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Research Papers

On the Stability of Rigid Multibody Systems With Applications to Robotic Grasping and Locomotion

[+] Author and Article Information
Péter L. Várkonyi

Department of Mechanics, Materials
and Structures,
Budapest University of Technology
and Economics,
Muegyetem rkp. 3,
Budapest 1111, Hungary
e-mail: vpeter@mit.bme.hu

1Corresponding author.

Manuscript received March 31, 2014; final manuscript received November 28, 2014; published online April 6, 2015. Assoc. Editor: Anupam Saxena.

J. Mechanisms Robotics 7(4), 041012 (Nov 01, 2015) (8 pages) Paper No: JMR-14-1074; doi: 10.1115/1.4029402 History: Received March 31, 2014; Revised November 28, 2014; Online April 06, 2015

This paper shows that the equilibria of a wide class of multibody systems with quasi-rigid, frictional, or frictionless supports correspond to local minima of their potential energy; hence they are stable against small perturbations of external forces. This is a generalization of a theorem by Howard and Kumar on the stability of a single rigid body held by a gripper. It is also demonstrated that ambiguous equilibria (those, which coexist with the possibility of accelerating motion) may be stable. These results help finding safe grasps on nonrigid objects and assessing the stability of quasi-static robots moving over complex terrains.

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Figures

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Fig. 1

Four examples of ambiguous equilibria. (a) A block between two planes, (b) a two-element open kinematic chain between two planes, (c) a rigid body resembling a person with a heavy backpack resting on a step-shaped terrain (proposed by Elon Rimon, personal communication), and (d) a two-wheeled planar robot on a slope with one wheel freely rotating and the other one blocked. The objects are subject to their own weights and contact forces, both denoted by solid arrows. The nonstatic solutions (arrows and dotted lines representing forces and displacements, respectively) are shown on the left side, and the static equilibria are illustrated on the right side. (a) is force-closed, however, the other three are not. (b) and (d) are statically determinate, whereas (a) and (c) are indeterminate.

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Fig. 2

An unstable (left) and a stable (right) system violating condition (iv). Their stabilities are determined by the curvatures at the contact point. G is the center of gravity.

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Fig. 3

(a) Initial configuration (solid line) and a general configuration (dashed line) of a model a planar biped standing on a stair. The weight of the body is 1, and the legs are weightless. (b) and (c) x and y values corresponding to equilibria path of the model with l1 = 4.03, l2 = 2.06, α10 = −1.70, α20 = −1.33 (b) and with l1 = 2.34, l2 = 0.94, α10 = −2.27, α20 = −1.01 (c). The corresponding values of α1, α2, and ε are not shown. Large and small circles represent stable and unstable equilibria, respectively. Light grey dots represent nonphysical solutions (negative ε or tensile contact forces). The shapes of the robot in the quasi-rigid equilibrium configurations (the initial configuration in both panels and together with a second one in panel (c)) are shown in dashed line.

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Fig. 4

(a) Forces acting on the biped. For the specific configuration plotted in the figure, and T = 0, the equilibrium equations dictate tensile support reaction at the right leg (dashed arrows). Hence the robot topples. However for appropriately chosen T and large enough friction coefficient, the toppling solution coexists with a static equilibrium. The corresponding support reactions are shown by solid arrows. (b) The stability properties of the biped depend on the location of the center of mass relative to the support points. For the detailed description of labels 1, 2, 3, and 4, see text.

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Fig. 5

Filled contour plot of the minimum of the friction coefficient (left) and the corresponding applied torque (right) for various positions of the center of mass. The support points are represented by black circles. Both functions are discontinuous at the vertical lines through the support points.

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