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Research Papers

A Task-Driven Approach to Optimal Synthesis of Planar Four-Bar Linkages for Extended Burmester Problem

[+] Author and Article Information
Shrinath Deshpande, Anurag Purwar

Computer-Aided Design and
Innovation Laboratory,
Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

1Corresponding author.

Manuscript received February 21, 2017; final manuscript received August 22, 2017; published online September 18, 2017. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 9(6), 061005 (Sep 18, 2017) (9 pages) Paper No: JMR-17-1048; doi: 10.1115/1.4037801 History: Received February 21, 2017; Revised August 22, 2017

The classic Burmester problem is concerned with computing dimensions of planar four-bar linkages consisting of all revolute joints for five-pose problems. We define extended Burmester problem as the one where all types of planar four-bars consisting of dyads of type RR, PR, RP, or PP (R: revolute, P: prismatic) and their dimensions need to be computed for n-geometric constraints, where a geometric constraint is an algebraically expressed constraint on the pose, pivots, or something equivalent. In addition, we extend it to linear, nonlinear, exact, and approximate constraints. This extension also includes the problems when there is no solution to the classic Burmester problem, but designers would still like to design a four-bar that may come closest to capturing their intent. Machine designers often grapple with such problems while designing linkage systems where the constraints are of different varieties and usually imprecise. In this paper, we present (1) a unified approach for solving the extended Burmester problem by showing that all linear and nonlinear constraints can be handled in a unified way without resorting to special cases, (2) in the event of no or unsatisfactory solutions to the synthesis problem, certain constraints can be relaxed, and (3) such constraints can be approximately satisfied by minimizing the algebraic fitting error using Lagrange multiplier method. We present a new algorithm, which solves new problems including optimal approximate synthesis of Burmester problem with no exact solutions.

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Figures

Grahic Jump Location
Fig. 1

Example 7.1: optimal four-bar mechanism that minimizes algebraic fitting error for third pose. Although second pose lies on different circuit, this Grashof type four-bar produces desired continuous motion from first to last pose.

Grahic Jump Location
Fig. 2

Example 7.2: five landing gear positions are shown where the third position can be relaxed. Allowed region for fixed pivots of mechanism is also shown.

Grahic Jump Location
Fig. 3

Example 7.2: image-space representation of intersection of third and fourth optimal constraint manifolds from Table 7. Also shown are five image points as dark spheres representing the five poses; four of them lie exactly on the intersection of the two surfaces while one is closest possible.

Grahic Jump Location
Fig. 4

Example 7.2: first and second dyads in Table 7 are combined to form the linkage shown. It can be clearly seen that coupler curve fairly approximates the third pose while fixed pivots are inside the allowed region.

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