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RESEARCH PAPERS

A Relaxation Method for Simulating the Kinematics of Compound Nonlinear Mechanisms

[+] Author and Article Information
Hod Lipson

Sibley School of Mechanical and Aerospace Engineering and Faculty of Computing and Information, Cornell University, Ithaca, NY 14853hod.lipson@cornell.edu

A computer, in those days, was a person hired to perform arithmetic calculations.

J. Mech. Des 128(4), 719-728 (Oct 14, 2005) (10 pages) doi:10.1115/1.2198255 History: Received December 10, 2004; Revised October 14, 2005

This paper describes a relaxation-based method for simulating 2D and 3D compound kinematic mechanisms. The relaxational process iteratively propagates node motions and degrees of freedom throughout a given kinematic mechanism. While relaxation methods were classically used to solve static problems, we show that the propagation of displacements during the calculation process itself reveals the kinematics of the structure. The method is slower than approaches based on solving simultaneous differential equations of motion, but provides several advantages: It achieves a higher level of accuracy, is more robust in handling transient singularities and degeneracies of the mechanism, and can handle more complex compound mechanisms with many links in multiple entangled kinematic chains. It also allows straightforward introduction of linkages with nonlinear behaviors such as wrapping strings, hydraulics, actuators, contacts, and other arbitrary responses. The basic simulation algorithm is presented, and a number of applications are provided including robotics, design, and biomechanics.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Pseudocode for the simple kinematic relaxation algorithm for a static pin-joined frame; (a) asynchronous, (b) synchronous

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Figure 2

Handling of a singular frame

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Figure 3

Handling of unstable equilibrium

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Figure 4

Simulation of a four-bar mechanism

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Figure 5

Test case 1: The Peaucellier mechanism (8 links, 1DoF) was simulated to accuracy 10−5

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Figure 6

Test case II: Snapshots from a kinematic simulation of 10 chained Stewart platforms. Shown 60 actuators (red) and 11 rigid platforms (blue)

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Figure 7

Errors during simulation of various linked Stewart platforms. (a) Maximum residual and absolute error versus iterations while simulating ten-linked platforms, (b) absolute error trends for a number of mechanism ranging from six Stewart platforms (S6) to 20 linked platforms (S20).

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Figure 8

Test case III: Random underconstrained mechanisms with 200 links and 100 degrees of freedom. (a) Initial state, (b) final actuated state; (c)–(f) progress of maximum residual force for ten random mechanisms (c) 100DoF, synchronous relaxation, (d) 50DoF, synchronous relaxation, (e) 100DoF, asynchronous relaxation (d) 50DoF, asynchronous relaxation.

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Figure 9

Some application examples

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