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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

Professor
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

Professor
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.

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Figures

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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.

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Fig. 3

Reference frames used to express the structural compliance matrices

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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.

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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.

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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

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Fig. 6

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

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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

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Fig. 10

The definition of vectors and scalars for mechanism I

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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

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Fig. 11

The definition of vectors and scalars for mechanism II

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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

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