Research Papers

Experimental Validation of Jacobian-Based Stiffness Analysis Method for Parallel Manipulators With Nonredundant Legs

[+] Author and Article Information
Antonius G. L. Hoevenaars

Department of Precision and
Microsystems Engineering,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands
e-mail: a.g.l.hoevenaars@tudelft.nl

Clément Gosselin

Department of Mechanical Engineering,
Laval University,
1065 Avenue de la médecine,
Québec, QC G1V 0A6, Canada

Patrice Lambert

Department of Precision and
Microsystems Engineering,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands

Just L. Herder

Department of Precision and
Microsystems Engineering,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands

Manuscript received June 30, 2015; final manuscript received November 24, 2015; published online March 7, 2016. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 8(4), 041002 (Mar 07, 2016) (10 pages) Paper No: JMR-15-1168; doi: 10.1115/1.4032204 History: Received June 30, 2015; Revised November 24, 2015

A complete stiffness analysis of a parallel manipulator considers the structural compliance of all elements, both in designed degrees-of-freedom (DoFs) and constrained DoFs, and also includes the effect of preloading. This paper presents the experimental validation of a Jacobian-based stiffness analysis method for parallel manipulators with nonredundant legs, which considers all those aspects, and which can be applied to limited-DoF parallel manipulators. The experimental validation was performed by comparing differential wrench measurements with predictions based on stiffness analyses with increasing levels of detail. For this purpose, two passive parallel mechanisms were designed, namely, a planar 3DoF mechanism and a spatial 1DoF mechanism. For these mechanisms, it was shown that a stiffness analysis becomes more accurate if preloading and structural compliance are considered.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Majou, F. , Gosselin, C. , Wenger, P. , and Chablat, D. , 2007, “ Parametric Stiffness Analysis of the Orthoglide,” Mech. Mach. Theory, 42(3), pp. 296–311. [CrossRef]
Pinto, C. , Corral, J. , Altuzarra, O. , and Hernández, A. , 2010, “ A Methodology for Static Stiffness Mapping in Lower Mobility Parallel Manipulators With Decoupled Motions,” Robotica, 28(5), pp. 719–735. [CrossRef]
Gosselin, C. , 1990, “ Stiffness Mapping for Parallel Manipulators,” IEEE Trans. Rob. Autom., 6(3), pp. 377–382. [CrossRef]
Quennouelle, C. , and Gosselin, C. , 2011, “ Kinematostatic Modeling of Compliant Parallel Mechanisms,” Meccanica, 46(1), pp. 155–169. [CrossRef]
Chen, S.-F. , and Kao, I. , 2000, “ Conservative Congruence Transformation for Joint and Cartesian Stiffness Matrices of Robotic Hands and Fingers,” Int. J. Rob. Res., 19(9), pp. 835–847. [CrossRef]
Merlet, J.-P. , and Gosselin, C. , 2008, “ Parallel Mechanisms and Robots,” Springer Handbook of Robotics, B. Siciliano and O. Khatib , eds., Springer, Berlin/Heidelberg, pp. 269–285.
Quennouelle, C. , and Gosselin, C. , 2009, “ A Quasi-Static Model for Planar Compliant Parallel Mechanisms,” ASME J. Mech. Rob., 1(2), p. 021012. [CrossRef]
Ahmad, A. , Andersson, K. , Sellgren, U. , and Khan, S. , 2012, “ A Stiffness Modeling Methodology for Simulation-Driven Design of Haptic Devices,” Eng. Comput., 30(1), pp. 125–141. [CrossRef]
Cheng, G. , Xu, P. , Yang, D. , and Liu, H. , 2013, “ Stiffness Analysis of a 3CPS Parallel Manipulator for Mirror Active Adjusting Platform in Segmented Telescope,” Rob. Comput. Integr. Manuf., 29(5), pp. 302–311. [CrossRef]
Joshi, S. A. , and Tsai, L.-W. , 2002, “ Jacobian Analysis of Limited-DOF Parallel Manipulators,” ASME J. Mech. Des., 124(2), pp. 254–258. [CrossRef]
Huang, T. , Liu, H. T. , and Chetwynd, D. G. , 2011, “ Generalized Jacobian Analysis of Lower Mobility Manipulators,” Mech. Mach. Theory, 46(6), pp. 831–844. [CrossRef]
Huang, T. , Zhao, X. , and Whitehouse, D. J. , 2002, “ Stiffness Estimation of a Tripod-Based Parallel Kinematic Machine,” IEEE Trans. Rob. Autom., 18(1), pp. 50–58. [CrossRef]
Wahle, M. , and Corves, B. , 2011, “ Stiffness Analysis of Clavel's DELTA Robot,” Intelligent Robotics and Applications (Lecture Notes in Computer Science), Vol. 7101, S. Jeschke, H. Liu, and D. Schilberg , eds., Springer, Berlin, Heidelberg, pp. 240–249.
Pham, H.-H. , and Chen, I.-M. , 2005, “ Stiffness Modeling of Flexure Parallel Mechanism,” Precis. Eng., 29(4), pp. 467–478. [CrossRef]
Zhang, D. , and Lang, S. Y. , 2004, “ Stiffness Modeling for a Class of Reconfigurable PKMs With Three to Five Degrees of Freedom,” J. Manuf. Syst., 23(4), pp. 316–327. [CrossRef]
Wang, Y. , Liu, H. , Huang, T. , and Chetwynd, D. G. , 2009, “ Stiffness Modeling of the Tricept Robot Using the Overall Jacobian Matrix,” J. Mech. Rob., 1(2), p. 021002. [CrossRef]
Li, Y. , and Xu, Q. , 2008, “ Stiffness Analysis for a 3-PUU Parallel Kinematic Machine,” Mech. Mach. Theory, 43(2), pp. 186–200. [CrossRef]
Pashkevich, A. , Chablat, D. , and Wenger, P. , 2009, “ Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44(5), pp. 966–982. [CrossRef]
Sung Kim, H. , and Lipkin, H. , 2014, “ Stiffness of Parallel Manipulators With Serially Connected Legs,” ASME J. Mech. Rob., 6(3), p. 031001. [CrossRef]
Hoevenaars, A. G. L. , Lambert, P. , and Herder, J. L. , 2015, “ Jacobian-Based Stiffness Analysis Method for Parallel Manipulators With Non-Redundant Legs,” Proc. Inst. Mech. Eng., Part C (in press).
Yi, B.-J. , Chung, G. B. , Na, H. Y. , Kim, W. K. , and Suh, I. H. , 2003, “ Design and Experiment of a 3-DOF Parallel Micromechanism Utilizing Flexure Hinges,” IEEE Trans. Rob. Autom., 19(4), pp. 604–612. [CrossRef]
Alici, G. , and Shirinzadeh, B. , 2005, “ Enhanced Stiffness Modeling, Identification and Characterization for Robot Manipulators,” IEEE Trans. Rob., 21(4), pp. 554–564. [CrossRef]
Trease, B. P. , Moon, Y.-M. , and Kota, S. , 2005, “ Design of Large-Displacement Compliant Joints,” ASME J. Mech. Des., 127(4), pp. 788–798. [CrossRef]
Murray, R. M. , Li, Z. , and Sastry, S. S. , 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
Kövecses, J. , and Angeles, J. , 2007, “ The Stiffness Matrix in Elastically Articulated Rigid-Body Systems,” Multibody Syst. Dyn., 18(2), pp. 169–184. [CrossRef]
Hoevenaars, A. G. L. , 2015, Wrench Measurements on a Planar, Passive, 3-DoF, 3-RPR Parallel Mechanism and a Spatial, Passive, 1-DoF, 3-RRR Parallel Mechanism, Dataset, TU Delft.
Griffis, M. , and Duffy, J. , 1993, “ Global Stiffness Modeling of a Class of Simple Compliant Couplings,” Mech. Mach. Theory, 28(2), pp. 207–224. [CrossRef]


Grahic Jump Location
Fig. 1

Mechanism I is a passive planar 3-RPR mechanism, where the interaction wrenches are the result of elongation/contraction of the linear springs, depending on the pose of the end-effector. Here, mechanism I is shown in pose I-d as introduced in Table 1, where the pose is determined by a position of reference frame E with respect to O and a rotation θ.

Grahic Jump Location
Fig. 2

Mechanism II is a 1DoF passive spatial 3-RRR mechanism, where the interaction wrenches are the result of elastic deformation of the compliant joints that make up the second revolute joint of each leg. The pose is determined by a position of reference frame E with respect to O.

Grahic Jump Location
Fig. 3

Reference frames used to express the structural compliance matrices

Grahic Jump Location
Fig. 4

(a) The end-effector was connected via the end-effector interface to the attachment beam, which is part of the inertial measurement frame. A caliper was used to control the position of the end-effector interface along (b) the X-axis, (c) the Y-axis, and (d) the Z-axis, while (e)screw holes in the end-effector interface allowed for a discrete rotation of 1/8 rad about the Z-axis. This figure illustrates this concept for mechanism II, where the caliper is outlined in white. The same concept holds for mechanism I, but because mechanism I is a planar mechanism it was not displaced along the Z-axis.

Grahic Jump Location
Fig. 5

To measure the interaction wrench, a wrench sensor is integrated between the end-effector body and the end-effector interface, which is rigidly connected to the inertial frame, here shown for mechanism I

Grahic Jump Location
Fig. 6

Two attachment beams were required to put mechanism I at a reference pose where θ≠0

Grahic Jump Location
Fig. 7

The end-effector reference frame E, the rotated measurement frame before imposing the displacement, A, and the rotated measurement frame after imposing a displacement in the X-direction, B, for mechanism I in pose I-d

Grahic Jump Location
Fig. 8

Box plots of the normalized error values for the stiffness models without (benchmark model) and with consideration of the effect of preloading, as obtained from measurements on mechanism I

Grahic Jump Location
Fig. 9

Box plots of the normalized error values for the stiffness model which only considers preloading and for the stiffness model which also considers structural compliance, as obtained from measurements on mechanism II

Grahic Jump Location
Fig. 10

The definition of vectors and scalars for mechanism I

Grahic Jump Location
Fig. 11

The definition of vectors and scalars for mechanism II



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