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Estimating How Сlose Parallel Mechanisms are to their Singularities by Differentiating Constraint Equations

Authors: Laryushkin P.A. Published: 18.02.2019
Published in issue: #1(124)/2019  

DOI: 10.18698/0236-3941-2019-1-71-83

 
Category: Mechanical Engineering and Machine Science | Chapter: Machine Science  
Keywords: parallel mechanisms, singularities, Jacobian matrix, constraint equations, screw theory

The paper presents an approach to computing velocity- and force-based criteria for determining how close a parallel mechanism is to its singularity by using inverse Jacobian matrix components derived via differentiating constraint equations. We show that, for this type of mechanisms, methods based on screw theory are not required to achieve the desired result; however, in this case certain assumptions are necessary. Let us consider a planar parallel mechanism. We limited our computation to linear output link velocity and a single force as external loading, which allows us to avoid normalising vectors containing quantities of dissimilar physical significance and to use this approach to analyse mechanisms with mixed degrees of freedom. Our computation yields maximum velocities and forces in actuated joints, which we compare to values computed for two velocity and external force vectors that have the same magnitude but whose direction is constant. We show that the approach suggested actually produces the maximum possible velocities and forces in actuated joints. At the same time, when the mechanism approaches its singularities, we note a significant increase in the deviation of the values computed via this method from those computed for output link velocities or external forces whose direction is constant

The study was conducted as part of a government scientific investigation assignment (grant no. 9.5309.2017/8.9)

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