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Research Papers

Determination and Stability Analysis of Equilibrium Configurations of Objects Suspended From Multiple Aerial Robots

[+] Author and Article Information
Qimi Jiang

GRASP Laboratory,  University of Pennsylvania, Philadelphia, PA 19104qimi@seas.upenn.edu

Vijay Kumar

GRASP Laboratory,  University of Pennsylvania, Philadelphia, PA 19104kumar@seas.upenn.edu

J. Mechanisms Robotics 4(2), 021005 (Apr 04, 2012) (21 pages) doi:10.1115/1.4005588 History: Received March 22, 2011; Accepted December 13, 2011; Published March 28, 2012; Online April 04, 2012

This work addresses the problem for determining the position and orientation of objects suspended with n cables from n aerial robots. This is actually the direct kinematics problem of the 3D cable system. First, the problem is formulated based on the static equilibrium condition. Then, an analytic algorithm based on resultant elimination is proposed to determine all possible equilibrium configurations of the planar 4-bar linkage. As the nonlinear system can be reduced to a polynomial equation in one unknown with a degree 8, this algorithm is more efficient than numerical search algorithms. Considering that the motion of a 3D cable system in its vertical planes of symmetry can be regarded as the motion of an equivalent planar 4-bar linkage, the proposed algorithm is used to solve the direct kinematics problem of objects suspended from multiple aerial robots. Case studies with three to six robots are conducted for demonstration. Then, approaches for stability analysis based on Hessian matrix are developed, and the stability of obtained equilibrium configurations is analyzed. Finally, experiments are conducted for validation.

Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

3D towing of a triangle payload with three aerial robots

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Figure 2

Object suspended from multiple aerial robots

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Figure 3

Object suspended from three robots

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Figure 4

Four-bar linkage (object suspended from two robots)

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Figure 5

Twelve equilibrium configurations of the planar 4-bar linkage

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Figure 6

Object suspended from three robots

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Figure 7

Four equilibrium configurations of an object suspended from three robots in the plane Q1 P1 PQ

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Figure 8

Object suspended from four robots

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Figure 9

Two equilibrium configurations of an object suspended from four robots in the plane Q1 P1 P3 Q3

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Figure 10

Four equilibrium configurations of an object suspended from four robots in the plane Q5 P5 P6 Q6

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Figure 11

The configuration in which every two opposite cables intersect each other for the case with four robots

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Figure 12

Object suspended from five robots

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Figure 13

Four equilibrium configurations of an object suspended from five robots in the plane Q1 P1 P6 Q6

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Figure 14

Object suspended from six robots

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Figure 15

Four equilibrium configurations of an object suspended from six robots in the plane Q1 P1 P4 Q4

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Figure 16

Four equilibrium configurations of an object suspended from six robots in the plane Q7 P7 P8 Q8

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Figure 17

The configuration in which every two opposite cables intersect each other for the case with six robots

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Figure 18

The used experimental device

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Figure 19

The experimental equilibrium configurations for the case with three cables in one vertical plane of symmetry

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Figure 20

Stable configurations obtained by experiments for the case with four cables in one vertical plane of symmetry

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