics, Automation, and Mechatronics, Singapore, Dec. 1–3, pp. 886–891.
Hamedi, J., and Zohoor, H., 2007, “Simulation and Optimization of the Rectangular Stewart Cable-Suspended Robot,” Proceedings of the 13th IASTED International Conference on Robotics and Applications, Wurzburg, Germany, Aug. 29–31, pp. 400–407.
Williams, R. L., II, Xin, M., and Bosscher, P., 2008, “Contour-Crafting-Cartesian-Cable Robot System Concepts: Workspace and Stiffness Comparisons,” ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, New York, NY, Aug. 3–6.
Salisbury, J. K., 1980, “Active Stiffness Control of a Manipulator in Cartesian Coordinates,” Proceedings of the 19th IEEE Conference on Decision and Control, Albuquerque, NM, pp. 87–97.
Pashkevich, A., Chablat, D., and Wenger, P., 2009, “Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44(5), pp. 966–982. [CrossRef]
Li, B., Yu, H., Deng, Z., Yang, X., and Hu, H., 2010, “Stiffness Modeling of a Family of 6 DOF Parallel Mechanisms With Three Limbs Based on Screw Theory,” J. Mech. Sci. Technol., 24(1), pp. 373–382. [CrossRef]
Portman, V. T., Chapsky, V. S., and Sheneor, Y., 2011, “Workspace of Parallel Kinematics Machines With Minimum Stiffness Limits: Collinear Stiffness Value Based Approach,” Mech. Mach. Theory, 49, pp. 67–86. [CrossRef]
Behzadipour, S., and Khajepour, A., 2006, “Stiffness of Cable-Based Parallel Manipulators With Application to Stability Analysis,” Trans. ASME J. Mech. Des., 128, pp. 303–310. [CrossRef]
Arsenault, M., 2011, “Stiffness Analysis of a 2 DOF Planar Tensegrity Mechanism,” Trans. ASME J. Mech. Rob., 3, p. 021011. [CrossRef]
Deblaise, D., Hernot, X., and Maurine, P., 2006, “A Systematic Analytical Method for PKM Stiffness Matrix Calculation,” Proceedings of the International Conference and Robotics and Automation, Orlando, FL.
Reinholtz, C. F., and Gokhale, D., 1987, “Design and Analysis of Variable Geometry Truss Robots,” 9th Annual Conference on Applied Mechanisms, Oklahoma State University, pp. 1–5.
Hesselroth, A. H., “Analytical Evaluation of the Stiffness Matrix for a Double Stewart Platform,” 2012, MSME Paper, School of Engineering, University of St. Thomas, St. Paul, MN.
Hennessey, M. P., Hessselroth, A. H., and Sturm, A., 2011, “Analytical Evaluation of the Stiffness Matrix for Various Cable-Driven Robots, Including the Double Stewart Platform,” Mathematical Physics Seminar, School of Mathematics, University of Minnesota, Minneapolis, MN.
Weinkauf, D., and Ramm, L., eds., 2012, Fukushima and Chernobyl Spawn Cable-Driven Robot Research, St. Thomas Engineer Magazine, St. Paul, MN, Spring, pp. 4–5.
Doebler, G. R., 2012, “PaR Tensile Truss for Nuclear Decontamination and Decommissioning—12467,” WM2012 Conference.


Grahic Jump Location
Fig. 4

Displaced middle and bottom platforms

Grahic Jump Location
Fig. 5

Coordinate frame transformations

Grahic Jump Location
Fig. 3

Top view of movable platforms

Grahic Jump Location
Fig. 2

Front view of initial configuration

Grahic Jump Location
Fig. 1

Three-dimensional platform representation of a double Stewart platform

Grahic Jump Location
Fig. 6

Weighted configuration front view

Grahic Jump Location
Fig. 7

Weight and application loading effect on cable vectors

Grahic Jump Location
Fig. 8

Coordinate frame shift due to weight

Grahic Jump Location
Fig. 9

Free body diagrams for middle and bottom platforms, including weight and small magnitude application force and moment loads

Grahic Jump Location
Fig. 10

Single Stewart platform model instrumented for yaw stiffness experiment

Grahic Jump Location
Fig. 11

Double Stewart platform model instrumented for yaw stiffness experiment




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In