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RESEARCH PAPERS

Computer Aided Synthesis of Piecewise Rational Motions for Planar 2R and 3R Robot Arms

[+] Author and Article Information
Zhe Jin

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300zjin@ic.sunysb.edu

Q. J. Ge

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300qiaode.Ge@stonybrook.edu

J. Mech. Des 129(10), 1031-1036 (Oct 02, 2006) (6 pages) doi:10.1115/1.2756082 History: Received February 14, 2006; Revised October 02, 2006

This paper deals with the problem of synthesizing piecewise rational motions of an object that satisfies kinematic constraints imposed by a planar robot arm with revolute joints. This paper brings together the kinematics of planar robot arms and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. Through the use of planar quaternions, it is shown that for the case of a planar 2R arm, the problem of rational motion synthesis can be reduced to that of circular interpolations in two separate planes and that for the case of a planar 3R arm, the problem can be reduced to a combination of circular interpolation in one plane and a constrained spline interpolation in a circular ring on another plane. Due to the limitation of circular interpolation, only C1 continuous rational motions are generated that satisfy the kinematic constraints exactly. For applications that require C2 continuous motions, this paper presents a method for generating C2 continuous motions that approximate the kinematic constraints for planar 2R and 3R robot arms.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Planar displacement

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Figure 2

A planar 2R robot arm

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Figure 3

A planar 3R robot arm

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Figure 4

Quadratic NURB circular arc

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Figure 5

Interpolation of points Ei (planar 2R robot arm)

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Figure 6

Interpolation of points Fi (planar 2R robot arm)

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Figure 7

An unconstrained C2 cubic B-spline interpolation of a set of points that violates the circular-ring constraint

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Figure 8

A constrained C2 cubic B-spline interpolation in a circular ring with the new point at the mid point δ=b∕2

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Figure 9

Diametral clearance of planar 2R robot arm

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Figure 10

Interpolation of points Ei (planar 3R robot arm)

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Figure 11

The initial interpolation of points Fi (unconstrained)

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Figure 12

Constrained interpolation of points Fi in a circular ring with width δ=0.002

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